Structural Road Accident Models

structural Road Accident Models

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structural Road Accident Models The International DRAG Family

Marc Gaudry Universite de Montreal, Montreal, Canada Universite Louis Pasteur, Strasbourg, France and

Sylvain Lassarre Institut National de Recherche sur les Transports et leur Securite, Arcueil, France

2000

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" Of making many books there is no end; and much study is a weariness of the flesh. " Ecclesiasticus, 12, 12.

To :

Andre Viel John Lawson Claude Dussault Michel Houee Alexander von Humboldt-Stiftung Centre National de la Recherche Scientifique willing to take research funding risks, Frank Haight willing to wait for a manuscript of 1984, sine quibus non.

Contents v

Contributing authors

xi

Foreword: on a manuscript of 1984 Sylvain Lassarre

xiii

Research support, and more Marc Gaudry, Sylvain Lassarre

xvii

PART I. NATIONAL AND REGIONAL MODELS 1 MULTIPLE LEVELS, DAMAGES, FORMS, MOMENTS AND VARIABLES IN ROAD ACCIDENT MODELS Marc Gaudry LL Introduction: the «Modening Quartet» in this book L2. Problem Formulation 1.2.1. A multilevel structure; a multidamage application 1.2.2. Perspectives on problem structure 1.3. The quantification of effects 1.3.1. From fixed to flexible mathematical form 1.3.2. From monotonic to multitonic forms: the case of alcohol Variables: multimoment, multivariate 1.4. 1.4.1. The dependent variable: from observations to moments 1.4.2. The explanatory variables: not a triad, but a quatrain 1.4.3. Is pregnancy a risk factor? 1.5. Other modelling dimensions of interest 1.5.1. Multidata 1.5.2. Multiple documentation of reference results 1.6. Conclusion: matching tools to questions 1.7. Post Scriptum: model acronyms 1.8. References 2 THE DRAG-2 MODEL FOR QUEBEC Frangois Fournier, Robert Simard 2.1. Introduction 2.2. The structure of the DRAG-2 model 2.2.1. A diagram of the model structure 2.2.2. Dependent variable graphs 2.2.3. The matrix of direct effects of independent variables over dependent variables 2.3. Results on mathematical form and particular variables 2.3.1. Econometric results 2.3.2. Results on elasticity 2.3.3. Other results: forecasts for the period of 1997-2004

1

1 2 2 6 9 9 14 19 19 24 27 29 29 30 31 31 32 37 37 38 38 40 45 49 50 51 63

vi STRUCTURAL ROAD ACCIDENT MODELS 2.4.

Other developments

3 THE SNUS-2.5 MODEL FOR GERMANY Ulrich Blum, Marc Gaudry 3.L Context 3.2. Structure of model 3.2.1. The dependent variables 3.2.2. Visual analysis of the dependent variables 3.2.3. Matrix of direct effects 3.3. Results and their interpretation 3.3.1. Statistical results 3.3.2. Economic results: overall specific results 3.3.3. Decomposition of the impact by variable: results common to other models 3.3.4. Results for other variables 3.4. Deriving other interesting results 3.4.1. The analysis of victims: direct, indirect and total elasticities 3.4.2. Multiple moments and their marginal rates of substitution 3.4.3. Marginal rates of substitution with comparable accident data 3.4.4. Marginal rates of substitution with disaggregated accident data 3.5. Policy implications 3.5.1. Higher prices save energy and lives 3.5.2. Risk substitution in terms of first moments 3.6. References 4 THE TRULS-1 MODEL FOR NORWAY Lasse Fridstrom 4.1. Introduction 4.2. Structure of the Model TRULS-1 4.2.1. Dependent variables: definitions and relations 4.2.2. Visual analysis of dependent variables 4.2.3. Matrix of direct effects 4.2.4. The casualty subset test 4.3. Results on form and selected explanatory variables 4.4. References 5 THE DRAG-STOCKHOLM-2 MODEL Goran Tegner, Ingvar Holmberg, Vesna Loncar-Lucassi, Christian Nilsson 5.1. Introduction 5.1.1. The Dennis agreement 5.1.2. The MAD-project 5.1.3. The concept of zero fatality 5.2. Structure of the DRAG model for Stockholm county 5.2.1. Introduction

66 67 67 68 68 69 70 73 73 73 75 79 82 82 84 88 92 94 94 94 94 97 97 98 99 100 107 108 111 126 127 127 127 128 12 8 128

Contents vii

5.3.

5.4.

5.5.

5.6. 5.7. 5.8.

5.9.

5.2.2. Dependent variables: definitions and relations 5.2.3. Visual analysis of dependent variables 5.2.4. Matrix of direct effects Model form and explanatory variables 5.3.1. Summary of econometric results 5.3.2. The demand for road use 5.3.3. Comparison between estimated and actual demand for road use 5.3.4. The contribution of road infrastructure to road traffic growth The Road accident frequency and gravity models 5.4.1. Economic activities 5.4.2. Quality of vehicle fleet 5.4.3. Road network data 5.4.4. Weather data 5.4.5. Intervention measures 5.4.6. Gasoline price The DRAG-Stockholm-2 model 5.5.1. The new model specification 5.5.2. Comparison of results between the "old" and "new" specification Comparison of actual and estimated accident risks Specific results on the DRAG-Stockholm model Points of interest and conclusion 5.8.1. Alcohol consumption: the J-shaped relationship 5.8.2. Medicine consumption 5.8.3. Pregnancy—a new risk factor 5.8.4. Conclusions References

6 THE TAG-1 MODEL FOR FRANCE Laurence Jaeger, Sylvain Lassarre 6.1. Introduction 6.2. Structuring the TAG model 6.3. Econometric form of the TAG model 6.4. The estimates produced by the TAG model 6.4.1. Model of road transport demand 6.4.2. Constructing a model of average speed 6.4.3. Analysis of the results by risk indicator 6.4.4. Analysis of results by explanatory factor 6.5. Conclusion 6.6. References 7 THE TRACS-CA MODEL FOR CALIFORNIA Patrick McCarthy lA. Introduction

129 129 13 3 13 4 134 134 13 9 139 140 141 141 142 142 142 142 143 143 143 146 148 149 150 151 151 154 154 157 15 7 159 163 167 168 170 173 175 181 182 185 185

viii STRUCTURAL ROAD ACCIDENT MODELS 7.2.

7.3.

7.4. 7.5.

TRACS-CA model structure 7.2.1. Exposure and crash losses 7.2.2. Historical trends 7.2.3. Determining variables included in the TRACS-CA structure Estimation results 7.3.1. Statistical summary 7,32. Common variable results 7.3.3. Specific variable results 7.3.4. Further results Discussion and future directions References

8 COMPARING SIX DRAG-TYPE MODELS Nicolas Chambron 8.1. Risk exposure 8.2. Driver behaviour 8.2.1. Speed 8.2.2. Seatbelt wearing 8.2.3. Consumption of alcohol 8.2.4. Consumption of medicines 8.3. Economic variables 8.3.1. Households' economic and financial situation 8.3.2. Fuel prices 8.3.3. Competing supply from public transport 8.4. Conclusion 8.5. References

186 186 188 191 194 194 195 198 200 200 203 205 206 209 209 212 214 216 217 217 220 222 222 224

PART II: OTHER MODELS AND ISSUES 9 THE ROAD, RISK, UNCERTAINTY AND SPEED Marc Gaudry, Karine Vernier 9.1. Risk, uncertainty and observed road accident outcomes 9.2. Model structure: simultaneity and perceived risk 9.3. Selected results: accident frequency and severity 9.4. Selected results: speed 9.5. Conclusion 9.6. References 10 THE RES MODEL BY ROAD TYPE IN FRANCE Ruth Bergel, Bernard Girard 10.1. Introduction 10.2. Structure of the model 10.2.1. General outline

225 225 226 230 233 235 235 237 237 23 8 238

Contents ix 10.2.2. The data base 10.2.3. Economic formulation 10.2.4. Econometric specification 10.2.5. Algorithm The Results 10.3.1. Tests of functional form 10.3.2. Measuring elasticities 10.3.3. Short and medium term simulations Conclusion References

239 242 243 244 245 245 246 248 248 249

11 POSTFACE AND PERSPECTIVES Sylvain Lassarre 11.1. Relevance of models for understanding the influence of risk factors 11.2. Outlook for research in constructing risk models 11.2.1. Data extraction 11.2.2. Adding levels to the structure 11.2.3. Breakdown of indicators by user and road types 11.2.4. Disaggregation by location or vehicle x driver 11.3. Relevance of the models for managing road safety 11.4. References

251

10.3.

10.4. 10.5.

252 25 8 258 259 260 260 260 262

PART III: ALGORITHMS AND DETAILED MODEL OUTPUTS 12 THE TRIO LEVEL-1.5 ALGORITHM FOR BC-GAUHESEQ REGRESSION Tran Liem, Marc Gaudry, Marcel Dagenais, Ulrich Blum 12.1. Introduction and statistical model 12.1.1. Introduction 12.1.2. Log-likelihood function 12.1.3. Computational aspects 12.1.4. Model types 12.1.5. Model estimation 12.2. Estimation results 12.2.1. Definitions of moments of the dependent variable 12.2.2. Derivatives and elasticities of the sample and expected values of the dependent variable 12.2.3. Derivatives and elasticities of the standard error of the dependent variable 12.2.4. Derivatives and elasticities of the skewness of the dependent variable 12.2.5. Ratios of derivatives of the moments of the dependent variable 12.2.6. Evaluation of moments, their derivatives, rates of substitution and elasticities 12.2.7. Student's t-statistics

263 263 265 270 275 276 278 278 283 288 293 296 301 305

X STRUCTURAL ROAD ACCIDENT MODELS

12.3.

12.4.

12.2.8. Goodness-of-fit measures Special options 12.3.1. Correlation matrix and table of variance-decomposition proportions 12.3.2. Analysis of heteroskedasticity of the residuals 12.3.3. Analysis of autocorrelation of the residuals 12.3.4. Forecasting: maximum likelihood and simulation forecasts References

13 THE IRPOSKML PROCEDURE OF ESTIMATION Lasse Fridstrom 13.1. Accident frequency model specification 13.2. Severity model specification 13.3. References 14 TURNING BOX-COX INCLUDING QUADRATIC FORMS IN REGRESSION Marc Gaudry, Ulrich Blum, Iran Liem 14.1. Model with two Box-Cox transformations on a same independent variable 14.1.1. Solution 14.1.2. First-order conditions 14.1.3. Second-order conditions 14.1.4. Special case: quadratic form 14.2. Model with powers A^ and A2 only on a same independent variable 14.2.1. First-order conditions 14.2.2. Second-order conditions 14.2.3. Special case: quadratic form 14.3. Two-step transformations on a same independent variable 14.4. References 15 APPENDIX 1. DETAILED MODEL OUTPUTS Marc Gaudry, Sylvain Lassarre

306 309 3 09 310 311 314 321 325 325 331 334 335 335 335 336 337 339 339 340 341 343 343 346 347

Contributing Authors xi SYLVAIN L A S S A R R E

CHRISTIAN NILSSON

INRETS, Arcueil www.inrets.fr [email protected]

Transek AB http ://www.transek. se christian@transek. se

MARC G A U D R Y

Universite de Montreal, Montreal Universite Louis Pasteur, Strasbourg www.crt.umontreal.ca/crt/AgoraJulesDupuit/ gaudry @crt. umontreal. ca FRANCOIS F O U R N I E R

Societe de I'assurance automobile du Quebec [email protected]

LAURENCE JAEGER

Universite de Haute-Alsace, Colmar laurence.j [email protected]

PATRICK MCCARTHY

Purdue University [email protected]

NICOLAS C H A M B R O N ROBERT S I M A R D

Societe de I'assurance automobile du Quebec [email protected] ULRICH BLUM

Technische Universitdt Dresden, Dresden [email protected] LASSEFRIDSTR0M Transportokonomisk institutt (T0I), Oslo www.toi.no lasse.fridstrom(a)toi.no GORAN TEGNER

Transek AB www.transek.se goran@transek. se

Federation fran9aise des societes d'assurances [email protected]

KARINE VERNIER

Gaz de France [email protected]

RUTH B E R G E L

INRETS, Arcueil [email protected]

BERNARD G I R A R D

Universite Paris I, Paris [email protected]

TRAN L I E M INGVARHOLMBERG

Gotegorgs Universitet [email protected]

Universite de Montreal, Montreal www.crt.umontreal.ca/crt/AgoraJulesDupuit/ [email protected] .ca

VESNA L O N C A R - L U C A S S I

Transek AB www.transek.se [email protected]

MARCEL DAGENAIS

Universite de Montreal, Montreal marcel. dagenais@umontreal. ca

This Page Intentionally Left Blank

Foreword: on a Manuscript of 1984 xiii

FOREWORD: ON A MANUSCRIPT OF 1984 Sylvain Lassarre

I became aware of the birth of the research stream presented in this book almost at the same time as Frank Haight, Editor-in-chief of Accident Analysis and Prevention, received the submitted manuscript describing the ancestor model, now called DRAG-1, in October^ 1984. That 220-page paper, written in French (Gaudry, 1984), formulated the road safety problem as a simultaneous equations model of demand for road use, safety and speed but, in the absence of data on speed, retained the reduced form equations of the system. These equations explained safety outcomes (victims injured and killed) through a multi-layer decomposition of the number of victims by category among exposure, frequency and severity effects. This innovative decomposition made it possible to test generally for the presence of risk substitution among the different dimensions of road safety, such as accidents of different categories and their severity. Substitution might occur if changes in some explanatory factor led to, say decreases in fatal accidents associated with increases in other accident severity categories, perhaps combined with partially offsetting changes in the severity (morbidity and mortality rates) of each accident category. Patterns of risk substitution explored in the original paper are analyzed further in this book, with the additional benefit of international comparisons, some of which are derived from second generation models obtained after painstaking improvements to first-cut data bases. At the same time, the first use of Box-Cox transformations in road safety analysis made it possible for instance to test for, and in this case to reject, the proportionality of accidents to vehicle kilometrage and to remove many uncertainties associated with fixed-form results, along the lines previously demonstrated by Gaudry and Wills (1978). In addition, the use of a multivariate monthly time series specification—^with many interesting ^ His letter of acknowledgement was received by Marc Gaudry, then on a sabbatical stay at the University of Karlsruhe, in November 1984.

xiv Structural Road Accident Models graphs demonstrating the great variability of many series on a monthly basis—favoured the joint inclusion of variables belonging to different classes of determinants and yielded a number of interesting results linking for instance the state of the economy, fuel prices, automobile insurance regimes and various laws and regulations to safety outcomes by category. The paper also ventured unusual and challenging results and conjectures on the role of alcohol, hours worked and pregnancy that pointed to needed research using other data sets, including less aggregate data. Such models, some based on count and discrete data, are found here. In his letter of October 2, 1985, accepting the paper for publication, Frank Haight stated: « Taking into consideration the length of the paper, which may run to as much as 150 printed pages, it may be necessary to publish it by sections in consecutive issues of the journal. » But it had to be translated first. And Marc Gaudry, named for a second time director of the Centre for Research on Transportation (CRT) of the University of Montreal, was immediately busy securing a large collective research grant for CRT and became somewhat overtaxed: he then failed to provide the necessary English translation of his manuscript and concentrated instead on algorithmic developments for the TRIO statistical software used by the growing number of colleagues, such as Ulrich Blum in Germany and Lasse Fridstrom in Norway, who had almost immediately (well before 1989) started developing DRAG-type approaches of their own. The result of these diffusion efforts presented in this book allow, perhaps for the first time, a multinational comparison of road safety results obtained within similarly structured multivariate approaches. This truly ambitious international activity, carried out within an active research network, has prompted great interest from the safety research and policy community, as indicated for instance by state-of-the-art analyses of DRAG network methodology and output by international committees (OECD, 1997; COST 329, 1999). But it is this book that provides the first thorough overview of the current state of the models (all of which being the object of ongoing work towards improved versions), of the estimation methods, as well as of the detailed results for the six models at the core of the network. The book also reports on other irmovative models based on the DRAG-type structure and estimated with Box-Cox transformations on variables. Part I. I have selected one feature drawn from each of the six models found in the first part of the book, both to whet the reader's appetite and to point to future research needs: • Ch. 1: a set of previously unpublished results on the shape of the curve linking (aggregate) alcohol consumption and accident frequency and severity by category raise the following question: would other less aggregate data sets exhibit such J-shaped effects if one looked for them instead of assuming monotonic shapes in tests? These results, determined within the multivariate structure of the DRAG-1 model, warrant urgent further examination;

Foreword: on a Manuscript of 1984 xv • Ch. 2: the second generation model DRAG-2, developped by the Quebec Automobile Insurance Board (SAAQ) to make official analyses and policy evaluations—^no other jurisdiction has an official model used, maintained and developped in such a continuous fashion—, produces forecasts of road fatalities using in particular an asymmetric (quasiquadratic) relationship between vehicle kilometrage and fatal accident frequency and severity. This interesting device gets around the lack of observations on congestion; • Ch. 3: the SNUS-2.5 model for Germany includes an original multimoment analysis of the empirical trade-offs among the first three moments of accident frequencies, with amazing similarities between Quebec and Germany. This occurs despite the fact—itself of certain interest for the understanding of safety behavior—^that the frequency distributions of fatal accidents of these two jurisdictions are strongly asymmetric in opposite directions; • Ch. 4: TRULS-1 results based on an extraordinary pooling of time series and cross sectional data for Norway (5016 observations!) include numerous interesting findings, for instance on the role of infrastructure or of pregnancy, the latter based on a comparison of subsets of drivers. These latter results have given rise to a multidisciplinary Norwegian research effort, starting in early 2000, to probe the issue further through an analysis of all road accidents by women in Norway over more than two decades; • Ch. 5: the results for the DRAG-Stockholm-2 model for the County of Stockholm provide evidence of the countereffectiveness of certain safety measures, as well as complementary evidence on the unexpected effects of alcohol and pregnancy found in Ch. 1 and 4, repectively. A model for the City of Stockholm is under development; • Ch. 6: the TAG-1 model for France, the only time series model presented that includes a speed equation, shows how speed on the intercity road network responds to various determinants, such as fuel prices; • Ch. 7: the youngest of the models, TRACS-CA for California, contains explorations of quadratic effects for a number of variables that raise many unanswered questions. The reader will find in Ch. 8 a comparison indicating closeness among many national results and pointing to new policy options: for instance, the important role of fuel prices as leading safety control variables, a result that should count as one of the important findings of this book. Part II. The second section of the book does not contain complete regional models. However, it presents a number of safety research innovations. I would note the following: • Ch. 9: in a simultaneous cross sectional analysis of safety outcomes and speed, the authors introduce Knight's famous distinction between (calculable) risk and (not calculable) uncertainty to test and account for functionally identifiable (non random) gaps between actual (realized) and controlled risk (represented by chosen speed). They also introduce a measure of expected risk based on random utility theory and isolate many fine effects of road design on safety and speed, with due regard to nonlinearities of the various

xvi Structural Road Accident Models responses; • Ch. 10: the analysis by type of road network presented for France, with Box-Cox transforms used on a subset of explanatory variables of a vector autoregressive (VAR) model, allows for a clear rejection of the popular linear VAR form and generally supports logarithmic (constant elasticity) forms but contains some evidence of finer (non constant) elasticities—implying the presence of saturation effects over time—for some individual variables. Part III. Exacting researchers requiring complete descriptions of estimation methods and statistics obtained for the various models presented will be well served by the third part of the book, extracted from TRIO software documentation (Gaudry et al., 1993). Also, Appendix 1 provides links to web sites presenting downloadable TRIO-generated TABLEX tables of results for those seeking to make comparisons with their own results, or simply wishing to analyze, for any equation, the exact elasticities for all variables of a model or, for any variable, the sign patterns found across all equations of a model. Readers should everywhere appreciate the use of elasticities to report on results for all variables, including qualitative (dummy) variables. One wishes that such standardized measures were used more frequently in order to empower readers to decide easily on the reasonableness of results. Despite this helpful elasticity-based presentation, the research and policy communities still have much left to digest: to quote Frank Haight again, the heroic efforts made here "go beyond the well-known formula devised by Reuben Smeed over fifty years ago and challenge us all to understand and apply the models reported on". As this occurs, I have no doubt that the approach to road safety documented in this book, with its emphasis on multiple-layer multivariate flexible-form specifications, will lead to an even larger family of DRAG-inspired models. REFERENCES COST 329 (1999). Models for Traffic and Safety Development and Interventions. Final Report of the Action, Directorate General for Transport, European Commission. Gaudry, M. (1984). DRAG, un modele de la Demande Routiere, des Accidents et de leur Gravite, applique au Quebec de 1956 a 1982. Publication CRTS59, Centre de recherche sur les transports, et Cahier #8432, Departement de sciences economiques, Universite de Montreal, 220 p. Gaudry, M. and M.J. Wills (1978). Estimating the Functional Form of Travel Demand Models. Transportation Research, 12, 4, 257-289. Gaudry, M. et al. (1993). Cur Cum TRIO? Publication CRT-901, Centre de recherche sur les transports, Universite de Montreal. OECD Road Transport Research (1997). Road Safety Principles and Models: Review of Descriptive, Predictive, Risk and Accident Consequence Models. OCDE/GD(97)153, OECD, Paris.

Research Support, and More xvii

RESEARCH SUPPORT, AND MORE

Marc Gaudry Sylvain Lassarre

RESEARCH AND ITS IMPLEMENTATION The long road to new states of the art. It is not possible to finance public interest research without backing from and some risk taking on the part of funding agencies, public agencies in particular. Indeed, some risk taking is necessarily involved in new methodologies producing new results, perhaps including unpopular results; also, significant amounts of money are necessary if modelling has to reach beyond the production of academically significant and publishable results to yield realistic and credible resuhs, always including forecasts. This book is no exception. Mixing colours. Although the Quebec Automobile Insurance Board (originally called RAAQ, now known as SAAQ^) provided the original and principal funding of the Quebec models presented in this book, the research network itself could not have developed without direct contributions of network partners and their government agencies to the construction of their own models and to the sustainance of the TRIO sofware used by all. In addition, general purpose funding of our linkage activities provided continuity and greatly helped, notably for the joint international supervision of doctoral work. So we provide in these pages a few words about the intertwining of funding streams, in case minor lessons might be drawn by readers about the helter-skelter research and policy process, and to give some well deserved thanks^ ^ Established by the Quebec government since March 1, 1978, as a state-run monopoly insurance entreprise for bodily damage claims arising from road accidents, the Regie later became the Societe de Vassurance automobile du Quebec. ^ In particular, Manuel Ramos and Annie Thuilot deserve our warmest thanks for the excellent job done in producing a first version of the camera-ready manuscript, finalized by Catherine

xviii Structural Road Accident Models BACKGROUND SUPPORT AND RISK TAKING IN QUEBEC'S ROAD SAFETY RESEARCH FUNDING Twelve years out of seventeen. During the last 17 years (1982-1999), the Quebec Automobile Insurance Board has provided a string of grants: firstly to the CRT^ university research project (1982-1984) towards the original DRAG-1 model, and then to implement it (1989-1991) and to insure in-house development of DRAG-2 at its Quebec City headquarters since 1991—^where support for model development is now oriented towards the continued use of the model as an official policy evaluation and forecasting tool^. But the SAAQ has also contributed since 1991 to continuing methodological work on two-moment analysis^ at CRT through its common research program with the Quebec Ministry of transport (MTQ), jointly run by the Quebec government's research funding agency (FCAR). The SAAQ also stepped in to provide basic support (1996-1998) for the DRAG network proper when funding intended for this purpose suddenly vanished from the group grant where it had been embedded since 1991. But this list does not do justice to a particular person, Andre Viel, whose understanding of modelling, willingness to take risks and foresightedness effectively determined many crucial outcomes. Risk taking by civil servants. In December 1982, Andre Viel, as head of research at RAAQ, accepted to fund an unsollicited research proposal. This proposal sugggested to develop an approach inspired by the successful three-level system (Gaudry, 1980) of aggregate structural Demand, Performance and Supply equations implemented by the Montreal transit authorities^ to explain and forecast monthly transit ridership. It also stated that flexible (Box-Cox) mathematical forms should be applied to variables in order to obtain credible results. But it did not include a clear idea of how « safety performance » would be formulated in the new model—beyond noticing the availability of data on road victims by category—and certainly gave no inkling of the interest of using within the model a clear distinction between the frequency (accidents) and the severity (victims per event) of accidents; neither did it raise more complex issues about the nature of driving behaviour, such as later arose by viewing accidents in a multi-moment framework. In effect, the RAAQ just took a chance on something proposed

Laplante. ^ DRAG-type safety research is now carried out at the Agora Jules Dupuit (AJD) of the Center for Research on Transportation (CRT), a joint research centre of Universite de Montreal, Ecole polytechnique and Ecole des hautes etudes commerdales, all located in Montreal. ^ In 1999, the administrative basis for development of DRAG-3 has effectively been set. ' See Ch. 1 and 3 for details. ^ And, to a lesser extent by those of Toronto as well.

Research Support, and More xix from outside, a form of risk taking ruled out in practice by the new system^ of mandated research topics applied since 1998 under the revised terms of the SAAQ-MTQ-FCAR program, or for that matter by the « tasks » under the various Fourth and Fifth Framework programme calls of the European Commission. But Andre Viel took other risks without which this research stream might have ended in a trickle: when RAAQ authorities ordered a reorientation of their 1985 joint research program with the Quebec government's FCAR research fund, with a view to excluding from it DRAGtype research because they disliked some of the results found in the report they had just received (Gaudry, 1984), he maintained a strict scientific and professional neutrality until these authorities changed their mind and decided five years later to implement the model in house. Fortunately, John Lawson, then director of road safety research at Transport Canada, had stepped in to provide minimal « survival» ftinding in the meantime. More recently, Claude Dussault, current head of research at SAAQ, similarly supported DRAG network activities when amounts intended for this purpose were arbitrarily « reassigned » within a multi-project group grant. This decision made a very large difference to the vitality of the international network.

OTHER SPECIFIC, JOINT AND COMMON FUNDING OF A RESEARCH NETWORK National databases. After 1984, it took about three years before the elaboration of two new models started at Karlsruhe and Oslo. Ulrich Blum and Lasse Fridstr0m then funded the construction of national databases from local sources, as did leaders of other modelling efforts more than six years later: Sylvain Lassarre in Paris, Goran Tegner in Stockholm and Patrick McCarthy in West Lafayette. Graduate students at the Master's (Diplomarbeit) and Ph. D. levels were involved in Canada, Germany and France. In all cases, high quality fully documented databases, like the DRAG-1 database (Gaudry et al, 1984), were constructed progressively over a period of years, allowing for successive generations of models. It is generally very difficult and expensive to finance work on new high quality national time

^ Although mandated research topics make it impossible for managers of large multi-project research grants subjected to inadequate supervision within the university to redistribute fimds to the advantage of their personal projects after these funds have been procured globally for a group of projects, one wishes that it had been possible to close the door to such « larceny » within universities without mandating topics. Project-specific funding approval would have sufficed to reestablish honesty by forbidding « redistribution » among projects.

XX Structural Road Accident Models series: for instance, derivation of monthly vehicle kilometrage from motor vehicle fuel sales and other indicators took more than one person-year for France—it is therefore gratifying that the resulting series and its methodology will become a permanent feature of the French national accounts in 2000, an outcome greatly facilitated by Michel Houee's encouragement. In Germany, the construction of vacation calendars by province required one man-year of work and the numerous statistical issues related to unification since 1989 still constrain the credible available data set. The youngest of the databases, for California, was constructed on a shoestring budget. In Belgium, The Netherlands and Israel it has not yet been possible to fiind the construction of a database, despite the interest and availability of research team leaders at Louvain, Delft and Ben-Gurion universities. Vested interests, often using only individual data of a cross-sectional nature, have objections to time series analysis and do not share our catholic perspective on the comparative advantages of different types of data. At other times, evaluators used to bivariate, or even to multivariate, linear relationships shy away from relationships determined without the comfortable straight]acket of fixed functional forms. Collaboration. Perhaps more difficult still is the financing of international collaboration. In this case. Marc Gaudry was fortunate to benefit from three sources of flexible funds. Firstly, throughout the whole 17-year period, from the Natural Sciences and Engineering Research Council of Canada (NSERCC): their program evaluates individual researchers every three years, with the emphasis on their output rather than on the contents of their proposals. They do not assume that, if you need a portable PC or a 24-hour trip to Karlsruhe in the middle of July for a thesis defence, they are a better judge of relevance that you are. It would be surprising if there were worldwide a more productive and cheaper to manage funding program than this Canadian NSERCC program. Secondly, since 1984, from the Alexander von Humboldt Foundation of Germany: in particular, the abillity to spend over many short stays in Germany his 1990 research prize award {Forschungspreis), and to combine it with a DFG guest professorship at Karlsruhe in 1993, was extremely helpftil, in view of the long-term nature of the DRAG collaboration projects. Thirdly, support in 1998 from the French National Centre for Scientific Research (CNRS^) through the tenure of a research position at BETA^ in Strasbourg was helpful in the same way: the great freedom associated with this position made a crucial difference to the coordination of activities necessary for this book, in particular those linked to the doctoral ^ Centre national de la recherche scientiflque. ^ Bureau d'economic theorique et appliquee, Universite Louis Pasteur and UMR CNRS 7522.

Research Support, and More xxi theses of Laurence Jaeger and Karine Vernier. Collaboration was also greatly helped by the membership of Lasse Fridstrom and Sylvain Lassarre in the road safety methodology committees of the OECD (1997) and the European Commission (COST 329, 1999). Conferences. Many researchers organized useful full day seminars on the DRAG approach: Roger Marche (at IRT/ONSER, in Arcueil, 1985), Sylvain Lassarre (in Strasbourg, 1993), Fran9ois-Pierre Dussault (at IRRST in Montreal, 1993) and Goran Tegner (in Borlange, 1996). Also, the series of well attended INRETS*^ road safety modelling seminars in Paris, financed by the Directorate for Safety and Circulation on Roads (DSCR^^) of the French ministry of transport (MELTT) since 1992 and organized by Sylvain Lassarre, provided salutary collaboration opportunities, as well as chances to expound upon work-in-progress and to produce proceedings^^. These seminars culminated in the international conference on DRAG-type models held in November 1998 in Paris, where first versions of the papers found in this book were presented, thanks to joint financial support from the DSCR, the SAAQ and the DFK (Swedish Foundation for Transportation Research). In view of the complexity and limited availability in English of most models, it was then decided to obtain better-coordinated second versions of all papers even if this certainly slowed down publication of the book. Common tools. All but one of the models presented here rely on some of the algorithms implemented within the fully documented TRIO software'I It has been possible to finish Version 2 of this program in 1993 (Gaudry et a/., 1993) with a large TRIO-DRAG contract funded by Transport Canada (1991-1993). Since then, all DRAG network participants have made important contributions to its maintenance, much larger than those made by other users*"^. However, one member, Ulrich Blum, made exceptional financial contributions to the maintenance of the program and even funded in 1998-1999, through a grant from the German

^" Institut national de la recherche sur les transports et leur securite, in Arcueil. ^^ Direction de la securite et de la circulation routieres, Ministere de Vequipement, du logement, du tourisme et des transports (MELTT). ^^ The series starts with Paradigme (1993). ^^ See Part III of this book, where two of these algorithms are partially documented. ^^ The TRIO user network currently has about 100 registered members worldwide.

xxii Structural Road Accident Models Research Foundation (DFG^^) at Dresden University, the extension to the third moment (see Ch. 3 and Ch. 12) of the two-moment analysis available in the LEVEL-1.4 algorithm*^ since 1991.

REFERENCES Carre J.R, S. Lassarre et M. Ramos eds. (1993). Modelisation de Vlnsecurite Routiere. INRETS, Arcueil. COST 329 (1999). Models for Traffic and Safety Development and Interventions. Final Report of the Action, Directorate General for Transport, European Commission. Gaudry, M. (1980). A Study of Aggregate Bi-Modal Urban Travel Supply, Demand, and Network Behavior Using Simultaneous Equations with Autoregressive Residuals. Transportation Research, B 14, 1-2, 29-58. Gaudry, M. (1984). DRAG, un modele de la Demande Routiere, des Accidents et de leur Gravite, applique au Quebec de 1956 a 1982. Publication CRT-359, Centre de recherche sur les transports, et Cahier #8432, Departement de sciences economiques, Universite de Montreal. Gaudry, M., Baldino, D. et T.C. Liem (1984). FRQ, un Mchier Routier Quebecois. Publication CRT-360, Centre de recherche sur les transports; Cahier #8433, Departement de sciences ecomiques, Universite de Montreal, 215 p. Gaudry, M. et al (1993). Cur Cum TRIO ? Publication CRT-901, Centre de recherche sur les transports, Universite de Montreal. OECD Road Transport Research (1997). Road Safety Principles and Models: Review of Descriptive, Predictive, Risk and Accident Consequence Models. OCDE/GD(97)153, 105 p. Paris.

'^ Deutsche ForschungsGemindschaft. ^^ See the updated description in Ch. 12 and the application in Ch. 3.

Multiple Dimensions in Road Accident Models 1

MULTIPLE LEVELS, DAMAGES, FORMS, MOMENTS AND VARIABLES IN ROAD ACCIDENT MODELS Marc Gaudry

1.1. INTRODUCTION: THE «MODELLING QUARTET» IN THIS BOOK^ The first part of this book contains a family of six models that explain the Demand for Road use, Accidents and their Gravity (DRAG), sharing a structure, use of flexible form regression analysis, calibration with monthly time series data defined over a country or region, and the establishment of a reference set of documented results. The purpose of this chapter is to introduce their common approach in terms of these four components that define a "Modelling Quartet" for any model: formulation, quantification of effects, data type and the expression of results. The focus on the six members of the family will also initiate to the two models presented in the second part of the book and to the algorithms found in the third part. Second generation models. The six national or regional models presented are all in effect second generation models partially documented at earlier stages of development for Quebec (DRAG-2 Foumier and Simard, 1997), Germany (SNUS-2; Gaudry and Blum, 1993), Norway (TRULS-1; Fridstr0m and Ingebrigtsen, 1991, Fridstr0m, 1997a, 1997b), Stockholm (DRAGStockholm-1; Tegner and Loncar-Lucassi, 1997), France (TAG-1; Jaeger, 1994, 1997, Jaeger 1

This chapter is based on a previous draft (Gaudry, 1998) written at the invitation of the third Annual Conference on Transportation, Traffic Safety and Health, Washington, D.C., December 2-3, 1997, organised by The Karolinska Institute, The World Health Organization, VOLVO and the U.S. Department of Transportation through The Bureau of Transport Statistics (BTS) and The National Highway Traffic Safety Administration (NHTSA). Model acronyms are defined in the Post Scriptum, Section 1.7 below.

2 Structural Road Accident Models et Lassarre, 1996, 1997a, 1997b) and California (TRACS-CA; McCarthy, 1998). Stylized facts about mature models. As stated in the Foreword, various aspects of the approach have also been partially documented (Gaudry, 1995a), examined and made the object of a technical annex in the OECD Committee RS6 report on Safety Theories, Models and Research Methodologies (OECD, 1997), and carefully studied by the COST 329 Committee on Models for Traffic and Road Safety Enhancement and Action (COST 329, 1999). This introductory chapter can therefore concentrate on stylized facts about the models, leaving general methodological points to these committees and detailed issues to the model-specific chapters. From stylized facts to perspectives. To provide perspectives on the models, we draw disproportionately, but not exclusively, from the initial model for Quebec, called DRAG-1 (Gaudry, 1984), and from its current version DRAG-2 due to the previous existence of an English version (Gaudry et al, 1996) of its summary presentation in French (Gaudry et al, 1993a, 1993b, 1993c). As the official model of the Quebec automobile insurance board (SAAQ), it is fully documented in French (Gaudry et al, 1993d, 1994a, 1994b, 1995a) in reports that cover 950 pages, are written for educated laymen and contain graphs of variables as well as detailed explanations. This abundant Quebec documentation only explains in part the perspectives presented here: they pertain to all DRAG-type models, and to many other models. In order to introduce both to the approach and to issues of importance, we state various "perspectives" on model dimensions: with respect to model formulation, we state 3 perspectives on model structure (PS-) and 2 perspectives on variables (PV-), both dependent and independent; with respect to quantification methods, we state 4 perspectives on the mathematical form (PF-) used to determine relationships; with respect to data types and the expression of model results, we state 1 perspective each, respectively denoted (PD-) and (PR-).

1.2. P R O B L E M F O R M U L A T I O N

1.2.1. A multilevel structure; a multidamage application Three-level transportation systems. In our approach, safety is a dimension of transportation system performance modelled as a third and explicit level between the classical supply and demand levels. Some years ago, we introduced (Gaudry, 1976, 1979) this 3-level structure to capture the fact that realized transportation service levels often differ from supplied service levels and we estimated a full system for an urban area (Gaudry, 1980). We called the resulting structure "Demand-Cost-Supply" to distinguish it from "Demand-Supply" structures of classical Economics. In that new structure, costs denote realized money, time or safety levels. Naturally, using the D-C-S system instead of the classical D-S system gave rise to new equilibria, such as the "Demand-Generalized Cosf equilibrium that differs from the "Demand-

Multiple Dimensions in Road Accident Models 3 Supply" equilibrium within the same 3-layer system. We then relabeled the D-C-S system as a D-P-S (Demand-Performance-Supply) system, added layers and changed the notation (Florian and Gaudry, 1980, 1983), to that used in Figure 1.1 to make it more accessible within the wide transportation subculture. Demand-Performance equilibria in networks. Here we want to focus on the middle level where, given the supply actions [S, T, F] undertaken and actual demand D, the performance level yields market-clearing money and service level conditions, including safety. The performance level determines actual queues, level of congestion and risk, as well as other forms of modal performance (effective capacity, occupancy or load factors and crowding, etc.) conditional on both actual demand and given supply actions. We neglect here the formal discussion of equilibrium conditions on P and F, C and T, as well as on D and S, that may allow for steady state Demand-Performance-Supply solutions. In addition, we refer the reader to the 1976 and 1979 papers cited above for detailed discussions of the car trip market where observed car flows and occupancy rates associated with vehicle network performance levels (particular Demand-Performance «network»equilibria) need not simultaneously imply the existence of Demand-Supply «market» equilibria for car trips within households. Naturally, some of the issues are defmitional. For instance, we have applied this three-level structure to the reestablishment of equilibrium in Centrally Planned Economies through black market prices and queues (Gaudry and Kowalski, 1990), distinguishing between free and regulated queues, to avoid the explicit modelling of disequilibrium in these economies, which yields very peculiar results such as the finding that the Polish economy exhibited excess supply most years between 1955 and 1980 (Portes et al, 1987)! Similar issues arise in modelling centrally-planned health care: the explicit representation of the performance level avoids silly regression work where it is found that state-ordered reductions in the supply of doctors are found to reduce health-care "costs" (due to a longer queue) and increases in the supply of doctors are found to increase them. DRAG application. One approach to the problem of explaining the number of road victims is to relate it, or its components (fatalities and injuries), directly to the demand for road use and to a set of other factors, as in figure 1.1. But the approach taken in DRAG is not so direct: rather, the number of victims is decomposed through an accounting identity into three elements, namely exposure, frequency and severity, which themselves become the objects to be explained. Thus, the number of victims VI is equal to the product of exposure (kilometres driven), accident frequency (accidents per kilometre) and the severity of accidents (victims per accident). This means that an explanation of the number of victims is effectively derived from the separate explanation of the three terms of the identity, as in the upper part of Figure 1.2. We note that the distinction between the three levels

4 Structural Road Accident Models matches the linguistic distinction between exposure activities and so-called auto protection activities (through "fail-safe" objects or behaviour) that influence accident frequency or socalled auto insurance («safe fail») activities that influence accident severity. (1.0) VICTIMS <- [DEMAND FOR ROAD USE, OTHER FACTORS] D

=

Dem(P, C,Y,A)

[P,C]

= Per(D, [ S , T , F ] )

[ S, T, F ] = Sup ( SO, RE , [(W ( S*, T* )], ST) ST = (P**,C**,D**)

Risk.

DEMAND PROCEDURE PERFORMANCE PROCEDURE SUPPLY ACTIONS PROCEDURE

with: D: market demand; P: out-of-pocket unit expenditures; C: levels of service; Y: consumer socio-economic characteristics and their budget; A: economic activity; S: quantity supplied; SO: supplier objectives; T: scheduled service levels; RE: regulatory environment; F: scheduled price, or fare; ST: suppliers' estimate of the state of the system; [W (.)]: set of minimum cost combinations for the reaUzation of any scheduled (S*, T*); D**, P**, C* denote realized values of demand, unit costs and service levels. Figure 1.1. Market Analysis: a three-level approach This figure also contains a set of subsidiary and unspecified equations. One equation can be thought of as determining the occupancy rate of vehicles jointly with the supply of vehicle-trips (observed as the demand for road use) and other equations as determining the characteristics of vehicles purchased and their state of maintenance, the desired speed and vigilance (these are complements) and driver ebriety (as related to fatigue, alcohol and medical drugs, for instance). Different models of the family have different subsidiary equations (e.g., the French model has an equation for speed, the Norwegian model an equation for vehicle ownership and the Quebec DRAG-1 model equations for aggregate alcohol consumption) and car ownership, but in most models there is no equation to explain the level of activities that influence individual risk: the absence of particular important components of the ARC vector in the DR-A-G equations proper transforms these structural equations to some extent into reduced form equations. Observed multi-damage risk substitution. Such a structure makes it possible to search for evidence of risk substitution among observed exposure, frequency and severity risk dimensions. For instance, snow, a factor included in the three groups of explanatory variables Xi, X2 and X3, might lead to less driving (DR decreases) and, at the reduced exposure level, to more accidents (A increases) but less severe accidents (G decreases): the net impact on the number of road victims results from the relative strength of these potentially offsetting effects.

Multiple Dimensions in Road Accident Models 5 D-P-S structure for the Transport system i D-P-S structure for Households or Firms >l Demand for vehicle travel <-[

]

(DVT) Motor vehicle occupancy

m

<—[

]

p

(MVO) Agent risk control (ARC)

<—[

]

Demand for road use


),

, FACTORS]Exposure risk

(DR) VI ^

Accident frequency

(XO <-[(DR, MVO), ARC, FACTORS] Frequency risk

(A)

Accident severity

(X2)

<-[(DR, MVO), ARC, FACTORS] \Severity risk

(G)

(X3)

Realized speed

^ [ DR ,

(V) Infrastructure services

<—[

, ARC , T ,

] ]

(T)

m //;6

1 where [ M = (M-C, M-M) = (Motor vehicle characteristics, Motor vehicle maintenance) ARC = 1V* = desired Velocity or speed (not to be confused with realized speed V) 1 B = Beh use Lc = Competence <- (Driver Quality (Age/Sex), Vigilance and Ebriety, Other) Figure 1.2. DRAG structure In addition, each dimension is broken up by subcategory: type of road use (gasoline or diesel), category of accident (fatal, injury, etc.) and measure of severity (mortality, morbidity). Such disaggregation means that patterns of substitution that are not too coarse can be detected: clearly, single-equation models (say on the number of fatalities) make it difficult to detect substitution because the substitute (say injuries) is not jointly considered. Peltzman (1975) was the first to explain three subcategories. Substitution can be observed in principle with respect to any explanatory variable found in accident models: prices, vehicle availability and characteristics, network characteristics (legal regimes, modal mix, weather, etc) consumer characteristics and activity levels or trip purposes (employment, shopping, etc?). And we shall come back below to the importance of distinguishing between multi-category observed or nominal damages and underlying utility-adjusted or real damages defined within a multimoment interpretation.

1

6 Structural Road Accident Models 1.2.2. Perspectives on problem structure The mystery of 1972-1973. In many of the OECD countries, the absolute number of road fatalities reached its most recent maximum in 1972-1973 (Israel, Belgium, France, Germany, The Netherlands, Finland, Canada, United States, New Zealand, etc.) or very close to this moment (Denmark, Sweden, Japan, others) and generally ahead of the first oil crisis (October 1973); in other countries of low motorization (Spain, Ireland, etc.) this did not happen and the absolute number of fatalities is still often increasing. This major turn-around led to selfgratulation in different countries, each attributing her good fortune to a different set of local measures (ranging from alcohol restraints to vehicle safety design improvements), but is still a major puzzle because vehicle kilometrage was barely affected by either the first or second oil shocks and has kept increasing rapidly since then. Such a major change, occurring in countries of comparable development around 1971-1973, points to a major structural change. What are the candidate explanations? Car occupancy and congestion are certainly prime candidates. Market equilibrium, car occupancy and Smeed's Law. The major simultaneous changes in economic performance—^the rate of productivity increase has a breakpoint (the rate of change is cut in half) in OECD countries and unemployment starts increasing in most EU countries— in 1973 suggests a structural change associated with the post-war baby boom of the late 1940's. Labour markets have to adjust to a lower equilibrium price and marginal productivity (a phenomenon difficult to capture with fixed-coefficient Cobb-Douglas or Constant Elasticity of Substitution production functions estimated without due care for this structural change). As car occupancy is not observed continuously, the best proxy variable is population per vehicle, which falls sharply in the early 1970's. In consequence, the occupancy rate must fall sharply. As it is impossible for a driver to compensate for risk in proportion to the number of occupants in the vehicle, the frequency of fatal accidents naturally falls, but the severity of fatal accidents falls even more: in Quebec, where it had been essentially constant for 20 years, it takes a 45degree turn in 1973 and has since fallen by more that 60 %; similarly, in Britain, the proportion of car occupant deaths in total road deaths, which had increased steadily since 1952, starts to fall in 1973. As the age-weighted risk index of the quality of drivers fell due to large increases in the share or relatively young drivers until about 1980, this factor cannot explain the structural change: it is then hard to resist the conclusion that the new car occupancy and associated modal mix changes were decisive in explaining the turning point: in effect, fewer people per car (and fewer pedestrians and cyclists sharing the road) atomises or internalises risk as drivers take risks more and more only for themselves or "for" other vehicles which carry fewer passengers on roads with fewer pedestrians. This view explains why Smeed's (1949) almost log-linear observed positive relationship between the number of road accident deaths per vehicle in a country and her population per vehicle fits the data well until the mid-70's and then over predicts fatalities (Fournier et

Multiple Dimensions in Road Accident Models 7 Simard, 1999) due to the absence of an occupancy term in the equation. We therefore venture: PS-1: In many advanced countries, we are approaching the point where car occupancy (of 1.1 persons per car) stops falling and the age-weighted risk index of the quality of drivers (linked to the U-shaped age-risk curve), improving for about 20 years, starts worsening due to population ageing. Both imply an upward pressure on road fatalities. However, there may be many more countries approaching the favourable turning point where vehicle occupancy and mode mix effects start to offset the impact of increased motorization on road fatalities. We do not know.

Indeed, it does not seem possible to make well-founded aggregate forecasts of the evolution of world road fatalities without working out how far each world area is from this desirable turning point. If that was not done in a recent study (Murray and Lopez, 1996) to forecast the rank order of the road traffic accidents among the top 15 causes of disability-adjusted life years (DALYs) losses, its results may be exceedingly pessimistic in suggesting a movement from rank 9 to rank 3 between 1990 and 2020. Network equilibrium and congestion. A second candidate, also difficult to observe, is congestion. Starting around 1968, the level and share of public investment in highways started to fall in Canada, the United States and some other OECD countries. As a consequence, the number of cars per kilometre of road started to increase rapidly (breaking trend) in 1970. There are almost no time-series of average speed over regions or countries, except in France, so there is no substitute for determining the effects of speed in an indirect way. This matters: for instance, apparent improvements in driver fatality rates per cohort, observed in the United States over the period 1975-1990 (Evans, 1993) could be entirely attributable to falling average speeds and (as I suspect) have nothing to do with driver experience. Clearly, one would expect the average speed to fall gradually as more roads become congested, a result naturally found in Mahmassani's network simulations. On the road link level, one would expect increased congestion to eventually cause a fall in the number of accidents and the resulting frequency curve to have the shape of an inverted U, as traffic increases from zero to gridlock. One might also hypothesise, as shown in Figure 1.3, that each type of accident would exhibit a different shape of its specific asymmetric inverted U. Some work on the relationship between the value of road damages and traffic volume by road link, done in Israel (Cohen, 1980), has found the very shape shown in part B of Figure 1.3. Although it is only normal that the aggregate curve should behave in similar fashion, it is however by no means clear where the turning point occurs by accident severity type for an average measure and a particular country or region.

8 Structural Road Accident Models To detect indirectly the effects of congestion in the aggregate network, a series of experiments performed in December 1992, as part of the tests of robustness of the DRAG-2 reference set of results, were designed to detect the presence of a symmetric inverted U (or "integer" quadratic form) in the relationship between total vehicle kilometrage in Quebec and the frequency of fatal, injury and material damage accidents. The results, that appeared later (Gaudry et al, 1995a), showed a strong quadratic effect on fatal accident, a weaker one on injury accidents and a weaker one still on material damage only accidents. Although these results were reported, quadratic forms were not incorporated in the reference set of equations until a second series of tests to update the model in 1997 included (symmetric) quadratic effects on the frequency of fatal accidents, the severity of fatal accidents (fatalities per fatal accident), and the number of persons killed. This 1997 model update (Foumier et Simard, 1997) therefore incorporated a form implying that, increasingly often in Quebec, the addition of vehicles reduces fatalities. More recently still, fiirther tests on these three outcomes (fatal accidents, the severity of fatal accidents and the number of persons killed) have shown that the quadratic effect was not symmetric (i.e. the maximum is not reached in the same way on both sides of the inverted U), a finding of decisive importance for the production of forecasts until 2004, as will be documented by the authors of Chapter 2. A. Volume-Delay curve

B. Volume-accident curve Average ; time i

Marginal Time

!5

1

1

<

>-i

a>

a (D

a

H

/'

Fatal '"i"^ / " ^^^^^^=C;^'^\;;^^\Material

\

y''^

/

/

/

.,.-:i^l^^--'"'^^

Number of vehicles per unit of tim

'o o cd

C+H

o

\-*.

(L>

^

1 Number of vehicles per unit of tim

The scale used in B corresponds only approximately with that used in A

Figure 1.3. Speed and accidents on a road link To some extent in parallel with that work, the first version of the Stockholm model in 1996 incorporated symmetric shapes with respect to vehicle kilometrage in the explanation of the total number of bodily injury accidents (the only frequency measure used in the first version of that model), and more complex shapes on the different measures of morbidity and mortality (the severity measures) used.

Multiple Dimensions in Road Accident Models 9 Moreover, in the French model of Chapter 6, the explicit use of a speed variable provides still more interesting results on the impact of increased traffic density at given average speeds. Whence another perspective on model structure: PS-2: Increased congestion and stop-go traffic in many countries are such that additional vehicles confer a positive safety externality on others, by reducing everyone's speed. This has already influenced the number of fatal accidents and should soon be of] decisive importance for the number of injury accidents, as additional vehicles reduce the absolute frequency and the severity of accidents in some of the countries. However, the location of this turning point within each country is not known. Risk substitution among observed outcomes. For many of the explanatory variables, all DRAG-type models show strong substitution among the different dimensions of risk and their subcategories, where substitution is defined either weakly as different marginal effects of a variable, including opposite signs across categories, or more strongly as different elasticities. Some types of variables, associated with policing and heavy fines, have strong across-theboard effects while others, such as gasoline and car maintenance prices, and weather, have stronger substitution effects. We therefore suggest: PS-3: It is fundamentally inadequate to talk about overall accident frequency, or even about the total number of road accident victims. Increased attention must be paid to the frequencies and severities by subcategory as many desirable measures increase the total number of accidents but reduce their average severity, with implications for human organ supply and hospital care costs.

1.3. T H E Q U A N T I F I C A T I O N O F E F F E C T S

1.3.1. From fixed to flexible mathematical form The problem of signs in multivariate regression. If models were univariate, it would naturally matter to obtain the correct functional form for the relationship between dependent and explanatory variable: as is obvious from Figure 1.5, the quality of the fit and any measure of impact (partial derivative, or elasticity) and statistical significance would be modified by use of the proper form. However, it is hard to imagine how a monotonic transformation of either or both variables could change the sign of the correlation. This situation is however radically changed in multivariate models because regressors are not generally orthogonal (uncorrelated) in any form: the sign obtained for a given variable depends on the covariances among regressors. In consequence, as all practitioners know, changing the form in which variables enter the regression can alter their signs. As finding the right form

10 Structural Road Accident Models variable by variable in discrete trials is a time-consuming combinatorial game, the Box-Cox transformation became the most widely used non-linear monotonic transformation of variables in applied work (Davidson and MacKinnon, 1993) because it includes the linear and multiplicative forms as special nested cases—^to say nothing of all other classical values of interest (square root, etc.) shown in Figures 1.4 and 1.5. Choosing desired resuks. We thoroughly documented the influence of form on regression signs (and elasticities) many years ago, both in transportation (Gaudry and Wills, 1978) and elsewhere (Blum and Gaudry, 1990), as recalled in Figures 1.6.A. and I.6.C. Despite this, many papers are written after searching "by hand" for the form that gives the desired results. For instance, in a recent study of the impact of air competition on the dispersion of prices charged by airlines, the authors (Borenstein and Rose, 1994) obtain positive and significant signs for 3 of their crucial variables (pertaining to the degree of competition) when the model is linear, but negative and significant signs when the model is multiplicative (log-linear). As indicated in Figure 1.6.B, they then accept the log-log model without formal tests of the appropriate form, preferring to ignore that such tests might well invalidate their preferred finding, perhaps by revealing a point between the linear and logarithmic cases, and insignificant results, or even a point not too far from the linear point or contrary to their expectationsl Figure 1.4: Box-Cox models of levels: 4 simple forms nested in Eq. (1.1)

E

a

SEMI-LOG AND UNEAR

INN/ERSE LOG AND LOG-LOG

My)-Q

X(X) =

X(X)=\ ?i(X) = 0

k(X) = 0

i ••* -I

SEMI-LOG

- /.(\i-;i,/-(X)~()

' "^ J

\

Multiple Dimensions in Road Accident Models 11 Figure 1.5. Classical shapes in linear regression

Fortunately, other authors scientifically document the changes in signs and find that, in credible models, the optimal form is often that which yields the most reasonable signs, for instance in Logit models of the type found in Chapter 9 (e.g. Fridstr0m and Madslien, 1995). We therefore propose this first perspective on form: PF-1: Models of untested fiinctional form are not credible, irrespective of the economic or other theory on which they are based. Such tests, applied to many current models using predetermined monotonic forms would show that the models are not robust. This is true for all data types used: revealed and stated preference data, experimental data.... Form and nonspherical residual errors. The regression algorithm used for all members models, called^ LEVEL-1.4 and available in thefiiUydocumented TRIO program (Gaudry et al, 1993), simultaneously accounts for the mathematical form of an equation with Box-Cox transformations on variables, multiple-order autocorrelation and heteroskedasticity of the residuals of a very general form (Gaudry and Dagenais, 1979). The method both distinguishes fiinctional form determination from obtaining constant variance residuals and strikes a balance between simple fixed form structural models and pure Box-Jenkins models, avoiding the pitfalls (Dagenais, 1994) of approaches that whitewash series containing errors in the variables—clearly the normal state of affairs with series generated by samples or models. Formally, the econometric specification, defined more completely in Chapter 12, is then:

(1.1)

(^„)

K

Uy )

k =1

" LEVEL-1.5, documented in Chapter 12, is the most recent version of this algorithm.

12 Structural Road Accident Models (1.2)

(1.3)

^K,) eXp Y^^rnZ,

^i = Z A V / + > ^ ,

with the Box-Cox transformation defined for any strictly positive y, Zm or XR variable as:

^^^z/A^O, (1.4)

xr-

lnX^z/A=0.

Old favorites. This definition makes it possible to explicate more formally Figure 1.4 because Equation (1.1) contains special cases, such as the multiplicative (log-log) regression model if the transformation coefficient/i = 0, the linear \i X= 1, a square power model if /I = 2, and so on. But, naturally, these are standard cases and there is no particular reason to obtain «classical» values. In fact, all member models in this book obtain equation-specific forms for (1.1) and autocorrelation schemes for (1.3). One model (Fridstrom, 1997a) derives and tests a theory of the form of heteroskedasticity (1.2) in a preliminary equation formulated to estimate DR from road link counts and fuel sales (most member models derive DR only from fuel sales) in the 19 Norwegian counties, but this background work to the TRULS-1 model is not described in Chapter 5. In most equations, it is generally the case that linear and logarithmic models are rejected by the data, with significant consequences for fit and elasticities. In particular, it is often possible to reject the accident frequency model that assumes proportionality between accidents and kilometrage in models that use a monotonic form for exposure DR. In models that use symmetric or asymmetric forms for exposure, a proportional model is not even meaningful. The considerable gains made by the analysis of the appropriate form for each of the equations belonging to DR, A or G levels demonstrate the losses of understanding involved in performing regression analyses directly on the number of road victims, as in Equation (1.0): even if all equations were log-log, there would be losses. Indeed, because VI = DR*A*G, regression coefficients obtained by regressing the number of victims VI (by category) on the explanatory variables would be equal to the sum of the individual coefficients obtained on the DR, A and G equations, but interesting information on the individual coefficients would be lost (as they clearly differ, even in sign!); for instance, a net coefficient of zero on the snow

Multiple Dimensions in Road Accident Models 13 Figure 1.6. Form and Signs in Models of Levels A . Transit Demand in Montreal

Gaudry and Wills, 1978; Results from Figure 19, Wait-time, Travel Time and Income Elasticities. y variable: monthly transit demand by schoolchildren 2

.„

1.5 .

«« -

1 .

'\

0.5 .

o

...«-

0 .

••••™

---m&:::::::::^

1-

-----

y^ / «,

-0.5 . -1 .

m^.

-1.5 _ -2 -2.5 -

LAMBDA

> 0

2

1 1

Wait-time

Travel-time

.^

Income 1

Comment: note how variables change signs. The optimal value is close to one: at that point, public transit is a superior good and loses clients if in-vehicle time increases. The opposite occurs with a logarithmic form. B. Airline Price Dispersion Market Structure, U.S.A. Borenstein and Rose, 1994; Model 2 from Table 3 Coefficients and t-statistics conditional on form y variable: X variables...

Gini ticket price dispersion index 4 Linear

3 Log-log

Optimal X 1

C. Household Savings, Germany Blum and Gaudry, 1990; Models from Tables 4 and 5 Elasticities and t-statistics conditional on form y variable : X variables

Annual household savings || 1 Linear

2 Log-log

5 Optimal X

a) Self employed Monopoly Duopoly Large-duopoly Small-duopoly Competitive Lambda (?i) Log-likelihood

+0,154 (+4,81)

-2,169 (-5,27)

n.c. n.c.

Social insur. I contributions |

-0,149 (-10,24)

(+2,86)

+0,054

-0,048 (-5,35)

+0,174

-2,033

n.c.

Lambda (A-)

1,00 Fixed

0,00 Fixed

+0,648

(+4,97)

(-9,46)

n.c.

Log-likelihood

65,477

-649,585

248,304

n.c.

-0,022

-0,117

(-2,77)

(-0,21)

n.c.

-0,017 (-1,89)

-0.067 (-1,10)

n.c. n.c.

Social insur. contributions

-0,962 (-2,97)

+0,245 (+0,28)

b) Low-income white-collar -0,836 (-2,41)

+0,172

-1.807

n.c.

Lambda (A,)

1,00 Fixed

0,00 Fixed

+0,748

(+7,16)

(-6,98)

n.c.

Log-likelihood

-139,074

-293,503

-126,687

1,00 Fixed

0,00 Fixed

n.c.

n.c.

n.c.

n.c.

Comment: No log-likelihood values are reported by the authors who arbitrarily choose Model (3) after stating that "the main qualitative results are robust to changes in functional form" (sic !).

Comment: The impact of social security contributions on household savings is expected to be negative, as it is in both cases when the optimal form is used. The optimal point is close to half way between the linear and logarithmic.

14 Structural Road Accident Models variable in the equation for fatalities hides strong and offsetting effects on DR (-), A (+) and G (-) components! But generally, the loss of valuable information can be much larger still, as the optimal forms differ among the equations, especially among those that belong to different levels of performance. For instance, none of the 9 equations contained in the DRAG-2 model are linear or log-linear. Hence we venture without risk the following complementary perspective:

PF-2: The availability of powerful algorithms to determine the flexible monotonic form of | effects in multivariate regression will help professionals overcome their insecurity at abandoning the predetermined linear or other presumed fixed forms used in road accident models. This should hold for all types of regression models: classical, Tobit, Poisson, Gamma, Logit, Probit... 1.3.2. From monotonic to multitonic forms: the case of alcohol. From Box-Cox to asymmetric quadratic forms in classical models. We mentioned above that asymmetric "quadratic" forms, namely forms in which the exponent 2 of a proper quadraticterm is replaced by any other real number (positive or negative, integer or not (except for the value 1)) were in current use to determine the effect of congestion within the last version of the DRAG-2 model. Such asymmetric forms, estimated with TRIO, are obtained by putting a Box-Cox transformation on the variable of interest. To see this without reading Chapter 14, consider the simple model where y and X are strictly positive variables:

(1.5)

/ ^ ^ = A + A ^ ^P.X^'^Uu.

and (X,y) and (X-x) denote Box-Cox transformations on the variables y and X, respectively. We realised in 1990 that setting the Box-Cox transformation on the X variable to 2 naturally yielded the normal symmetric (integer = 2) quadratic form but could yield any asymmetric Ushaped form if this transformation was allowed to take any value (except 1), as long as the two regression coefficients were of opposite signs (with the conditions for a maximum or minimum reversing themselves depending on whether the value of the Box-Cox transformation on X was smaller or greater than 1) and independently from the value of the Box-Cox transformation on the dependent variable. Figure 1.8 illustrates various departures from symmetry, with a simple case of a linear dependent variable. We immediately used this property of Box-Cox transformations to study the role of alcohol within the DRAG-1 model, but only recently wrote a note (Gaudry, 1996), the background work for Chapter 14, to document this point, long known among TRIO users. The resulting shapes can be J-shaped, as found in Figure 1.9 which shall be further discussed below.

Multiple Dimensions in Road Accident Models 15 In 1993, we also carried out (still unpublished) dose-response tests on two anti-asthma drugs, using clinical data from a Montreal-area firm on the response to these drugs. The form apparently mandated by the U.S. Food and Drug Administration at the time was shown to be dominated by the more general asymmetric forms estimated with model (1.1) which implied different shapes of the response curve for both drugs than the mandated shape, and in consequence different maxima and areas under the curve. In probabilistic models with response asymmetry. The method can also apply to probabilistic models such as the Box-Cox Logit used in Chapter 9. But B-C Logit forms, both symmetric and not, are shown without dips or «quadratic» effects in Figure 1.7. We thus propose: PF-3:

U-shaped or J-shaped curves need not be symmetric and can be easily compared in nested Box-Cox fashion with linear and curvilinear forms, whether the dependent variable be an unrestricted level, a probability, or a share.

First alcohol results, 1983-1984 and the Driver mix conjecture. During the development of the DRAG-1 model in 1982-84, I rapidly found negative correlations between total alcohol consumption (or consumption by type such as wine, spirits, beer, cider) and some measures of accident frequency or severity, as indicated in the final report to the Quebec Automobile Insurance Board (Gaudry, 1984; Gaudry et aL, 1984). Some of these results, shown without measures of statistical reliability in Table 1, were particularly significant and robust: notably the drop in fatal accident frequency and severity associated with increases in total alcohol consumption and all results associated with wine. Interestingly, both of these results remained, but at a lower level of statistical significance in the DRAG-2 model (Gaudry et aL, 1995a). The key to my provisional acceptance of these early results, obtained with all the limitations of aggregate data, was an examination of the disaggregate results obtained in the famous Grand Rapids study (Borkenstein et aL, 1964, 1974), where the relative probability, as summarised in Goldberg and Havard (1968), appears to be J-shaped. The fact that relative probability falls with moderate consumption is acknowledged in the Borkenstein et aL study. However, little is made of it and the literature that follows concentrates exclusively on the strongly rising part, with occasional exceptions, including one of Borkenstein's co-authors (Zylman, 1968) and a few others, with an emphasis on sampling (AUsop, 1966) or on the frequency of drinking (Hurst, 1973). I came to state a Driver-Mix Conjecture, worded approximately in the 1984 report and more precisely in a summary of selected results (Gaudry, 1989): Driver-Mix

Conjecture, «When aggregate consumption of alcohol increases from

relatively

low per capita levels, the overwhelming majority [of drivers] drink a little, which reduces their risk below that prevailing when they do not drink—perhaps

because they compensate or are

less aggressive; this reduction may more than offset the increase in risk for the minority [of drivers] who drink more heavily, depending on their relative proportion on the road».

16 Structural Road Accident Models

Pl(X,)f

0

2

4

6

8

10

12

5^^

Figure 1.7. Linear-Logit vs Box-Cox-Logit Second alcohol results, 1990-1991. Results obtained soon afterwards in New Zealand (Scott et ai, 1987) and Norway (Fridstrom et al., 1989) were both partly consistent with those of DRAG-1. Further searches of individual data sets and results unearthed quite a few results on accidents or on the physiological response (blood pressure) to low levels of alcohol consumption consistent with the idea that—^to restate the conjecture more formally—,«if the individual risk curve is J-shaped, the net effect of higher alcohol consumption will depend on its distribution among drivers» (Gaudry, 1993). But, as Figure 1.9 makes obvious, observations that are truly «quadratic» can be fitted quite well with monotonic functions. Clearly, quadratic effects are hard to detect with monotonic functions unless samples are precisely segmented to obtain a piece-wise (often linear) approximation. The results of my formal tests of form, summarised in Table 1.2 (Gaudry, 1993) differ from those of Table 1.1 in that both injuries and fatalities now decrease with higher alcohol consumption. In addition, tests on the first version of the model for Germany (with monotonic forms) have yielded small (but not very significant) decreases in all accident frequencies with increased beer consumption (Gaudry and Blum, 1993), and the first full version of the model for Stockholm exhibited very significant (symmetric) U-shaped effects of alcohol consumption on the frequency of bodily-damage accidents (the only measure of frequency considered in this first version) and less significant U-shaped effects on the severity of all categories of bodilydamage accidents (Tegner and Loncar-Lucassi, 1996). Considering all this, including the comparative advantages of aggregate and disaggregate data, well summarised in Fridstr0m (1999), and keeping in mind that there is a project at NHTSA in Washington, involving R. Compton and R. Blomberg, to replicate the Borkenstein study at two sites (Long Beach, CA and Fairfax, VA), it may well be time to heed the advice given to us on the phone in November 1996 by R. Crowther (who now lives in Toronto), one of Borkenstein's co-authors who believes that the «dip» in their study was real:

Multiple Dimensions in Road Accident Models 17 PF-4: «Marc, the Borkenstein study «dip» is real: it is urgent to revisit the problem using more advanced statistical tools.» R. Crowther to M. Gaudry, November 1996.

30

1

1

1

20 10 ^

\ ^ >^ > / \ / ,' >^ 1

! I i

0 g

1

1

1

!

1

1 \

>v

1 \ Lambda2 =1.5

1

i

1

1

r

^

!

!

'

\

\

iS.

1

1

N.

^

N-- - ^ 1

' \ LarVda2 =

-10 -20 I

!

-30

L

L

-40

1

! \

1

Lambcla2 = 2.5

J

1

\-~-,^--

1 |.

^

-50

\

J

[

A

8

9

10

Figure 1.8. From symmetric (>^ = 2) to asymmetric Qw "^ 2) quadratic forms

Relative L Accident Probability

•

l\:i::.

y

^ Blood Alcohol Concentration

/

y^

1

l:;:;:i:l Observations

1

1 Monotonlc transfornnation

1—-\ Linear

h—H True form

DUFigure n i . 9 . Linear, curvilinear and asymmetric quadratic shapes

18 Structural Road Accident Models Table 1.1. DRAG-1 results with Box-Cox forms on alcohol (1982-1984)* With Box-Cox transformation on

Explanatory variable Total alcohol

Spirits

+

+

1

Wine

Beer

Cider

1

Effect on Accident frequency with Material damages only

+

-

+

Injuries

+

Fatalities Total accidents Accident severity Injured per bodily damage accident Killed per bodily damage accident Derived effect on road victims Injured Killed

+

+

+

-

+

+

-

+

+

-

+

+

0 0

Total victims (injured + killed)

1

+

-

+

0

-

0

1 * The form used on the alcohol variable is the same as that used on other variables of the equation in question. None of the optimal forms are linear or log-linear: the parameter values are between those two special cases.

Table 1.2. DRAG-1 results with nonmonotonic forms on alcohol (1990-1991)* !

Explanatory variable

Asymmetric quadratic form on Effect on

sign

shape

1 Beer

Wine

Spirits

Total alcohol

sign

shape

+

n

sign

shape

sign

1

shape 1

Accident frequency with

-

Material damages only Injuries

-

u

+

/

-

u

+

n

-

u

-

w

+

n

u

+

n

u

Fatalities

u Total accidents

-

\j

1 Accident severity Injured per bodily damage accident

n

\

Killed per bodily damage accident Derived effect on road victims Injured Killed 1

Total victims (injured + killed)

u

-

+

-

+

-

-

n

u

-

1 * The alcohol variable is used both linearly and with a specific Box-Cox transformation distinct from the transformation used on the other variables of the equation. The symbols u and n , denote that a quadratic form was accepted, instead of a monotonic form, but does not distinguish between symmetric and asymmetric cases. The sign is calculated with both terms considered and represents the net effect of the variable (using the sum of the two elasticities in question), all derivatives being evaluated at the sample means of all variables.

Multiple Dimensions in Road Accident Models 19 1.4. VARIABLES: MULTIMOMENT, MULTIVARIATE L4.1. The dependent variable: from observations to moments Multiple moments. But the psychology of risk is not as simple as we have implicitly assumed all along. To the extent that people are trying to achieve a combination of objectives through the control of a single instrument, observed variations in the level of this instrument are just tools to reach the multiple objectives. For instance, there is no doubt that in purchasing financial assets, people care about their return (expected value), their risk (variability, often represented by variance or standard error) and the asymmetry of risk (downside or upside variability, representable by skewness) or asymmetry of the frequency distribution of the return. More formally, the expected utility of the random prospect gi, ..., gj, ...gn of positive or negative gains or returns gi depends "not only on its average, but also on its distribution as a whole about its average", as stated by AUais (Allais, 1987) who also formulates the utility corresponding to the monetary value Fof a random prospect, for someone with wealth C, as: (1.6-A)

u(C, F) = u + R (|i2, ..., ^P,...,),

where u is the mathematical expectation of the Ui (the cardinal utilities corresponding to the different gains gi), while jiip denotes the moment of order p of these utilities u\, and the ratio (1.6-B)

r=R/u «can be considered as an index of the propensity for risk: for r = 0, the behaviour is Bernoullian [only the first moment matters] , for r positive, there is a propensity for risk and for r negative there is a propensity for security''. In effect, Allais (op. cit.) «adds to the Bernoullian formulation a specific term characterising the propensity to risk which takes account of the distribution as a whole...».

In this perspective, one can derive from any model explaining an observed variable y sample measures (say the partial derivative or the elasticity of y with respect to any explanatory variable Xk) and measures that recognise the fact that y is a random variable (e.g. the regression has a random error), for instance the partial derivative or elasticity of the median, or of the expected value, or of the standard error, of y. Consider for example in some more detail the notion of elasticity (although we shall also present below similar computations from partial derivatives), to which we shall come back later. Elasticities and multiple moments. The notion of elasticity invented by Alfred Marshall in Palermo in 1882 may be defined in a number of ways. Its intuitive Palermitan form is simply the ratio of the percentage change in a left-hand-side variable y to the percentage change in a

20 Structural Road Accident Models right-hand-side variable Xk belonging to a general function such as (1.7)

y ^

f ( x , , . . . , X , , . . . X ^ ),

namely (1.8)

%Ay % A X,

Ay/

^f

AX

Ay

X 100

•^Z r Xk X 100

Xk

AX

where y ' and Xk are the reference levels of y and Xk . As percentages are involved in the ratio, an elasticity is without units. However, it has a sign that may be positive or negative depending on the direction of the effect associated with Xk . As the expression (1.8) is written in terms of discrete changes in Xk and y, it is called an arc measure of elasticity. Quite naturally, one prefers the point measure associated with infinitesimal changes provided by the calculus, and a more formal writing to recognise the fact that (1.7) may be a non-linear function. The first, intuitive, notion, the sample measure based simply on model fitting parameters, is: (1.9)

ri ( y , x , ) =

x^

5y 5 X,

y

x j , , y \ X',

where the vertical line simply means that the derivative and the reference levels are «evaluated at» y = y ' and X,. = X|^ for the variables of interest and at X^ = X ' for other righthand-variables belonging to (1.7). This expression makes it clear that, even if the function (1.7) of interest is linear, the value of the elasticity depends on the reference levels and is therefore not constant. More formally, drawing from Dagenais et al. (1987), where one can find a general discussion that also deals formally with the special cases of dummy or categorical explanatory variables, the measures in terms of the first two moments are, for the expected value and for the standard error, respectively: (1.10)

r| ( E ( y ) , X, ) =

£(y )

X,

a X,

E(y)

Xi; , [E(y)]', XJ

and (1.11)

ri (a( y ), X, ) =

do(y

a X,

)

X,

a(y) Xi; , [a(y)]', XJ

Application to accident frequency, severity and victims. The idea that people "trade" between different moments is very appealing, and should be no less interesting in the analysis of

Multiple Dimensions in Road Accident Models 21 transportation accidents than in explaining financial asset choice, where the increased complexity of this necessary realism has kept authors unduly shy in empirical work. As a minimum, we state that people's choices will reveal their empirical trade-offs. As the normal distribution has an expected value that depends only on the mathematical expectation and the variance of the random variable in question, the obvious place to start, if the data are aggregate enough that a normal distribution of the residuals makes sense, is the couple (Expected value, variance): we simply assume that drivers adjust their behaviour in order to obtain the Expected value and Standard error of the risk dimension of interest. We have for this purpose included in TRIO procedures that compute the three partial derivatives and the three elasticities found in (1.9)-(1.10)-(1.11) for a very general model with Box-Cox transformations on dependent and independent variables, multiple-order autocorrelation and heteroskedasticity according to (1.1) to (1-4). We originally performed these tests on DRAG-1 equations in 1991-1993, but we present here selected results derived from the most recent DRAG-2 model, the model in current use by the Quebec Automobile Insurance Board (SAAQ) to make forecasts (Fournier et Simard, 1999). We show in Table 1.3 the results for the accident frequencies, severities and victims (the latter obtained from their own equations, not derived from the products of the frequencies and severities). In Table 1.3, we first note that the sample and expected value elasticities are almost identical (if the model were linear and had no Box-Cox transformations, they would be). We then ask whether a single-(first)-moment model would be sufficient and conclude that the second moment also matters because all elasticity ratios are different from zero. But then, is the tradeoff rate between one unit of expected value against one unit of variance equal to one, as for instance the Poisson model assumes? Because we are working with the standard error, the trade-off rate in terms of variance cannot be easily derived from the marginal rate of substitution listed. However, the equality of the first two moments, and a unitary trade-off rate, would then imply an elasticity ratio of 0,50 because the elasticity of the standard error equals half of the elasticity of the variance: in Table 1.3, this occurs for only one of the measures of severity (the number of injured persons per bodily injury accident), implying rejection of a Poisson restriction in all other cases. As the sign of this ratio constitutes a measure of Allais' coefficient of risk propensity, we generally note a propensity for risk (ratios are positive) except in the case of mortality (killed per fatal accident) which exhibits a propensity for security: when drivers increase their expected severity measure, they simultaneously decrease its variability. It is then reasonable and credible to state:

22 Structural Road Accident Models PV-1: Single-moment accident models that do not consider trade-offs with higher moments should be displaced by multiple-moment models. A Poisson trade-off or MRS equal to one between the expected value and the variance is demonstrably too restrictive.

Table 1.3. Two-moment own substitution, DRAG-2 results (November 1997) Accident category Dependent variable y—>

Victims

Severity

Material 1 Injury 1 Fatal Morbidity 1 Mortality

1

Injured 1 Killed

Sample monthly values mean|Liofy

8129

2189

78,3

1.25

1.13

3124

91,2 1

standard error cj of y

4985

1355

51,3

0,28

0,85

1909

61.3

0,05

0,05

0,08

0,01

-0,01

0,05

0,08 1

Marginal rate of substitution MRS ratio partial derivatives aVfi' Elasticity w.r.t. temperature variable 1. sample value of y (Eq. 1.9)

-0,27

-0,05

0,20

-0,01

-0,04

-0,04

-0.18

2. expected value of y (Eq. 1.10)

-0,27

-0,05

0,19

-0,01

-0,03

-0,04

-0.17

3. standard error of y (Eq. 1.11)

-0,19

-0,04

0,12

-0,01

-0,01

-0,03

-0.10

4. Allais'r = (3)/(2) (Eq. 1.6-B)

0,71

0,72

0,60

0,50

-0,24

0,77

0,61

Student's t-statistic of P coefficient

(14,79) 1 (2,99) 1 (5.40) (-1,38) 1 (-1,45) (-2,37) 1 (4,37) 1 •Sample monthly values over the period December 1956-December 1996 (481 observations). 1 •Marginal rates of substitution (M.R.S.) are the ratios of the partial derivative of the standard deviation (square root of the variance) of the dependent variable y to the partial derivative of the Expected value of the dependent variable y, drawn from Equations (l.lO)-(l.ll), i.e. evaluated at the means of all variables including the means of the standard deviations and expected values of individual observations for y. •Elasticities used are those with respect to the variable temperature, a very important variable in many models, including those with many explanatory variables. This variable exhibits an interesting sign pattern indicating substitution among the different risk categories; the tstatistics of the underlying regression coefficient are conditional upon the estimated value of the Box-Cox transformations of each equation. •The three elasticity measures, sample, expected value and standard deviation, are computed according to Equations (1.9)-(1.10)-(1.11), respectively. •The ratio of the elasticity of the standard error to that of the expected value is an indicator of Allais' r coefficient of risk propensity, as is the ratio of partial derivatives. The ratio of elasticities also allows to test whether the first two moments are equal: if they were, the ratio would equal 0,50. Conditional t-statistic

Of moments in general. The fact that one can compute empirical utility trade-offs among moments opens the door to certainty-equivalence units, constructed as «generalised costs» in transportation analysis: in that measurable sense of a utility construct, the intuitive notion of

Multiple Dimensions in Road Accident Models 23 «homeostasis» (Wilde, 1982) might be made meaningful. For instance, some safety measures, such as the addition of driver information, may reduce accident variance but increase total accidents because the drivers re-establish the subjective certainty equivalence of their acts. In addition, there might also exist for the consumer another conceptually identifiable gain to the addition of this safety information if one could find the analogs of the substitution and income effect components of microeconomics. A decomposition problem of the same type must be definable, but with utility functions defined directly over moments. Puritans, liberals, and The Law. In this respect, various legal developments can be contrasted in terms of the distinction between observations and moments. Consider the «criminalization» of drinking and driving. It involves moving away from observed accidents by particular individuals (liable if there is an accident, irrespective of true cause) to the expectation (correct or not, that is another problem, as Section 1.3.2 makes clear) of an accident without due regard for individual variance (liable even if there is no accident, irrespective of inter-individual variability). Contrast this with the movement away from class (e.g. age or sex) insurance rating based on expected value towards giving due weight to variance (through experience rating). In the first case, one is moving from observed guilt to expected guilt; in the other, from expected guilt to variance of guilt. In the first case, Puritans naturally feel guilty irrespective of actual behaviour and, in the second. Liberals naturally feel free irrespective of expected bondage. But we are surrounded by two-moment trade-offs: travel in different planes by royal family members going on the same trip abroad...; strict vs. limited liability firms; average GNP growth in income per capita v^*. inequality in income per capita; efficiency through deregulation of industry at the cost of variability (busts and booms). And of course there are other moments, such as the third: nuclear deterrence with few and small war incidents vs. the «peace dividend)) of reduced asymmetry but greater mean and variance of the number of wars... And what of portfolios? In Table 1.3, we only compute the trade-offs for each risk dimension (3 frequencies, 2 severities, 2 classes of victims) between the moments of 3. given dimension: in addition to these own trade-offs, all cross trade-offs also exist. As these trade-offs among the two moments of all dimensions are assumed to be independent of direction, the resulting triangular matrix for the complete multimoment multi-damage portfolio, of dimension 14 by 14, contains (n^-n)/2 = 91 rates of substitution. As this is large, we show in Table 1.4 only the portion pertaining to accident frequency. The lightly shaded values are the own MRS also found in Table 1.3. The rest is read in the usual fashion. For instance one reads that drivers behave as if they were indifferent between 3,17 expected material damage accidents per expected injury accident or 11,88 material damage accidents per fatal accident. These bold and italicised values are a first way of understanding the trade-offs involved. But, if one looks at the second moment, one notes that drivers would «give up)) fewer material damage «spread)) (standard error) units per fatal accident spread unit (7,50) than they are willing to give up

24 Structural Road Accident Models material damage expected value units per fatal accident spread unit (158,33). Quite a complex set of trade-offs, especially with third moments (see Chapter 2) but still simple by the standards of financial market portfolios. Table 1. 4. Two-moment, three-good marginal rates of substitution, DRAG-2 (Nov. 1997) Accident frequency

[d (moment i)/ d X]/ [d (moment j)/ d X]

Material Accident frequency category Material

i /j a 1^

Injury

Fatal

a

a 1,00

category Fatal

Injury

M-

a

^

a

0,05

3,21

0,15

7,50

0.56

1,00

67,86

3J7

158,33

11,88

1,00

0,05

2,33

0,18

1,00

50,00

3,75

1,00

0,08

a

^

1,00

To an economist, much in the behaviour of nature also looks suspiciously like portfolio choice: (i) the idea of evolution by group selection, whereby individuals collaborate socially for the benefit of the group, compete for territory and status and are programmed to regulate their numbers (in opposition to the idea that individuals simply compete for food and seek to increase their numbers); (ii) the idea that complex ecosystems have different chances to fall apart than simpler ones, thereby giving a foundation for biodiversity; (iii) some biological traits appear more commonly than they ought to appear if they were the result of a simple random mutation (bi-polar disorder (manic depression); homosexuality), which suggests that an expected value/variance trade-off occurs to maintain the certainty-equivalence of the portfolio...

1.4.2. The explanatory variables: not a triad, but a quatrain No accidents without activities. We want to point out interesting results related to the casual and obvious triad «driver, vehicle, infrastructure)), to which we add a fourth term «Activity)) (the variable A in Figure 1.1). Activity is fundamental to accident analysis because the basic level of activities and their composition determine the derived total and modal demands for transport and the occupancy rates of vehicles by persons and goods: this demand «scales)) the triad (which is implicitly defined at constant activity levels). All models in this book use variables that can be classified among these four classes: e.g.

Multiple Dimensions in Road Accident Models 25 Activities: employment level and composition by industry branch, shopping, vacation or income or output measures are «activities» that determine total or modal demands. The DRAG2 model in Chapter 2 contains an extremely refined representation of the structure of the demand for freight transport and an excellent representation of the structure of the demand (trip purposes) for passengers. Vehicles: vehicle availability and characteristics (size, equipment) both matter. Concerning the latter for instance, the Stockholm model relates accident frequency to the number of cars with registered brake errors at annual inspections (Tegner and Loncar-Lucassi, 1996; see also Graph 5.11 below). Infrastructure: With the exception of the Stockholm model of Chapter 5, physical infrastructure variables are few in most of the models, but the model for Norway in Chapter 4 contains many indirect (financial) indicators of infrastructure quality. However, weather variables are of major interest in all models and, once DR has been controlled for, effectively characterise infrastructure. Drivers: many factors influence driver behaviour. The relevant list, as demonstrated in this book, certainly includes fuel prices, legal insurance regulatory regimes, infrastructure access and penalty regimes and general competence linked to age, sex, ebriety, fatigue and, we suggest, pregnancy. About factors affecting drivers. Consider prices and regulations first. The models generally show that automotive fuel and vehicle maintenance prices tend to reduce all accident frequencies and severities. The first result suggests that higher fuel prices lead to slower speeds, as demonstrated in Chapter 6 for highway speeds. The second may be linked to the compensating prudence associated with having a car in bad condition (see previous paragraph on infrastructure). Concerning regulations, the reader will find many results related to safety belts. Concerning insurance regimes, the reader interested in detailed analysis of the impact of the establishment of a strict no-fault system of automobile insurance for bodily damages (without experiencerating) in Quebec should read papers on estimates found with the DRAG-1 model. Summarily stated, this law had disastrous effects (Gaudry, 1991), as it led to a very large increase in the share of young drivers, male in particular. This effect exists in the DRAG-2 model, but it is hidden within the variable ((Proportion of drivers aged I6-24» of Figure 2.2 below, and has not been separately identified. Concerning competence, we want to say something about pregnancy, but must before remind the reader of an interpretation difficulty specific to aggregate data in the case of explanatory

26 Structural Road Accident Models variables that pertain to a subset of the population. When variables such as prices or weather pertain to everyone, the risks of spurious correlation is quite different from what it is when variables pertain to a subset of the dependent variable. In the case of pregnancy, this risk can be reduced greatly if data are available on subsets of that variable. Fridstr0m (1997b) defines to this end informal subset tests that we present intuitively, before addressing the case of the pregnancy variable: interested readers can get the formal treatment in Section 4.2.4 below. The issue would be the same for many other competence-determining factors (in addition to alcohol already mentioned), such as hours worked"^ and the consumption of medical drugs. Totals and subsets: Fridstr0m's test. Consider the case of a variable T defined as the sum of components, for instance: (1.12)

T = Ti + T2 + T3

from which one can define shares, for any component c: (1.13)

Pc = Te/T

and remember that we are considering the impact of changes in Xk on the total T. In order to decide whether such impacts are due to a true causal relafionship, it may be possible to do supplementary regressions of the set of variables used to explain T on some component of T. As we have some a priori knowledge on the extent to which any particular Xk affects specific components, there may be ways of exploiting such knowledge. For the sake of clarity, Fridstrom does not discuss this problem in terms of partial derivatives with respect to Xk but instead conveniently starts from the known (Oum et al., 1992) rewriting of Equation (1.13) in elasticity form, which is valid under the usual conditions for the existence of a consistent aggregate: (1.14)

T] (T) = h (TO ] pi+ [11 (T2) ] P2 + h (T3) ] P3

which is simply a sum of elasticities weighted by shares. He then considers cases where one component (say T3) is known to be equal to zero. In his first series of tests of interest here, he tests an ordering of the remaining elasticities under the maintained hypothesis that one of the elasticities (say rj (T2 ) ) is equal to zero, which leaves two possibilities, if the regression signs 3 The variable "average hours worked" used in DRAG-1, where increases in hours worked implied a substitution from light accidents (material damage only accidents fell significantly) towards more numerous and more severe personal injury accidents, was not retained in DRAG-2 due to the fact that it was constructed only from manufacturing sector employment data. However, there is little doubt that fatigue is relevant, notably for night shift workers: for instance, Novak and Auvil-Novak (1996) report that 95,5% of their sample of shiftwork nurses had either closely escaped or had an had a road accident in the previous 12 months.

Multiple Dimensions in Road Accident Models 27 are correct and different from 0: (i) whether | r| (T) | is greater than | ri (Tc) |, called the affirmative subset test; (ii) whether | rj (Ti ) | is greater than [ rj (Tc*) = 0 ], called the complement subset test. In the first case, the test amounts to asking the very intuitive question "is the elasticity of the component c greater than the elasticity of the total"? In the second case, the test amounts to a comparison of the elasticity of the remaining component with that which is assumed to be equal to zero.

1.4.3. Is pregnancy a risk factor? DRAG-1 results and Conjecture 7. In the DRAG-1 model for Quebec, population pregnancy rates were found to have a strong influence on accident counts, in particular at the beginning of pregnancy: the first two-month pregnancy rates appeared to increase bodily injury accidents with an average elasticity of about 0,30 and the severity of fatal accidents considerably, notably during the second month of pregnancy (patterns for subsequent months were quite mixed in terms of values and significance, exposure effects no doubt blurring any effect on risk). I tentatively attributed this finding to the very strong alterations in the hormonal balance during pregnancy, stating Conjecture 7 of the report: Conjecture 7. "an increased ratio of oestrogen hormones to progesterone that is not compensated by a sufficient increase in androgen hormones reduces the ability to perform learnt mechanical tasks». (1984, op. cit., p. 149). Related results. In addition to explaining my results obtained from aggregate data for Quebec as a whole. Conjecture 7 had the advantage of explaining British results (Skegg et ah, 1979) on the influence of drugs on the probability of accident obtained with disaggregate (individual) data. In that study, oral contraceptives were found to be the category of drugs that most increased accident probability: they increased relative risk 5.6 times—^sedatives and tranquillisers increased it 5.2 times and drugs in general 2.0 times. As oral contraceptives increase the ratio of estrogens to progestogens by much more than an order of magnitude (we computed an average ratio of 33 for 24 brands available in 1978) and create a state of permanent pregnancy, they should be expected to affect the performance of tasks. My conjecture, inspired by results obtained on rats (Zuckerman, 1952), was not formulated with precision: one could in principle separately identify the effects of levels and of ratios of the various hormones with Box-Cox transformations applied to a general regression formulation. Other data? My medical adviser Dr. B. Leduc, responsible for gynaecology in the system of University of Montreal hospitals (processing about 15 000 pregnancies per year) was very interested in 1984 in testing this hypothesis with individual data, but the complexity and costs

28 Structural Road Accident Models of the task, as well as other factors, led me to temporarily abandon the project. At some point, the head of the psychology department at the French national research institute (INRETS) considered using a forthcoming simulator at Arcueil (near Paris) to explore the idea, but the simulator is not yet in place. Furthermore, during the development of the DRAG-2 model in 1990 or 1991, as series were improved and the sample size greatly increased, pregnancy rate tests yielded even stronger elasticities (some reaching 0,35,1 believe) than those of DRAG-1, with very high levels of statistical significance: so we decided to provisionally exclude these variables from the reference model, intending to study the problem later as a model variant. Fridstr0m's 1997 results. The situation was dormant until the summer of 1997 when Fridstrom (1997b), using his splendid data, provided major new evidence on this issue in TRULS-1. See Graph 4.14 in chapter 4. His data contains first quarter pregnancy rates of women aged 18-44 for the 19 counties of Norway over 22 years, i.e. 5016 monthly observations. Because of the great variability of this variable, and in view of the relatively small size of the counties and consequent size of the dependent variables (see below Graphs 4.6 and 4.7 for injury accidents and car occupants injured), these data are, so to say, halfway between «aggregate» and «disaggregate» data, without being count data. His results are reproduced in Table 1.5 in a form slightly different from that of Figure 4.11 below. In that table, the difference between the Column AR, for the reference model, and the others designed for subsets, pertains to the sample size, reduced by 20% in Columns A to D due to data availability on the dependent variables used in Columns B to D. One should first note that the elasticities in Columns AR and A are of the same order of magnitude as those found for early pregnancy rates in Quebec. More importantly, Fridstrom remarks that a comparison of Columns B and C indicates that the pregnancy variable passes the affirmative subset test because the elasticity for the subset (itself associated with an acceptable t-statistic) is larger than that for the total set. He also remarks that a comparison of Columns C and D indicates that it also passes the complement subset test because the elasticity for the subset of involved women is much larger that that for all other drivers, itself naturally not different from zero, as interaction between those potentially affected by pregnancy and the complementary set of other drivers is not expected to amount to much. It therefore seems appropriate to venture a specific perspective on this variable, in view of the potential practical importance of the topic: PV-2: «We do suggest that further research be done on the subject [of the influence of] pregnancy on accidents], preferably relying on disaggregate data». L. Fridstrom (op. cit., Ch. 6, November 21, 1997).

Multiple Dimensions in Road Accident Models 29 Table 1.5. Fridstr0m's (1997b) tests for pregnancy (from his Tables 6.1.2 and 6.4) Column

AR

Dependent variable Injury accidents

A Injury accidents

B Injured car occupants

C

D

Accident involved Killed or injured female car car drivers except women 18-40 drivers 18-40

Explanatory variables • Women

pregnant

in

first

quarter per woman 18-44 • elasticity • conditional t-statistic

0,17

0,20

0,14

0,32

0,02

(3,24)

(3,53)

(1,78)

(2,23)

(0,25)

• ... Sample size Period

5016

4104

4104

4104

4104

1973-94

1977-94

1977-94

1977-94

1977-94

This is a topic about which little appears to be known in the road accident research community"^. One place to start is the work on the menstrual cycle. For instance, Liskey (1972) linked the menstrual cycle and accidents, but unfortunately did not compare the accident rates between birth control pill users and non-users. He indicated to us in a letter in 1989 that he dropped pill users from the sample and only considered non-users. Related work has been done by Silverman and Eals (1992) who noted dramatic improvements on mental-rotational tests (a sort of visual task) while women were menstruating, with the highest scores whenever their oestrogen levels were at the lowest, and by Hampson and Kimura (1988) on systematic performance fluctuations on various skills across the menstrual cycle.

1.5. OTHER MODELLING DIMENSIONS OF INTEREST 1.5.1. Multidata Knowledge of the changing structure of the economy turns out to be as important as speed and (incorrectly observed) vehicle occupancy rates. The problem is complicated because one may classify data types into 9 classes, as indicated in Table 1.6. In principle, data should be neutral, but they are not. Modelling efforts are not uniformly distributed across data types. Moreover each type tends to have comparative advantages, with aggregate time-series data having comparative advantages in forecasting but comparative disadvantages in explanation, because of the reduced variance of observations. At the other end, individual data offer much understanding, but have to deal explicitly with all idiosyncratic factors that generally somehow ^ On June 5, 1998, Goran Tegner, Vesna Loncar-Lucassi and Christian Nilsson, of Transek AB, found results consistent with those described here-linking pregnancy and road accident frequency and severity-in earlier specifications of their model for Stockholm (Chapter 5).

30 Structural Road Accident Models cancel out at the aggregate level. Monthly time-series, for instance, strike a good balance between availability of data on all explanatory factors and variance of data. Higher aggregation kills variance; lower aggregation voids the structural explanation in aggregate models: indeed, no country currently produces economic forecasts with disaggregate data or models. Table 1.6. Data types in transportation demand and safety analysis Type of aggregation Time dimension Cross-section Time-series Pooled Aggregate Count Disaggregate In addition, as there is no explicit spatial dimension in Table 1.6, it is clear that some effort at working out the regional data needs is also long overdue. International comparisons make it possible to obtain sampling variance on usual variables and to discern the influence of cultural factors (necessarily of more limited variance locally) for instance on various forms of regulation. Our perspective on data in this book is therefore:

PD-1: As safety is a superior good and the demand for it increases rapidly with income, there is a need for a serious and long-range data requirement exercise. Requirements should combine data types and specify their regional and national dimensions. 1.5.2. Multiple documentation of reference results We have stated above the importance of computing the impact of variables on the different moments of the dependent variable. There are model types where that is a very difficult task to accomplish: for the Logit model, for instance, we are not aware of such derivations anywhere. Elasticities. There is however a computation that is within reach, but may not be in the interest, of most researchers: results that are not expressed as elasticities are almost impossible for the reader to decipher. Imagine how many articles would have been refused if, in addition to "statistically significant coefficients", they had been forced to produce elasticities that anyone can understand. Combined with the requirement that no result of untested functional form should be presented, it is clear that much time would be saved by editors and readers alike! We therefore state: PR-1: Save everyone a great amount of time by always requiring that results be expressed as elasticities, as in Eq (1.9)-(1.11) for models of levels, or as the equivalent probability points in disaggregate Logit models (see Chapter 9 on this). The reader will then be able to distinguish «statistically significant)) academic results from credible results.

Multiple Dimensions in Road Accident Models 31 Forecasts. All models belonging to the DRAG family tend to use a reference set of results. In all cases, such results naturally include «backcasts» in various forms. But recently, the Quebec Automobile Insurance Board has started to forecast with its model, as Chapter 2 will show. Conditional forecasts also form a part of model results and should provide a mechanism to evaluate models and decide whether they stand or not. 1.6. CONCLUSION: MATCHING TOOLS TO QUESTIONS People who ask complex questions cannot expect simplistic answers. The modelling «quartet» of theory (structure and variables), quantification (form and stochastic considerations), data and documentation (of the relevance) of results necessarily involves much work that is made considerably easier by international collaboration, as we hope that the reader will concur. However, this collaboration merely eases the sophisticated computer modelling tasks that are necessary to complement—and often do much better than—our intuition about explaining the past or forecasting the future in a way compatible with the refinement of questions that are asked. There is a natural development of complexity from the fixed-form single-damage model of Smeed (1949) to the fixed-form multi-damage model of Peltzman (1975), towards the multitonic, multi-level, multi-damage, multi-moment models presented in this book. This increasing complexity is required as a sensible way to match tools to the sophistication of the questions asked by policy makers. With it, there is less chance that our reach will exceed our grasp. It is hoped that this book is will make a contribution to this matching, and perhaps provoke more work within the outlined perspectives.

1.7. POST SCRIPTUM: MODEL ACRONYMS We state birth dates associated with the various models. A birth date requires that the model be gelled in a reference form and documented. 1984 Demande Routiere, Accident et Gravite Demande Routiere, Accident et Gravite 1993 1993 StrassenverkehrsNachfrage, Unfalle und ihre Schwere StrassenverkehrsNachfrage, Unfalle und ihre Schwere 1998 1991 TRULS-1 TRafikk, ULykker og deres Skadegrad DRAG-Stockholm-1 Demand for Road use, Accidents and their Gravity in Stockholm County 1996 TAG-1 Transports routiers. Accidents et Gravite 1997. TRACS-CA Traffic Risk And Crash Severity-CAlifornia 1998

DRAG-1

DRAG-2 SNUS-1 SNUS-2

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Multiple Dimensions in Road Accident Models 35 I'essence et au diesel. Gaudry, M., F. Fournier et R. Simard (1995a). DRAG-2 un modele econometrique applique au kilometrage, aux accidents et leur gravite entre 1957 et 1989 au Quebec. Societe de I'assurance automobile du Quebec. Partie 4. Application du modele aux accidents, a leur gravite et aux victimes de la route. Gaudry, M., F. Fournier et R. Simard (1995b). DRAG-2 un modele econometrique applique au kilometrage, aux accidents et leur gravite entre 1957 et 1989 au Quebec. Societe de I'assurance automobile du Quebec Partie 5. Synthese des resultats. Gaudry, M., F. Fournier et R. Simard (1996). A V4 summary of the DRAG-2 road accident model within the context of the international DRAG research network. Publication CRT-9627, Centre de recherche sur les transports, Universite de Montreal. Gaudry, M. and J.S. Kowalski (1990). Demand-performance-supply equilibria in centrally plarmed economies. Communist Economies, 2, 2, 205-221. Gaudry, M. and M.J. Wills (1978). Estimating the functional form of travel demand models. Transportation Research, 12, 4, 257-289. Goldberg, L. et J.D.J. Havard (1968). Alcool et Medicaments. Recherches sur la securite routiere, OCDE, Paris. Hampson, E. and D. Kimura (1988). Reciprocal effects of hormonal fluctuations on human motor perceptual-spatial skills. Behavioral Neuroscience 102, 3, 456-459. Hurst, P.M. (1973). Epidemiological Aspects of Alcohol in Driver Crashes and Citations. Project NR 309-033/4-27-72 (441), NTIS, Springfield, Virginia. Jaeger, L. (1994). Methodologie d'estimation du kilometrage mensuel des vehicules routiers en France. Rapport d'etape. Bureau d'economie theorique et appliquee, Universite Louis Pasteur, Strasbourg. Jaeger, L. (1997). L'evaluation du risque dans le systeme des transports routiers par le developpement du modele TAG. These de Doctorat de Sciences Economiques, Faculte des Sciences Economiques et de Gestion, Universite Louis Pasteur, Strasbourg. Jaeger, L. and S. Lassarre (1996). The TAG-1 Road Safety Model: First Results for France. Presentation made at the Oslo COST 329, Meeting on Road Safety, June. Jaeger, L. et S. Lassarre (1997a). Pour une modelisation de devolution de Vinsecurite routiere: Estimation du kilometrage mensuel en France de 1957 a 1993: Methodologie et resultats. Rapport DERA n° 9709, Institut national de la recherche sur les transports et leur securite, Arcueil, France. Jaeger, L. et S. Lassarre (1997b). Developpement d'un modele econometrique du risque routier. Institut National de la Recherche sur les Transports et leur Securite, Arcueil. Liskey, N. L. (1972). Accidents—rhythmic threat to females. Accident Analysis and Prevention, 4, 1-11. McCarthy, P. (1998). TRAVAL-1: a First Model of Traffic Volume and Accident Losses for California. Forthcoming, Krannert Graduate School of Management, Purdue University, Indiana.

36 Structural Road Accident Models Murray, C. J. L. and A. D. Lopez (1996). The Global Burden of Disease-Summary. The Harvard School of Public Health, Harvard University Press. Novak, R.D. and S.E. Auvil-Novak, (1996). Focus group evaluation of night nurse shiftwork difficulties and coping strategies. Chronobiology International, 13, 457-463. OECD Road Transport Research (1997). Road Safety Principles and Models: Review of Descriptive, Predictive, Risk and Accident Consequence Models. OCDE/GD(97)153, OCDE-OECD, Paris. Oum, T. H., Waters II, W. G., and J.-S. Yong (1992). Concepts of price elasticities of transport demand and recent empirical estimates. Journal of Transport Economics and Policy, 139154. Peltzman, S. (1975). The effects of automobile safety regulation. Journal of Political Economy, 83, 4, 677-725. Portes, R., R. Quandt, D. Winter and S. Yeo (1987). Macroeconomic planning and disequilibrium estimates for Poland, 1955-1980. Econometrica 55, 1, 19-41. Scott, G., G. Pittams and N. Derby (1987/ Regression Model of New Zealand Road Casualty Data: Results of a Preliminary Investigation. Transport Research Section, Ministry of Transport. Silverman, I. and M. Eals (1992). quoted in Time, January 20, p. 42. Smeed, R.J. (1949). Some statistical aspects of road safety research. Journal of the Royal Statistical Society Series A, Part I, 1-34. Skegg, D.C.G., S.M. Richards and R. Doll (1979). Minor tranquillizers and accidents. British Medical Journal 1, 917-919. Tegner, G. and V. Loncar-Lucassi (1996/ Tidsseriemodeller over trafik- och olycksutvecklingen. Transek AB, Stockholm. Tegner, G. and V. Loncar-Lucassi (1997). Demand for Road Use, Accidents and their Gravity in Stockholm: Measurement and analysis of the Dennis Package. Transek AB, Stockholm. Wilde, T. J. (1982). The theory of risk-homeostasis: implications for safety and health. Risk Analysis 2, 209-225 and 209-258. Zuckerman, S. (1952). The influence of sex hormones on the performance of learned responses. In: Hormones Psychology and Behaviour and Steroid Hormones Administration, pp34-44, Ciba Foundation Colloquia on Endocrinology, J. and A. Churchill Ltd, London. Zyman, R. (1968). Accidents, alcohol and single-cause explanations. Quarterly Journal of Studies on Alcohol, Supplement No. 4, 213-233. .

The DRAG-2 Model for Quebec 37

THE D R A G - 2 MODEL FOR QUEBEC Frangois Fournier Robert Simard

2.1. INTRODUCTION Development of a highway infrastructure plays a major role in any economy, including Quebec's, by improving transportation of people and goods. Development does, however, come at a cost, namely air and noise pollution and road accidents. Road accidents directly affect the Societe de 1'assurance automobile du Quebec (SAAQ) at two levels. The SAAQ is the agency set up by the Government of Quebec in March 1978 to oversee its new no-fault public insurance scheme for bodily injuries sustained by Quebecers in road accidents. A few years later, given its financial responsibility to Quebecers, in 1980 the government transferred to the SAAQ the mandate of controlling access to the road network through regulation of drivers and vehicles and promotion of road safety. To fulfil these mandates, the SAAQ must focus on minimizing bodily injuries from road accidents by promoting appropriate safety measures. It is therefore essential that the SAAQ have the necessary tools to better understand all aspects of road safety. To better understand trends in road accidents, in 1983-1984 the SAAQ granted funding to professor Marc Gaudry at the transportation research centre of the University of Montreal, under its road safety research program, to develop an effective means of analysis. This initiative led to version 1 of the DRAG econometric model. Given the valuable results obtained and the potential for this analysis tool, the SAAQ requested that professor Gaudry assist in implementing the model, which led to development of the DRAG-2 model. The SAAQ needed to obtain the following information: firstly, identify factors that influence changes in distances travelled in a vehicle, accidents and victims, as well as measurements on the direction, intensity and certainty of that impact and, secondly, a means of forecasting changes in distances travelled, the number of accidents and victims over the coming years.

38 Structural Road Accident Models 2.2. THE STRUCTURE OF THE D R A G - 2 MODEL This section of the chapter details the structure of the DRAG-2 model developed in Quebec, which essentially comprises two levels. Level one comprises two equations to explain exposure to the risk of accident based on distance travelled. At level two, seven equations are used to explain the frequency of accidents, their severity and the number of victims. Then, the nine dependent variables were considered using graphs to better illustrate changes in each variable in Quebec during the period covered by the model. Part three is devoted to the matrix of independent variables. Part four examines certain characteristics of the Quebec model. 2.2.1. A diagram of the model structure Level one, which focuses on exposure to risk, contains two equations, one for distance travelled by gas-powered vehicles, the other for distance travelled by diesel-powered vehicles. Monthly data on these two types of distances are obtained using the following information. Firstly, source information is based on fuel sales of gas and diesel expressed in litres, which are then associated with energy efficiency of vehicles using these fuels. In Quebec, automobiles consume approximately 90% of gas and diesel is used in a similar proportion by trucks. Vehicle energy efficiency, expressed in litres per 100 kilometres, has improved considerably over the period in the model. The influence of Quebec's cold winters was also taken into account, since it dramatically reduces engine efficiency. Lastly, changes in types of vehicles on the road were considered. The model tries to explain monthly changes in distances travelled using a number of independent variables that pertain to transportation costs, vehicles on the road, weather conditions, legislation in effect, etc. The second level contains the model's seven equations, which explain the frequency of accidents, their severity and the number of victims. Three categories are used to describe the frequency of accidents: fatal accidents, accidents with injuries and accidents with property damage only. Severity is determined on the basis of mortality, that is, the ratio of the number of victims killed compared to the number of fatal accidents, and morbidity, which is defined as the ratio of victims injured compared to the number of accidents, in other words, the total number of fatal accidents or bodily-injury accidents. Lastly, victims are differentiated as killed or injured. The model attempts to explain changes to each of the seven dependent variables during the period in question by using a certain number of independent variables, including total distance travelled, automobile characteristics, road safety legislation, etc. The overall structure of the model appears in diagram 2.1.

The DRAG-2 Model for Quebec 39 Diagram 2.1 INDEPENDENT VARIABLES > > > > > > >

Prices Vehicles on the road Licence holders Economic conditions Legislation Weather conditions Reasons for travel

DEPENDENT VARIABLES

> Distance travelled by gaspowered vehicles -•

> Distance travelled by dieselpowered vehicles

L... __.. > Etc. DEPENDENT VARIABLES

INDEPENDENT VARIABLES > > > > > > > > >

Total distance (gas + diesel) Prices Automobile characteristics Road safety legislation Weather conditions Economic conditions Licence holders Reporting of accidents Etc.

> Frequency of accidents • Fatal • Bodily injury • Property damage only ->

> Severity • Mortality • Morbidity > Victims • Killed • Injured

J

Data have been aggregated in order to reflect the situation in Quebec. In 1998, the total population of Quebec was 7.5 million, 4.4 million of which held a driver's licence. There were also 4.5 million vehicles on the road. The model was built on a monthly basis, which established a balance between the desired range of fluctuation of information and its availability. A per month basis also enabled a large number of observations. As for the period covered by the model, the first specification defined by professor Gaudry pertained to the period of December 1956 to December 1982. When the Societe de I'assurance automobile du Quebec first began using the model, information already available which covered the period of December 1956 to December 1989 was used. The model was however later updated and a number of variables were added and therefore the results presented refer to the model built for the period of December 1956 to December 1993, which include a total of 445 observations.

40 Structural Road Accident Models As regards mathematical calculations, each of the equations includes "A," Box-Cox transformations. For a given equation, the same " I " is used for independent variables and the dependent variable. The value of "A,", however, is different for each equation. In the case of frequency of fatal accidents, mortality and number of victims killed, a number of "X" are used, which will be described later in this document. Lastly, an error autocorrelation structure was introduced in each of the model's nine equations. In general, approximately three "p" were used in each equation and the autocorrelation coefficients were always generally of the order less than 15. 2.2.2. Dependent variable graphs In this section, graphs are presented showing the monthly changes in data of the nine dependent variables for the period of December 1956 to December 1993. One notes that each variable changes according to its own conditions. The purpose of building a model is therefore to explain the changes of each variable. Distance travelled by gas-powered vehicles. Graph 2.1 details changes in distances travelled by gas-powered vehicles. One notes instantly that distances travelled rose significantly between 1957 and the early 1980s, followed by sharp drop in 1982 due to a severe economic recession. The distance travelled showed steady growth up to the early 1990s, but thereafter, grew at a slower pace. Indeed, on a monthly basis, one notes a factor of almost 10 between the lowest value of approximately 700 million kilometres in the late 1950s and the highest value of almost 6.5 billion kilometres at the end of the period. On a per-year basis, this translates into a fivefold increase in distances travelled by gas-powered vehicles in Quebec between 1957 and 1993, that is, from 12 to 66 billion kilometres. Lastly, one also notes that in each year, the distance travelled is higher during the summer months of June, July and August than during the winter. Distance travelled by diesel-powered vehicles. Changes in distances travelled by dieselpowered vehicles appear in graph 2.2. The period between 1957 and the early 1970s shows small growth. During the 1970s, this increase grew sharply. As in the case of distance travelled by gas-powered vehicles, the 1982 recession caused distances travelled to dip slightly. Since then, growth in diesel consumption has been rapid. Beginning in the early 1980s, however, one must consider that trucks gradually converted to diesel instead of gas. Lastly, as with the case gas, distances travelled by diesel-powered vehicles are generally higher during the summer months of June, July and August. Accidents with property damage only. Graph 2.3 shows a steady rise in the number of accidents involving property damage alone, from the 1950s to the early 1970s. After a noticeable decline of the 1970s, numbers climbed steadily through the late 1970s. From this period up to 1993, numbers generally fell with a low during the recession in 1982 and 1983.

The DRAG-2 Model for Quebec 41 From one month to another there are sharp fluctuations, showing that property-damage accidents peak in the winter months of December, January or February.

G R A P H 2.1 D I S T A N C E T R A V E L L E D BY G A S - P O W E R E D V E H I C L E S IN Q U E B E C , 1 9 5 6 - 1 9 9 3

^

,.^^

,V

^

5


G R A P H 2.2 D I S T A N C E T R A V E L L E D B Y D I E S E L P O W E R E D V E H I C L E S IN Q U E B E C , 1 9 5 6 - 1 9 9 3 8.E + 08 ^ CO

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Bodily-injury accidents. As graph 2.4 reveals, the number of accidents with bodily injuries grew steadily between 1957 and 1974 except in 1970. The years 1975-1977 marked a decline. This period was followed by an increase, which peaked in 1980. Following a decline, which reached its lowest point in 1982, numbers grew up to 1985. Since then, there has been a downward trend. As for monthly fluctuations, they are very noticeable. In any given year, the number of bodily-injury accidents peaks in summer, particularly June, July and August. Fatal accidents. Graph 2.5 clearly shows an increase in the number of fatal accidents from the late 1950s to 1973, when it reached a peak. In the twenty years from 1973 to 1993, the trend shows a sharp decline in numbers. However, as with other categories of accidents, one notes

42 Structural Road Accident Models very significant monthly fluctuations showing that the number of fatal accidents peaks in summer and bottoms out in winter. GRAPH

2.3 P R O P E R T Y - D A M A G E A C C I D E N T S Q U E B E C , 1956-1993

IN

20000

MONTH

GRAPH

2.4 B O D I L Y - I N J U R Y A C C I D E N T S Q U E B E C , 1956-1993

IN

Morbidity. Morbidity is defined as the ratio between the number of victims injured and the number of accidents including fatal accidents and bodily-injury accidents. As graph 2.6 shows, the ratio constantly changes but within a very small interval. This ratio is generally between 1.3 and 1.5, except for the month of May 1968, where it was over 2. One also notes a slight downward trend for the entire period. The exceptional nature of the observation for May 1968 is attributed to imprecise data on the number of bodily-injury accidents. A "dummy" variable was therefore included in the relevant equations of the model to compensate for the effects this incorrect data had on results. Mortality. The ratio of the number of victims killed over the number of fatal accidents is the definition of mortality. Graph 2.7 shows that this ratio fluctuates constantly, but is somewhere in a relatively narrow range of 1.05 and 1.25 in most cases. As with morbidity, one notes a slight downward trend in this ratio. The high ratios, which peak at 1.42 in August 1978, are

The DRAG-2 Model for Quebec 43 explained by a single accident that caused the death of numerous individuals. As with morbidity, we have taken these events into account in relevant equations by inserting a "dummy" variable.

250 .

G R A P H 2.5 F A T A L A C C I D E N T S 1956-1993

IN

QUEBEC,

LU

m

^•N

^^

^\

^

G R A P H 2.6 V I C T I M S I N J U R E D P E R A C C I D E N T Q U E B E C ( M O R B I D I T Y ) , 1 956-1 993

IN

2.2 2.0 ^ 1.8 1 .

Lf^^

o-^'










MONTH

Victims injured. As one might expect, graph 2.8 shows that changes in the number of victims injured is largely related to the number of bodily-injury accidents. Numbers increase at the beginning of the period in question, December 1956, up to the mid-1970s, except for a dip in 1970. Between 1975 and 1977, their numbers declined before beginning to climb again and peaking in 1979. Thereafter, numbers fell sharply to their lowest point in 1982, before rising again to a high point in 1985. In the following eight years, there was a downward trend. Lastly, one notes that there were large variations from month to month and that numbers peaked in summer. Victims killed. Graph 2.9 reveals changes in the number of victims killed. The graph clearly shows two distinct movements. Firstly, one notes that the number of deaths increased up to 1973, when it peaked. In the 20 years that followed, between 1973 and 1993, the trend showed

44 Structural Road Accident Models a clear decline in numbers. One also notes significant fluctuations from month to month, with a higher number of deaths in the summer and a lower one in winter. GRAPH o o

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^

1 .30 1 .20

O

1 .10

i

1 .00

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2.7 F A T A L I T I E S P E R F A T A L A C C I D E N T Q U E B E C (MORTALITY), 1956-1993

o-^' o-^ MONTH

GRAPH

2.8 V I C T I M S I N J U R E D 1956-1993

IN

QUEBEC,

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4000 2000

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M ONTH G R A P H 2.9 V I C T I M S K I L L E D IN 1956-1993

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The DRAG-2 Model for Quebec 45 2.2.3. The matrix of direct effects of independent variables over dependent variables The purpose of this section is to present the independent variables used in Quebec's DRAG-2 model to explain changes in each of the nine dependent variables over the 445 months from December 1956 to December 1993. Table 2.1 summarizes this approach. Table 2.1 FACTOR

EXAMPLE

Exposure

Distance travelled

Prices

Fuel

Total vehicles on Number

SAFETY RECORD

DISTANCE ESSENCE

DIESEL

V V

V V

ACCIDENTS

SEVERITY

VICTIMS 1

V V V

V V

v

V V V

v

V

V

road

V

Vehicle

Proportion of small

characteristics

automobiles

Laws,

Seatbelt

V

V

^l

V

V

Strikes

Police

Weather

Temperature

V V

V V

V V

V V

Road system

Proportion of

V V V

Drivers

Young

Medication and

Bill "0.08"

V V

V

V V

V V

V V

Work

V

^}

V

V

V

Unemployed

V

V

V

V

1 Administration

New form

V

Roadside check

Activities

V

V V

Isolated cases

Tragedies

1 Aggregation

Work days

V

v v

V V V V

V V V V

regulations, and various measures

highways

1 alcohol Reasons for travel Economic 1 conditions

V

Exposure to risk (2 var.). The total distance travelled is undoubtedly the most important variable for explaining changes in the number of accidents, the degree of severity and the number of victims of these accidents. In the model, we also inserted another variable identical to the first, but which is affected by a different "X" of the Box-Cox transformations. We shall examine this feature of the DRAG-2 model in a later section.

46 Structural Road Accident Models Prices (4 var.). At the outset, we used three prices that might affect the distance travelled by gas-powered vehicles: gas prices, vehicle maintenance costs and public transit costs. Diesel fuel prices applied in the case of distance travelled by diesel-powered vehicles. Based on results of the influence fluctuations in gas prices have on distance travelled by gaspowered vehicles, gas prices changes may also have an effect on the number of accidents, their severity and number of victims, particularly if drivers change their speed to conserve gas. For example, drivers may reduce distances travelled following an increase in gas prices and, at that reduced distance travelled, they may drive slower to save gas. This also applies to vehicle maintenance costs. Number of vehicles on the road and their characteristics (8 var.). Three variables were used for the distance travelled by gas-powered vehicles: changes in the number of gas-powered vehicles, the number of motorcycles and mopeds on the road and changes in the proportion of small automobiles. For the distance travelled by diesel-powered vehicles, we took into account changes in the number of diesel-powered vehicles on the road. In the case of frequency of accidents, their severity and victims, we first examined changes in the number of motorcycles on the road considering their vulnerability to accidents and the fact that this variable is not related to total distances travelled. We also took into account changes in the proportion of small automobiles because they are more vulnerable when involved in accidents and the possible phenomenon of behavioural feedback. We also examined the effect of the gradual entry onto the market of three safety features: airbags, daytime running lights and the center high mounted stop lamp. Lastly, we developed a variable to represent changes in the number of persons per vehicle. Road safety laws, excluding laws governing alcohol (6 var.). We do not believe this type of legislation has a significant effect on distances travelled. It was used primarily as a means of verifying the hypothesis that in the instance of distance travelled by gas-powered vehicles, we examined the introduction of a demerit point system in 1973, mandatory seatbelt legislation, which was introduced at the same time as the reduction of speed limits on major highways in 1976, the new Highway Safety Code in 1982 as well as its amendments in 1987 and 1991 and, lastly, speed limits on Quebec highways between 1970 and 1977, which used to be 70 m/h, the highest limit during the 1957-1993 period. For distance travelled by diesel-powered vehicles, we took into account only the Highway Safety Code variable. As regards the frequency of accidents, their severity and victims, as in the case of distance travelled, the following were considered: the demerit point system, legislation governing seatbelts and traffic speed and the new Highway Safety Code. We also took into account introduction of two laws governing motorcycles, namely mandatory helmet and compulsory use of headlights at all time.

The DRAG-2 Model for Quebec 47 Truck legislation (5 var.).The principal purpose or effect of introducing legislation was an increase in the distance travelled by diesel-powered trucks. In 1973, the Quebec Transportation Commission was created and a blue dye was added to heating oil. There was deregulation in the trucking industry in 1988 and regulatory control of the weight and size of trucks in 1972. In 1980, for tax evasion reasons, red dye was added to heating oil to increase diesel fuel consumption. We also examined the SAAQ's takeover of control of transportation safety for trucks. The purpose of these activities was to influence the frequency of accidents, their severity and victims. Given the safety measures imposed, control of trucking may affect the distance travelled by diesel-powered vehicles. Road system (1 var.). We took into account changes in the proportion of distance in kilometres of highways per 100 kilometres of road. We examined how growth of the highway system compared to growth of the ordinary road system increased the distance travelled by dieselpowered vehicles. Weather conditions (5 var.). With regard to weather conditions, we used the five following elements: length of day, temperature, number of hours of sunlight during day, rain and snow. These five variables were used in the equations pertaining to frequency of accidents, their severity and victims. For distance travelled by gas-powered vehicles, all of these variables were used, except temperature. For distance travelled by diesel-powered vehicles, the length of day was not considered. Drivers (2 var.). The first variable examined was the number of driver's licence holders on a per vehicle basis. This variable was used in the equation for distance travelled by gas-powered vehicles. The proportion of young drivers between age 16 and 24 compared to the total number of driver's licence holders was used in all the equations pertaining to the frequency of accidents, their severity and victims. Alcohol and medication (6 var.). We first examined the effect of four laws governing drinking and driving on equations for frequency of accidents, their severity and victims. These laws include introducing a breathalyser test in 1969, lowering the legal age for alcohol consumption from age 21 to 18 in 1971, harsher penalties under the Criminal Code for impaired driving in 1985 and extending the driver's licence suspension period for conviction of impaired driving in 1986. We also examined alcohol consumption per adult in all the equations related to frequency of accidents, their severity and victims, as well as equations for distances travelled by dieselpowered vehicles. Lastly, we included consumption of medication on a per adult basis in the two distance travelled equations and in the seven equations regarding accidents, their severity and victims.

48 Structural Road Accident Models Reasons for travel (21 var.). We began by examining distance travelled by gas-powered vehicles. We noted the seven following reasons for travel: work, shopping, recreation, construction, home deliveries, agriculture and vacation. As regards distance travelled by diesel-powered vehicles, we took into account manufacturer deliveries, forestry activity, home deliveries, construction, engineering works, agriculture, construction of large hydroelectric dams, public transit and vacation. Lastly, for equations pertaining to accidents, their severity and victims, we considered the ratio of reason for travel indicator to data on total distance travelled. We therefore note that the relative importance of the reason grows if the ratio computed increases between two observations in a row. This ratio was therefore established for the six following reasons: work, shopping, vacation, forestry activity, construction and public transit. Economic conditions (1 var.). One variable was used for this analysis, namely the number of unemployed on a per driver's licence holder basis. This variable was included in the equation pertaining to distance travelled by gas-powered vehicles and in all of the equations pertaining to accidents, their severity and victims. Legislation governing automobile insurance (2 var.). We examined legislation introducing compulsory liability insurance for use of an automobile, which came into force in 1961. We also took into account the effect of Quebec's new public automobile insurance scheme introduced in 1978. This new no-fault compensation scheme covered all bodily injuries sustained by Quebecers in a road accident. At the same time a new accident report form was introduced. These two variables were included in all of the equations pertaining to accidents, their severity and victims, as well as the equation for distance travelled by gas-powered vehicles. Strikes and work slowdowns (4 var.). This category includes strikes and work slowdowns of Quebec's provincial police force the Surete du Quebec, Montreal Urban Community police, public transit employees and Montreal road maintenance employees during the winter. We examined the influence of these four variables on all the equations of the model, including the two distance travelled equations, the three categories of accidents, the two degrees of severity and two categories of victims. Reporting of accidents (4 var.). In 1962, important measures were taken to improve the quality of accident reporting. These efforts were enormous since the severity of the accidents was high. One would therefore, expect an effect on the number of police reports on accidents, their severity and victims. The actual number of accidents should not be affected by this change. At the same time, however, the minimum age for a driver's licence was lowered from age 18 to 16. This change should undoubtedly have a bearing on the number of accidents, their severity and victims, as well as the distance travelled by gas-powered vehicles. This variable therefore combines two very different effects.

The DRAG-2 Model for Quebec 49 In June 1979, a joint accident report was introduced for drivers involved in accidents with only minor property damage, since they could complete the form themselves acknowledging that they were involved in an accident to receive compensation from their respective insurance companies. Only one dependent variable included this explanatory variable, namely the variable for frequency of accidents with property damage only. One would expect their real number to remain unchanged, and that the number reported by police would decline. On a similar matter, the minimum limit of property damage necessary to warrant writing an accident report has changed over the years. One would therefore expect these changes to affect only the number of property-damage accidents reported by police. Their actual number, however, should not be affected by this change. Lastly, the accident report form was changed again in 1988. First, two sections were created on the form so that an accident with only property damage takes less time to complete. Secondly, the amount of property damage necessary to warrant writing an accident report was increased. One would therefore expect the number of bodily injury accidents and property-damage accidents to decline as a result of these changes. In addition to these two categories of accidents, we examined the effect of these changes on morbidity and the number of victims injured. We did not examine their impact on fatal accidents, mortality and victims killed since in principle these changes would have no effect. Major events (2 var.). We took into account the effect of major events such as the Montreal World Fair in 1967 and the 1976 Summer Olympic Games in Montreal on the distance travelled by gas-powered vehicles. Isolated cases (3 var.). Firstly, as indicated in the section above, data on the number of accidents with injuries for the month of May 1968 is incorrect. We therefore introduced a variable into this equation to compensate for the effect of this erroneous data and also into equations on morbidity and injured victims. In addition, prior to 1994, two serious tragedies occurred in Quebec, one in August 1978, the other in July 1993. These two tragedies caused 40 and 20 deaths respectively. To take into account the effect these tragedies had on the number of victims killed and mortality, we introduced two variables recognizing these events in both equations. Aggregation variables (3 var.). In each of the nine equations of the model, we introduced the three following aggregation variables: number of workdays, number of Saturdays and number of Sundays and holidays. 2.3. RESULTS ON MATHEMATICAL FORM AND PARTICULAR VARIABLES In this section, we shall begin by examining the overall econometric results that pertain to the model as whole. We shall then select a few variables of interest and present the results of elasticity with respect to dependent variables.

50 Structural Road Accident Models 2.3.1. Econometric results Among overall econometric results, we are interested in the distribution of the number of variables included in each equation of the model based on the "Student t" value of each. We shall also examine the autocorrelation structure by presenting the number of "p" coefficients used in each equation, the Box-Cox transformation "?^" values, the "Pseudo-R^" value obtained from each equation and, the number of observations used out of the 445 observations available. Observations that were not used correspond to the autocorrelation coefficient of the highest order. Furthermore, no heteroscedasticity parameter was used in the model. Results appear in table 2.2. We shall first examine the two equations pertaining to distance travelled, which include over 30 variables. In each case, almost half of the variables have a "Student f higher than 1, which means that a large number of variables have a low "Student t". It must be specified that a number of variables were included in these equations to make them similar to equations on accidents and victims. These low "Student t" values were therefore predictable. The "PseudoR^" is over 99% in each case, which indicates that the model very accurately reproduces actual data. The autocorrelation structure is comprised of five coefficients in each case. Lastly, the "X" value of Box-Cox transformations is approximately 0.2 in both cases, which indicates a closeness with the logarithmic function because this is obtained with the value of 0. Table 2.2 Elements examined

Diesel PDO Injury

Gas

Severity

Accidents

Distance Number of «X» variables

37

32

48

47

46

Number of «t» > 2

14

8

18

18

17

3

8

11

14

10

20

16

19

15

19

Number of 1 < «t» < 2 Number of «t» < 1

Victims

Morb. Mort.

Fatal

Killed

Inj.

48

47

48

8

4

16

16

12

13

17

16

27

31

14

16

0

0

3

1

47

Heteroskedasticity parameters

0

0

0

0

0

0

0

Autocorrelation parameters p

5

5

3

4

2

2

2

0.174 0.241

0.279

0.360

0.539 1.567 0.201

0.367

0.360

0.539 1.567 0.201

0.367

Value of?, (y) Value of A, (x)

0.213 0.213

0.174 0.241

0.279

Fixed value of A, (km)

1

1

1

Fixed value ofk (km^)

2

2

2

Pseudo-R^ (E)

445

445

Number of observations used

432

431

0.696 0.331 0.964

0.889

0.968

0.898

445

445

445

445

445

445

445

432

433

439

433

434

433

439

0.994 0.993 0.963

Total number of observations

The DRAG-2 Model for Quebec 51 With regard to the three categories of accidents, as with the two equations on victims, one notes a higher proportion of variables for which the "Student f value is higher than 1. The "Pseudo-R^" value is over 96% for property-damage accidents, bodily-injury accidents and victims injured. For fatal accidents and victims killed, it was almost 90% in each case. Therefore the model reproduces accurately the actual data for these five years. The autocorrelation structure varies somewhat, but is comprised of approximately three coefficients in each case. Lastly, the "?i" value of Box-Cox transformations varies between 0.20 and 0.28, which once again approaches the logarithmic form, except for fatal accidents and victims killed. In these instances, the "X" affecting the dependent variable and all of the independent variables is 0.36. The distance travelled, however, is examined in the following manner: first, a variable takes into account the distance travelled in a linear form (k=\) and another variable represents square distance travelled (X=2). As for morbidity and mortality, we obtained very different overall results. The proportion of non-important variables is higher. This result could be anticipated based on the graphs showing these variables. Indeed, the values fluctuated in a very small interval and therefore it became very probable that the number of significant variables would be relatively small. As in the case above, these variables were used nonetheless, in order to ensure homogeneity of the entire model. As for the "Pseudo-R^", it was only 33% in the case of mortality and 69% for morbidity. The autocorrelation structure is comprised of two coefficients in each case. Lastly, the "?t" value of Box-Cox transformations was 0.539 for morbidity and therefore midway between the logarithmic and linear form. For mortality, the "X" affecting the dependent variable and all of the independent variables is very high because it is 1.567. By analogy with fatal accidents and victims killed, the distance travelled is considered using a variable in a linear form (k=\) and another one representing square distance travelled (X,=2). 2.3.2. Results on elasticity In this part of the chapter, the results on direct elasticity are presented for a number of variables in the model with respect to all the dependent variables in question. Section 1 shows the results for six common variables in the models of the DRAG family. Section 2 is devoted to six other variables including some that are unique to Quebec. All results on direct elasticity appear in the two figures at the end of each section. A) Results for six variables common to all models These six variables are as follows: total distance travelled, gas prices, temperature, legislation to reduce highway speed limits, which was introduced during the same month as mandatory use of seatbelts, alcohol consumption and retail sales.

52 Structural Road Accident Models Total distance travelled. Graph 2.10 shows changes in total distances travelled. One notes first from the graph that distances increased significantly between 1957 and the early 1980s. After a fall in 1982, total distance travelled increased again significantly up to the early 1990s, following which it continued to increase but at a slower pace. One also notes that in any given year, the distance travelled is much higher during the summer months of June, July and August than during winter months. As indicated above, results on elasticity appear in the figure at the end of this section. A 10% increase in distance travelled would result in a 6.0% increase in property-damage accidents, a 7.8% increase in accidents with injuries and 7.2%) increase in victims injured. These three results are accompanied by a "Student t" of more than 2, which indicates a strong probability that the actual effect is something other than 0. As defined above, for morbidity a reduction of 0.8%) is obtained, which is accompanied by average certainty, which translates into a "Student f of between 1 and 2. G R A P H 2.10 T O T A L D I S T A N C E T R A V E L L E D Q U E B E C , 1956-1993

IN

[2 6.E + 09 a: ly

4.E + 09

2.E + 09 O.E + 00

<^

<6^

q)^

o-"' o-"'
Before presenting the results for fatal accidents, mortality and victims killed, the procedure utilized must first be explained. During the previous specification of the model, for which the observation period ended in 1989, we detected the possibility that the influence of distance travelled on these three variables may decrease and eventually contribute to a decline with a later increase in distance travelled. A quadratic form in the shape of an inverted "U" may represent this effect. During the update that is the subject of this document, this possibility manifested itself again but in a slightly more noticeable manner. We shall therefore present the results obtained with the following specification of the model: firstly, the variable representing total distance travelled with an assigned ">." Box-Cox transformation value of 1, that is, in a linear form, to which we assigned a second variable representing total distance travelled, but with a "A." value of 2, that is, in a quadratic form. Based on this specification of the model, a 10% increase in distance travelled in its linear form would result in a 15.1%) increase in fatal accidents with a 7.1%) decline for distance travelled in its quadratic form. The results for victims killed are similar to those for fatal accidents. We

The DRAG-2 Model for Quebec 53 would therefore obtain an 18.2% increase for distance travelled in its linear form to which one associates an 8.3% decline for distance travelled in its quadratic form. These four results are accompanied by a very high "Student f, indicating that in all likelihood the actual effect is something other than 0. Lastly, for mortality, we obtain a 1.4% increase for distance travelled in its linear form reduced by 0.1%) for distance travelled in its quadratic form. These results, however, are accompanied by a very small "Student t". Gas prices. Graph 2.11 shows changes in gas prices in constant dollars required to travel one kilometre. By excluding monthly fluctuations, one notes first that gas prices declined slightly in Quebec from the late 1950s up to the first oil crisis in 1973. Afterwards, they remained relatively stable before climbing in the early 1980s. Gas prices then declined considerably due to its relatively stable price in current dollars combined with more efficient gas engines. The abrupt monthly fluctuations observed are evidence of increased fuel consumption during winter months due to Quebec's cold weather. G R A P H 2.11 R E A L P R I C E OF G A S O L I N E P E R K I L O M E T R E IN Q U E B E C , 1 9 5 6 - 1 9 9 3




MONTH

With regard to results on elasticity, in addition to the effect this factor has on accidents and victims, we examined first its influence on distance travelled by gas-powered vehicles. As one might expect, results show that this factor has a significant influence on distance travelled by gas-powered vehicles. Therefore, a 10% increase in gas prices per kilometre would cause distance travelled by gas-powered vehicles to decline 4.6%. This result is also accompanied by a very high "Student f. As for the effect of this factor on the total number of road accidents, a 10% increase in this gas prices would cause property-damage accidents to decline a small 0.2%, but would mean significant declines of 3.9% for accidents with injuries and 4.4% for fatal accidents, since these results are accompanied by a "Student t" of more than 2. The effect of this factor on morbidity and mortality is very small, respectively an increase of 0.1% and a decline of 0.2%, but these results are subject to a very small "Student f. Lastly, for victims injured and victims killed, we obtain results very similar to those obtained for accidents of similar severity. Therefore, a 10%

54 Structural Road Accident Models price hike would means a 3.9% decrease in the number of victims injured and a 4.9% drop in the number of victims killed; these results are accompanied by a very high "Student t". Temperature. Graph 2.12 shows monthly changes in average temperatures in Quebec. The values obtained reflect the very large temperature fluctuations experienced throughout the year. Data is based on the minimum and maximum daily temperatures in Quebec City and Montreal. In Quebec, the hottest temperatures are in July and August, with temperatures of 70 degrees Fahrenheit and the coldest temperatures are generally in January and February, with temperatures of approximately 10 degrees Fahrenheit. As one might expect, there are slight fluctuations from year to year. GRAPH

2.12 A V E R A G E

TEMPERATURE

(FAHRENHEIT),

IN

QUEBEC

1956-1993

on o

< r "" MONTH

In addition to the influence of temperature on accidents and victims, we considered this factor in the distance travelled by diesel-powered vehicles. One notes that a very similar variable, hours of daylight, was used for distance travelled by a gas-powered vehicle. As one might expect, results show that a 10% increase in temperature, expressed in degrees Fahrenheit, would increase distance travelled by diesel-powered vehicles by 0.3%. This result, however, is accompanied by a "Student t" of average certainty. As for the effect of temperature on the total number of road accidents, we obtained the following results marked by very significant effects and accompanied by a very high "Student t". Therefore, a 10% increase in temperature expressed in degrees Fahrenheit would result in a significant decline of 2.7%) in property-damage accidents. This result is consistent with the reality in Quebec where the number of property-damage accidents peaks in winter. A 10% increase in temperature would also result in fewer accidents with injuries, but to a lesser extent, since it would be 0.5%) and the number of fatal accidents would increase 2.2%). This result is also consistent with expectations since the number of fatal accidents in Quebec is highest in the

The DRAG-2 Model for Quebec 55 Temperature has very Httle effect on morbidity and mortality rates, which show respective declines of 0.2% and 0.3%. These results, however, are accompanied by a "Student t" of between 1 and 2. Lastly, for victims injured or victims killed, we obtain very similar results to those obtained for accidents of corresponding severity. Therefore, a 10% increase in temperature would bring the number of victim's injured down 0.5% and cause the number of victims killed to rise 2.0%. These results are accompanied by a very high "Student f. Legislation to reduce highway speed limits, which was introduced during the same month as mandatory seatbelt use. Because this is a "dummy" variable, the corresponding graph is not presented. What must be specified is that this variable was assigned a value of 0 from December 1956 to July 1976 and a value of 1 thereafter. In addition to the effect these two legislations have on accidents and victims, we inserted this variable in the equation for distance travelled for gas-powered vehicles. As one might expect, results show that the combined effect of these two legislations, which both came into force in August 1976, had very little effect on distance travelled by gaspowered vehicles, more specifically a 0.2%o increase. The "Student f, however, is so small that in all likelihood, the actual effect is zero. These two legislations, however, did significantly affect on the number of road accidents. The number of accidents with property damage only increased 2.0%) but this result is accompanied by a very small "Student f, which limits the scope of the result. With regard to accidents with injuries and fatal accidents, these two legislations caused respective declines of 9.4% and 14.7%, each of which has a "Student f of more than 2. The effect of these two legislations on morbidity and mortality is very small, with respective declines of 0.8%) and 2.0%). These results, however, are accompanied by a "Student t' of less than 1. Lastly, for victims injured or victims killed, we obtained results very similar to those for accidents of corresponding severity; namely declines of 9.1% for victims injured and 17.8% for victims killed. These results were accompanied by a "Student f of more than 2. Alcohol consumption per adult. Initially, we were looking for a variable to represent the percentage of individuals who drive their vehicle while impaired. Such a measurement, available each month since 1957, however, does not exist in Quebec. We therefore had to proceed in another manner. We used instead sales of products containing alcohol, which we broke down into figures of pure alcohol per adult to represent this phenomenon, based on the following hypotheses. Firstly, suppose that monthly fluctuations in alcohol purchases for the entire population would also reflect drunk driving rates for the subgroup of the population that includes drivers. We also hypothesized that alcohol products purchased during a particular month were also consumed during the same month. A third element was changes in consumption of different types of alcohol products and its effect on road safety. Over the last few years, wine

56 Structural Road Accident Models consumption in Quebec has grown significantly, while beer consumption has leveled off and spirits has declined. The results that follow must therefore be interpreted carefully. Graph 2.13 shows monthly changes in consumption of pure alcohol per adult. Aside from monthly fluctuations, the overall trend for consumption of pure alcohol per adult shows an increase from 1957 to the late 1970s, which falls thereafter. One also notes significant monthly fluctuations which peak in December and fall sharply in January and February, which reflects the fact that the source data used to build this variable are based on sales of alcohol products. GRAPH

2.1 3 P U R E A L C O H O L C O N S U M P T I O N A D U L T IN Q U E B E C , 1 9 5 6 - 1 9 9 3

\0mw^^^w

0 . 2 0 X^

<^

^

PER

^

^ MONTH

As with other variables, results on elasticity appear in the figure at the end of this section. In addition to its effect on accidents and victims, we examined this factor on distance travelled by diesel-powered vehicles. Results show that a 10% increase in consumption of pure alcohol per adult would have very little effect on distance travelled by diesel-powered vehicles, a drop of 0.2%. The "Student f', however, is so small that the actual effect is likely zero. However, a 10%) increase in consumption of pure alcohol per adult would cause the number of accidents with property damage alone to increase 1,0%. This result is accompanied by a high "Student f of more than 2. Bodily-injury accidents and fatal accidents would decline 0.3%). The "Student f, however, is very small in both cases, which significantly limits the scope of this result. The effect of a 10% increase in consumption of pure alcohol per adult would translate into a relatively modest increase of 0.1% for morbidity and 0.9% drop in mortality, since the decline in mortality is accompanied by a "Student t" of more than 2. Lastly, there would be essentially little effect on victims injured, but this result is accompanied by a very small "Student t". For victims killed, there would be a 1.3%) drop and the "Student t" is higher than 1 but lower than 2. Considering the results obtained and the changes over the last few years in consumption of the main alcohol products, more specifically wine, beer and spirits, combined with the

The DRAG-2 Model for Quebec 57 characteristics of the consumers of these products, in particular with regard to driving an automobile, we conducted an additional exercise in the previous version of the model, which ended in December 1989. Results showed in particular that increased beer consumption would result in higher numbers of victims killed but increased wine consumption would result in a decline in the number of victims injured and victims killed. Given the limits to building this variable, results must be interpreted with care and additional work must be carried out to better understand the effect of increased alcohol consumption on the number of road accidents in Quebec. Retail sales. We used retail sales in constant dollars weighted by an index to translate the number of trips per dollar spent, which we then converted to an employment basis. The resulting variable was used to assess the prevalence of shopping as the reason for travel in the distance travelled by gas-powered vehicles. Retail sales data were also used to estimate the proportion home deliveries by truck as the reason for travel represent in distance travelled by diesel-powered vehicles. Lastly, by building a ratio using retail sales as the numerator and total distance travelled as the denominator, we set out to determine the effect of this ratio, as a substitute to shopping as the reason in total distance travelled, on the number of accidents and number of victims. Graph 2.14 shows the monthly changes of the variable used to represent shopping as the reason for travel in distance travelled by gas-powered vehicles. If we examine the general trend, we note first relative stability from the late 1950s to the late 1970s. Thereafter, figures rise sharply up to the mid-1980s and then remain relatively stable. Each year, however, one notes abrupt fluctuations from month to month, which peak in December. G R A P H 2.14 S H O P P I N G - R E L A T E D T R I P S E M P L O Y E E IN Q U E B E C , 1 9 5 6 - 1 9 9 3

PER

6000 lii

S

4000

Therefore, a 10% increase in this variable, which is used to determine the proportion of shopping as the reason for distance travelled by gas-powered vehicles, would increase distance travelled by 2.4%, which has a "Student t" of much more than 2. For the corresponding variable, which is used to determine the effect of home deliveries by diesel-powered trucks as

58 Structural Road Accident Models the reason for travel, we note that a 10% increase in the value of this variable would cause a 0.4% increase in the distance travelled by diesel-powered vehicles, but this result is accompanied by a very small "Student t".

FIGURE 2.1 ELASTICITIES - COMMON VARIABLES PERCENTAGE VARIATION OF THE DEPENDANT VARIABLE ASSOCIATED WITH A 10 % INCREASE (OR ENFORCEMENT OF LEGISLATION) IN THE INDEPENDENT VARIABLE

Retail sales Total alcohol per adult Speed limit and compulsory seat belt use

Temperature (Fahrenheit) Real cost of gas per km Total distance travelled (square) Total distance travelled

PERCENTAGE

As for the variable representing a substitute for the proportion of shopping in total distance travelled, we note that a 10%) increase in this measure would result in a 0.4%) drop in the number of accidents with property damage only, but this result is accompanied by a very small "Student t". However, for bodily-injury accidents and fatal accidents, we obtain respective increases of 1.6% and 1.4%. The first result is accompanied by a "Student t" of more than 2 and for the second, the " Student t" is between 1 and 2.

The DRAG-2 Model for Quebec 59 A 10% increase in this measurement would mean a 0.5% decline in morbidity, but a 0.7% increase in mortality and these two results are accompanied by a "Student t" of between 1 and 2. Lastly, the effect obtained for victims corresponds essentially to overall effects on the frequency of accidents and their severity, that is, respective increases of 1.2% for victims injured and 2.2% for victims killed, since these results are also accompanied by a "Student t" of between 1 and 2. B) Results for six other variables Although not necessarily specific to Quebec, these six variables pertain to vehicle maintenance costs, the number of motorcycles and mopeds on the road, the new Highway Safety Code, the proportion of young drivers age 16 to 24, unemployment, and introduction of Bill "C-19", which attributed harsher penalties for impaired driving. To make the presentation of data easier, no graphs are presented on monthly changes of these variables. The principal characteristics of these changes, however, will be described for each variable. Results of elasticity appear in the figure at the end of this section. Vehicle maintenance costs. This is more precisely an index of maintenance costs for vehicles expressed in constant dollars. Changes in this variable are characterized by growth between the late 1950s and mid-1970s. A decline at the end of the 1970s is followed by a resurgence in the early 1980s. A downward trend was noted in the ten years that followed. There are also obvious monthly fluctuations. As one might expect, the results show that a 10% increase in vehicle maintenance costs would result in a 0.9% decrease in distance travelled by gas-powered vehicles. This result, however, is accompanied by a small "Student t". With regard to the influence of this factor on the number of road accidents, we note that an increase in vehicle maintenance costs would translate into a decline in total road accidents. Therefore, a 10%) increase in maintenance costs would result in a decline of 3.8% for propertydamage accidents and 4.4% for bodily-injury accidents and fatal accidents. These results are accompanied by a "Student f of more than 2 for the first two categories and between 1 and 2 for fatal accidents. The effect of a 10% increase in maintenance costs, would translate into declines in morbidity and mortality of 0.7% and 1.5% respectively. These results are also accompanied by a "Student t" of between 1 and 2. Lastly, the number of victims would decline 5.5% for injuries, with a "Student t" higher than 2 and 4.4% for death with a "Student t" of between 1 and 2. Number of motorcycles and mopeds per adult. This variable was built on the basis of the number of motorcycles and mopeds registered for road use in Quebec. This number of vehicles was computed on a per adult basis. Lastly, we took into account that these vehicles are not used during the winter, between November and March, that there is only partial use in the spring and

60 Structural Road Accident Models fall and maximum use in summer. From the late 1950s to the late 1970s, there was strong growth in the numbers of motorcycles and mopeds per adult. There was, however, a decline after 1983. As one might expect, the results show that a 10% increase in the number of motorcycles and mopeds per adult would have a marginal effect on distances travelled by gas-powered vehicles, a decline of 0.1%. However, despite this minimal effect, the "Student t" is between 1 and 2. As for the effect of this factor on the number of road accidents, as one might expect, an increase in the number of motorcycles and mopeds per adult would cause an increase in the number of road accidents. Therefore, a 10% increase in this number would lead to a 0.2%) increase in the number of accidents with property damage only, 0.9% for bodily-injury accidents and 0.8%) for fatal accidents. Despite these effects, which may appear minimal, the "Student t" that accompanies these results is between 1 and 2 for property-damage accidents and well over 2 for the two categories of most severe accidents. The effect of a 10% increase in the number of motorcycles and mopeds per adult would result in a 0.1%) drop in morbidity and a 0.1%) increase in mortality. Despite these marginal effects, the "Student t" is between 1 and 2. Lastly, the number of victims would increase respectively 0.9%) for victims with injuries and 0.8%) for deaths, since these results are accompanied by a "Student f of well over 2. This means that if the number of motorcycles and mopeds were to double, an 8%) increase in total deaths could be expected. This result is consistent with expectations since, for the last five years of the period in question, from 1989 to 1993, motorcyclists and moped riders represented 7%) of total deaths on Quebec roads. The new Highway Safety Code. This intervening variable was included in the model to determine the effect of introducing the new Highway Safety Code in 1982 and its amendments in 1987 and 1991. This is therefore a "dummy" variable. As one might expect, results show that the new Highway Safety Code had very little effect on total distances travelled. Distance travelled by gas-powered vehicles increased 1.1 %o and distance travelled by diesel-powered vehicles increased 0.7%). The "Student t" values associated with these results, however, are very small and therefore their effects are uncertain. As for the effect of the new Highway Safety Code on the number of road accidents, as one might expect, it resulted in a significant drop in the number of road accidents. There were sharp declines of 20.9%) among property-damage accidents, 24.3% for bodily-injury accidents and 13.9%) for fatal accidents. These effects are accompanied by a "Student t" of more than 2 for the first two categories of accidents, and a "Student t" of between 1 and 2 for fatal accidents.

The DRAG-2 Model for Quebec 61 The effect of introducing a new Highway Safety Code on morbidity resulted in a 2.4% drop but mortality increased 3.4%. The "Student t" accompanying these results are very small. Lastly, the number of victims declined respectively 24.4% for injuries and 11.3% for deaths, and these results were accompanied by a "Student t" of well over 2 in the first instance, but between 1 and 2 for the latter. The proportion of drivers age 16 to 24. We wanted to determine the effect of a change in the proportion of young drivers age 16 to 24 compared to the total number of drivers on the number of road accidents. This proportion grew significantly in the period from the end of the 1950s up to 1977, from 12% to 21%. Over the last few years, however, this proportion has constantly declined and returned to 12% in 1993, which reflects an aging population. As one might expect, an increase in the numbers of young drivers results in a rise in the number of road accidents. Therefore, a 10% increase in this proportion would result in a 1.5% increase in the number of property-damage accidents, 4.1% for bodily-injury accidents and 8.1% for fatal accidents. One notes that the effect obtained for fatal accidents is higher than that anticipated. Furthermore, despite these important effects, the "Student f accompanying these results is relatively small in the first case, of average certainty in the second case and slightly more than 2 in the case of fatal accidents. However, the effect of a 10% increase in the proportion of young drivers age 16 to 24 would result in respective declines of 1.1% and 0.8% for morbidity and mortality. The "Student t" is nonetheless between 1 and 2 for morbidity, but very small for mortality. Lastly, the number of victims would increase respectively 3.7% for injuries and 6.5% for deaths, since these results are accompanied by a "Student f of between 1 and 2. As in the case of fatal accidents, the effect obtained is higher than anticipated in the case of victims killed. The number of unemployed per driver's licence. We wanted to include an indicator of economic conditions in the model. We therefore built a ratio of the number of unemployed compared the number of people with driver's licences. This ratio was higher prior to 1965. Since then, however, the ratio has fluctuated significantly in accordance with economic cycles. Therefore, although the number of people holding a driver's licence continued to increase, the number of unemployed fell between 1983 and 1990 before climbing again. Abrupt monthly changes were also noted. As one might expect, we note that an increase in the ratio of the number of unemployed per driver's licence would result in a decline in the distance travelled by gas-powered vehicles. Therefore a 10%) increase in this ratio would result in a 1.2% decrease in the distance travelled by gas-powered vehicles. This result is also accompanied by a very high "Student t". The results on accidents and victims are also consistent with expectations. Therefore, a 10% increase in the ratio of the number of unemployed per driver's licence would result in a 1.2% decline in accidents with property damage only and bodily-injury accidents and a decline of

62 Stmctural Road Accident Models 1.1% in fatal accidents. Furthermore, the "Student t" accompanying these results is higher than 2 for the first two categories of accidents and between 1 and 2 for fatal accidents.

FIGURE 2.2 ELASTICITIES - SPECIFIC VARIABLES PERCENTAGE VARIATION OF THE DEPENDANT VARIABLE ASSOCIATED WITH A 10 % INCREASE (OR ENFORCEMENT OF LEGISLATION) IN THE INDEPENDENT VARIABLE

I I Victims killed Bill C-19 Number of unemployed per drive

• Victims injured

H Mortality 5

Proportion of drivers age 16-24

• Morbidity Highway Safety Code (1982, 87 and 91)

D Fatal accidents

Number of motorcycles and mopeds per adult

@ Bodily-injury accidents

^P.D.O. accidents Real cost of vehicle maintenance

3 Distance travelled by diesel-powered vehicles i Distance travelled by gasoline-powered vehicles! -30

-25

-20

-15

-10

-5

PERCENTAGE

However, the effect of a 10% increase in this ratio would be a slight drop of 0.2% in morbidity since the "Student t" associated with this result is between 1 and 2. For mortality, the effect is essentially zero but this result is accompanied by a very small "Student t". Lastly, the number of victims would decline respectively 1.5% for injuries and 1.1% for deaths, since these results

The DRAG-2 Model for Quebec 63 are accompanied by a "Student t" of more than 2 in the first case and between 1 and 2 in the second case. Criminal Code C-19: harsher penalties for driving under the influence of alcohol. This variable was included in the model to determine the effect of introducing new provisions in the Criminal Code for stiffer penalties for driving under the influence of alcohol. This law came into force in December 1985. It is therefore another intervening variable. As one might expect, results show that this legislation had a significant effect on the number of road accidents. The number of accidents with property damage only declined 4.4%. However, the "Student f associated with this result is very small. This legislation also resulted in a 15.0% decline in the number of accidents with injuries and 28.8%) for fatal accidents and the "Student t" is higher than 2 in both cases. The effect of this legislation on morbidity is a slight decrease of 1.4%) and a 1.0%o increase for mortality. However, the "Student t" accompanying these results is very small. Lastly, the number of victims would have declined respectively 16.2% for injuries and 30.1%) for deaths. These results are accompanied by a "Student f higher than 2 in both cases. 2.3.3. Other results: Forecasts for the period of 1997-2004 Following this specification of the model, we advanced to the next phase, which consisted of obtaining forecasts. This section details the methodology used and shows an example of the forecasts obtained. A) The methodology Obtaining forecasts using an econometric model from the DRAG family requires the combination of two blocks of information. Firstly, parameter values for the model must be available, as well as data for independent variables covering the forecast period. We began by using all of the parameters obtained following specification of the model for the period of December 1956 to December 1993. For each independent variable, we consulted appropriate reference sources to ensure that their monthly extension up to December 2004 was the best possible quality. The "TRIO" software program used was helpful in merging coefficients with data from independent variables. Therefore, by applying data from independent variables for the future to the parameters of the model, we obtained a first series of forecasts. The results were realistic and acceptable for the two types of distances travelled, property-damage accidents, bodily-injury accidents and victims injured. However, the forecasts obtained for the number of fatal accidents and numbers of deaths were particularly small and it therefore appeared unlikely that these numbers would decline to that extent in the near future. We therefore looked to identify a reason to explain the small numbers forecast for fatal accidents and victims killed.

64 Structural Road Accident Models The results of a search revealed that the quadratic aspect, measured by the model, of the influence of distance travelled on these accidents and victims was the most important factor. Indeed, with the model used, the distance travelled began to enter into an area where its influence on the number of fatal accidents and deaths reaches a maximum. Therefore, since the distance travelled should continue to increase in the coming years, which would place us in the part of the curve where an increase in distance travelled should cause a decline in fatal accidents and deaths, based on a quadratic form. We therefore used actual data available for the years 1994, 1995 and 1996 to re-estimate all of the parameters of all the model's equations with this new data. We conducted this work using the same variables, the same autocorrelation structure and the same number of "?i" for BoxCox transformations. Therefore, in addition to examining 36 other observations, the only other change made to the model was to free the second "X" associated with distance travelled, which initially had a value of 2. The freeing of this constraint allowed the inverted "U"-shape to be asymmetrical. It may be that one or more new variables should have also been included in the model. However, since this was the first forecast exercise, we deemed it best to devote all the time available to obtaining forecasts as quickly as possible, and at a later date, to examine the effect of the possible absence of certain variables on all the aspects and results of the model. The results obtained confirmed the influence in an inverted "U"-shape for the distance travelled on fatal accidents and victims killed. However, the value of distance travelled where the risk of fatal accident peaks is offset to the right, therefore it is higher. Furthermore, the descending portion of the curve now shows a much less inclined slope then that obtained previously. As a result of these two changes, we obtain forecasts of fatal accidents and deaths that are much higher than those obtained with the previous model. Furthermore, all of the other parameters and results of elasticity that pertain to each of the independent variables used in the different equations of the model essentially remained unchanged. Therefore, the forecasts obtained change very little for all of the dependent variables, except for fatal accidents and victims killed, which now have higher values. This verification served to finalize the process of obtaining forecasts. In the section that follows, we shall detail, by means of illustration, forecasts obtained for victims killed. B) Forecasts for the number of deaths for the period of 1997-2004 The results presented refer to the basic scenario used. We shall comment later on the influence of some changes to the basic scenario on the forecasts obtained. They were also produced on a monthly basis. On this matter, we note significant fluctuations and therefore the predicted number of deaths is smaller during the winter months than during the summer.

The DRAG-2 Model for Quebec 65 Table 2.3 shows the yearly changes to forecasts of the number of victims killed for the 19972004 period. Furthermore, in order to give a little more perspective to these forecasts, the table includes observations for the 10-year period of 1987-1996. This table reveals that the downward trend noted since the beginning of the 1990s should continue until 2004. Compared to the 877 deaths in 1996, there are an expected 779 in 2000. This number would decrease to 720 in 2004, representing a 17.9% decline from 1996. These forecasts therefore indicate that the downward trend in the number of deaths that began approximately 25 years ago in Quebec can be expected to continue in the coming years. Table 2.3 Yearly changes in number of victims killed, in Quebec between 1987 and 2004 Actual data (1987-1996) and forecasts (1997-2004)

DATA

YEAR

VrCTIMS

1987

1 116

A

1988

1091

C

1989

1 141

T

1990

1 085

U

1991

1 012

A

1992

981

L

1993

982

1994

827

1995

883

1996

877

F

1997

835

O

1998

808

R

1999

787

E

2000

779

C

2001

761

A

2002

746

S

2003

733

T

2004

720

Source:

Direction de la planification et de la statistique, SAAQ, September 1998

66 Structural Road Accident Models We also examined other scenarios pertaining to expected changes to some independent variables. These variables should be important and subject to changes that may differ significantly from the basic scenario. The results of this exercise revealed that forecasts for the number of deaths obtained vary significantly from those of the basic scenario. Lastly, it must be specified that these forecasts do not include the effects of Bill 12 introduced in 1997, which should also contribute to a decline in the number of deaths. This legislation contains provisions for graduated access to a driver's licence for new drivers and harsher penalties for impaired driving or driving under suspension. This legislation was the subject of substantial media attention in the months leading up to its introduction. Indeed, the actual number of road accidents between 1997 and 1998 was respectively 805 and 717 victims killed. Therefore, the actual number of deaths in 1998 is already lower than the forecast for 2004. In all likelihood, the legislation of Bill 12 accounts for a large part of this decline in 1997 and 1998.

2.4. OTHER DEVELOPMENTS

The first complete cycle was conducted on the basis of development of the DRAG-2 model. However, the completion of this work required other developments of the model. For instance, from January 1978, bodily-injury accidents and the corresponding victims are now distinguished according to severe or minor bodily injuries. The availability of this information therefore represents enormous potential for improving the model by adding equations that take into account these two categories of injuries. The influence of distance travelled on accidents could also be further examined. The results obtained show that this influence, especially for the most severe accidents, would not only be not directly proportional to the distance travelled but where its true shape is that of an inverted "U", may even have a positive, zero or negative influence on these accidents, depending on the distance travelled. One possible explanation to this phenomenon is growing traffic congestion. Given its importance, the influence of fluctuations in distances travelled on road accidents and victims should be the subject of a separate examination. Lastly, one must also consider the influence of new factors on the number of road accidents. We might for example take into account recent progress in trauma surgery, new rates for driver's licences based on demerit points, which has produced its full effect since the fall of 1994, and the influence of introducing Bill 12 in June and December 1997.

The SNUS-2.5 Model for Germany 67

THE SNUS-2.5 MODEL FOR GERMANY

Ulrich Blum Marc Gaudry 3.1. CONTEXT* The present article presents an improved and refined version of the SNUS-1 model (Gaudry and Blum 1993) documented only in French. The greatest difficulty faced in the development of the model did not have to do with structure - the multilevel structure is straightforward - but with the specification of the employment activity variable, due to the specifics of the German economy, and with the proper formulation of the role of vehicle stocks in the road demand models. Moreover, we consider the following aspects to be special in the context of an analysis of Germany: (i) there exist no general speed limits on motorways, i.e. about 70% allow unlimited speed today, and in the Sixties, when our analysis starts, this share was even higher; (ii) the country is large compared with other regions were the DRAG-methodology is employed, and it possesses high car ownership levels and an important car industry that sees the German infrastructure as an appropriate testing ground; (iii) Germany is poly-central, its infrastructure re-

1 Acknowledgements. This research was made possible by support from the DFG {Deutsche Forschungs-Gemeinschaft), the Alexander von Humboldt-Stiftung of Germany, the National Sciences and Engineering Research Council of Canada (NSERCC) and by Marc Gaudry's tenure as a 1998 Centre National de la Recherche Scientifique (CNRS) researcher at BETA, Universite Louis Pasteur and UMR CNRS 7522. The authors are grateful to many others who helped with the data base, estimations and interpretations, including Andreas Althoff, Benoit Brillon, Francine Dufort, Stephane Gelgoot, Fran9ois Fournier, Pierre Langlois, Falk Kalus, Sylvie Mallet, Robert Simard and Liem Tran. Earlier versions of this chapter were presented in 1998 at the September BASt-conference on road safety held at Bergisch Gladbach and at the November 26-27 international conference "Za modelisation de I'insecurite routiere par I'approche DRAG/ThQ DRAG approach to road safety modelling" held in Paris under the auspices of the Institut National de Recherche sur les Transports et leur Securite (INRETS), The Swedish Foundation for Research in Transportation (KFB) and the Societe de Vassurance automobile du Quebec (SAAQ) which has supported the development of the international DRAG network since 1995.

68 Structural Road Accident Models sembles a grid, whereas France's is almost a hub-and-spoke system, as compared for instance to Norway's line; (iv) unification is not yet included because of lagging data availability and, thus, problems to compensate for the structural break in data series. In terms of econometric analysis, we were led to apply the TRIO-LEVEL algorithm (see Ch. 12) in new ways, both in the analysis of the functional forms and the evaluations of multimoment determinations of the models. Our analysis is primarily in terms of the first moment (expected value) until Section 3.4.2 where it is extended to higher moments. Up to that point, we will report principally on elasticities, without stating the t-statistics associated with model parameters: these can be found in Appendix 3.1.

3.2. S T R U C T U R E O F M O D E L

3.2.1. The dependent variables As shown in Diagram 3.1, inspired by Jaeger (1998), no speed variable or congestion data are available in the German context. As they cannot be observed directly, some interpretation of results will require taking them into account. German data do not make it possible to distinguish between injury and fatal accidents despite the fact that the number of injured and killed victims are recorded. PERFORMANCE

3 4.

OF THE TRANSPORT

SYSTEM

SYSTEM

CHARACTERISTICS

VEHICLES q u a n t i t y quality

D e m a n d for road us K i l o m e t r a g e g a s o l i n e vehi. K i l o m e t r a g e diesel v e h i c l e

I N F R A S T R U C T U R E

(^ ACCIDENT FREQUENCY N. h L i g h t m a t e r i a l d a m a g e a c c i d e n t s 10 S e v e r e m a t e r i a l d a m a g e a c c i d e n t s 3 Total m a t e r i a l d a m a g e a c c i d e n t s ( 3 + 4 ) 11 B o d i l y d a m a g e a c c i d e n t s 1 2 T o t a l a c c i d e n t s ( 5 + 6 )

V

\

/ ^

3

M o r b dily M o r b b o d i l M o r b

4

M (

J2

(^ 5 6 7. 8

Inju Inju Inju Kill

r r r e

i d i t d a m i d i t y d i d i t

ed ed ed d

killed

p e r

D R I V E R S

.1 u a 1 i t y

J

ACCIDENT S E V E R I T Y " ^ y light lightly injured per bo a g e a c c i d e n t y s e v e r e s e v e r e l y injured pei a m a g e a c c i d e n t y injured p e r bodily d a m a g e

ty

K

b o d i l y

VICTIM S lightly (6 x 8) s e v e r e l y (6 x 9) total (6 x 1 0 ) ( 6 x 1 1 )

d a m a g e

ETC I s tr a t i

The SNUS-2.5 Model for Germany 69 3.2.2. Visual analysis of the dependent variables 3.2.2.1.Road demand. As Appendix 1 makes clear, road demand was expressed by the kilometres driven with gasoline (KMBL) and diesel vehicles (KMDL). The original variables, monthly gasoline and diesel consumption, were transformed by dividing them by their specific consumption rates. These two variables taken together are used in the accident equations as measure of exposure (KML). In the severity equations, a gasoline car use index was computed that captures the kilometrage driven by gasoline cars relative to that of all cars (GCUI).

^ Graph 3.1. Road demand: kilometres driven by gasoline and by diesel vehicles 3.2.2.2. Accident Frequency. Accidents were available in two categories with their respective subcategories and aggregates: (i) accidents with light material damage (ULSS) and severe material damage (USSS) according to a delimitation of 1 000 DM; from 1983 onwards 3 000 DM both add up to accidents with material damage (USS); (ii) accident with personal damage (UPS); if this type of accident is observed, a parallel material damage is not counted; (iii) total accidents as the sum of personal and material damage (UG=USS+UPS).

Graph 3.2. Structure of road accidents by category

70 Structural Road Accident Models 3.2.2.3. Accident severity. The morbidity and mortality rates are defined as: (i) number of persons with light bodily damage (MBL) and with severe bodily damage (MBS) per bodily damage accident; they can be added (MB=MBL+MBS); (ii) number of persons killed per bodily damage accident (MO); 1 .60 1 .40 1 .20 1 .00

0.80 0 .60 0 .40

0.20 0.00

i k i l l e d p e r s o n s p e r a c c id e n t w i t h b o d i l y d a m a g e s i s e v e r e l y i n j u r e d p e r s o n s p e r a c c id e n t w ith b o d i l y d a m a g e s llightly injured persons p e r a c c i d e n t w i t h bodily d a m a g e s

Graph 3.3. Severity of road accidents by category 3.2.3. Matrix of direct effects General overview. Table 3.1 shows the relationships between the dependent variables and the 12 categories of explanatory variables. In particular, note in the full list of Appendix 3.1 that: (i) exposure was included with two variables derived from the dependent variables of road demand (KML and GCUI); (ii) prices: we used the real price of gasoline per kilometre, i.e. corrected for fuel efficiency, (RBPNSC) and the real price of diesel fuel per litre (RPBD) and the weighted combined price per litre (RPNSD); (iii) quantity of motor vehicle: this included the stock per employee for gasoline (PKWBPE) and diesel (PKWDPE) cars, for both their squared values as well; (iv) characteristics of motor vehicles: the belt usage rate was included (GAQC); this variable was corrected as observations started only in 1975 (with observed levels around 40%) so that we smoothed in earlier values with an geometric function; (v) laws, regulation and police surveillance: this group includes the alcohol limit established in 1973 as a dummy (P08); (vi) network and time service levels: speed limits, especially those enforced during the time of the oil crisis (HHG and NHG7374), are included; (vii) infrastructure and weather characteristics of the network: four types of weather variables were used, and the city of Frankfurt was taken as a reference base for Germany, rainy days (RF) and precipitation (NIF), temperature (TFF) and sunshine (SSDF); (viii) general characteristics of consumers: these include the establishing of provisional driving licenses in 1987 (FSP); (ix) ebriety or vigilance of consumers: the level of retail sales in drugstores per adult (REUAERW) and the production of beer (BIERPE). This variable is more related to consumption than in the case of other alcoholic beverages, and is also used as an activity variable to model individual mobility;

The SNUS-2.5 Model for Germany 71 (x)Jinal and intermediate activities: final and intermediate economic activities play an important role for road use and were described by five variables. Besides long distance truck transportation (FVLKWPE), and sales and overnight stays per employee (REUNBPE, UENGPE), variables were constructed to capture the differentiated pattern of work attendance and free days in Germany. The complex holiday structure that varies among provinces is used to calculate an employee presence index and an index of free days (EPIFT, FRTMFG). Income does not play a role in the model once final and intermediate activities are accounted for; (xi) et cetera with respect to administrative rules: the change of the material damage classification after 12/82 was accounted for by a variable (SSSKIC) which was set to DM 1 000, —for the first 180 observations and DM 3 O00,for the observations thereafter. This series was then divided by the Consumer price Index; (xii) et cetera with respect to aggregation: as monthly data were used, the differences in the length of the months and the number of Saturdays, Sundays and holidays was captured with three aggregation variables (AT, ST, SF). Table 3.1. Matrix of direct effects of SNUS-2.5

Exposure Prices of fuel or other prices Motor vehicle quantity and characteristics Laws and regulations; safety measures Network or time service levels of road modes Infrastructure and weather

Demand for road use Accident Accident Victims frequency severity (exposure) D AI V

V V V V V

General driver characteristics Ebriety or vigilance Activities intermediate or final; income

V V

Administrative Aggregation, seasonal, trends

V

V

V

D D

<

V V V V V V

4 V

V

V V V V V

D D D D D D D D

In practice, for reasons of multicolinearity, many of the variables were expressed in ratios, for instance per-employee ratios in the road demand functions or in per-km ratios in the accident and the severity models. Some variables of specific interest to Germany, and not shown in the discussion of results below, deserve comment: the stock of cars, the employment presence index and the sales in pharmacies. Stock of cars and fuel efficiency. We clearly see in Graph 3.4 that vehicle ownership has increased tremendously over time with a tendency towards saturation in the gasoline fleet whereas diesel cars are still picking up. Germany was a latecomer in the private use of diesel cars.

72 Structural Road Accident Models

e h i c l e s p e r Em p l o y e

Graph 3.4. Vehicle ownership per employee in Germany Employment presence. Long-term changes in German employment are rather small because of institutional regulations; their development over time is rather smooth and only has a limited explanatory variance. More important are fluctuations in worker presence because of the rather complicated holiday scheme in Germany where each province follows a distinctly different pattern, as shown in Graph 3.5. This is especially during summer where the six-week school holidays of each of the provinces revolve over a time span of three months. Retail sales in drugstores. In Graph 3.6 it is interesting to note both the strongly seasonal pattern of drug sales and the strong upper trend since 1982. The strong links with accidents and their severity shown in the Appendix imply increasing problems of management not independent from those requiring attention with an aging population. There is no age structure population in SNUS-2.5.

C^


^^

^^

^ Scr

N^

N''

^

^ ^'

<^ <^'

r

Graph 3.5. Employment presence index

C5%

^ <^'

N^^

O^'

^f

The SNUS-2.5 Model for Germany 73

Graph 3.6. Retail sales in drugstores per adult 3.3. RESULTS AND THEIR INTERPRETATION 3.3.1. Statistical results Each model is reported in a column of Appendix 3.1. The first two columns relate to road demand, columns 3 to 7 to the number of accidents and columns 8 to 11 to the severity of accidents. In seven out of the eleven equations, the same transformations were applied to the endogenous and all exogenous variables. In case of the remaining four, only one Lambda on all exogenous could be applied -a further breakdown did not provide any statistical gain. A strong heteroskedasticity could only be found in the case of the MBL equation. Autocorrelation of multiple orders was present in all of our series - in all of our equations, a first order was found, a strong autocorrelation structure around orders 4 and 11/12. All R^ are very high. Multicollinearity was eliminated using the Belsley test by excluding affected variables or expressing them as ratios. We also summarize in Table 3.2 the overall results. It is easy to see how models with fixed forms would not have been acceptable.

3.3.2. Economic results: overall specific results Road demand. Figures 3.1 and 3.2 summarize important results not discussed in detail below. If we analyze the impact of the gasoline and the diesel fleets, we see that additional cars first increase and later decrease road demand, i.e. the structure of the quadratic function is concave. The case of the diesel vehicles involves a straightforward quadratic form with coefficients

74 Structural Road Accident Models fi 1=81,6 > 0 and fi2= -357,5 <0 describing a maximum at 0,114 vehicles per employee, which is well above the average of that series (0,06) - in fact, this maximum was surpassed only from June 1986 onwards. To understand how additional diesel cars could reduce demand it has to be remembered that the dependent variable is a transformation of litres into kilometres that does not take into account the increased fuel efficiency of the diesel engine and that additional vehicles are often small cars. Table 3.2. On functional form, stochastic specification, and other summary statistics Demand

Severity

Frequency

KMLB

KMLD

ULSS

USSS

uss

UPS

UG

MBL

MBS

MB

MO

X variables:

18

16

21

21

20

20

20

19

20

20

20

•

n. of t-stat. (2
10

6

10

12

8

13

10

6

8

6

5

•

n. of t-stat. (l
3

5

8

2

6

3

3

5

4

10

10

•

n. of t-stat. (0
5

5

3

7

6

4

7

8

8

4

5

Heteroskedasticity*

0

0

0

0

0

0

0

1

0

0

0

Autocorrelation **

9

2

4

5

4

5

4

5

5

5

4

Form •

^(y)

0,309

0,091

0,081

0,125

0,233

2,214

0,388

-3,198

0,973

-2,722

0,528

•

^ (X,)

0,309

0,091

0,081

0,125

0,233

1,200

0,388

0,947

2,183

0,276

0,528

•

^ (x^)

-3,296

Significance (opt)]

-3896,3

-4256,3

-2484,9 -2147,7 -2538,2 -2193,8 -2568,9

737,6

901,0

789,7

1219,4

•

[LL at i

[LLatX=l(lin.)]

-3916,1

-4311,8

-2515,6 -2209,0 -2556,9 -2206,9 -2575,7

708,5

898,4

781,6

1214,2

•

[LL at ?i=0(log.)]

-3908,4

-4264,2

-2485,4

-2149,0 -2540,0 -2240,1 -2574,3

712,6

892,4

784,1

1211,7

264

264

264

264

264

264 264 Sample (1/68-12/89) number of parameters ** number of rhos

264

264

264

264

As for gasoline cars, the function used^ involves X^^^ and [X^j . It can be shown (Gaudry et al., 2000) that a maximum occurs if /?, • (A -1) • X^~' +2-13^-{2-X-\)-

X'^"' < 0, a condi-

tion that is met in our case with coefficients ^7= -93 > 0 and^2"^ 4,4 <0 because of the hyperbolic transformation with /I =-3,3 describing a maximum at x=\-P^I2-p.^\^

= 0,49 cars per

employee. This maximum, however, lies well below the average of 0,800 cars per employee — the maximum was surpassed from April 1970 onwards.

2 The addition of the squared term and the use of a/L specific to both increased the loglikelihood from -3900,08 to -3896,33. In the above-mentioned diesel equation, an attempt to use a similar form yielded /l= 0,915 and demonstrated that the simple quadratic form term used was sufficient.

The SNUS-2.5 Model for Germany 75

1

All A ctivitie s (1-3) 0 ve nig ht S t a y s R e t a il S a le s w

F o o d and C l o t h i n g

L 0 n 3 D is ta n c e

1 '

r "

••'

1

(3) (2)

T r u e c i n g (1 ) 1

1

1

"S

1

tock*2

S took

-0,60

-0.40

1 1 -0,20

1 P rice

0,00

0,20

0,40

0,60

0,80

1,00

E la s tic itie s

Figure 3.1. Decomposition of road demand (diesel) by important variables This means that, in both the diesel and the gasoline vehicle stock cases, marginal additions are associated with reduced kilometrage. Further work is needed to understand the extent to which unmeasured congestion and population aging contribute to this result, in addition to factors already mentioned. 1

All A c t i / i t i e s (1-5)

F etai

Sales Food and C lothing (4) Long

>

1

1

m

O v e r n i g h ; Stays (5)

ro

1

1

1

u c k i n g (3) Distance T Beer (2)

1

1 e s e n c e (1) =1 imployee P Stock*2 1

1 -0 ,60

-0,40

-0,20

-

Stock Price

0,00

0,20

0,40

0,60

0 80

Elasticities

Figure 3.2. Decomposition of road demand (gasoline) by important variables 3.3.3. Decomposition of the impact by variable: results common to other models Exposure. Exposure, shown above on Graph 3.1, has differentiated impacts, as shown in Figure 3.3. As an endogenous variable in the accident frequency models, we see increased impacts on severe material damage, and a shift to higher risks in the severity models. Fuel price. The evolution of fuel prices is shown in Graph 3.7, and the impact of the two (combined) prices on accident frequency and severity shown in Figure 3.4 is of major interest as it implies an inference on risk taking of higher prices despite relatively low elasticities of the

76 Structural Road Accident Models demand for road use itself.

1

• ^

Mortali ty (MO) Morbidi ty ( M B t

•a

c

CD W 0) O

c

(U CT Q) LL C Q) T3 O

1

1

1

Total Damac e (UG)" Corporal Damag J (UPS)" Material Damag 3 (USS)-

Severe Material Damage (USSS)"^

1

1

Severe M orbidity Light Morbidit; 1 (MBL)~ ] Se verities


w

1

1 1

1

Light Material Damage (ULSS)

p 1

Freqi lencies

<

-D,10

0,00

0,10

0,20

0,30

0,40

0,50

0,60

Elastici ties

Figure 3.3. Impact of exposure on road accidents

1,60 1.40 1.20 1.00 0,80 0,60 0.40

.^ f^

,/ -real price of gasoline (Normai/Super)

.^ real price of Diesel}

Graph 3.7. Real Prices of Fuel Temperature. As shown in Figure 3.5, rising temperatures imply strong decreases in material damage accidents and strong increases in the frequency of bodily injury accidents and in morbidity and mortality rates, to say nothing of the strong increase in road demand. Percentage of belt use. Figure 3.6 indicates that belt use reduces both the frequency and the severity of bodily injury accidents, increasing the frequency of material damage accidents.

The SNUS-2.5 Model for Germany 77 M o r t a l i t y (MU) Morbidity (KMB) • ^ S e v e r e Morbidity (IVBS) Light Morbidity (IVBL) Severities' [nage (UG) H Corporal Damage (UPS) Material Damage (USS)

0) •o o

Severe l\ilaterial Danhage (USSS) Frequencies

Light Mateijial Damage (ULSS) Diesel Kilohneters Gasoline Kilometers (KMBL)| Road Demand

-0,20

-0,15

-0,05

-0,10

105

Elasticities Se

Figure 3.4. Impact of fuel prices on road demand and road accidents M V r t d l i t y (MO)

t n ^ E

S e v e r e M o r b i d ty ( M B S ) J Light Morbidity

(M^L)

Severities T o t a l D a m a g e (UGJ C o r p o r a l Dam: ge ( U P S ) J

H Material Damage ( USS)

CHZ L i g h t Mait e r i a l D a n j a g e ( U L S P) Frequencies Diesel Kilometrds

(KMDL)

Gasoline Kilometrds

(KMBL)

Road D e m a n d -0,04

-0,02

Elasticities

Figure 3.5. Impact of temperature on road demand and road accidents Beer consumption. As shown in Figure 3.7, beer consumption is both a social activity variable, thereby increasing road demand, and a factor changing the frequency and severity mix of accidents, but the effect on the increase in bodily injury accidents is larger than that on mortality, implying an increased number of fatalities. Our results are consistent with the view that most of those that drink and drive have only consumed little and compensate to prevent accidents, while those who drink a lot may increase their risk.

78 Structural Road Accident Models

L^

T

I. . . i

-. -

Mortality

(MO)

Morbidity

(MB)

Severe Morb dity Light Morbidity

[

(MBS) (MBL)

Severities •D

O Total Damage [ r

(UG)

. - . • . • . • . • . • . • . • .

M a t e r i a l 1) a m a g e

(UPS)

(USS)

Sevi ire M a t e r i a l D amage ( U S S S ; Li )ht M a t e r i a l D a m a g e

1

C o r p o r a l D ar l a g e

•

(ULSS)

^

.•- . . .

Frequencies -0,2 0

-0,10

0,0 0

0,10

0,20

Elasticities

0,30

Figure 3.6. Impact of belt use on road accidents Food and clothing. Figure 3.8 shows that shopping trips are relatively dangerous, perhaps because this trip purpose involves higher occupancy rates of vehicles than, say work trips, thereby increasing the frequency of bodily injury accidents three times as much as that of material damage accidents; these effects are not fully offset by reduced mortality rates. Mortality (MO)

q

M o r b i d i t y (M B) ere

M orbidlity

(MBS)

c:

Light

M o r b i d 11 y ( M B ll

Total

Darrtage

Severities

rporaI

Damage

(UPS)

(UG

] Material

p amage

Severe

M |a t e r I a I

Light

M at

rial

( U J S) Dan age

D a m a le

(USS

(ULSS)

equencies

Gasolirje

Kilometers d -0,10

(KMBL) Demand 0,00

Elasticities

Figure 3.7. Impact of beer consumption on road demand and road accidents

4)

The SNUS-2.5 Model for Germany 79

M ortalit / ( M O )

(

M o r b i d ity

(MB)"

u

M o r b id it y ( M B S ) ^

Severe

M orbidity

Ligh

(MBL)

S( j v e r i t i e s (A

Tc tal C o r p o r i\

O

D am a ge

M a t e r i i 1 D a m a g (; Material Damage

Severe Light

Damage

Materia

(U G)""

. . . 1

1

D am ag e (U P S ) ~

1

1

(USS)

(USSS)

"

(U L S S ) ~

1

1

1

1

1

1

2 1

'

]

F r e q u en c i e s

D iesel

K i lorn e t e r s ( K M D L ) "

Gasoline

K i lorn e t e r s ( K M B L) ~

. • .

1

1

0,10

0,20

.

.

.

1

R o a d C) e m a n d -0 ,20

-0,10

0,00

0,30

0,40

0 50

Elasticities

Figure 3.8. Impact of demand for food and clothing on road demand and road accidents

3.3.4. Results for other variables Sunshine. As shown in Figure 3.9, the effects of sunshine on the frequency and the severity of accidents are startlingly strong. It may be that sunshine reduces visibility in ways that are insufficiently compensated by drivers. Mo rtality

(MO) Morbidity

c

>evere

Morbidity

(MB)

(MBS) Light

Morb idity

(MBL)

Severities Total o

Dam age .

•

s

^ 1

.

T " " " "

1

^"'^

^^

.

Material Severe Light

.

D esel

Kilometres Kilome tres R )ad

-0 , 0 6

-0,04

-0,02

.

D image Ma terial

Mate rial

Frequencies

Gas ollne

(UG) .

.

.

-

(USS

.

.1

)

D a m a je

(USSS)

Damage (ULSS)

(KMDL (KMBL

.1

Demand 0,00

0,02

0,04

Elasticities

Figure 3.9. Impact in sunshine on road demand and road accidents

0 06

80 Structural Road Accident Models Rainy days and precipitation. Figures 3. 10 and 3.11 show the distinct but closely related effects of the presence of rain and of the amount of rain per day. Including both variables in the model allows us to account for distributive effects of rainfall over the month, but the presence of rain has larger proportionate impacts than the amount of rain.

0.00

0.01

0.02

Elasticities

Figure 3.10. Impact of rainy days on road demand and road accidents Speed limits 1973-1974. As shown in Figure 3.12, the imposition of speed limits in the aftermath of the October 1973 first oil shock had beneficial effects on all dependent variables.

M^r t a 1 i t v/ ttAn\

1

1

CD M o r b i d i t y

(MB)

E H S e v e r e M o r b i d i t y (MBS') Light Morbidity (MBL) IIH

\

\ Severities | CO

T o t a l Da mage (UG) UH O

C o r p o r a l Danl a g e ( U P S )

^

^l a t e r i a l

Danr age ( U S S ) igp

fm

(IfSS'^)

I

L i g h t M a t e r i a l Dam£ ige ( U L S S ) ] F requencies

1

1 D i e s e l K i l on e t r e s (KMC >L)

G a s o ine K i l o m e t r e s ( K M B L ) E H Road D e m a n d -c ,010

-0,005

0,000

0,005

0,010

0 015

Elasticities

Figure 3.11. Impact of mm of rain on road demand and road accidents

0 020

The SNUS-2.5 Model for Germany 81 Blood alcohol limit imposed in 1973. It is clear in Figure 3.13 that the 0,8 % blood alcohol concentration (BAC) limit imposed on October 1, 1973 reduced driving, accidents and their severity. 1.

.

.

1

1

]•

o

-^ 1 . 1

^

-

'

1

.

•

1

"

1

.

•

.

•

.

•

1.

.

1

. . . . . . .

1

.

.

1

•

.

•

.

•

.

•

.

•

.

•

.

•

.

1

.

.

.

•

.

•

.

•

.

•

.

.

•

-

.

1

.

Morbidity

Total

.'

.

.

.

Damage

Corporal

-

(MBS) (MBL)

(UG)

Damage

(UPS)

Material Damage (USS) Severe Material Damage

1

1

.

•

1

1

.

Morbidity

ities

1

1 ''

1

•

(MB)

Liglit

r

(J

(MO)

Severe

1

Seve

Mortality Morbidity

.

Light

Material

Damage

(USSS) (ULSS)

F equencies Diesel f.'." Road . .

Kilometres

Gasoline

(KMDL)

Kilometres

(KMBL)

Demand

.-Q,D4

•°%i a s t i c i i i e s

Figure 3.12. Impact of 1973-74 speed limits on road demand and road accidents Retail Sales in Drugstores. In passing, we note that accident risks are heavily polarized because of significant increases of light morbidity and of mortality rates. M o r t a l i t y (MO) M o r b i d i t y (MB) Morbidity (MBS) 1 . . . . . . Severe r. . . . . . L i g h t M o r b i d i t y ( M B L ) Severities

r. . .

I . . . . .

.

(0 •D U '^

1

1. . . . 1

1

•

.

•

.

•

.

•

.

.

•

.

•

.

•

.

1 1 . . . . . . . . . . . . . . . . . . . . . . 1 1

1

1

1

1

•

.

0,08

0,06

.

•

1

• "1

0,04

-

1

• "

,

1 . . .

1

1

0,10

•

\

•

0, D2

.

.

•

.

•

.

•

.

T o t a l Damage (UG) Corporal Damage (UPS) Material Damage (USS) - Severe Material Damage - (USSD) Light Material Damage Frequencies Diesel Kilometres (KMDL) Gasoline Kilometres (KMBL) Road D e m a n d 0,0 0 .

. . . . . .

.

-

-

'

Elasticities

Figure 3.13. Impact of blood alcohol limit on road demand and road accidents

82 Structural Road Accident Models 3.4. D E R I V I N G O T H E R I N T E R E S T I N G R E S U L T S

3.4.1. The analysis of victims: direct, indirect and total elasticities The inquiry into victims will first concentrate on the impact of alcohol; all other variables can be analysed in the same way. We will derive the impact on injured persons of both changes in beer consumption and BAG limits, taking due account of the indirect effects rippling from the demand equation to the frequency and the severity equation. Finally, we will add a corresponding analysis of the impact of fuel price. Let us then determine how the change of one exogenous passes through the different layers of the model. Take, for instance, beer consumption in Figure 3.14: it increases road demand (elasticity of +0,204) and road demand increases corporal damages {+0,197), giving a total effect of 0,0402. The direct effect of beer consumption on corporal damage frequencies is 0,103, so it adds up to a gross corporal damage frequency elasticity of 0,1432 (= 0,0402 + 0,103). Furthermore, given the impact of beer consumption on road demand (+0,204) and of road demand on morbidity (-0,018), the total indirect effect is -0,0037; accounting for the direct effect of beer consumption (-0,014), we obtain a total morbidity (i.e. injured per accident) of0,0177 (== -0,0037 - 0,014). The total elasticity for victims is the sum of the total frequency and morbidity elasticities, i.e. 0,1255 (= 0,1432 - 0,0177).

(10)

Total

Injured

Corporal

Victims

D a m a g e

(5+9)

Frequency

(3 + 4 ) (4 ) C o r p o r a l (3)

D a m a g e

Directly

from

Beer

Corporal

D a m a g e

via

Road-[

D e m a n d (2)

Corporal

(1)

Road

from

Beer

D a m a g e

D e m a n d

(1*2)

from Road D e m and from -0,05

Beer 0,00

0,05 0,10 Elasticities

Figure 3.14. Direct, indirect and total corporal impacts of beer consumption

The SNUS-2.5 Model for Germany 83 These derivations are indicated in Figure 3.14 for beer consumption, in Figure 3.15 for the August 31, 1973 blood alcohol concentration limit, and in Figure 3.16 for the gasoline price. The interested reader can compare in these graphs the direct effects, the indirect and the total effects. (10)

Injured Victims

(9) T o t a l Accident (8 )

M o r b i d ity (7 + 8)

M o r b i d ity

( 5 + 9) = Injured

Directly from

(7 ) M o r b i d ] t y v i a R o a d U A l c o h o l L Im i t ( 1 * 6 ) (6)

M o r b id it y f r o m

R o aa

pe

A Ic 0 h 0 I

Limit

D e m a n d ff r 0 m

Demand

0) O

(5)

T o t a l C o r p o r a l Di m a g e Frequency (3+4) Directly

H ( 3 ) C 0 r p 0 ra I Demand (2)

Corporal

from

Alcohol

Damag e from

L m it

Road

D a m a g e via

fi

Road

(1*2)

Demand (1 ) R o a d 0,00

D emand 0,05

Elasticities

from

Alcohol

llimit

0,10

Figure 3.15. Direct, indirect and total corporal impacts of the alcohol limit

(1 0) I n j u r e d V i c t i n i s ( 5 + 9 ) (9) T o t a l M o r b i d i t y = I n j u r e d per Accid] (7 + 8) (8) M o r b i d i t y D i r e d tly f r o m P r i c e (7) M o r b i d i t y via Ro|ad Demand f r o m Price (1*6) I

(6) M o r b i d i t y f r o m Road Demand (5) T o t a l C o r p o r a l JDamage F r e q u e n c y

3+-4)

(4) C o r p o r a l Damage D i r e c t l y f r o m pricj (3) C o r p o r a l Damabe via Road Demand f r o m Price^ ( 1 * 2 ) 2) C o r p o r a l Damage f r o m Road Demand (1) Road Demand f r o m G a s o l i n e 0,00 Elasticities

Figure 3.16. Direct, indirect and total corporal impacts of the fuel price

0,20

84 Structural Road Accident Models 3.4.2. Multiple moments and their marginal rates of substitution Observations and moments. All discussions above focus on « explaining y ». But if drivers are trying to achieve a combination of risk objectives through control of y, observations on y are just about the tool, about a sort of derived demand, that should reveal an underlying mechanism at work. In the analysis of financial returns of assets, for instance, it is believed that at least the first and second moments of y (return) are of interest to investors. But the introduction of such a wedge between « explaining observations » on a variable and « explaining moments » of that variable in effect constitutes a major change, both conceptually and in terms of required computations. Firstly, except in a linear model, « explaining y » is not the same as « explaining |a (y) », the first moment or expected value of y, even if this distinction is ignored in conversation. More generally, if drivers care about many features of the occurrence of accidents, and in effect care about the very shape of the accident probability distribution, then regression models should be constructed, and their measures of « quality of fit» defined, not just in terms of « observed y », although this remains of some interest, but in terms of the first moment and of higher moments: the moments of y themselves have become the objects of explanation. This requirement is a tall order for model specification because intuition about road safety is generally only about the first moment (the expected value) and only about accident frequency (not about severity rates). In consequence, model building is, at best, implicitly oriented towards fitting only the first moment: to be more precise, it is usually explicitly oriented only towards fitting the « observed y »—and towards interpreting the results in terms of that single dimension. It is also a tall computational order to obtain all derivatives, elasticities and marginal rates of substitution pertaining to all moments of y. For instance, discrete choice models are not yet concerned with densities of the choice probabilities because the task is extremely difficult. Moments, utility maximisation and local trade-offs. Clearly, as observed accident frequency distributions are not normal (Gaussian), there would be an a priori case for thinking that an underlying multimoment mechanism seeking « desired » values in moments is at work. The moments of interest, in addition to the first, should include the variability of the accident probability, as measured by its variance (or, more conveniently, by the standard error a (y)), and whether the accident risk is skewed « downwards » or « upwards », as measured by a (y), the asymmetry coefficient (that can be negative (to the left) or positive (to the right)). The fourth moment, kurtosis, that tells us about the concentration of observations, its « flatness » or « peakedness » about the mode of the distribution, might also be of eventual interest. But the nature of the underlying utility function is difficult to hypothesise with moments higher than the second and dubious even for the second because implied quadratic utility functions are not without their problems, well summarised in Machina and Rotchschild (1987).

The SNUS-2.5 Model for Germany 85 The theory of expected utiUty used in financial analysis, for instance the well-known capital asset pricing model (CAPM), allows expected utility to depend only on two such moments because nobody really believes that the mathematical expectation of asset return is the only moment that matters to utility. In consequence, the usual investor with positive marginal utility [5U/6w ) 0]of wealth w, diminishing as wealth increases [s^U/^w^ < O], has long been shown to display positive preference direction for the first moment and negative preference direction for the second moment. But the formalization of underlying mechanisms, and the measurement of trade-offs or marginal rates of substitution among moments, has proven very difficult for moments higher than the second. Although Kraus and Litzenberger (1976) have implied positive preference direction for the normalized third moment and Scott and Horvath (1980) appear to have shown more generally that the investor's preference direction is positive (negative) for positive (negative) values of every odd central moment and negative for every even central moment, these results have recently been thrown into doubt by Brockett and Garven (1998) who have constructed counterexamples showing that this convenient rule of preference direction is false and that the ceteris paribus conditions assumed by these demonstrations are logically impossible since equality of higher order central moments implies the total equality of the distributions involved. In view of this last result that, for any commonly used utility function, moment preferences do not match up with a sequence of expected utility derivatives, we adopt the simple-minded view that observed choices reveal underlying local utility trade-offs and that it will eventually be possible to construct acceptable analytical mechanisms of utility maximisation consistent with them. Moments in accident analysis. Analyses of road safety treating accidents of various categories as a portfolio took a long time to appear. Despite the intuitive appeal of this view, we are not aware of any published accident moment studies outside of the DRAG research network. In other domains of application, such as the analysis of political events, formal statistical concerns with over dispersion and under dispersion within the Poisson model (King, 1989) are relevant, for the two-moment case at least, but do not formally treat the accident probability as a twomoment trade-off problem. Within the DRAG network, formal calculations (partial derivatives and elasticities) of observed y, \i (y) and a (y) are fully documented and available in the Tablex tables of TRIO since 1993: the computations are not trivial, for equations (1) to (3) with Box-Cox transformations, heteroskedasticity of a general form and multiple-order serial autocorrelation. Two-moments tests of derivatives, elasticities and of rates of substitution (also called Allais' r (AUais, 1987) coefficient) using these program features had started in 1990-1991 and had been duly reported to the

86 Structural Road Accident Models funding agency (SAAQ, 1991) and at various seminars in 1993 and 1994 on the basis of DRAG-1 results (presenting in particular effects of the snowfall variable). But the first generally available manuscript on these two-moment tests (presenting in particular effects of the temperature variable), written after a long, and perhaps unnecessary, wait for the longer series (481 observations, instead of 313) DRAG-2 results produced in 1997 to generate forecasts (Foumier et Simard, 1999), was completed only recently (Gaudry, 1998). That paper shows the fitted implicit trade-offs or marginal rates of substitution between the expected value |j, and the standard error a of accident frequencies, severities and victims by category, as well as the cross-category trade-offs among the accident frequencies of DRAG-2. The estimates turned out to be extremely close to those of the DRAG-1 model, despite the vastly increased sample size (481 instead of 313 observations) of the most advanced model. Also, the very reasonable values found implied a rejection of a Poisson assumption for all equations but one: the Poisson assumes that the first two moments of a dependent variable are equal, so that that the marginal rate of substitution between a unit of expected value and a unit of variance is one. There is of course no reason why behavioural rates of substitution among moments of a given dependent variable should be so restricted in accident analysis, or elsewhere for that matter. From two to three moments: more locally revealed trade-offs. Here we want to extend the analysis to the third moment a, called asymmetry, having augmented the 1993 algorithm previously limited to the first two moments. As a is adjusted by division by the third power of a, it is without dimensions and can naturally be positive or negative, according to whether the distribution has a tail to the right or to the left: by convention, this distribution is said to be « noticeably skewed » if a is greater than one-half in absolute value. We therefore extend to the third moment the hypothesis that drivers do locally adjust separately and independently (in non-Poisson fashion) among the moments of accident frequencies by category, and across categories. Their utility simply depends on the mathematical expectation, standard error and asymmetry of the accident probability: with revealed local trade-offs, it should be possible to construct certainty equivalence measures expressing these risk dimensions in terms of a numeraire about which it is meaningfiil to inquire whether drivers maintain it at a constant « homeostatic » level or not. Trade-offs and units. The marginal rates of substitution among moments, defined as ratios of partial derivatives of moments of y with respect to independent variables X,,, do not depend on the variable considered: [sfmom i)/5Xj,]/[5(moni j)/5Xj^] = 5(moni i)/5(moni j). They are indeed the same for all independent variables of an equation—even though the partial derivative of any moment with respect to a particular independent variable (say snow or tempera-

The SNUS-2.5 Model for Germany 87 ture) naturally depends on the variable considered. Note however on this point that, given the derivative with respect to the first moment, 5fj,/5X,^, and the Jacobians from )i to a and from L| L to y, we can deduce the derivatives da/dX^^ and dy/d X^, respectively. But, as we consider ratios involving changes in [i, a and a, we must remember that the first two moments have dimensions but that the third one does not: a

=

E[(y - ECy)]

, a

whereas \x = Efyjhas the same units as

= E[(y - E(y)] /a"^ has no dimension. Although this hetero-

geneity of dimensions severely limits our intuitive understanding of ratios computed between the first two and the third moment—^they are therefore not reported on in tables—, the signs of those trade-offs remain interpretable. The interpretation of signs. As in any ratio, the sign of the marginal rate of substitution depends on the sign of the elements, in this case partial derivatives: it will be positive if the derivatives of the two moments have the same sign and negative if they have opposite signs. Table 3 therefore presents expectations concerning the signs of the marginal rates of substitution among the first three moments. Table 3. Expectations of signs of marginal rates of substitution among moments Two patterns of marginal rates of substitution d (mom. i)/5 (mom. j) [A]. Riscophobe [B]. Riscophile i\j

JL

^

1

a

1

a1

_^

- 1

Assumed MRS Derived MRS

1 1 The [A] pattern of risk aversion for a financial (FIN) asset or a road accident (ACQ means that. dvi(Y)lda(Y)

(FIN): greater uncertainty a is traded against higher return (i; ) 0 (ACC): more uncertainty a is traded against higher expected probability JLI.

(FIN): increased upside risk (decreased downside risk) a is accepted against lower expected return \i; (ACC): higher upside accident risk a (decreased downside risk) is accepted against lower expected accident probability |LI. (FIN): increased upside risk (decreased downside risk) a is accepted against du(Y)/da(Y) < 0 lower uncertainty of return a ; (ACC): higher upside accident risk (decreased downside risk) is accepted against lower uncertainty (higher certainty) c of the accident probability. This result follows from the first two. These defined patterns are feasible (Table 6 in Liem et al., 2000). Riscophilia and riscophobia are de fined by assuming particular signs for the marginal rates of substitution (MRS) between the first andl other two moments. The MRS between the second and third moments is derived from these, whence! its' negative sign in both cases.

dv.(Y)lda(Y) ( 0

If accidents are objects of interest, and conceived as a « bad », one would expect the sequence of signs to be reversed, but the ratios or marginal rates of substitution between any pair of

88 Structural Road Accident Models moments to be unaffected in sign. This is exactly what is indicated in Table 3, where a reference pattern [A] is defined for a risk-averse individual, with a justification for each sign provided both for the case of a return from an hypothetical financial asset (a « good ») and for the case of a road accident (a « bad »). The psychological key to the understanding of this patter for accidents is to note that, in contrast with the case of the financial return from an asset, a less certain (higher a) accident is preferred to a more certain accident (lower a) and that downward risk (asymmetry to the left, i.e. a negative a) is preferred to upward risk (asymmetry to the right, i.e. a positive a). But this mutatis mutandis converse preference direction has no impact on the signs of the marginal rates of substitution among moments. Pattern [B] defines a pattern for a risk-lover. Poisson trade-offs of 1 between the first two moments and trade-offs of 0 associated with straight-line horizontal indifference curves in the two-moment risk neutral formulation (Tobin, 1965) are special cases. Germany, Quebec and speed limits. To perform our multimoment analysis, we limit ourselves to a discussion of frequencies and neglect other levels of the model structure, such as severity, due to the large amount of information to be reported and to the importance of making a comparison of Germany with Quebec. We compare both the data and the results of SNUS-2.5 and DRAG-1 accident frequency equations, as these models contain a common 15-year period, 1968-1982, within their respective longer monthly time series samples. A crucial difference between these data sets pertains to speed limits: there is no speed limit on more than three quarters of the 12 000 km German autoroute network. As compared to a fully regulated network, this « high end» freedom should affect trade-offs, both among moments of a given accident category and by implication across accident categories, for accidents most likely to occur at high speeds. It has been noted (Praxenthaler, 1987) that, over the period 1970-1986, both the autoroute share and the autoroute frequency per vehicle-kilometre (relative to the frequency for the total German road network) of injury accidents increased. 3.4.3. Marginal rates of substitution with comparable accident data Comparable data but contrasting asymmetries. Germany and Quebec have almost identical definitions of material damage accidents (variable USS for Germany and variable MA for Quebec) and of bodily damage accidents (variable UPS for Germany and COR for Quebec), covering both injury and fatal accidents. There are some minor differences in the definition of « material damage accidents » between these samples, because in Quebec the introduction of a self-reporting mechanism (during the last 3,5 years of the sample) for material damage accidents reduced the number of these accidents reported to the police, the common source of the accident data for both regions: although considered in the model, this may slightly influence both the measured moments and the estimated marginal trade-offs. The frequency distributions are shown in Figure 17, where the observed third moment asymmetry coefficients (other mo-

The SNUS-2.5 Model for Germany 89 ment measurements are found in Table 4) are shown in the subtitles. It is striking that the observed third moment is strongly negative in Germany for bodily damage accidents (UPS) and positive otherwise, both in Germany (USS) and Quebec (MA, COR). This is not surprising in the sense that, with unregulated speeds, one expects a strong tail to the left (for UPS, a = 0,66) as drivers aim for a sharply dropping fatal accident probability at high speeds. In speedregulated environments, where drivers are « forced to prudence » tails to the right are expected, for both material and bodily damage accidents, to the extent that some individuals will not respect the law but that the overwhelming majority will reduce speed. Note also that, in Germany, the asymmetry to the right of the frequency of material damage accidents is tampered by a second peak that may also be related to speed limits: it is as if one observed a mixed distribution consisting in one underlying distribution, asymmetric to the right, dominating another weaker distribution, asymmetric to the left. Table 4. Marginal rates of substitution among moments with comparable data Bodily damage accidents Material damage accidents | 1 d (mom. i)/ d (mom. j) 11. Results Germany USS 11 UPS SNUS-2.5 a a i\j a 1 M1 ^ ^ + 124 + 1 + 23 7,46 1^ USS

+ 1

a a

^^1

1^ UPS a a 1 Sample moment value 103 617 24 149 1 Fitted value at the means 103 453 5 844 1 Mean of fitted values 103 616 23 406 12 . Results Quebec MA DRAG-1 a i\j |Li + 1 ^ + 15 MA

_ 1 -... 1 1

+ 1

a ^ a

9 121 8 998 9 220

3 478 695 3 374

+

7

+ ...

-

...

1 "^^

-16 + 1

-... 1 -... 1 +... j '^- 1

-... 1

~+l

0,388 1 1 0,127 1 1 0,362 1 1 1

29 717 29 710 29 706

1^

a 1 ~.. \ 1 +6,18

-... 1 +^ 1

COR a Sample moment value 1 Fitted value at the means 1 Mean of fitted values

1

- 0,46

|

1

1

+0,38

1 1 ^^

0,656 0,186

2 552 2 504

0,490 JJ

2 551

4 724 1442 4 528

- 0,661 -0,185 - 0,757

COR

1

a + 112 +

7

-...

+T8 + 1 973 178 946

a

-... 1 -... 1 +... 1 -... 1

1~^\-••• 1 0,443 0,164

0,366 1

90 Structural Road Accident Models Own and cross trade-offs. Table 4 presents the trade-offs within each accident category, as well as the cross category trade-offs, for both Germany and Quebec. Because the lower triangular parts of the matrices present the inverse of the upper triangular part, we do not show all possible rates of substitution. Model fit. The first thing to note in Table 4 is that the first moment is better modelled that the second or third moments (positive or negative), as a comparison between observed and fitted moment values, evaluated either at the sample means of the variables or by the mean of the moment values fitted for individual observations, shows. From our earlier remarks emphasizing the « first moment minded » focus of modelling practice, this is hardly a surprise.

Figure 3.17. Accident frequency distributions in Germany and Quebec Similar rates of substitution among moments. A second thing to notice in Table 4 is the amazing closeness of the estimated trade-offs both between the first two moments of a given accident category and across accident categories, abstracting for signs found in the case of bodily injury accidents (UPS) in Germany. For instance, in both Germany and Quebec, drivers behave as if they were willing to increase the probability of material damage accidents by about 20 units (15 in DRAG-1, about 21 in DRAG-2 (see GAUDRY, 1998) and 23 in SNUS-2) in order

The SNUS-2.5 Model for Germany 91 to gain an increase in the uncertainty (a decrease in the certainty) of these material accidents of 1 unit (of standard error). All of these rates are very far from unity, the value assumed to hold in Poisson models. And they will accept about 115-124 more material damage accidents to obtain an increase in the uncertainty (a decrease in the certainty) of bodily damage accidents of 1 unit (of standard error). It is not a surprise that bodily damage accidents are « worth » more than material damage accidents. The signs of rates of substitution. The third thing to notice is that all accident types in Germany and Quebec share the same sign pattern of substitution among moments, as indicated in the shaded areas of Part 1 and Part 2 of the table, except for bodily damage accidents (UPS) in Germany. To clarify the meaning of these signs, we use expectations defined in Table 3. Interestingly enough, the own-moment pattern of bodily damage accidents for Germany (UPS v^. UPS in Part 1) exhibits riscophilia, or type [B], and all 3 other patterns exhibit riscophobia, or type [A]. This means that speed limit regulations somehow force drivers to be riscophobes, at least for the component of their behaviour that is the object of police enforcement (aimed at the first moment, but affecting the third): forced into right-tail skewness—^to a strong reduction in a (the values are 0,388 for Germany and, respectively 0,656 and 0,443 for Quebec) and in ji—, they compensate in part for this utility loss by increasing a. Table 5. Own elasticities of substitution among moments with comparable data r| [(mom. i)/(mom. j)]

Material damage accidents

Bodily damage accidents

|

1 Germany SNUS-2.5

USS

UPS

1

i\j

p + 1,00

USS

a + 1,30

pn,oo^

a a

Quebec DRAG-1

1

f^ + 1,00

-2,99

a

a

a -0,81 +1,00

UPS

+ 1,00

a(*)

-0,47 1 + 0,59 1 + 1,00 1

COR

MA + 1,00

MA

a -3,94

+ 1,23

-5,13

+ 1,00

-4,16

+ 1,00 COR

1

+ 1,29

-4,23

+ 1,00

-3,27 1 + 1,00 1

+ 1,00

(*) Sign inversions fourid by comparing with Table 4, Part 1, are due to the negative a.

Elasticities of substitution. Our interpretation is therefore that drivers are riscophiles and that speed limits constrain them by acting principally on the first and third moments of the accident probability. If that is true, one would expect the presence of pent-up tension as drivers, forced into a comer solution, are « kept honest », at least in a first and third moment sense, by the law. As this is a constrained equilibrium, one would expect drivers so restrained to be quite sensi-

92 Structural Road Accident Models tive and ready to re-establish their desired risk certainty equivalent utility. Some evidence to that effect is found in the analysis of rifmom i/mom j), the own elasticities of substitution found in Table 5. Elasticities measure the sensitivity of the rates of substitution to changes in conditions. We note (in the shaded area) that, in the case of the variable most affected by free speeds (UPS in Germany), the elasticities are indeed much smaller than in the 3 other cases (USS in Germany as well as MA and COR in Quebec), especially for rates of substitution involving the third moment. 3.4.4. Marginal rates of substitution with disaggregated accident data Disaggregation into different categories. The closeness of results obtained for Germany and Quebec is made possible by the existence of comparable data. However, in each case it is possible, as in Table 6, to disaggregate one of the series, but not the corresponding one. The German series on material damage accidents USS can be split between light and severe damage (ULSS and USSS) events, but this is not possible for the corresponding Quebec series MA. In Quebec, the series on bodily damage accidents COR can be split between injury and fatal (NM and MO) events, but this is not possible for the corresponding German series UPS. One would expect the disaggregated series to yield the same sign patterns as those obtained with their totals, but very different marginal rates of substitution. This is verified in Table 6. Note, for instance that in Quebec the previous marginal rate of substitution between first moments was 6,18 material damages accident per corporal damage accident in Table 4, whereas the component rates in Table 6 are 5,87 material damage accidents per injury accidents and 73 455 material damage accidents per fatal accident. As expected, the « bumpers vs. limb » rate is much lower than the « bumpers vs. life » rate. And the trade-off measured with an aggregate number differs from the trade-off against its components. In Germany, the marginal rate of substitution between first moments, previously -7,46 material damage accidents per bodily injury is now, in terms of light material damage accidents, 1,80 per severe damage accident and 1,04 per bodily injury accident: having a « bumper v^. bumper » rate higher than the « bumper vs. limb » rate naturally depends on how expensive the marginal « expensive bumpers » are and on how trivial the marginal « small injuries » that may dominate the bodily damage series are. Although the trade-offs measured with aggregates were almost identical with those of Quebec, trade-offs among their components are again different. The absence of comparable disaggregated series unfortunately prevents us from effecting a complete comparison between the results for Germany and Quebec.

The SNUS-2.5 Model for Germany 93 Table 6. Marginal rates of substitution amon g moments with disaggregated data 11. Germany SNUS-2.5 1\1 D ULSS

D

- ... 1

+1 11

D

+ ... 11

~-

1 "^^

n

+ 21

- ... +1 ...

^1 1 + 0,02 1

1 +0,58 1 - 5 8 8

nn 1 - ... 1

D

1 +1 f

D

UPS

Bodily damage

ULSS (light) 1 1 USSS (severe) 1 UPS D D 1 D D D 1 D 1 D + 18 +1 - ... 1 1 + 1,80 + 38 1 + 1,04 1 -17,5 + 1 1 +0,09 + 2 1 + 0,05 1 - 0,9 D

D

USSS

1 1

Material damage (USS)

|a(m.i)/5(m.j)

D

- 27 + ... -16 + 1

D

Sample values 79 787 1 Fitted at means 79 510 1 Mean of fitted 79 768 12. Quebec DRAG-1 I\j MA

22 389 4 781 21739

MA D +1

D + 15 + 1

• D

1° -... 1 -... 1

+ 1

D

1 °

- ... 1 - ... 1

1 "~ -

+... 1 +... 1

-... 1 +... 1 +... 1

-... 1

+... 1

-... 1 +1

-... 1 -... 1

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+ 18

1 "^1

D D

1+ 0,10 1 +1

D

1 Sample values 1 Fitted at means 1 Mean of fitted

1 1

+ ...

D

MO

D

0,821 1 1 23 842 9 026 0,884 1 1 29 717 1 4 724 -0,661 0,158 1 1 23 850 1290 0,138 1 129 710 1 1442 -0,185 0,815 1 123 813 8 832 0,876 1 1 29 706 1 4 528 -0,757 1 11 NM (injury) | MO (fatal) 1 D D D 1 D 1 D 1 1 +73455 1+7538 1 + 5,87 + 110 1 +4600 1 + 472 1 + 0,36 + 6 + 1 1 1 - ... + ... 1 +... 1 1 ~ •••

D

NM

1

9 121 8 998 9 220

3 478 695 3 374

2 445 2 375 0,490 1 j _ 2 444 0,656 0,186

939 174 913

0,474 0,158 0,389 [^

107 99 107

42 13

1 39

-... 1 -... 1 "^1 0,237 0,292

0,010 1

Multiple moment choices and homeostasis. However, it is quite clear overall that drivers do make multimoment choices, so that any attempt to determine whether they maintain their utility by adjusting only one moment, such as the first, cannot be a meaningful way of testing a « constant risk » or « constant expected utility » assumption: such an assumption should be formulated as a « constant certainty equivalence of risk » to be amenable to tests, and the tests at least apply to marginal rates of substitution among the moments of an accident category and across categories.

94 Structural Road Accident Models 3.5. P O L I C Y I M P L I C A T I O N S

3.5.1. Higher prices save energy and lives Our results show that road demand is rather inelastic with respect to fuel prices-which limits the potentials of pure pricing strategies if a reduction of mobility is politically wanted. Increased prices of mobility also reduce the number of accidents and their severity. This effect can be partly offset by activity variables that have a positive impact on road demand and on material damage. Within the category of activities, compensatory effects can be found especially with respect to bodily damage and mortality.

3.5.2. Risk substitution in terms of first moments Considering the results obtained in terms of first moments of the various frequency and severity categories, we find strong evidence of risk substitution in the case of belt usage. The alcohol limit imposed in August of 1973 has considerably reduced the number of accidents of all categories. Differentiated weather conditions also lead to a complex pattern of risk compensation that may also be explained by changes in speed and congestion. Speed limits have rather complicated impacts on accidents-contrary to much colloquial and political wisdom.

3.6. REFERENCES Allais, M. (1987). Allais Paradox. In: The New Palgrave Dictionary of Economics, (Eatwell, J., Milgate, M. and P. Newman, eds), 1, pp. 80-82, The Macmillan Press, London. Blum, U., G. Foos and M. Gaudry (1988). Aggregate Time Series Gasoline Demand Models: Review of the Literature and New Evidence or West Germany. Transportation Research A, 22, 75 - 88. Blum, U. and M. Gaudry (1992). Verbessertes Nachfragemodell ftir den StraBenverkehr in Deutschland. Internationales Verkehrswesen, 44, 1/2 , 30-35. Brockett, P. L. and J.R. Garven (1998). A Reexamination of the Relationship between Preferences and Moment Orderings by Rational Risk Averse Investors. Geneva Papers on Risk and Insurance Theory. Dagenais, M.G., M. Gaudry, M.J.I, and T.C. Liem (1987). Urban Travel Demand: the Impact of Box-Cox-Transformations with Non Spherical Residual Errors. Transportation Research B, 21, 443-477. FCAR (Fonds pour la formation des chercheurs et I'aide a la recherche) (1991). FCAR-67.

The SNUS-2.5 Model for Germany 95 Rapport d'activite sur les 10 mois d'operation du Programme multidimensionnel

en securite

routiere. Projet 5.2. Estimations dans le cadre du modele DRAG, Tache 1. Fournier, F. and R. Simard (1999). Modele econometrique DRAG-2: Previsions sur le kilometrage, le nombre d'accidents

et de victimes pour

la periode

1997-2004.

Societe de

rassurance automobile du Quebec. Fridstr0m, L. and S. Ingebrigtsen (1991). An Aggregate Accident Model Based on Pooled, Regional Time-Series Data. Accident Analysis and Prevention, 23, 5, 363-378. Fridstr0m, L., J. Ifver, S. Ingebrigtsen, R. Kulmala, and L.K. Thomsen (1993/ Explaining the Variation of Road Accident Counts. NORD35, Nordic Council of Ministers. Fridstr0m, L. (1999). Econometric Models of Road Use, Accident and Road Investment

Deci-

sions. Institute of Transport Economics, Oslo. Gaudry, M. (1984). DRAG, un modele de la demande routiere, des accidents et leur gravite, applique au Quebec de 1956-1982. Publication CRT-359, Centre de recherche sur les transports, et Cahier #8432, Departement de sciences economiques, Universite de Montreal. Gaudry M. (1998). Some Perspective on the DRAG Approach and Family of National Road Safety Models. (H.V. Hoist., A. Nygren, and A. E. Andersson, eds). Transportation,

Traffic

Safety and Health, III, 123-168. Gaudry, M., et al (1993). Cur cum TRIO? Publication CRT 901, Centre de recherche sur les transports, Universite de Montreal. Gaudry, M. and U. Blum (1993). Une presentation breve du modele SNUS-1 (StraBenverkehrsNachfrage, Unfalle und ihre Schwere). In: Modelisation de I'insecurite routiere, (J.R. Carre, S. Lassarre et M. Ramos eds.). Tome 1, pp. 37-44, INRETS, Arcueil. Gaudry, M., U. Blum and T. Liem (2000). Turning Box-Cox including quadratic forms in regression In: Structural Road Accident Models: The International DRAG Family (Gaudry, M. and Sylvain Lassarre Eds.), Chap. 14, pp 337-348, Elsevier Science, Oxford. Gaudry, M. and M. Dagenais (1979). Heteroscedasticity and the Use of Box-Cox Transformations. Economics Letters, ll'i, 225-229. Gaudry, M., F. Fournier, and R. Simard (1993). Application du modele econometrique DRAG2 a la frequence des accidents au Quebec selon differentes gravites. Compte-Rendu Vllleme Conference

Canadienne Multidisciplinaire

sur la Securite Routiere,

de la

Saskatoon,

163-176. Also in (1994), Modelisation de I'insecurite routiere, (J.R. Carre, S. Lassarre et M. Ramos eds.),Tome 1, pp. 41-54, INRETS, Arcueil. Gaudry, M. and M. J. Wills (1978). Estimating the Functional Form of Travel Demand Models. Transportation Research, 12, 4, 257-289. Hakim, S., D. Shefer and A.S. Hakkert (1991). A Critical Review of Macro Models for Road Accidents. Accident Analysis and Prevention, 23, 5, 379-400. Jaeger L. (1998). Developpement

d'un modele explicatif des accidents de la route en France

sur une base mensuelle de 1956 a 1993. These de doctorat, Universite Louis Pasteur, Strasbourg.

96 Structural Road Accident Models King, G. (1989). Variance Specification in Event Models: From Restrictive Assumptions to a Generalized Estimator. American Journal of Political Science, 33, 3, 762-784. Kraus, A. and R. Litzengerger (1976). Skewness Preference and the Valuation of Risk Assets. Journal of Finance, 31, 4, 1085-1079. Lassarre, S. (1994). Cadrage methodologique d'une modelisationpour un suivi de I'insecurite routiere. Synthese INRETS No. 26, Institut National de Recherche sur les Transports et leur Securite, Arcueil. Liem, T., M. Gaudry, M. Dagenais, and U. Blum (2000). LEVEL: The L-1.5 program for BCGAUHESEQ (Box-Cox Generalized AUtoregressive HEteroskedastic Single EQuation^ regression and multimoment analysis. In: Structural Road Accident Models: the International DRAG Family, (Gaudry, M. and S. Lassarre, eds.), Elsevier Science, Oxford. Machina, M. J. and M. Rothschild (1987). Risk. In: The New Palgrave Dictionary of Economics. (J. Eatwell, M. Milgate and P. Newman, eds), 4, pp. 201-206, The Macmillan Press, London. Oppe, S. (1991). The Development of Traffic and Traffic Safety in six Developed Countries. Accident Analysis and Prevention, 23, 5, 401-412. Praxenthaler, H.(1987). Strafiensicherheitsforschung in der Bundesrepublik Deutschland. Paper presented at Goteborg, 20 p. Scott, R.C. and P.A. Horvath (1980). On the Direction of Preference for Moments of Higher Order than the Variance. Journal of Finance, 35, 4, 915-919. Spanos, A. (1987-88). Error Autocorrelation Revisited: the AR(1) Case. Econometric Reviews, 6, 2, 285-294. Tobin, J. (1965). The Theory of Portfolio Selection. In: The Theory of Interest Rates, (Hahn and Brechling, eds.), pp. 3-51, St. Martin's Press, New York. Wegman, F.C.M., M.P.M. Mathijssen and M.J. Koornstra, eds. (1989). Voor alle veiligheid: bijdragen aan de bevordering van de verkeersveiligheid. SDU uitgverij, 's-Gravenhage. Wilde, G. (1982). The Theory of Risk-Homeostasis: Implications for Safety and Health. Risk Analysis, 2, 209-255.

The TRULS-1 model for Norway 97

THE T R U L S - 1 MODEL FOR NORWAY Lasse Fridstr0m

4.1. INTRODUCTION The TRULS-1 model for Norway was developed at the Institute of Transport Economics (Transport0konomisk institutt - T0I) in Oslo, as part of the author's PhD thesis submitted to the University of Oslo in April 1999. Ancestor studies of the TRULS-1 model include the generalized Poisson regression models estimated by Fridstr0m and Ingebrigtsen (1991) and by Fridstrom et al (1995). The latter study is a four-country analysis for Denmark, Finland, Norway and Sweden, focusing on the role of random variation, exposure, weather and daylight conditions in explaining casualty counts. It shows that, when account is taken of the inevitable (objective) random variation present in accident counts, there is a limit to the obtainable goodness-of-fit. When the data set consists of many small accident counts, this limit could well drop below 50 per cent, as judged by the ordinary R^ measure. In the Norwegian data by province and month, randomness accounts for a full 80 per cent of the total variation in fatal accidents, yielding a maximal obtainable R^ of about 0.20. Fridstr0m et al. (1995) argue that accident counts are generally Poisson distributed and develop, based on this assumption, specialized goodness-of-fit measures for systematic variation ( R^, say), in which the unexplainable random disturbance part has been filtered out. A perfectly specified and estimated accident model would be characterized by Rj=\ and an overfitted model by Rl>l. In the four-country analysis, exposure, weather and daylight are sufficient to explain 70-90 per cent of the systematic variation in road fatalities between provinces and months.

98

Structural Road Accident Models

4.2. STRUCTURE OF THE MODEL T R U L S - 1 Neither of the two above-mentioned studies attempts to explain exposure. The TRULS-1 model does. In fact, compared to most DRAG-type models it includes two additional layers of exposure explanation or prediction, viz. (i) aggregate car ownership and (ii) a decomposition between light and heavy vehicle road use, adding to the set of econometric equations. Also, while most DRAG-type models use the fuel sales as a (rather imperfect) measure of the traffic volume, in the TRULS-1 model light and heavy vehicle exposure measures are based on (iii) a submodel designed to "purge" the fuel sales figures of most nuisance factors affecting the number of vehicle kilometres driven per unit of fuel sold. These nuisance factors include vehicle fuel economy, aggregate area-wide vehicle mix, weather conditions, and fuel hoarding due to certain calendar events or price fluctuations. Fourth, the TRULS-1 model includes a small (iv) submodel of seat belt use, in which periodic road-side sample surveys are exploited to estimate the effects of legislative changes and economic incentives in this area and to impute urban and rural seat belt use rates for the entire sample. A further point at which the TRULS-1 model differs from other members of the DRAG family, is by the estimation of (v) separate equations for various subsets of casualties (car occupants, seat belt non-users, pedestrians, heavy vehicle crashes, etc). These equations are meant to shed further light on the causal mechanism governing accidents and severity. In order to avoid, to the largest possible degree, spurious correlation and omitted variable biases, certain specificity tests (called casualty subset tests) are developed and applied (see section 4.2.4). Building on the above-mentioned study by Fridstrom et al. (1995), the TRULS-1 model starts from an assumption that casualty counts in general follow a (generalized) Poisson distribution. To enhance efficiency, the accident equations are therefore specified with (vi) a disturbance variance assumption approximately consistent with the Poisson law. To this end, we develop a special statistical procedure, termed Iterative Reweighted POisson-SKedastic Maximum Likelihood (IRPOSKML), for use within the standard DRAG-type modelling software (see Chapter 13). Finally, the TRULS-1 model is the only DRAG-type model so far being based (vii) on pooled cross-section/time-series data. Other DRAG family models rely exclusively on time-series. The TRULS-1 data are monthly observations pertaining to all counties (provinces) of Norway. The period of observation is 1973-1994, yielding a total of 5016 units of observation (19 counties x 264 months). Although the data structure is such as to allow for specialized panel data estimation techniques, a homogeneity constraint, forcing cross-sectional and time-series effects to be identical, is imposed throughout.

The TRULS-1 model for Norway 99 4.2.1. Dependent variables: definitions and relations The structure and interdependencies between endogenous variables in the TRULS-1 model are shown in Diagram 4.1. Single line arrows between boxes indicate use as explanatory variables, while double line arrows are used to indicate that certain dependent variable relations, rather than being estimated directly, are derived from a combination of other relations. There are, in the full model, 13 separate dependent variables (equations): (1) Aggregate car ownership; (2) Overall road use (total vehicle kilometres); (3) Heavy vehicle road use (kilometres driven by vehicles with at least 20 passenger seats or more than 1 ton's carrying capacity); (4) Urban seat belt use; (5) Rural seat belt use; (6) Injury accidents; (7) Car occupants injured; (8) Motorcycle occupants injured; (9) Bicyclists injured; (10) Pedestrians injured; (11) Severity2'. severely injured per injury accident; (12) Severityy. dangerously injured per injury accident (data from 1977 onwards only); (13) Severity/^: fatalities per injury accident ("mortality") The severity categories are defined cumulatively, meaning that any degree of severity always includes the even more severe cases. Thus, "severitys" includes the fatal injuries, and "severity2" includes all cases in the "severitys" category. Models for the '"severityx" category - slightly injured per (injury) accident - are inestimable since the relationship between this category and the number of injury accidents is one of nearly perfect correlation. Norwegian road accident statistics unfortunately do not provide data on accidents with only material damage. In the TRULS-1 model, predictions on the number of severe, dangerous, and fatal injuries are derived by multiplying relations 11 through 13, respectively, by the number of injury accidents (relation 6). Note that the submodel (iii) used to purge the fuel sales statistics of the "nuisance" factors affecting the number of vehicle kilometres driven per unit of fuel sold, is not shown in the diagram, as it is only an auxiliary relation (a measurement model) which does not form part of the model's recursive causal structure. Nor does the diagram show the dynamic, partial adjustment structure of the car ownership model. A vehicle pool is an inert matter, comparable to a human population, although with generally higher rates of turnover and shorter life expectancy. The stock of cars registered within a given geographic unit changes from one year to the next, in response to the flows of (i) new car acquisition ("births"), (ii) used car sales ("migration"), and (iii) scrapping ("deaths"). Given the very high level of purchase tax imposed on automobiles in Norway, used cars can be sold abroad only at very substantial losses. Thus, the only important downward adjustment mechanism operating at the macro level is scrapping, something which also involves heavy losses unless the car is old enough to have lost most of its market value. Hence, in the aggregate, car owners can be expected to adjust only slowly to changes in economic variables.

100 Structural Road Accident Models

Exogenous variables

Seat belt use: 4. Urban 5. Rural

Aggregate car ownership: 1. Passenger cars registered

Road use: 2. Total (light + heavy) vehicle kms 3. Heavy vehicle kms

Victims (estimated): 7. Car occupants injured 8. MC occupants injured 9. Bicyclists injured 10. Pedestrians injured

Victims (derived): 14. (Slightly injured - neglected) 15. Severely injured: 6x11 16. Dangerously injured: 6x12 17. Fatalities: 6x13

Accident frequency: 6. Injury accidents (inci fatal accidents)

Severity: 11. Severely injured per injury accident 12. Dangerously injured per injury accident 13. Killed per injury accident

Diagram 4.1. Recursive structure of the model TRULS-1. We therefore model car ownership as a partial adjustment process, implying that the aggregate car stock, when subject to exogenous shocks, adjusts only slowly towards its new long-term equilibrium.

4.2.2. Visual analysis of dependent variables Graphs 4.1 through 4.13 are scatter plots in which the dependent variables are plotted against time, measured in months since the start of the TRULS-1 observation period. Since there are 19 counties in the data set, there are generally 19 points visible for each unit of time along the horizontal axis. This clearly limits the transparency of the data set, when all counties are included in the plot. The graphs do bear testimony, however, of the large amount of variation present in the data set, which can be exploited for purposes of econometric analysis.

The TRULS-1 model for Norway 101

Graph 4.1. Aggregate car ownership by county and month. Norway 1973-94

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rood

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trio Graph 4.2. Aggregate road use by county and month. Norway 1973-94

102 Structural Road Accident Models

Graph 4.3. Heavy vehicle road use by county and month. Norway 1973-94

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The TRULS-1 model for Norway 103 R^rctl

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trio Graph 4.6. Injury accidents by county and month. Norway 1973-94

104 Structural Road Accident Models

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The TRULS-1 model for Norway 105 B J c y c l » s I: s. « n j vi r e d

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106 Structural Road Accident Models Severity

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The TRULS-1 model for Norway 107 ;> I' d e g r e e A : mo r t ci

trio Graph 4.13. Severity4: fatalities per injury accident, by county and month. Norway 1973-94 As can be seen from Graphs 4.6 through 4.10, the casualty counts upon which the TRULS-1 model is based are generally quite small, including numerous zeros. To avoid numerical problems and ensure that the casualty counts have finite variance under the Poisson assumption, a small (Box-Tukey) constant (a = OA) is added to all counts prior to Box-Cox transformation.

4.2.3. Matrix of direct effects While Diagram 4.1 contains dependent variables only, in Table 4.1 we provide an overview of (broad categories of) independent variables entering then model. Note that only direct effects are ticked off in this table. In general, the total effect of an independent variable on - say - accident frequency, will be a mixture of direct and indirect effects, as channelled though the recursive system pictured in Diagram 4.1. For instance, the interest level has a direct effect on car ownership only. However, since car ownership affects road use, which in turn affects accidents, interest rates may turn out as an important indirect determinant of road casualties. The tracing of such effects is the very purpose of our recursive, multi-layer modelling approach.

108 Structural Road Accident Models Table 4.1. Independent variables in the model TRULS-1. Direct effect upon (dependent variable) Independent variable

Vehicle Car ownership kms

Seatbelt use

1

Exposure Prices

V

Vehicle stock and characteristics

V V

Laws and regulations Fines and penalties Public transportation supply Road infrastructure

V V

V V ^l

V V

Road maintenance Daylight Weather conditions Population Income Interest rates Taxes

V V V V

Geographic characteristics

V ^l V V

V

V V V

V

V V

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^l

Reporting routines Access to alcohol Information Randomness and measurement errors

Accidents

Victims

~

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~

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V V

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V V V V V V

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4.2.4. The casualty subset test Omitted variable bias is an important source of error in any econometric study. Whenever a regressor is correlated with the collection of explanatory variables not included in the model, the effect due to the excluded variables tends to be ascribed to the included one, inflating (or deflating) the coefficient of the latter. Any statistically significant effect found may thus, in principle, be due either (i) to a true causal relationship or (ii) to some kind of spurious correlation, or, indeed, to a combination of the two. The number of factors influencing casualty counts is notoriously quite large. It is inconceivable that any econometric model would encompass all of them. Some factors are quite general, potentially influencing the frequency of (virtually) all types of accidents or victims, while other factors may be assumed to affect only certain subsets of casualties. To exploit our a priori knowledge or assumptions about such relationships we introduce the following specificity tests.

The TRULS-1 model for Norway 109

Definition 4.1: Casualty subset tests. Let^, B, C and D denote four sets of casualties (accidents or victims) such that (4.1)

5 n C = 5 n D = C n D = 0 and 5 u C u i ) = yi,

IQB, C and D are disjoint, exhaustive subsets of ^, not all of them necessarily non-empty. Let (4.2)

Y^^E[y,\x),

Y,^ = E[y,\x) and Y,^ ^ E[y,>\x)

Y,^^E[y,\x),

denote the expected number of each type of casualties, conditional on a set of independent variables JC = [xj ^2....]'. Also, denote by //I ox

(4.3)

^Ax

^i

^Bx

^i

^Cx

^i

^ i

^ix

^/

Ycx

A

^Dx

^i

^ i

Yo^

f^. = — ^ — ^ , ^ ^ . = — ^ - i - , ^^,.=_x^_L_ and Sj^i = — ^ _ ^ ^/

'^Ax

the partial elasticities of 7 ^ , P^^, Y^.^ and Y^y^ with respect to some element x,. of JC. Note that, by definition, (4.4) S^, = S,,.Sj,^ + £c.S,.^ + £^.S,,^ , where (4.5)

s„^ . ^

> 0, , , , ^ | i > 0 and

^Ax

^Ax

s,^A>Q ^Ax

denote the share of casualties belonging to subsets B, C and D, respectively. Suppose that D = 0 and that we want to test a hypothesis of the form (4.6) or

H;.- e,,>s,,>0

(4.7)

H;:

= e,,

^ , , < ^ , , < 0 = ^,,

in other words that x- has a larger positive (negative) effect on the number of casualties within subset B, a smaller negative (positive) effect on the total number of casualties (set A), and a zero effect on casualties of type C. Let £^., £j^., ^(.., and i^,. denote empirical sample estimates corresponding to the theoretical elasticities ^^,, Sj^^, ^^.., and £j^j, respectively. The hypothesis H^ (or H^ ) is said to pass the affirmative casualty subset test as applied to B versus A if and only if (4.8)

^g. > £^. > 0 (in case Yi\) or £j^. < ^^. < 0 (in case H^).

It is said to pass the complement casualty subset test as applied to B versus C if and only if (4.9)

£j^. > £^,. « 0 (in case H^

or £^. < ^f,,. « 0 (in case H^).

Alternatively, assume that C = 0 and consider the hypotheses (4.10) H;:

£,,>0>£^,

110 Structural Road Accident Models or (4.11) H-,:

^,,<0<^,,,

Hypothesis H2 (or H^ ) is said to pass the converse (opposite) casualty subset test as applied to B versus D if and only if (4.12) i,j. > 0 > Sjyi (in case H2)

or

Sj^. < 0 < 6,^^ (in case Hj).

The logic of these tests is illustrated by the following examples. Example 4.1: Let A denote the set of all road users injured, B the set of car occupants injured, C the set of non-occupants injured. D is an empty subset. Also, let x^ denote the rate of seat belt non-usQ. Clearly, in this case one expects hypothesis H^ to hold. If the total number of road victims goes up as a result of reduced seat belt use (increased non-use), one should ceteris paribus - be able to observe a stronger (relative) effect on car occupants (B) than on road injuries in general (^4). This is the affirmative casualty subset test, confimiing the impact of the safety measure by narrowing in on its specific target group. One should, however, not see any effect of seat belt (non-)use on bicyclist and pedestrian injuries ( Q - unless, of course, car drivers adapt in the way maintained by Peltzman (1975), exposing non-occupants to higher risk. This is the complement casualty subset test, comparing the effect on the target group to the effect on its complement subset. Example 4.2: Let A denote the set of car occupants injured, B the set of car occupants injured while wearing a seat belt, and D the set of car occupants injured while not wearing a seat belt. C is empty. As in the previous example, let x. denote the rate of seat belt non-use. In this case one expects hypothesis i/j to hold: increased seat belt non-use should be positively related to the number of non-users injured, but negatively related to the number of seat belt users injured, simply because of the exposure effects. This is the converse (or opposite) casualty subset test, checking if the risk factor in question has the expected converse (opposite) effect on a suitably defined subset of the casualties. More seat belt use should - ceteris paribus - mean more seat belt users injured, even if the injury risk is much lower than in the non-user group. At this stage the reader may want to ask what is the point of "testing" such entirely trivial relationships. It is this: If our seat belt variable does not pass the complement casualty subset test as applied to car occupants versus non-occupants, but shows, e g, a clearly significant, positive partial elasticity of non-occupant injuries with respect to seat beh non-use, there is reason to suspect omitted variable bias, probably inflating the effect of the seat belt variable on its target group (car occupant injuries) as well.

The TRULS-1 model for Norway 111 An even stronger indication of such bias is conveyed if our hypothesis fails to pass the converse casualty subset test as applied to seat belt users versus non-users. One may note that our casualty subset tests are not set up as formal statistical significance tests. Only point estimates are compared, and pragmatic conclusions are drawn on the basis of their relative magnitudes. This is so because in most practical applications, one would not possess the relevant covariance estimates needed to perform, e g, the asymptotic Wald test. Nor would comparable likelihood statistics be available, since casualty subset tests are generally based on separate, identical regressions explaining different dependent variables. Only when a single elasticity is to be tested against a zero (or constant) alternative will we have enough information to perform a significance test. In some cases, however, the zero alternative (in the complement casualty subset test) must be regarded as only approximate, such as when a risk or safety factor has a diluted effect even outside its main "target group". This will rarely apply to severity reducing (or increasing) factors, but quite frequently to accident reducing (or increasing) variables, since the latter will have spill over effects to other road user groups involved in bipartite or multipartite accidents. For instance, measures to reduce the accident risk of young drivers have a primary effect (if any) on this particular age group, but presumably also a diluted effect on the average risk experienced by other road users. In this case, therefore, one should not expect the effect observable within the complement subset to be exactly zero. 4.3. RESULTS ON FORM AND SELECTED EXPLANATORY VARIABLES. In table 4.2, we summarize certain characteristics of the TRULS-1 model. All equations except the severity relations are specified with a logarithmic transformation (A-(y) = 0) on the dependent variable. Exposure. In figure 4.1, we show selected results on exposurefi-omthe TRULS-1 model, in the form of compound (total) elasticities as evaluated at the sub-sample means for the last year of observation (1994) (228 units of observation - 19 counties x 12 months). The compound elasticities are computed by accumulation through the recursive chain of effects: car ownership, road use, seat belt use, accidents, victims, and severity, as pictured in diagram 4.1. That is, whenever applicable, the effect channelled through increased (or decreased) equilibrium car ownership is included in the road use elasticity, the effect coming through increased road use is included in the accidents or victims elasticity, and the effect on accident frequency is included in the fatalities elasticity.

112 Structural Road Accident Models The partial effect on road use, given car ownership (i e, the "short term" effect) is given by the difference between the two compound elasticities. And the partial effect on accident frequency, given road use (i e, the effect on "risk"), is given by the difference between the compound effect on road use and the compound effect on accidents. Table 4.2. The TRULS-1 model - functional form, stochastic specification, and other summary statistics Equation (see section 4.2. T):

2

1 ~24"

Independent variables (incl constant): number of t-statistics (: 2 < i t i ) number of t-statistics (1 :<|t| < 2) number of t-statistics (0<: | t | < l ) Heteroskedasticity parameters Autocorrelation parameters

23 1 0

r1

7

4

5

42"~jr

T

6"

47"_ _

31 5

29 4

6

8

6 0 0

6 0 0

30 8 9

2~

2~

2

2

0

0

0

0

3

6

r 0r 2r

8

9

_~ 4 7

24 6

27 16

18

5

i

r

25 6 16

i

10

12

13

47"~l8~ ^~A6

48

9 12 25

7 20 21

i

r

27 6 14

r

11

14 10 24

r

2

2

2

2

0

0

0

0 .547 .436 .350

2

2

2

Form* ^(V|/)

0

^ ( ^ . ) - .226

^fe) ^fe)

0

8.324 -.316 .912 .689 2.199 1.904 3.128 .372 1.729 .215

.591 .445

2 1.829 .249 3.212 5.305

^(^4)

2.320 3.876 4.038

^(^-s)

-1.20 -.757

^(^6)

-.013 .226 2.668 -.613 .014 .036 .190 .247

^(^v)

1.437 -.772 .174

^(^8)

-2.53 .464 -1.55 -.176 -3.94 .702 •-.005 -.769

^(^9) ^ (^10)

Sample size

0

1.069 -.939

-1

.284

-1

-1

-1

-1

1 4.277 --.1013.762 3.312 -1

-1

-1

-1

-1

(6.099

^(^,1)

.035--.345

^ (^.2)

-.485 -•1.27 418 5016 5016 350 350 5016 5016 5016 5016 5016 5016 4104 5016

* The independent variable Box-Cox parameters %X\), Xxj), ... , respectively, refer to (1) income, (2) fuel cost, (3) weather, (4) daylight, (5) seat belt non-use, (6) traffic density, (7) alcohol shops, (8) bars and restaurants, (9) per capita car ownership, (10) relative diesel cost of sea mode, (11) seat belt installation rate, and (12) real value of seat belt ticket.

The injury accident frequency has an elasticity of 0.911 with respect to the total volume of motor vehicle road use {vehicle kilometres). That is, injury accidents increase almost in proportion to the traffic volume, other things being equal. This elasticity applies, however, only on the condition that the traffic density, defined as vehicle kilometres driven per kilometre road length, is kept constant. In other words, the elasticities with respect to traffic volume implicitly assume that the road network is being extended at a rate corresponding exactly to the traffic growth, so that the ratio of vehicle kilometres to road kilometres remains unchanged. In

The TRULS-1 model for Norway 113 injury accident elasticity of approximately 0.50 (= 0.911 - 0.414) is calculable for 1994. An increase in traffic density tends, in other words, to dampen the (otherwise near-proportionate) effect of larger traffic volumes, as measured in vehicle kilometres. Heavy vehicles (i.e., vehicles with more than 1 ton's carrying capacity or more than 20 passenger seats) are more dangerous than private cars. The larger the heavy vehicle share of the traffic volume, the higher the injury accident frequency. However, the number of car occupants injured does not increase with the heavy vehicle share. This may reflect the fact the truck driver himself is well protected and usually escapes the accident without (severe) injuries. Heavy vehicles appear to be particularly dangerous to two-wheelers, while car occupant injuries become less frequent when a large share of the traffic does not consist of private cars. Motorcycle exposure has a clear effect on motorcycle accidents but on account of its small share of the exposure, a quite modest effect on the overall accident frequency \ In table 4.3, we show the result of a specificity (casualty subset) test (see section 4.2.4) obtained by dividing the total set of injury accidents (A) into two disjoint sets, those which involve heavy vehicles (B) and those which do not (C) ' One notes that the heavy vehicle traffic shavQ passes the affirmative casualty subset test as applied to B vs. A, in that this independent variable has a much stronger impact on subset B than on the total set A. It also passes the converse casualty subset test as applied to B versus C, in that the latter subset exhibits a negative elasticity. The heavy vehicle traffic share negatively affects the number of accidents not involving heavy vehicles, if at all. Enhanced public transportation services tend to reduce the use of private cars and hence the total number of vehicle kilometres. A one per cent increase in the density of bus service lowers the overall traffic volume by an estimated 0.062 per cent. However, this is not sufficient to offset the exposure effect of the bus service: injury accidents increase by 0.212 per cent, i e by 0.264 {= 0.212 + 0.062) per cent as reckoned per vehicle kilometre. This effect is due primarily to more pedestrians being injured, presumably on their way to or from the bus stop, but even car occupants and two-wheelers are exposed to a somewhat higher risk owing to the bus service.

1 The "elasticity" computed for motorcycle exposure should not be interpreted literally, since the independent variable used is only a proxy, which appears to capture bicyclist exposure as well. 2 These indicators are available only for a certain sub-sample, viz the 1973-86 period. This is why the elasticity for all injury accidents in table 4.3 is different from the one reported in figure 4.1, which is based in the entire TRULS-1 sample (1973-94).

114 Structural Road Accident Models Here, again, a casualty subset test is available (table 4.3). Buses are heavy vehicles and should be expected to affect subset B (injury accidents involving heavy vehicles) comparatively more than the complement C or the total A. This is confirmed by the affirmative casualty subset test as applied to B versus A. It is also confirmed by the complement casualty subset test as applied to B versus C, although a certain diluted effect is left even in subset C, as these accidents include, e g, cases in which pedestrians are hit by a car on their way to or from the bus stop.

S t r e e tc a r/s u b w a y density

-0.505

•ilPlliilllllilliillllH^ P T I

^

1 .1 9 04

-0.24 5 I I ; I ; i | 0.2 4 0

-0.2 5 3 ~

^ 0 . 7 56 10.110 :>>>>>:t 0 . 3 6 2 fessa 0 . 1 4 9

Bus service density -0.062

MC

P

y m

0.2 1 2

0.034 _ 0.254 0.2 08 ,0.001

exposure (proxy)

P 0.026

H

Heavy veliicle tra ffic s h a r e

-0.972

-0.146

I

o .1 0 5

H

^

• P e d e s t r i a n s in u re d ms i c y c l s t s i n j u r e d S M C 0 C C U pa n t s in u re d a C a r 0 C C U pa n t s in u re d • In ju ry a c e d e n t s H T ra ffic V 0 1u m e

^

o 604 l i H i U m i i l i i

Traffic density , . , , ( v e h k m s p r k m

^0.012

_ _ . _ psgaass • P -0.319 -0.4 1 4 fTi . " " ! " ' ! " " ' ' o'.ooo"

road)

IllliillllllllllllllU

Motor vehicle k ilo m e te rs

1.109 "1.079

• ••('•••drdl'tfwi^B'rfWWWrfarM

S a 0.9 6 2 7 ; 7 ^ r ! ' . ! T ! . ! . ! . ' ! ' • !|o 911

- 1 .0

-0 .5

0.0

0.5

1.0

1 .5

E la s t i c i t y

Figure 4.1. The TRULS-1 model for Norway. Compound elasticities with respect to exposure as of 1994. Injury accidents and victims by road user category. Income effects. In figures 4.2 and 4.3, we show income and price effects, respectively, by road user categories. The TRULS-1 model shows an income elasticity of equilibrium car ownership of 1.18 and a (long term) income elasticity of road use of 1.61, as evaluated in 1994. The short-term income elasticity of road use is, in other words, 0.43 (= 1.61 - 1.18). In the long run, however.

The TRULS-1 model for Norway 115 households respond to increased income by increasing vehicle ownership, yielding a more than proportionate increase in road use demand. This translates into an approximately 0.8 per cent increase in injury accidents for each percentage change in real personal gross income. The number of car occupant injuries increase at an even higher rate, while the number of pedestrian injuries is relatively insensitive to income. Corporate income has an almost negligible effect on road use as well as on accidents. Table 4.3. Casualty subset models for accidents with or without heavy vehicles. Elasticities evaluated at sub-sample means, with t-statistics in parentheses. Selected results, 1973-86 sub-sample. Independent variable

Dependent variable All injury accidents A

Injury accidents involving heavy vehicles B

Injury accidents not involving heavy vehicles C

Heavy vehicle share of traffic

0.182

0.688

volume

(2.89)

(3.35)

-0.033 (-.460)

Density of public bus service (annual veh kms pr road km)

0.255 (8.15)

0.649 (6.63)

0.082 (1.77)

• Pedestrians injured E B i c y c l i s t s injured S M C occupants injured H C a r occupants injured

Corporate income

• injury accidents HTraffic volume H c a r ownership

IJJJllllil.ll.l.lll.ttllillltlMtll.' Personal income ' r S ' 1' I ' ! ' i ' I ' I ' ! ' I ' I ' I S

0.0

0.2

0.4

0.6

jo.79

0.8

1.0

1.2

1.4

1.6

1 .{

Elasticity

Figure 4.2. The TRULS-1 model for Norway. Compound income elasticities as of 1994. Injury accidents and victims by road user category. Price effects. A most important price variable (under Norwegian conditions) is the current rate of interest, which strongly affects the equilibrium car ownership and hence road use, accidents

116 Structural Road Accident Models and victims. For car ownership and road use its elasticity is estimated at close to -0.4, translating into a -0.19 injury accidents elasticity.

Subway/streetcar fares

TQ,137

-0.226 E -0.174

Fuel price

-0.257

D Pedestrians injured

mhti

Q Bicyclists injured ^MC occupants injured HCar occupants injured •3Injury accidents

-0.148

^Traffic volume

0.019

Tax a d v a n t a g e of car

^ Car ownership 164

o w n e r s h i p interest cost

I n t e r e s t cost of cars

-0.5

-0.374 m

-0.043 I 0 222|f[||f|||||[|fmilH

-0.271

-0.4

-0.3

H

. I. I . I. I . I . I . I

-0.2

-0.1

0.0

0.1

0.2

0.3

Elasticity Figure 4.3. The TRULS-1 model for Norway. Compound price elasticities as of 1994. Injury accidents and victims by road user category. The tax advantage due to interest payment deductibility works in the opposite direction, dampening the effect of increased interest rates. The fuel price elasticity as of 1994 is estimated at -0.257 for overall road use (vehicle kilometres). More than half of this effect (-0.148) is due to reduced (equilibrium) car ownership. Some households no longer find it worthwhile to keep a(n extra) car when its use becomes too expensive. In the short run, when car ownership is constant, the price elasticity is only -0.109 (= -0.257 + 0.148). Note, however, that the fuel price elasticity increases strongly with the initial price level, as witnessed by its very high Box-Cox parameter (8.32, see table 4.2). Obviously, the fuel price effects on road use translate into similar effects on traffic casualties. Public transportation fares have a modest, but clearly significant cross-price effect on motor vehicle road use and hence also on accidents and fatalities, although not for pedestrians. Injury

The TRULS-1 model for Norway 117 accidents may be expected to increase by 0.04 per cent for each per cent increase in the streetcar/subway fares. Weather. In Norway, injury accidents become less frequent when the ground is covered by snow (figure 4.4). We believe this is due to the fact that a certain layer of snow serves to reflect light and hence strongly enhances visibility at night. The risk reduction is larger the deeper the snow is. This is probably a snowdrift effect. The formation of snowdrifts along the roadside serves to reduce the frequency of single vehicle injury accidents, as they prevent cars from leaving the road and/or dampen the shock whenever a car is straying aside. On the other hand, snowdrifts tend to limit the road space and may thus increase the risk of head-on collisions, as when cars are thrown back into the road after hitting the snowdrift. During days with snowfall, however, the injury accident frequency goes up. At the same time, severity is reduced sufficiently to more than offset the increase in injury accident frequency, at least as far as fatalities are concerned. This is most probably a risk compensation effect: motorists reduce their speed on slippery surface, perhaps not quite enough to keep the injury accident frequency constant, but certainly enough to strongly reduce the consequence once an accident does occur. Does it matter how much snow is falling? One might imagine that heavy snowfall creates a particularly risky traffic situation. The variable ''heavy snowfalF is defined as the percentage of snowfall days during which the precipitation exceeds 5 millimetres (in water form). This effect, too, is generally positive for all road user groups, although too small to be statistically significant. An even clearer example of behavioural adaptation is seen in ih.Q frost variable. The monthly number of days with temperatures dropping below zero has a negative (i e, favourable) effect on the accident toll, especially on the most severe injuries. Comparing the two-wheeler injury models to the pedestrian and car occupant injury models, one notes, however, a much stronger, negative effect for bicyclist and motorcyclists (figure 4.5). This suggests that part of the frost effect found in the main model may be due to a reduction in two-wheeler exposure, not entirely controlled for through our MC exposure proxy. Yet, it is interesting to note that even for car occupants, the estimated effect is negative. When the temperature drops below freezing at night, but rises above 0 °C during the day, certain particularly hazardous road surface conditions may arise. If snow melts during the day, wetting the road surface and forming a cap of ice at night, road users risk being surprised by some extremely slippery patches on a road surface that generally appears clear and dry, suitable for considerable speed. The 'Hce cap risk'' variable measures the percentage of frost days during which the maximum temperature is above freezing. Its elasticity generally has the expected positive sign.

118 Structural Road Accident Models

I c e - c a p risk ( p e r cent frost days w ith t h a w )

D a y s w ith f r o s t

II 0 0 4 0 ^0.056 J 0.01 3 i 0.000 -0.252 -0.23 1

llllllllllllllllllllllllllli P

0.009

• • 0.024 [[10 .0 0 9

H e a v y sn ow fa II (per cent)

1]0.005 jo.0 0 0

0.0 10

D a y s w ith s n 0 wfa II

Illilllill 0 .0 7 6 -0.072 -0.102

D a y s w Ith ra in fa II

•

Fatalities

mDangerously s S e v e rely inju

S n ow d e pth (cm s)

i n j u re d red

03 I n j u ry a c c i d e n t s B T r a f f i c v o lu m e

Mean snow depth > 0 -0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

E lasticity

Figure 4.4. The TRULS-1 model for Norway. Weather effects as of 1994. Accidents and victims by severity. Rainfall has a seemingly negative (i e, favourable) effect on the injury accident count. Again it appears, however, that the effect is due mainly to reduced exposure among the unprotected road users, especially bicyclists. For car occupants, the effect is virtually zero (figure 4.5). Daylight. In figures 4.6 and 4.7 we show how the (lack of) daylight ("darkness") during various parts of the day affects risk. These graphs differ from the previous ones in that only direct effects on casualties are incorporated in the elasticities. That is, the (seasonally and regionally conditioned) association between daylight and traffic volume is not taken account of; the graphs show casualty elasticities given motor vehicle road use. Lack of daylight during the ordinary working hours (9 a m to 3 p m)^ does not have noticeable effects on the accident 3 This variable has non-zero values during the winter months in the northernmost counties.

The TRULS-1 model for Norway 119 frequency or severity. The effect of darkness during the traffic peak hour period (7 to 9 a m and 3 to 5 p m) does, however, have a clearly significant impact on risk, especially for pedestrians. For bicyclists, the estimated association is negative ("favourable") - presumably an exposure effect.

Ice-cap risk (per cent frost days with thaw)

Days with frost

-0.355 rifiiiiiiii^^^^^^^^ -0.342 t

-0.119

(,'iV???i.

J 0.001 \J.\

-0.005

Heavy snowfall (per cent)

B 0.009

IL

0.009

k

l o 006 .005 0.000

Days with snowfall

0.013 0.004

-0.044

frrryy^''^

-0.025 -0.116

Days with rainfall

Piiiii -0.058 -0.011 -0.024

i

° Pedestrians injured ^ Bicyclists injured ^ MC occupants injured ^ Car occupants injured •^Injury accidents s Traffic volume

-0.269 [iilliiliiyiiHiiiiitiflifiiiiil

Snow depth (cms)

-0.178

^^^^>>::
-0.042

!•'•'(

immm

Mean snow depth > 0

-0.103 E -0.076 -0.078 -0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

Elasticity

Figure 4.5. The TRULS-1 model for Norway. Weather effects as of 1994. Accidents and victims by road user category. An even stronger effect is due to dark evenings (5 to 11 p m). Again, the largest risk increase applies to pedestrians, while two-wheelers are probably subject to reduced exposure and hence also to a lower accident toll. Car occupant injuries are significantly more frequent when the evenings are dark. The length of the twilight period does not, in general, have any significant

reaching 360 (minutes per day) in the county of Finnmark in December.

120 Structural Road Accident Models impact on casualty rates, except for bicyclists and pedestrians. Here the effect is negative (favourable), when the amount of daylight is controlled for. 0.000 0.000

Working hour darkness

• Fatalities \n Dangercxjsly injured

-0.029 E

S Severely injured m Injury accidents

Rush hour darkness

Evening darkness

Twilight (min per day)

0.3

0.4

Elasticity Figure 4.6. The TRULS-1 model for Norway. Partial daylight effects, conditional on motor vehicle road use. Accidents and victims by severity. ho.007 ro.oo2 0.000 0.000 0.000

W orklng hour darkness

• Pedestrians injured E B i c y c l i s t s injured S M C o c c u p a n t s injured

]0.111 R u s h hour darkness

Evening darkness

S

Car o c c u p a n t s injured

E] Injury a c c i d e n t s

o.073|TTTmt

-o.093[

r^^a°o°By T w i l i g h t ( m i n per day)

-o.osoj 0 096ntTTTTTTTl Jj0.004 -0.003(|

Elasticity Figure 4.7. The TRULS-1 model for Norway. Partial daylight effects, conditional on motor vehicle road use. Injury accidents and victims by road user category.

TheTRULS-1 model for Norway 121 Seat belts. Seat belts are an effective injury countermeasure (figure 4.8) A 10 per cent increase in the number of car drivers not wearing the belt (from - say - the 12 per cent rate estimated in 1994 to 13.2 per cent) will increase the number of car occupant injuries by some 3 per cent and the number of fatalities by some 0.6 per cent. It appears that seat belts are more effective in preventing less severe injuries than in saving lives. In the TRULS-1 model, we find no sign that seat belts give rise to behavioural adaptation on the part of car drivers, in such a way as to represent an increased hazard to pedestrians, as was once suggested by Peltzman (1975).

Fatalities ^Dangerously injured pSeverely injured pinjury accidents Car occupants injured

0.12

^

Non-use of seat belt

"iii:;:::!:;:;:;:;

:;:;:.:.:

0.191 1 I. I

S'i'i-X';';';';':':';';';';';-:';';';!;';':';

0.235 0.302

0.10

0.15

Elasticity

0.20

0.30

Figure 4.8. The TRULS-1 model for Norway. Seat belt effects as of 1994 A more powerful, converse casualty subset test for the effect of seat belt use is reported in table 4.4. Since 1977, the accident report forms include information as to whether or not car occupants were wearing a seat belt at the time of the crash. Thus, in column A, we show partial result from a model explaining the number of car occupants injured while wearing a belt. In column B, we have regressed the number of car occupants injured while not wearing a belt on the same independent variable. For comparison, we have reestimated the models for all car occupant injuries and for all injury accidents on the 1977-94 sub-sample (columns C and D, respectively). Table 4.4. Casualty subset tests for seat belt users vs. non-users. Elasticities with respect to seat belt non-use share, evaluated at sample means, with t-statistics in parentheses. 1977-94 sub-sample. Car occupants injured while wearing belt A -.082 (-4.04)

Car occupants injured while not wearing belt B .380 (10.91)

Car occupants injured C .194 (6.97)

Injury accidents D .118 (5.99)

122 Structural Road Accident Models Unless the seat belt effect found in the main model is due to spurious correlation, we expect to find a negative elasticity in column A, a. positive one in column C, and an even larger, positive one in column B. Seat belt non-use should decrease the number of injuries among seat belt wearers, while a larger than average increase should be found among non-wearers (example 4.2 of section 4.2.4 above).

Real value of ticket for

• Fatalities mDangerously injured HSeverely injured Elnjury accidents HCar occupants injured • Seat belt use

not wearing belt

Elasticity Figure 4.9. The TRULS-1 model for Norway. Seat belt ticket elasticities as of 1994. This is exactly what our converse casualty subset test reveals, all coefficients being highly significant with the expected relative size and sign. By combining the elasticities found in the seat belt submodel with the elasticities shown in figure 4.8, we are able to calculate the estimated effect of increasing the (real value of the) ticket fine for not wearing a safety belt. This ticket runs at NOK"^ 500 as of 1994. A 10 per cent increase in this fine corresponds, as of 1994, to a 1.3 per cent increase in the rate of seat belt use, i e from 88 to 89.2 per cent. This corresponds to an almost 10 per cent decrease in the rate of non-use (from 12 to 10.8 per cent), which translates into a 2.8 per cent decrease in the number of car occupant injuries and a 2.2 per cent reduction in the total number of injury accidents. The gradual reduction of the real value of the ticket due to inflation will, by assumption, have opposite effects. Access to alcohol. Access to alcohol is more severely regulated in Norway than in most other western industrialized countries. Wine and liquor are sold only from state monopoly stores, generally found only in larger townships, and even beer sales are subject to licensing by the municipal assembly. Restaurants also need a central or local government license in order to serve alcoholic beverage.

4 As of 1994, NOK 1 = app. US$ 0.15.

The TRULS-1 model for Norway 123 More than half the counties have less than one alcohol outlet (shop) per 3 000 square kilometres. Even beer sales have been heavily restricted in some counties, although more so in the 1970s and early -80s than at present. A few municipalities still maintain an absolute ban on any kind of alcoholic beverage being served or sold. In the TRULS-1 model we decompose the availability of various forms of alcohol into six parts. By assumption, alcohol availability does not affect car ownership or road use. One (''alcohol outlets'') measures the total number of shops per 1000 inhabitants. A second one ^strong beer outlets - share'') measures the percentage of shops allowed to sell beverage stronger than lager beer (4.5 per cent alcohol by volume). A third variable Chard liquor outlets - share") measures the percentage of these, in turn, that are wine/liquor stores. In figure 4.10, the alcohol outlet effects come out strikingly consistent, yielding positive casualty elasticities for every degree of severity, with respect to every type of alcohol. Judged by these estimates, the restrictive Norwegian alcohol policy has helped prevent a certain number of road accidents and fatalities. Population. Car ownership and road use increase near-proportionately with the size (or density) of the population, other things^ being equal. Injury accident and casualties increase less than road use, owing to the traffic density effect (figure 4.11). Unemployment has a small, but highly significant, negative effect on road use, and an additional, barely significant effect on casualties. Our final population variable is the share of women in childbearing age that are pregnant in the first quarter (Graph 4.14)^. 19 counties 1973-94.In the main (injury accident frequency) model, the first quarter^ pregnancy rate comes out with a clearly significant, unfavourable effect on the injury accident frequency (an elasticity of 0.18 and a t-ratio of 3.38, as evaluated at the total sample mean for 1973-94), but not on the number of very serious or fatal injuries. In the TRULS-1 model, we are able to perform certain specificity tests (casualty subset tests) to check this result, by means of data pertaining to the 1977-1994 sub-sample period.

5 To be specific, the road network, public transportation supply, price levels and per capita income are assumed constant, but car ownership and road use are not. 6 This variable was calculated by summing up childbirths 6 to 8 months ahead, and then dividing this sum by the number of women aged 18-44. Pregnancies ending in (induced) abortion are hence not taken into account, but these represent no more than 20 per cent of the pregnancies and are unlikely to systematically distort the figures. 7 We limit our attention to pregnancies in the first quarter because in later stages of pregnancy, women may be expected to reduce their mobility, at least as car drivers. Thus there is a likely exposure effect present, blurring any effect on risk.

124 Structural Road Accident Models • 0 005

Hard liquor outlets (share)

• Fatalities CDDangerously injured H S e v e r e l y injured DDlnjury accidents

10.019

Strong beer outlets (share)

0.051 30.036 0.025

0.198

Alcohol outlets

llllllllllllllllllllllllllllllllllllllllllllllllllllllll 0.198 t. I . I . i. I. I. I. t. t. I . I ]

0.05

0.1

0.15

0.2

Elasticity

Figure 4.10. The TRULS-1 model for Norway. Partial alcohol availability effects as of 1994. Outlets. Accidents and victims by severity.

1 St q u a r t e r p r e g n a n c y rate

• Fatalities DDangerously injured B3Severely injured B i n j u r y accidents HTraffic volume WCaT ownership

-0.102 -0.072]

TJfffl o.Goq o.ooq

10.507

l i l l l i i i i i l i III |il|||||||u.ci/;5 S^S^SSS^^SSSS^SS^^SSSS

Population density

I. I . I . i . I . I . I . I . t . } . I . I

I . I . I . I . I r7T-T-r-io.656 . t . I . t . I

0.893 -0.023

Unemployment

1

-0.079 rrnr

"°o°034^N -0.021 g 0.000 0.4

0.6

Elasticity

Figure 4.11. The TRULS-1 model for Norway. Compound elasticities with respect to sociodemographic indicators, as of 1994. Accidents and victims by severity.

When we regress the number of accident involved (injured or non-injured) female car drivers aged 18-40 on the same variables, we obtain an elasticity of no less than 0.31, while in the complementary model for killed or injured car drivers except females aged 18-40, a zero effect (0.045) is found. Thus, the pregnancy variable passes the ''affirmative " casualty subset test as

The TRULS-1 model for Norway 125 applied to the subset of accident involved female drivers in the relevant age (versus all car occupant injuries), as well as the ''complement" casualty subset test as applied to all car drivers except those potentially affected by pregnancy. These tests, aggregate as they are, still do not represent conclusive evidence as to the possibly increased accident risk during pregnancy. We do not know if the increased accident frequency among female car drivers in childbearing aged is actually due to pregnant or to non-pregnant women. But the fact that no similar risk increase is found for the collection of all other road users is apt to strengthen our suspicion of an effect not being due to spurious correlation 1st

pregnane y

_, , ^ ^ Mo n t h s p o© se d s i TRIO PROJECT: DATE: 97 05 10 USER: lef

[)«?<: e m b e r

TRULS - an econometric modeJ ofroaduse, accidents and Iheir swrerify

Graph 4.14. The TRULS-1 model for Norway. First quarter pregnancy rates.

1 B/c*

trio

126 Structural Road Accident Models Killed and injured car drivers ©
Q) 0)

" "

car drivers aged 1840 Total car occupants killed or injured Total injury accidents

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 4.12. The TRULS-1 model for Norway. Estimated effects of pregnancy on four casualty subsets, with 95 per cent approximate confidence intervals. 1977-1994 sub-sample.

4.4. REFERENCES Fridstrom, L. (1999). Econometric models of road use, accidents, and road investment decisions. Ph D dissertation submitted to the Institute of Economics, University of Oslo. Fridstrom, L., J. Ifver, S. Ingebrigtsen, R. Kulmala and L. K. Thomsen (1995). Measuring the contribution of randomness, exposure, weather and daylight to the variation in road accident counts. Accident Analysis & Prevention, 27, 1-20. Fridstrom, L. and S. Ingebrigtsen (1991). An aggregate accident model based on pooled, regional time-series data. Accident Analysis & Prevention, 23, 363-378. Peltzman, S. C. (1975). The effects of automobile safety regulation. Journal of Political Economy, 83, 677-725.

The DRAG-Stockholm-2 Model 127

THE DRAG-STOCKHOLM-2 MODEL Goran Tegner Ingvar Holmberg Vesna Loncar-Lucassi Christian Nilsson 5.1. INTRODUCTION The present study is based on an investigation of traffic safety in the Stockholm Region, which essentially consists of the Stockholm County. It is based on two specific projects, namely an evaluation of the Dennis Traffic Agreement and the MAD-project, where MAD stands for "Measurement and Analysis of the Dennis Agreement". The study may also be seen as a comment to the general traffic safety project "the Zero Vision" recently launched by the National Swedish Road Administration. This program aims at reducing the number of road accidents to a minimum but also to eventually reduce the number of traffic deaths and severe accidents to zero.

5.1.1. The Dennis Agreement The Stockholm Traffic Agreement was originally signed in January 1991, between three political parties in the City and County of Stockholm. The Agreement is usually called "The Dermis Traffic Agreemenf after its prime negotiator, the Director of the National Bank of Sweden at that time, Bengt Dennis. The Agreement was confirmed a second time in September 1992. The Dennis Traffic Agreement of Stockholm contains the following three types of measures to solve the urban transportation planning problems in the Stockholm region: (i) a Ring Road system with one completed Inner Ring and one horse-shoe-shaped Outer Ring; (ii) a public transport commuter rail upgrading and a tangential light rail line; (iii) a system of vehicle tolls at the Inner Ring and on the Outer Western by-pass to reduce road traffic and to finance the road program. The Agreement called for the expenditure of SEK 36 billion on the traffic system over a 15-year period. Road construction would be wholly financed by tolls on vehicular traffic. The public transportation projects were to be financed jointly by the County and the Swedish Government. The Government would also provide an investment grant of

128 Structural Road Accident Models SEK 3.8 billion for the enhancement of public transportation. The Swedish Government repudiated the Stockholm Traffic Agreement early in February 1997.

5.1.2. The MAD-project During the course of implementation of the Dennis Traffic Agreement, i.e. 1993-2005 (according to the original time schedule) the regional authorities in the Stockholm Region launched an ambitious plan to monitor the impacts of the three major components of the traffic package: (i) the urban road by-passes; (ii) the public transport rail; (iii) the road user cordon toll system. In 1996 two consulting firms, Transek AB and Inregia AB, were given the task to carry out the so-called MAD-project, where MAD stands for "Measurement and Analysis of the Dennis Agreement". The aim of the study was to present a description as well as an analytical tool for assessing the past development of road traffic, accidents, environmental pollution and regional development in the Stockholm region during the last 25-year period. Such an analytical tool would be used as an ex-post monitoring tool, for following up the future development of the urban environment, and especially the impacts of the various transportation projects and traffic policy measures decided upon. An international seminar was held in October 1996, where preliminary results from the MAD-project were presented and discussed.

5.1.3. The concept of zero fataHty In 1996 the Swedish Government launched the concept of zero fatality as a goal of Swedish national transport policy. In the long run nobody should be killed or severely injured due to traffic accidents on the Swedish road network. The road transport system should be designed and adapted to the requirements of this zero fatality concept.

5.2. STRUCTURE OF THE DRAG MODEL FOR STOCKHOLM COUNTY 5.2.1. Introduction Daily traffic is the aggregate result of daily decisions to travel (or not to travel), taken by a large number of people. Components of such decision are where to go when to go, by which 1 The Board of Regional Planning and Urban Transportation, Stockholm County Council, the Stockholm branch offices of the National Swedish Rail Administration, the National Swedish Road Administration, respectively; the Greater Stockholm Local Transportation Company (SL) and the City of Stockholm.

The DRAG-Stockholm-2 Model 129 mode of transport, at which time of the day and by which route. Traffic produces benefits and costs to both users and non-users of the road-system. A common framework for discussing the problems of movement of people and goods is in terms of externalities, i.e. those costs or disutilities from mainly (but not only) motorized vehicular traffic. The most common types of problems associated with motorized traffic are: (i) lack of traffic safety; (ii) environmental pollution exhausts; (iii) traffic noise; (iv) visual intrusion. The list might be more elaborate. However, there are some common reflections to be made from various cost-benefit analyses carried out both at the local, regional and national levels of transportation planning. The most dominant disutility from road and rail traffic is, in most cases, the traffic safety costs. These may be expressed in terms of losses of human lives, severely injured or lightly injured victims from various types of accidents. When assessing the most important features of major transport projects or packages comprising combinations of transport projects, it is of a crucial importance to be able to calculate and evaluate the impacts of such projects and packages. In this context, the Metropolitan Stockholm region has a long lasting tradition of accomplishing rather ambitious impact studies.

5.2.2. Dependent variables: definitions and relations The model of traffic demand and accidents is estimated on aggregate time series data for the whole of Stockholm County with a total population of 1.7 million (Dec. 1995). The idea is to explain both traffic volumes, as measured by vehicle-kilometres (vkms), and road accidents by a wide spectrum of explanatory variables obtained from a database of monthly data. In our application of the DRAG approach for the Stockholm County, the following time-series models have been estimated on a monthly basis for the period 1970-1995: (a) an EXPOSURE model of total road mileage (vehicle-kilometres) for gasoline driven passenger cars; (b) a FREQUENCY model of the total number of injury and fatal road accidents; (c) a SEVERITY model of the number of: (i) lightly injured per road accident; (ii) severely injured per road accident; (iii) fatalities per road accident. A database mainly consisting of monthly data for one hundred variables were collected for the 25-year period 1970-1995, i.e. 300 observations. However, it was not possible to define all variables for the Stockholm County, which means that some variables are represented by national data. The general structure of the models is shown in the following diagram 5.1.

5.2.3. Visual analysis of dependent variables 5.2.3.1. Road traffic growth. As a measure of exposure, road traffic growth over the past 25 years has been analyzed in terms of vehicle-kilometres driven. This time series has been

130 Structural Road Accident Models calculated from records of gasoline sales per month within the Stockholm County. For the years 1970-1974 and 1995 no data were available and therefore had to be estimated from a multiple regression relationship between total gasoline sales in Sweden, Gross National and Regional Product and the County's share of the total population in Sweden.

CHARACTERISTICS O F PERFORMANCE O F THE

\

TRA1VSPORT SYSTEM

VEHICLES Car « quality : No. of passenger cars

SPEED

:'

ENVIRONMENT

THE TRANSPORT SYSTEM

(INJURY ACCIDENTS ]

1

[GRAVITY RATE 1

1

ROAD USERS Behaviour variables Car occupancy Drugs and medicine )

ROAD INFRASTRUCTURE Motorway share New road links

Road safety regulations Special e v e n t ;

NJ

t

CASUALTIES

Diagram 5.1. General structure of the model In the calculation of the total road traffic proper consideration has been taken of the fuel efficiency of passenger cars according to both the long-term trend (fuel efficiency over time) and short-term variations related to variations in temperature between winter and summer season^. The figure 5.1 illustrates the monthly variation in total gasoline consumption. The estimated traffic production by gasoline driven cars has increased from 6 to 11 billion vehicle-kilometres annually, or by 88 percent, which corresponds to an annual growth rate of 2 The following temperature corrections have been undertaken: a) If temperature > 8.8 centigrade => no correction; b) if 0 < temperature < 8.8 centigrade, the correction factor is = 1.16 - 0.018*temperature; c) if temperature < 0 centigrade, then the correction factor is: 1.16 0.009*temperature. (Source: SAAQ.)

The DRAG-Stockholm-2 Model 131 2.5 percent during the 25-year period 1970-1995. This road traffic growth pattern is illustrated in figure 5.2. Growth in car traffic has shown a considerable variation over the past 25 years. Periods of rapid growth were replaced by periods of decline caused by external circumstances. Growth of car traffic was fairly slow in the 1970s (a total growth by 22 percent). This was an effect of, among other things, the first oil price crisis accompanied by a rationing of gasoline in January 1974. Car traffic was down to the 1972 level or reduced by almost 6 percent in one year. In 1975, wages went up substantially and car traffic again started to grow. Liter/10 kilometre

Figure 5.1. Monthly variation in fuel consumption corrected for temperature 1970-1995 Vehicle-kms ('0000s) per month

hr i^fiAnhA11-fr f[rf\ ,1\ii\AAA1f r P 1 n 1 Air Ahn np P 1 p \k f\A AP AhVf 1 1 A

A

A

V

^ (\ /

1 1A , ^

1 '4

h

i'

r\

i

/

V

Figure 5.2. Road traffic growth per month in the Stockholm County 1970-1995

1

132 Structural Road Accident Models During the decade 1980-89 car traffic grow by more than 55 percent corresponding to an average of almost 5 percent per year, which must be considered an exceptionally high growth rate. The prolonged recession in the Swedish economy in the early 1990s reduced car traffic by about 17 percent over a period of three years. In the last years growth has been resumed leading to an increase by about 12 percent in the years 1993-95. In addition to the long-term trend, there is a substantial seasonal variation - some 50-60 percent between peak and off-peak months (summer-winter). 5.2.3.2. Traffic accidents. The number of police-reported road traffic accidents has shown a variation around 2,000 to 2,800 accidents per year in the period 1970-1995. The total number has only declined by 1 percent during the last 25-year period. Moreover, the number of fatalities and injured persons has declined by 3 percent during the same period. However, the severity of road traffic accidents has changed dramatically-the number of fatalities has gone down from some 150 to only 50 persons per year, i.e. a reduction by 65 percent in 25 years. To this comes a decline in the number of severely injured persons from some 900 in 1970 to 450 in 1995—a reduction by 50 percent. On the other hand, the number of light injuries has increased from approximately 2,600 to 3,000 persons per year, or by 18 percent. The overall accident pattern is shown in the figure below. NQ of accidents

JarvTO

Jan-73

Jan-76

Jafv79

Jar>€2

Jar>85

Jarv88

Jarv91

Jan-94

Figure 5.3. Light injuries, severe injuries and fataUties in traffic accidents in the Stockholm County 1970-1995

The DRAG-Stockholm-2 Model 133 5.2.4. Matrix of direct effects The following table gives an overview of all independent variables used in the models of road use and accidents by severity. Table 5.1. Independent variables in the Stockholm DRAG Model Independent variable Economic Activities Employment Retail sales Leisure activities Population Tourism Vacation activity The Vehicle Fleet No. of cars in use Vehicle-kilometres Car remarks Break errors Car occupancy Prices & Public Transport Gasoline prices Public transport improvement Road Network & Restrictions New road links Motorways Parking restrictions Temporary speed limits Use of safety belt Useof MC-helmet Use of headlights Legal alcohol limit Climate & Calendar Average temperature Daylight Weather conditions No. of snow days per month No. of workdays per month Special Events Gasoline rationing The Kuwait War Health Medicine prescriptions Pregnancies Randomness and measurement errors

Vehicle kms

V V V V V

Direct effect \ipon Severe Road Light accidents injuries injuries

Fatalities

V V V

V V V

V V V

V V V

V V

V

V

V

V V V V

V V V V

V V V V

V V V V

V V V V

V V V V V V V V

V V V V V V V

V V V V V V V V

V V V V V V V V

V V V

V V V V V

V

V

V

V V V

V V V

V V V

V V

V

V

V

V

^

V V V

V V V

V V V

V V V

V

V V A/

134 Structural Road Accident Models 5.3. MODEL FORM AND EXPLANATORY VARIABLES 5.3.1. Summary of econometric results The following table summarizes the main results from the estimation of a large number of different model variants. Traffic demand as estimated by vehicle-kilometres is explained by 20 factors, the majority of which were statistically significant. For accidents and their consequences, a model with 29 factors was considered the best choice. In this case, however, the statistical significance was less pronounced. Most of the parameters were significant at the 10-percent level. Autocorrelation was accounted for by three factors representing peeks at months 3, 9 and 12 while no terms representing heteroscedasticity were used in the models. Moreover, the performance of the model was improved when a Box-Cox transformation was introduced. Table 5.2. Function form, stochastic specification and other summary statistics. Dependent variable

Vehiclekilometres

Road accidents

Light injuries

Severe injuries

Fatalities

X variables: number of t-statistics ( 2 < t) number of t-statistics (2 < t < 1) number of t-statistics (0< t < 1)

20

29

29

29

29

15 1

10 10

3 13

4 12

2 12

4

9

13

13

15

Heteroskedasticity (number of parameters)

0

0

0

0

0

Autocorrelation (number of rhos)

3

3

3

3

3

0.53 0.53 L51

0.24 0.24 2.00

0.13 0.13 2.00

0.53 0.53 2.00

0.57 0.57 2.00

-2878 300

-1282 288

389 288

508 288

825 288

Form ^ (y) MXi) ^(X2)

Log likelihood at optimum form Sample size

5.3.2. The demand for road use Traffic demand, as estimated by vehicle-kilometres, is explained by 20 factors, the majority of which were statistically significant. The following section summarizes the results for some specific variables judged to be of greatest importance for the growth in traffic demand.

The DRAG-Stockholm-2 Model 135 5.3.2.1. Economic activities. The population of the Stockholm County increased from 1.5 million to 1.7 million in the period 1970-1995 or by around 17 percent. The labour market during the same period increased by 11 percent or from a total of 765,000 to 865,000. However, there was a very rapid increase until mid-1990 when the total increase as from 1970 reached 25 percent. This increase was due both to a growth in female labour participation rates and in employment in public and private service. As a consequence of the severe recession in Sweden in the early 1990s, the total number of economically active people declined drastically or by 12 percent between September 1990 and January 1994. Since then employment again has increased to more than 900,000. The quarterly variation in the labour force is illustrated in the following figure.

1 000 000 Total labor force

800 000 Reduction: Sept-90 - Jan 94: -115.000 j 600 000 Increase from 1970 to Sep-90: +186.400 (25%) 400 000

200 000

Jan-70

Jan-73

Jan-76

Jan-79

Jan-82

Jan-85

Jan-88

Jan-91

Jan-94

Figure 5.4. Quarterly variation in the labour force 1970-1995

The estimation of the model shows that the employment^ variable is the "locomotive" among the explanatory variables. A 10 percent increase in the number of employed is estimated to increase the number of vehicle-kilometres by 15 percent. Increased employment produces higher personal income, which in turn leads to a higher rate of car ownership and this in turn governs much of the activities in the urban area, such as private consumption and leisure activities. A one-percent population growth, without any increase in employment, on the other hand, reduces car traffic by 0.4 percent. To summarize employed people use the car, while others use public transport or the walk/bike mode. The demand for transport is a derived demand. The demand for all those activities creates the demand for mobility, not income per 3 In the model estimation, employment activity is measured by number of employed times share of workdays per weekday.

136 Structural Road Accident Models se. That is why we have not used a household variable or a variable representing personal income, but measured the demand via five activity variables. Another important factor determining the volume traffic is retail sales. Retail sales exhibit an extremely regular pattern with a very high peak towards the end of the year followed by a deep through during the first two months of the year. This pattern is clearly demonstrated in the following figure, which clearly reflects the importance of Christmas for retail trade:

Mkr/month 45 000

40 000 35 000 30 000 25 000

lUHI Ul 111111

Jan-76

Jan-79

20 000

k^mijM^

m i !:'•: :^ Tismr

|IL i\[ Jll

1 ll

15 000 10 000 5 000

Jan-70

Jan-73

Jan-82

Jan-85

Jan-88

Jan-91

Jan-94

Figure 5.5. Monthly variation in total retail sales 1970-1996

As real retail sales increases by 10 percent, road traffic increases by 2.7 percent. This indicator measures the effect of shopping trips on total road traffic demand. The corresponding elasticity of 0.27 for Stockholm may be compared to a similar result obtained in both Quebec, Canada, and in Germany. In Quebec the corresponding elasticity has been estimated at 0.25 and in Germany at 0.24; thus, the findings in Canada, Germany and Sweden are highly consistent with each other (Gaudry et al, 1993-1995). 5.3.2.2. The car park. The car park in the Stockholm County, which has grown by 52 percent from 385,000 cars in use in 1970 to 587,000 cars in use in 1995, has contributed to an increase in the road traffic volume. However, our analysis shows that the elasticity is 0.54 only. We have also introduced a non-linear effect in the model implying that at extremely high loads on the road network, i.e. during the summer and holiday periods, road usage may decline. This result implies that congestion itself may cause a negative effect of -0.25 on road traffic usage. Another explanation could be that, when car ownership increases above a certain level, it is the

The DRAG-Stockholm-2 Model 137 number of cars per household that increases and the second, third car and so on, is used less frequently than the first car. Such a limited effect of the size of the car park on road usage has also been obtained in a similar DRAG model for Germany. In this, case gasoline demand was estimated (Blum et al., 1988) for the period 1968 to 1983 (monthly data). The elasticity was estimated at 0.11, which is even lower than our Stockholm result of 0.29 (on average). Given a certain activity level and other things equal, the size of the car park does not have any decisive effect on car usage. An increase in the number of leisure days by 10 percent is estimated to increase road traffic by 7.6 percent, according to the combined effect of leisure per se and of the number of holidays. If, on the other hand, a one-percent change in the number of workdays is transferred to a onepercent increase in the number of holidays, the net effect will be a 3.5 percent reduction in total car traffic volumes. 5.3.2.3. The road network and the traffic system. Changes in the road network might affect road traffic in several different ways. New urban land is 'produced' or opened up by connecting new residential or commercial areas by new roads. An increase in road traffic is obviously one effect of such a change since the new road facility opens up the newly developed areas for mobility. Another effect may be recognized as an improvement in quality e.g. when a new more convenient road/street replaces a low quality road/street. The new road may be a more direct connection between different origins and destinations or it might be of a higher quality by means of wider or more lanes allowing higher speeds, which would result in a reduction in car trip times. Thus the new road link might attract more traffic. As a third effect, new road links might reduce total vehicle mileage in the entire urban area. This would be the case, when new road links replaces old congested bottlenecks. To avoid a loss in travel time, car drivers might have been used to reschedule their routes to other less direct routes resulting in the production of excessive vehicle-kilometres. With a new road link, where the old bottleneck is eliminated, road users can take the shortest route not only in time but also in distance and as a consequence, overall vehicle-kilometres are reduced. Traffic zoning, on the other hand, might lead to an increase in the overall road traffic volumes, due to the fact that it forces car users to divert from the shortest path and drive a longer route. During the time period 1970-1995 a total of 35 different road projects have been identified. Using dummy variables we have modelled this dual impact of new road links namely that some new road links have in reality replaced old bottlenecks leading to a negative effect on overall vehicle-kilometres while other road projects have contributed to an increase in road mileage.

13 8 Structural Road Accident Models

We have analysed their impact on total road mileage using a two-way approach. In one alternative we constructed a common "quasi-dummy variable", which takes on the value 1 if it is a real new road link, and the value 0.5 if it is only an entrance ramp. Adding another link gives the value 2 and so on. From the diagram below, it may be seen that ten new road projects were completed in the 1970s. The next ten road projects were opened for traffic during the next 12-13 year period; while an additional five new road projects have been put in operation in more recent years. Number of new road links (cumulative) 30

Figure 5.6. The introduction of new road links in the Stockholm County 1970-1995 (cumulated numbers) Our analysis shows that a one-percent increase in the number of new road links may cause a 0.26 percent increase in traffic volume. For traffic zoning an effect amounting to a 9 percent increase in the overall road traffic volume has been identified, although this variable is not significant. It should therefore be interpreted with caution, and this phenomenon should be further investigated. 5.3.2.4. Parking restrictions and temporary speed limits. One might argue that enhanced parking restrictions in the city centre ought to lead to a reduction in overall road traffic volumes. The reason would be that parking restrictions functions like an increase in the cost of car use. Our findings reveal, on the contrary, that the car drivers, in order to avoid higher parking fees and penalties, probably drive more kilometres just to avoid being charged. After 25 years of a continued sharpened parking policy in the city centre, road traffic has grown by 4.6 percent in the whole area. However, this effect is not statistically significant and must be interpreted with some caution.

The DRAG-Stockholm-2 Model 139 Temporary speed limits on urban motorways (110 km/h reduced to 90 km/h during summer 1979 and from the summer 1989 to the spring 1992) have contributed to a reduction in overall road traffic by 2 percent. This effect, too, is not significant and should be interpreted with caution. 5.3.2.5. Public transport. We have not been able to find any significant impact on road traffic volumes from the noticeable enlargement of the metro network in Stockholm (which took place in the 1970s and early 1980s). This does not mean that there was no such influence, only that it has not been possible to demonstrate such an influence by means of our time-series model. Other public transport improvements (such as new bus networks and new bus terminals) seem to have a minor impact on car traffic-an elasticity of -0.05 is found. This means that such measures might have contributed to a reduction in car traffic volume by five percent during the period 1970-1995, although this effect is not significantly significant. 5.3.2.6. Gasoline price. Gasoline price affects the demand for road traffic as measured by vehicle-kilometres. When the gasoline price is increased by 10 percent in real terms, car traffic is reduced by 2.8 percent according to the estimated average price elasticity of-2.8 during the 25-year period. This direct gasoline price elasticity must regarded to be of a considerable magnitude. In other time series models, e.g. the one for Quebec and in the above-mentioned German study price-elasticities of the same order of magnitude have been reported, viz. -0.25 and -0.28 respectively.

5.3.3. Comparison between estimated and actual demand for road use The performance of the traffic growth model is shown below, where observed and estimated traffic production according to the model are compared on a yearly basis. The overall correspondence between observed and estimated vehicle-kilometres is quite good. For single years the deviations between estimated and actual values vary between 0 and 3 percent, with very few exceptions: (i) in 1973 the model underestimates the observed road mileage by 6.5 percent; (ii) in 1989 the model underestimates the observed road mileage by 5.2 percent. In both cases the time period could be characterized by an exceptional increase in road traffic over the past years.

5.3.4. The contribution of road infrastructure to road traffic growth The most dominant single contributor to road traffic expansion in the Stockholm County during the last 25 years according to our model estimates seem to be leisure and shopping

140 Structural Road Accident Models activities. These activities explain one third of the increase in the number of vehicle-kilometres produced. Vehicle-kms ('0000s) per month 1 200 000

1

g Observed vkms

"1

n

r

1 000 000

800 000 J

|-| 1

600 000 J

400 000 -LI

200 000 -LI 0

\M.

1971

1973

-U^

1975

_L|J 1977

- H - ' -!+• ± f j -'+• - H - " -H-i - 4 ^ 1979 1981 1983 1985

-L|-' J^J-H

-Lf^ M * ' l ^ - M - * 1989

1991

1993

1995

Figure 5.7. Comparison between observed and estimated road vehicle-kilometres Rising employment adds an additional 11 percent to the total. The 50 percent increase in the car park (cars in use) contributes with a 15 percent increase in the number of vehiclekilometres. Parking restrictions and a somewhat warmer climate together adds another 8 percent. Factors that have contributed to a decline in road mileage are rising gasoline prices, improved public transport services and a rising proportion of non-employed persons (during the last five years). The road infrastructure factor, measured by the quasi-dummy-variable "new road links" seems to have contributed to the total growth of car traffic by one fourth (+26%). However, it is important to realize that this measure of the ex-post impacts of road infrastructure is an indirect way of monitoring its impact. With this quasi-dummy variable we have tried to grasp the real travel time gains produced by the new road facility. As we do not have access to complete travel time matrices for each year of the entire time period, this has been a proxy method for such an ambitious approach. 5.4. T H E ROAD ACCIDENT FREQUENCY AND GRAVITY MODELS A model with 29 factors was considered the best choice for explaining the monthly variation in the number of road accidents with injuries or deaths. Some of these variables are presented more in detail in the following sections.

The DRAG-Stockholm-2 Model 141 5.4.1. Economic activities The number of bodily injury accidents doesn't seem to be proportional to the exposure in terms of vehicle-kilometres driven. An elasticity on the number of road accidents with injuries of 1.8 due to the number of vehicle kilometres is found. However, in congested situations (late 1980s, summer months) the number of accidents tends to be reduced, probably due to lower speeds. The severity of road accidents, on the other hand, seems to follow an opposite pattern. First, at low or modest amounts of road-traffic, increases in the number of vehicle-kilometres driven tend to lead to a reduction in the number of light and severe injuries as well as of fatalities. Second, when congested situations become more frequent at least the proportion of severe injuries and fatalities seem to increase^

&ffly».<MiM

33%

Leisure & shopping

26%

New links Cars in use Employment Parking restrictions Climate Tourists/veh.km Non-empl. population Public transport improvement Real Petrol price -15%

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

35%

Figure 5.8. Factors contributing to road-traffic growth in the Stockholm County 1970-1995 Employment and shopping activities increase the number of road accidents with injuries, while the severity of these accidents becomes less pronounced when employment activities increase. Vacation activities and tourism seem to reduce both the exposure and the severity of road accidents, probably due to less time constraints and stress among drivers and pedestrians.

5.4.2. Quality of vehicle fleet Inferior cars, i.e. cars with a higher proportion of remarks from the annual inspections, increase 4 These findings do not seem logical. Therefore we have tested another model formulation in the DRAG Stockholm-2 Model, see section 5.3.4 below.

142 Structural Road Accident Models the number of accidents, while the frequency of brake errors seem to lead to a more cautious driver behaviour, and thus, to a reduction in accidents and their severity. This may be interpreted as a risk-compensating behaviour. With more people in each car, the number of accidents and their severity increases substantially. Over time, however, car occupancy has fallen, which has contributed to a reduction in the number of accidents.

5.4.3. Road network data New road links slightly increases the number of accidents (probably due to higher speeds), but reduces severe injuries and fatalities substantially. New and better roads are thus safer than other roads. A speed limit on the primary road network, in this case a reduction from 110 to 90 km/h, reduces the number of accidents and also of severe injuries. 5.4.4. Weather data Weather conditions do have a certain impact on the accident pattern. The number of accidents seems to increase in months where: (i) average temperature is higher than normal (more people exposed); (ii) rain and snow limits the sight of the driver; (iii) sunlight reduces the concentration of the driver on driving. A decrease in the number of accidents, on the other hand, could be noticed in months where: (i) the first snowfall of the winter season occurs make drivers more cautious; (ii) more daylight hours facilitates for the drivers to see unprotected pedestrians and cyclists; (iii) extremely cold weather slows down vehicle speeds.

5.4.5. Intervention measures An increased use of safety belts has a significant positive impact on road safety. Also the legal use of headlights during daytime has shown to be a positive intervention measure to reduce accidents and their gravity. The same impact is found from the increased use of motorcycle helmets.

5.4.6. Gasoline price In a special model variant we have tested the influence of gasoline price on the number of accidents and their severity. The hypothesis is that a higher gasoline price will cause the cardrivers to drive more carefully (slowly) in order to compensate for higher driving costs.

The DRAG-Stockholm-2 Model 143

Our findings indicate that these tendencies are fairly weak. The elasticity of the gasoline price with respect to the number of accidents is -0.8, although not statistically significant. This would imply that a one-percent increase of the gasoline price would lead to 0.8 percent decrease in the number of accidents. The effect on severity as well is inconclusive, as the elasticity is positive and not significant. The elasticity with respect to fatalities, on the other hand, is positive and significant at the 5 % level. These results are not all too surprising since all data pertain to a purely urban road network 5.5. THE DRAG-STOCKHOLM-2 MODEL 5.5.1. The new model specification The alternative model DRAG-Stockholm-2 was developed in the fall of 1997 and spring 1998 with the aims to improving the model system in several respects: (i) to include not only gasoline vehicle-kilometres but also diesel vehicle-kilometres (i.e. buses and trucks) in the exposure model, in order to improve the realism of the DRAG-model; (ii) to test a few new explanatory variables, in order to test the robustness of the original model, but also to try to find other interesting results; (iii) to test a partially new specification, both of the frequency model (from one single model over accident risk to three differentiated models), as well as for the severity models. Road accidents occur for many reasons, one being the exposure to motorized traffic. In our previous DRAG-Stockholm-1 model we had access to only kilometres produced by gasoline powered vehicles. The database was later enlarged by a time-series of kilometres produced by diesel-powered vehicles, i.e. trucks and buses for the same time-period. The first model specification implied only one single model for the accident risks, viz. the number of bodilyinjured persons per bodily injury accident (accidents involving injury or death). The new diCdidiQnX frequency model is defined in terms of three sub-models: (i) lightly injured persons per bodily injury accidents; (ii) severely injured persons per severe and fatal accidents; (iii) fatalities per fatal accident. The following table shows some of the results obtained from estimates of the parameters according to the new specification.

5.5.2. Comparison of results between the "old" and "new" specification However these three new frequency models show a non-plausible U-shape according for the vehicle-kilometre variable for severe accidents and fatalities. The new model specification lead to an improvement of the overall performance for the second model. The pseudo-R indicates that the model may explain almost 80 % of the total monthly variation.

144 Structural Road Accident Models The exposure variable—vehicle-kilometres—is strengthened when diesel vehicles are included. A greater share of heavy vehicles—diesel-share—^reduces accident risks. A 10 percent higher share of heavy vehicles on the roads seems to reduce the number of light accidents by 1 % and the number of fatalities by 2 %, while its impact on severe accidents is not modelled with any significant accuracy. Table 5.3. Comparison of the results from the old and new model specification Model

Model Variant A selection of Explanatory factors Vehicle kms per month without congestion Vehicle kms per month with congestion Diesel-share (share of diesel vehicles of total fleet in %) Employment activity LAMl 1 Share of motorway length state network New road links opened (dummy variable) Seatbelt LAMl Headlight 1 Model Performance 1 Lambda 1-value (t-testO;l) 1 Pseudo-R2 1 Log-Likelihood 1 No. of observations 1 Estimated parameters

DRAG-Sthlm-2

DRAG-Sthlm-2

DRAG-Sthlm-2

Road accidents with injuries and deaths

Frequency of Light injury accidents

Frequency of Severe injury accidents

Frequency of Fatalities

Gasoline vehicles

Gas+diesel vehicles

Gas+diesel vehicles

Gas+diesel vehicles

accbc5:8

accbc2:7

accbc2:8

accbc2:9

DRAG-Sthlm-1

Elasticity (t-value) 1.83 (2.83) -0.70 (-1.73)

2.68 (2.88) -1.14 (-1.93) -0.10 (-1.17)

0.03 (0.03) 0.26 (0.36) -0.01 (-0.08)

-2.91 (-0.99) 2.65 (1.34) -0.21 (-0.61)

0.43 (2.05) -0.03 (-0.16) 0.09 (0.62) -0.50 (-S.32) 0.101 (2.18)

0.72 (2.50) 0.23 (1.23) -0.02 (-0.09) -0.98 (-4.94) 0.14 2.36)

-0.03 (-0.08) -0.81 (-2.91) 0.03 (0.13) 0.41 (1.51) 0.10 (1.22)

-0.95 (-1.06) 0.47 (0.68) -0.28 (-0.35) -0.76 (-0.94) -0.25 (-1.39)

0.235 (1.22; -4.30) 0.753 -1282.3 288 36

0.34 (2.83; -5.53) 0.799 -1218.3 288 37

0.35 (2.41;-4.46) 0.662 -1007.6 288 37

1 1 1

1

1

0.61 (6.04;-3.81) 0.389 -681.0 288 37

The new model contains substantially more information about the road standard factors. More motorways (freeways) seem to contribute to fewer accidents. However, the number of light injury accidents tends to increase (probably due to the speed factor), while the number of severe injury accidents is substantially reduced; 10 % more motorways, seem to reduce these accidents by 8 %. Motorway crashes might also cause more fatalities when they occur. Opening up new urban road links in the Stockholm region might lead to fewer fatalities. Taken

\

The DRAG-Stockholm-2 Model 145 together, all such new road constructions, have contributed to reducing the number of fatalities by 27 % during the last 25-year period (However, this impact is not significant). The traffic intervention measures - the use of seatbelts and of headlight during daytime—is also shown to have interesting impacts on road traffic safety. The use of seatbelts has an even stronger positive effect on the reduction of light injury accidents with an elasticity of almost 1.0; and of-0.77 on fatalities—indeed an efficient traffic safety measure well worth enforcing. The use of headlights during daytime involves a moment of risk compensation, as stated earlier; the motorized vehicle drivers feel safer and drive faster. However, its impact on the number of fatalities is strongly positive (elasticity is -0,25). As regards the severity (gravity) models we succeeded in obtaining the correct inverted U-shaped relationship between total vehicle-kilometres and the dependent variables. This means that at a low or modest level of road traffic, the severity of accident increase as the traffic grows. But at a certain threshold level—when substantial congestion occurs—speed drops, which causes the severity of accidents to decline. Table 5.4. Comparison of the results from the old and new model specification

Elasticity of vehicle-km on: Light accidents Severe accidents Fatal accidents Elasticity of (vehicle-km)"^ on: Light accidents Severe accidents Fatal accidents Functional form (structure) Light accidents Severe accidents Fatal accidents Model performance Pseudo-R2 Light accidents Severe accidents Fatal accidents

Severity models "old" specification DRAG-1 Elasticity (t-value)

Severity models "new" specification DRAG-2 Elasticity {t-value)

0.41 (1,52) -2.03 (-1,95) -3.68 (-1,31)

0.45 (1.04) 0.23(0.72) 0.342(0.78)

-022(1.31) 1.11 (1.69) 2.61 (1.44)

-0,26 (-0.97) -0,22 (-0.90) -0,19 (-0.59)

Inverted U-shaped U-shaped U-shaped

Inverted U-shaped Inverted U-shaped Inverted U-shaped

0.572 0.706 0.422

0.576 0.032 0,002

146 Structural Road Accident Models Unfortunately the overall performance of the "new" model specification is very poor. The pseudo-R^ measure drops from 71 % and 42 %, respectively, for the models of severe accidents and fatalities to 3 % and 0,2 % for the new specification. The reason for this is simply that the variation in the new specification of the dependent variable is too small to produce reliable results. For example, the number of fatalities per fatal accident has a mean of 1.0 and with a diminutive variation. Our conclusion therefore is that the "old" specification of the dependent variable is superior, even if the vehicle-kilometre variable gets the wrong U-shaped structure. The lack of good speed data prevents us from establishing the perfect relationship in this respect.

5.6. COMPARISON OF ACTUAL AND ESTIMATED ACCIDENT RISKS The research project carried out for the Stockholm County shows that it is possible to identify some 30 different explanatory factors that influence the accident risk and the severity of road traffic accidents and fatalities. The following four diagrams illustrates the behaviour of the various models: 500 • Estimated 000

•c* jserveu ru fi ri T i r

500 000

11 n J nllrlTlJ iJlJ IJlJ

_n

rTiri

1

p

1—1-

500 000 500

1 "|''|H|'"|H| 1 l|l l|ll|ll|H|l l|ll|l' I ' ^ M 1970

1973

1976

1979

1982

" l l l^Afi- l+iXfLl^l ^ I ' ^ M

1985

1988

1991

I j i I+Li f j 1994

Figure 5.9. Estimated and observed number of accidents with injured persons in the Stockholm County 1970-1995

The DRAG-Stocldiolm-2 Model 147

n

• Estimated

.

BjObserved

n 1

r| "1

1

k-# k-l

Lf

Lf

•4.

Li.

Lf

•-f

L.

UL

Lf-

H-

Lf-

Lf-

Figure 5.10. Estimated and observed number of accidents with light injuries in the Stockholm County 1970-1995

•

p

Estimated

1 HObserved

1

\

1

1

1

"1 1

IH

M-• • - H -

1970

••-H

1973

1976

•-H1979

M+i • "

11 • - H - ' - H - • • - H - M_|_L ' - M -

1982

1985

1988

-^-H-

-•-H- • • - H 1991

1

'H—'

1994

Figure 5.11. Estimated and observed number of accidents with severe injuries in the Stockholm County 1970 - 1995 180 160

• Estimated

140

• Observed

120 100

1982

I

1985

g

Figure 5.12. Estimated and observed number of accidents with fatalities in the Stockholm County 1970 - 1995

148 Structural Road Accident Models 5.7. SPECIFIC RESULTS ON THE DRAG-STOCKHOLM MODEL In the table below the most important factors are summarized: Table 5.5. Factors with an important impact on the number of injury accidents (from the DRAG-Stockholm-1 model) Factor

Road traffic vehicle-kilometres Use of safety belts Number of employed per vehicle-km Number of remarks per inspected car Share of daylight hours per day Medical consumption (no of recipes/person) New road links

Average elasticity * 1970-1995 with respect to the total number of accidents +1.8 -0.5 +0.4 +0.4 -0.3 +0.3 +0.1

Average elasticity 1970-1995 with respect to the number of fatalities -1.8 -1.2 -0.9 +0.4 -0.9 +0.8 -0.25

* The elasticities indicate how much the number of accidents of a certain type changes as a result of a one-percent change in the explanatory variable.

Our results indicate that the number of road accidents in an urbanized area like the Stockholm County increases much more than proportionally to the amount of road-traffic (elasticity: +1.8). Maybe this can be explained by a rapid growth in the number of potential conflicts caused by an increase in total vehicle-kilometres. The number of fatalities seems to be reduced as well, probably due to reduced speeds. The quality of the vehicle fleet—measured here in terms of the number of remarks per inspected car—also points to an important factor that influences both the number of accidents and the severity of the accidents. Medical consumption—measured as the number of sold recipes per person and month—seems to have a major impact on road traffic accidents according to our findings. If supported by micro-studies, this result indicates an important factor to be dealt with in order to reduce road accidents and their severity. Another interesting result is the impact of new road links in Stockholm County during the last 25 years. Both the number of severely injured people and the number of fatalities are estimated to have been reduced by 25 percent as an effect of some 35 new road links during the 25-year period. An impressive amount of various traffic safety measures have been implemented during the last 25 years in Sweden (and elsewhere). One such example is the use of safety belts in the cars. A 10 percent increase in the use of safety belts is estimated to lead to a reduction in the number of road accidents with personal injuries by 5 percent and a reduction in the number of

The DRAG-Stockholm-2 Model 149 fatalities by 12 percent according to our estimates. A summary of the contribution from various explanatory factors to the number of road accidents with person injuries in Stockholm County 1970-1995 is presented in diagram 5.13. Traffic safety measures

Other factors

Road i n fra s t r u c t u re

-17%

-i%ri

Car Remarks & occupancy R o a d traffic vehkms

Figure 5.13. Contribution from various factors to the number of road accidents with injuries in the Stockholm County 1970 - 1995 To sum up, one could argue that all the traffic safety measures that have been implemented during the last quarter of the century have been necessary to balance the increase in the number of accidents caused by an increase in road traffic volumes at the same time period. This is also clearly illustrated above, where the magnitude of the two main factors exactly even out (- 54 %, +54 %). An improved quality of the vehicle fleet has contributed to a decrease in the accident rate. The road infrastructure is shown to have a slight positive impact, in terms of a minor accident reduction. However, the most positive impact is a substantial decrease in the number of severely injured people and in the number of fatalities. This is caused by new and better road links in the urban area. 5.8. POINTS OF INTEREST AND CONCLUSION In another special model variant we have tested the influence on the number of person injuries and fatalities of alcohol and medicine consumption. Both these factors have a substantial impact on traffic accidents. The following results were obtained in the model analysis:

150 Structural Road Accident Models Table 5.6. Effect of alcohol consumption and recipe prescriptions on injuries and fatalities Factor

Elasticity*

Alcohol consumption in litres/month and per vehicle-kilometre

- 0.834 (-2,30)

(Alcohol consumption) in litres/month and per vehicle-kilometre

+ 0.45 (1.83)

Number of recipes sold per person and month

+ 0.259 (1.71)

(*)

t-values in parentheses

5.8.1. Alcohol consumption: the J-shaped relationship Data from the Stockholm region show that low consumption levels of alcohol seem to reduce the number of personal injuries (light and severe). At higher levels of consumption the accident risk augments rapidly. Comparable results have been obtained in Quebec, Canada, where this has reopened the question of the true form of the function, an issue discussed in Chapter 1. 5.8.1.1. The Grand Rapids Study. In one of the most comprehensive and ambitious field studies ever on the relationship between car driving and alcohol consumption, the so called Grand Rapids-study (Borkenstein et al., 1964), a total of 6,000 road traffic accidents were examined during one year (July 1962-June 1963). For every road traffic accident, information was collected and reported on month, week, and day, exact time of day and street address. Furthermore, the drivers involved in the accidents were interviewed and their blood alcohol level was tested directly at the spot of the accident, A very ambitious effort was made to construct a control group consisting of 7,590 people based on personal interviews. The control group was composed so as to resemble the "accident group" as closely as possible. The following factors were controlled for in the study: sex, age, education, income, occupation, miles driven per year, and drinking habits. The importance of the Grand Rapids study lies in the fact that it formed the basis for the introduction of a limit on blood alcohol concentration (BAC) in most countries. The major findings in this study are: (i) Of all road traffic accidents, 83.4 percent was caused by car drivers with zero blood alcohol concentration (BAC); (ii) Road traffic accidents were clearly underrepresented among car drivers with a BAC between 0.0 and 0.3 per thousand; (iii) When the BAC exceeded 0.4 per thousand, the accident frequency sharply increased; (iv) The

The DRAG-Stockholm-2 Model 151 accident frequency among car drivers with a zero BAG was higher compared to those with a small amount of alcohol (BAG: 0.1 -0.3 per thousand); (v) The lowest accident frequency among car drivers was registered among drivers with regular drinking habits (once per week or more often) and the highest among those who use alcohol more seldom. The Grand Rapids Study is a micro study where the results support the results from our aggregate time series model, described above.

5.8.2. Medicine consumption The consumption of medicine (drugs sold on prescription only) is found to have a substantial and devastating impact on accidents with person injuries and fatalities. A 10 percent increase in the quantity of drugs sold leads to an increase in the number of road traffic accidents with injured persons by almost 3 percent; in light injured by 4 percent; in severe accidents by 8 percent and in fatalities by 13 percent. The relationship found in the Stockholm Gounty has been found in Quebec as well. The elasticity of 0.26 for the number of accidents in our study can be compared to an elasticity of 0.22 the Quebec study. Also the impact of medicine consumption on the accident severity seems to be confirmed by the Quebec studies. Both our results and the Quebec results are statistically significant with respect to the number of severely injured persons and for the number of fatalities. Using aggregate time-series data representing for the entire Stockholm Gounty for both the total number of accidents and the broad spectrum of explanatory variables is a highly indirect way of catching cause and effect. We do not know if an increase in the overall drug consumption also means that the car drivers as a group consume more medicine. Gar drivers can consume a proportionate amount of medicine, but-at the same time-avoid driving such days of the month. By using the aggregate time-series model we have traced a possible relationship but the model results do not tell us the whole truth about the causal relationship. These presented indicative results are very interesting, indeed. Therefore, we recommend a deeper study these relationships possibly in the form of a longitudinal micro-study of a sample of medicine and non-medicine users and their car driving habits. Diagram 5.14 illustrates these relationships.

5.8.3. Pregnancy—a new risk factor Recent research findings from other DRAG-type models strongly indicate that the number of

152 Structural Road Accident Models pregnant women might be an important risk factor per se in road traffic accidents. Therefore, we have tested this variable in the DRAG model for the Stockholm County. The number of births has shown a considerable variation in recent decades and the seasonal variation is well known. This variation is reflected in the number of pregnancies as illustrated in diagram 5.15. Accident

No. of Light Injured

No.of Severe injured

No. of Fatal deaths

20 .

1 15,5

15

^ H 1 2 , 7

8.8

10 5.4

4,5

5

8,3

4^4

^i-e.-.l

0 -5 10 -

-8 3 -10,6

15 20 -

• Alcohol low cons. (BAC< 0,3 o/oo)

25 -

H Alcohol higher cons.

30 -

[--]l\/le(Jicine (drug pre sen ption

-17,0

-27,0

Figure 5.14. Effects on number of accidents and their severity of a 10 percent change in alcohol and medicine consumption The variation in the number of pregnant women is as high as 75 percent between the highest and lowest recorded period. The seasonal variation between different months during a year is of the magnitude of 18 percent, with a peak in August-September. Diagram 5.16 shows the monthly variation in the number of pregnancies. Its influence on traffic accidents could possibly be due to the substantial hormonal changes especially during the first three months of pregnancy. The following results were obtained from the new version of the DRAG-Stockholm-2 model. The figure shows the impact from pregnancy for all the four types of sub-models. The share of pregnant women among all women in fertile ages (18-44 years) is usually rather small, around 1.25 percent. The results are really alarming: a 1 percent increase in pregnancy might lead to an increase in the number of accidents by 3.2 percent and by 6 percent in the number of fatalities in this group. Gaudry obtained similar results as early as in 1984. Fridstr0m, has found the same type of relationship for Norway, on the basis of 5 016 independent observations for the period 1973-1994 (both time-series and cross-section data). Pregnancy might be an important, but so far, unexplored accident risk factor. Therefore, it

The DRAG-Stockholm-2 Model 153 would be wise to check this type of result by a more elaborate micro-study at the individual level on a large sample of pregnant women. The study ought to be carried out by multidisciplinary team, with experts with a competence in medicine, traffic safety and mathematics/statistics. If these results should prove to be robust, they are alarming

000 -

000 -

Number of women

ji

Jan-70

i

Jan-73

Jan-76

Jan-79

Jan-82

Jan-85

Jan-1

Jan-91

Jan-94

Figure 5.15. Monthly variation in the number of pregnant women (1^^ three months) 19701995 in the Stockholm County

Number of pregnant women per vehicle-kilometer / , U U -|

6,00

6,00 5,00 4,00 -

3,60 o,^u

2,70

3,00 2,00 1,00 0,00 No of Accidents

No of Lightly injured

No of severely injured

No of fatalities

Figure 5.16. Effects on the number of accidents and their severity from a 10 percent change in the number of pregnancies

154 Structural Road Accident Models 5.8.4. Conclusions The following conclusions could be drawn from the DRAG-Stockholm Model 1 and 2 research activities: certain human activity and behavioural factors act in the direction of more accidents on our roads. Active traffic safety and intervention measures are aimed at counteracting and reducing both the number of accidents, and especially to limit the severity of accidents that occur: (i) A broad spectrum of factors influence the accident pattern; (ii) Road mobility and activities increases both road accident risk and severity; (iii) Traffic Safety Intervention— reduces accident risk and severity. The impact on accident risk and the severity of accidents of various traffic safety measures and intervention can thus not easily be predicted assuming a constant activity level in the society (outside the traffic sector) or from simplistic before and after studies as new safety measures are introduced. The explanation of the nature of accidents and their gravity (severity) lies in the understanding of the interactive process between human behaviour and activities, the vehicles and the road network and its infrastructure in terms of its physical, administrative, legal and surveillance performance. The interaction between activities, human behaviour and traffic safety intervention has to be fully recognized: (i) one such interaction is the risk compensating behaviour, that has been revealed by this Stockholm model exercise; (ii) another important finding is the non-linear relationship between cause and effect that has been revealed by this research; (iii) a third discovery might be the fact that one explanatory factor might have an impact of a certain direction (positive or negative) on one accident type - say on the total number of accidents, while at the same time having a totally different impact on the number of fatalities. 5.9. REFERENCES Blum, U.C, G. Foos and M. Gaudry (1988). Aggregate Time-Series Gasoline Demand Models: Review of the Literature and New Evidence for West Germany. Transportation Research A, 22A, 2, 75-88. Borkenstein, R.F., R.F. Crowther, R.P. Shumati, W.B. Ziel and R. Zylman (1964). The Role of the Drinking Driver in Traffic Accidents. (The Grand Rapids Study), Blutalkohol, 11, Supplement 1. Fridstrom, L. (1998). TRULS: An econometric model of road use, accidents, and their severity. Paper presented at the 8^^ World Conference on Transport Research (WCTR), Antwerp, July 12-17. Gaudry, M (1984). DRAG, un modele de la Demande Routiere, des Accidents et de leur Gravite, applique au Quebec de 1956 a 1982. Publication 359, Centre de recherche sur les transports (CRT), Universite de Montreal.

The DRAG-Stockholm-2 Model 155 Gaudry, M. (1989). Responsibility for Accidents: Relevant Results Selected from the DRAG Model. Canadian Business Law Journal\Revue Canadienne de Droit de Commerce, 16, 1, 21-33. Gaudry, M. (1991). Measuring the Effects of the No-Fault 1978 Quebec Automobile Act with the DRAG Model. In: Contributions to Insurance Economics. (G. Dionne, ed.), pp. 471498, Kluwer Academic Publishers. Gaudry, M. (1993a). Cur Cum TRIO? Publication CRT-901, Centre de recherche sur les transports, Universite de Montreal. Gaudry, M. (1993b). Le modele DRAG: elements pertinents au monde du travail—une expertise exploratoire. Publication CRT-948, Centre de recherche sur les transports, Universite de Montreal. Gaudry, M. (1995a). Road Safety Modelling: the DRAG Approach and Emerging Research Network. Newsletter of the World Conference on Transport Research Society, 8, 1, 23. Gaudry, M. (1995b). Is the Alcohol Road Accident Risk Curve J-Shaped? Killam Fellowship Application, The Canada Council. Gaudry, M. (1997a). The DRAG Approach and Research Network. App. C in: Road Safety Principles and Models: Review of Descriptive, Predictive, Risk and Accident Consequence Models. (1997), Report IRRD 892483, OECD/GD(97)153, Organisation for Economic Cooperation and Development, pp. 99-103, Paris. Gaudry, M. (1997b). Some Perspectives on the DRAG Approach and Family of National Road Safety Models. The third Annual Conference on Transportation, Trafic Safety and Health, Washington, D.C. Gaudry, M. et U. Blum (1993). Une presentation breve du modele SNUS-1. In: Modelisation de I'Insecurite Routiere, (Carre, J.R, S. Lassarre et M. Ramos, eds.). Tome 1, pp. 37-43, INRETS, Arcueil. Gaudry, M., F. Fournier and R. Simard (1993). Application of Econometric Model DRAG-2 to Road Travel Demand in Quebec. Proceedings Canadian Multidisciplinary Road Safety Conference VIII, June 14-16, 1993, Saskatoon, Saskatchewan. Gaudry, M., F. Fournier et R. Simard (1993). Applications du modele DRAG-2: impact des taux d'utilisation de la ceinture de securite sur les accidents de la route. Maladies chroniques au Canada/Chronic Diseases in Canada, Sante et bien-etre Canada, Ottawa. Gaudry, M., F. Fournier et R. Simard (1993, 1994, 1995). DRAG-2 un modele econometrique applique au kilometrage, aux accidents et leur gravite entre 1957 et 1989 au Quebec. Societe de I'assurance automobile du Quebec, 995 pages. Tomes I-V. Gaudry, M. and S. Lassarre (2000). Structural Road Accident Models: The International DRAG Family. Elsevier Science Publishers, Oxford. Tegner, G. (1996). The Use of Time-series Models for Planning and Monitoring Public Transport - Some Swedish Experiences. Transportation Days, Aalborg University. Tegner, G. (1994). The Dennis Traffic Agreement - A Coherent Transport Strategy for a Better Environment in the Stockholm Metropolitan Region. International Workshop under the

156 Structural Road Accident Models aegis of European Parliament, Scientific and Technological Options Assessment (STOA): The Technological City - ideas and experiments in urban organisation of mobility, transport, production and services. Brussels. Tegner, G. (1996). Infrastructure-induced Mobility-Some Swedish Evidence. ECMT Round Table!05. 0ECD,?2ivis. Tegner, G. and V. Loncar-Lucassi (1997). Demand for Road Use, Accidents and their Gravity in Stockholm: Measurement and analysis of the Dennis Package. Transek AB, Stockholm. Tegner, G. and V. Loncar-Lucassi (1997). Time-series Models for Urban Road Traffic and Accidents in Stockholm. Denmark Transportation Days. Aalborg University. Tegner, G., V. Loncar-Lucassi, M. Vesna. (1997). An Analysis of Urban Road Traffic Safety in Stockholm - The use of aggregate time-series models with the TRIO programme. Proceedings the European Transport Forum Annual Meeting, London.

The TAG-1 Model for France 157

T H E T A G - 1 MODEL FOR FRANCE

Laurence Jaeger Sylvain Lassarre

6.1. INTRODUCTION When researchers seek to study the influence on traffic risk of factors related to mobility, the economy and road safety policies, they tend to use models that are based on the exploitation either of cross sections of a panel of regions or sites, or of chronological series (armual, monthly or daily). We directed our research at this second type of model, which is useful for: (i)evaluating the effects of road safety measures at a national level that are becoming increasingly complex; this complexity comes from spatial and temporal variations in populations affected by the measures (points system for driving licences, technical inspections of vehicles, and so on); (ii)basing the interpretation of medium and long-term traffic risk on those of mobility and the economy. More specifically, our objective was to develop an econometric model of traffic risk known as TAG, for Traffic, Accident and Gravity (Jaeger, 1998), starting from the DRAG (Road Demand, Accident and Gravity) model conceived in Quebec, which remains a world class reference (Gaudry, 1984). We adapted the structure of this model to the specific conditions of traffic risk in France and added to it by building in speed as an indicator of driver behaviour. This adaptation consists, for example, of introducing variables relating to motorised twowheelers, or to the stock of small cars, or to traffic shares according to road network categories, which are peculiar to French conditions. The addition of a fourth dimension related to behavioural risk is an innovation in this kind of model, made possible by the availability of a long series on speeds driven, as measured by surveys on interurban roads. Risk analysis is classically based on the triad (OECD, 1997):

158 Structural Road Accident Models EXPOSURE

-^

ACCIDENTS

->

CASUALTIES

using risk indicators such as accident rates per vehicle-kilometre or the severity rate in terms of the numbers killed per injury accident. The usual traffic risk indicator from the transport standpoint is the accident rate per billion vehicle-kilometres. It is implicitly assumed that the number of accidents is proportional to risk exposure as measured by the mileage driven, which is the usual measure of risk exposure on a road network. This proportionality has been brought empirically into question by the many microscopic models of accident risk that are valid for one driver and also for a section of road or an intersection. That is why it is preferable to make the number of accidents depend on a non-linear function of the mileage driven and let the data determine the form of the relationship. On the other hand, one can use the rate of casualties (minor, serious and deaths injuries) per injury accident as an indicator of gravity, since the total number of casualties is a random sum of the number over accidents of random variables, which are the number of casualties in each accident. The interesting variable is the number of casualties per injury accident, and we have tried to construct a model of the mean number, which is expressed as a rate of casualties per injury accident. We expanded this risk chart by adding in the behaviour-offence-penalty sequence, which enables the system of road traffic enforcement and control to be studied. More particularly, the behaviour-accident and offence-accident relationships will serve to complete the battery of evaluation criteria based on risk indicators by bringing in behaviour indicators related to speed, drunk-driving, and so on. Exposure

Accidents

Casualties

Behaviour

Offences

Penalties

The fact that, by integrating intermediate, behaviour-related indicators, the focus is no longer on the result as measured by the risk of involvement in an accident or the risk of being injured or killed, makes it possible to evaluate more complex patterns of the impact of safety measures, which may bring into play risk compensation mechanisms, as regards the wearing of safety belts, for instance. In this spirit, we integrated speed into the model so as to take account of behavioural risk. We chose average speed as the indicator, both for theoretical reasons, knowing that it is an important measure of the performance of a transport system and a key factor in driving risk.

The TAG-1 Model for France 159 and for practical reasons, because of the readier availability of this statistical series, at least for the inter-urban network. Other aspects of behaviour are linked to risk, such as drinking alcohol before taking the wheel or using protective devices like safety belts and crash helmets. They interact with speed, and the problem is to identify to what extent, since the phenomenon of compensating behaviour leading to an increase in speed may reduce, if not wipe out, the additional safety benefit afforded by wearing a seat belt. In the first part, we describe the structure of the model, which was established on the basis of an analysis of the road transport system as a whole, as well as the determination of risk indicators and the identification of the risk factors that influence the risk of being involved or killed in a traffic accident. This analysis is followed by a brief presentation of the econometric formulation and the estimation method. We will comment on the development of the different indicators, the total mileage travelled, the average speed driven, the number of accidents, their degree of severity and the number of road victims. The last part is devoted to an estimation of the different indicators linked to road risk, namely total mileage, average speed, the two categories of accident (fatal and non-fatal personal injury), the three levels of severity (minor, major and fatal) and, by deduction, the three categories of road victim (slightly injured, seriously injured and killed).

6.2. STRUCTURING THE TAG MODEL The production of traffic risk follows a complex process that has to be analysed through a systemic approach. Road safety problems have traditionally been viewed as the result of malfunctions in the road transport system, and more specifically as the result of combinations of faults and errors on the part of its three components: vehicle - driver - infrastructure. The road transport system comprising these three components is integrated in a broader environment (figure 8), where it interacts with other governmental, economic, demographic and climatic systems. These have an impact on the performance of the road traffic system either directly, or indirectly through their effect on the characteristics of vehicles, road users and the road infrastructure (Lassarre, 1992). Among the indicators of road system performance are those which reflect the functioning of the system, such as total mileage driven and the speed practised, and those reflecting system malfunctions, such as the number of personal injury accidents and severity rates. TAG is structured so as to explain the damage (the victims of road accidents) as a function of exposure to risk (mileage driven), risky behaviour (average inter-urban speed), the risk of an injury

160 Structural Road Accident Models accident (number of injury accidents) and the risk of injury (number of casualties per injury accident) (Jaeger and Lassarre, 1998).

CHARACTERISTICS OF PERFORMANCE OF THE TRANSPORT SYSTEM

TOTAL MILEAGE

L » ^ AVERAGE SPEED

\

THE TRANSPORT SYSTEM

ENVIRONMENT

*

VEHICLES

•HGVs share •diesel engine sh

ROAD USERS

•behaviour variables •young driver share... INJURY A C C I D E N T S

GRAVITY RATE ROAD INFRASTRUCTURE

•motorway traffic share • main road traffic share

T Figure 1. Chart of interaction between the road transport system and its environment In other words, the model consists of four layers related to four risk dimensions (figure 1): (i) risk exposure measured according to the number of kilometres travelled; (ii) risk behaviour measured in terms of the average speed driven on the inter-urban network; (iii) injury accident frequency by number divided into fatal and non-fatal; (iv) accident gravity in terms of the rate of fatalities, minor injuries and serious injuries per injury accident. Each of these dimensions becomes an element to be explained. As a result, for each traffic risk dimension we have to identify the set of risk factors, which may influence the safety performance of the system. All the factors linked to the characteristics of vehicles, drivers and the road infrastructure are

The TAG-1 Model for France 161 regarded as factors internal to the system. The measuring and quantifying of these factors is done within the road system. For example, the proportion of front-seat passengers in private cars who fasten their safety belts is estimated by means of roadside surveys, as are the share of heavy goods vehicles (HGVs) in total traffic. Conversely, elements related to the system environment are regarded as external factors. Most of the time they have an indirect effect on the performance of the road transport system through their impact on its three components, but they also sometimes have a direct effect, for example weather conditions or prices. On the other hand, other factors related to the age structure of the population or to economic activity act through the intermediary of the transport system's components. For example, the proportion of young drivers depends on the share of young people of driving age in the total population. In the language of statistical models, these external factors are proxy variables which are substituted for internal factors of the system that cannot be measured, or only at too high a cost. Thanks to this systemic approach, which divides the system into three components - vehicle / driver / road - one can identify the fundamental factors that produce traffic risk, which are related to the characteristics of vehicles, drivers and the infrastructure. The main explanatory factors for road transport demand, average speed, the incidence of road accidents and their degree of gravity are classified by subject (table 1): vehicle characteristics with the composition of the vehicle stock, driver characteristics with behavioural variables, the characteristics of the road infrastructure according to network categories, the economic system incorporating prices, employment and unemployment, household consumption and vacations, linked to the different modes of personal travel, industrial activity for the transport of goods, the climatic system represented by the main climatic variables, and regulatory measures included under the heading of the main legislation governing road safety. The very different technical characteristics not just between categories of vehicle but also within vehicle categories - mass, speed, potential damage to structures on impact, power induce differentials of risk. Apart from speed, other aspects of driver behaviour in terms of protective measures or the consumption of toxic substances may be more or less dangerous. The wearing of safety belts by the occupants of private cars, and the use of protective devices for children enable injuries to be avoided or reduced in the event of a collision. Since no figures are available for the proportion of motorists driving under the influence of alcohol or intoxicated by other substances (drugs or tranquillisers), we used other intermediate variables, such as alcohol consumption and more particularly taxed wine consumption As regards the characteristics of the road infrastructure, we confined ourselves to factors that were representative of the share of traffic on motorway networks and on main roads. It is hard to integrate factors related to road characteristics (bends, intersections, surface, visibility

162 Structural Road Accident Models distances, platform, and so on) at an aggregate level. Table 1. Classification of all explanatory variables integrated in the TAG model.

rcHARACTiiyysTics

of vehicles

t. Stock

• PC/HGV/motorised twowheeler breakdown 1

of driven.

2. Characteristics

• (Proportion of young adults) 1 • Rate of seat belt wearing 1 • (Taxed wine 1 consumption)

h^V'^-::m^:
3. Behaviour variables

h:^::^':^~'^mmm^m

of the infrastructure

4. Networks

WZ%'kr-'-''^^iS^^^^ '-'\;'--"\''^-;^^:''-^^"-; 1/', ^P;y„., ;,;-/; J,:^?/':

• Share of traffic on motorways and autoroutes,

Demographic system

5. Population

• Proportion of young adults

Economic system

6. Price

• Price of fuel per kilometre

1 ,~/'7/"';<^'-\-" • ;v-;?i,j;

on main roads

• Price of a car,..

•"•\"7"vEXTi:Ri^",t::;'''

ijSiSlii

Government syst

1 1 1

7. Unemployment

• Proportion of unemployed 1

8. Reasons for people's journeys

• • • • •

9. Reasons for goods

Climatic system \i'l^0'C^%^rj:^Ait^'. '<^^-«-

1 1 1

transport 10. Road safety laws

11. Climatic variables

Working population 1 Household consumption Vacations Taxed wine consumption 1 Industrial activity

• Mandatory technical inspections,..

1

• Average temperature • Snow,...

1

In addition, insofar as they affect the characteristics of vehicles, road users and the road infrastructure, the governmental, economic, demographic and climatic systems are in interaction with the traffic system. The economic system also affects the mobility of people and goods through the intermediary of prices and revenues. l\iQ price of fuel, the prices of cars as well as the prices of public road, rail and air transport have a direct impact on road transport demand and hence an indirect impact on the accident toll. A large proportion of home-toworkplace journeys depend on the level of employment as measured by the working population and the unemployment rate. Families travel to enjoy leisure activities or to purchase foodstuffs. Household consumption thus has an influence on the number of journeys linked to shopping and recreation. Lastly, holidays generate seasonal journeys, which are tending to become more scattered through the year instead of being concentrated in the summer months. Industrial activity has a direct impact on road transport demand for moving goods, a direct impact on driving risk because of the risk differential between different categories of road vehicle and an

The TAG-1 Model for France 163 indirect effect on driving risk through the mileage driven. Regulatory measures in the area of road safety concern every element of the traffic system - the vehicle, the driver and the network. The government has a dual role to play by, on the one hand, instituting laws and regulations and, on the other hand, enforcing them through information campaigns, inspections and penalties. Preventive measures applied to all users and primarily to drivers have to be distinguished from enforcement measures, which are targeted at offending drivers, although these two aspects are often complementary. The fundamental element is the extensive legislation relating to the behaviour of road users, which covers the fight against excessive speed and drinking and driving, as well as the mandatory wearing of safety belts by car users and crash helmets by riders of motorised two-wheelers. At the same time, other regulatory measures of a so-called technical nature have been introduced which relate to infrastructures and vehicles. Although completely uncontrollable, weather conditions have a direct influence on accidents and their degree of gravity, as well as an indirect effect through road transport demand.

6.3. ECONOMETRIC FORM OF THE TAG MODEL The TAG model is structured according to a system of seven non-linear, simultaneous equations, in which an endogenous variable of one equation appears as an explanatory variable of another equation. In the context of the problems we are investigating, we opted for a recursive model, which expresses the total mileage as yJ^, the average inter-urban speed as j^^, the number of fatal and non-fatal road accidents as y^^, y^2t^ the fatal, serious and light severity rate as j^^^, j^^,,^y^^^, and the explanatory variables as x^^ (i = 1,.., k): Yi t = f(Xi h ; Ui t)

y3it = Kyit,y20 y^it^^yu^yit^

. , Xj

y4ir = f(yu,y2/.

. , Xj

y^2t =

Kyu^y2t,

. , Xj

y4u =

Kyu,y2t^

., X:

h---'.^A2t)

The random variables «,/ (i = 1,..., 7) are white noise. The interest of the TAG structure lies in particular in its capacity to, on the one hand, identify the direct and indirect effects of the explanatory variables of the road accident toll (through risk exposure and average speed) and, on the other hand, analyse the substitution or compensation

164 Structural Road Accident Models effects between the numbers of fatal and non-fatal injury accidents, or between the numbers of dead and injured. The model thus enables a more complete interpretation to be made of the system generating traffic risk than a simple model focusing on a single indicator. If we take the example of snow, the number of days with snowfall may reduce the number of kilometres driven as well as the average speed , and at this level of reduced exposure and speedy increase the number of injury accidents while reducing the gravity of injury accidents. The impact of different factors on the number of road accidents will be directly and indirectly estimated on a ''ceteris paribus'' basis (through road usage and average speed). As far as possible, we have retained the same explanatory factors in the equations in order to be able to illustrate the direct and indirect effects and a possible substitution effect between fatal and nonfatal personal injury accidents, or between degrees of severity (minor, major and fatal). To gain an understanding of the development of road accidents and their severity over the past four decades, we have to consider, on the one hand, their direct impact, which is to say the impact of an explanatory factor on accidents, their severity and the number of victims while maintaining all the other factors constant, and on the other, their indirect impact, which is to say the impact of a factor on mileage and/or speed, and consequently on accidents, their severity and the number of victims. The starting-point is the monthly statistical series for the total mileage, average speed and the numbers of accidents and casualties from 1967 to 1993 (making a total of 324 observations). Figures for total mileage, which is to say the number of kilometres travelled by all road vehicles on the French road network, are not available on a monthly basis for such a long period. It was thus necessary to develop a methodology for calculating the total mileage travelled on the whole of the French road network by all road vehicles, including foreignregistered as well as French-registered ones, on the basis of petrol and diesel sales (Jaeger and Lassarre, 1997). There is an increase over the whole period, with a very marked seasonal element. The average harmonic speed' of all motorised vehicles on all inter-urban road networks at a national level was calculated ex post by aggregating the data from surveys carried out on different road networks (outside urban areas) on behalf of the National Interministerial Road Safety Observatory (ONISR - Observatoire National Interministeriel de Securite Routiere). The ' The harmonic average speed is preferable to the arithmetical average because of its relationships with other fundamental traffic variables, namely flow and concentration as once pointed out by Lassarre and Page. The degree to which the speed limit is exceeded could also be used. As there is a strong correlation between these three indicators, it is sufficient to include only one of them in the model.

The TAG-1 Model for France 165 effects of the speed limits imposed in 1973 and of the first energy crisis in 1974 are very visible. Personal injury road accidents, which are to say those having caused at least one death or injury, and the casualties' degree of gravity are recorded in the Analytical Reports of Personal Injury Road Accidents (BAAC - Bulletins d'Analyse des Accidents Corporels de la Circulation). The definition of killed and injured was changed on 1 January 1967, which forced us to use that as the starting-point for our series. In trend terms, the injury rate is relatively stable, while the rates of serious injuries and fatalities both decline under the effect of the major measures introduced in 1973, but then diverge from 1982 onwards. This marked transfer of severity from serious to fatal remains to be explained. Each of the seven equations of the TAG model is a multiple regression of w observations with a flexible, functional form of the dependent variable Y and the explanatory variables X. A generalised structure of heteroskedasticity and autocorrelation of the residuals can be expressed as in equations (1.1) to (1.4). For all our estimations, we utilised the L-1.5 algorithm of the TRIO software program described in Chapter 12. It enables us to set or estimate the parameters of the Box-Cox transformations applied to the dependent variable and the groups of independent variables. In addition, this program produces a joint estimation of the parameters of the chosen self-correlation structure and the different parameters associated with the heteroskedasticity form of errors. One of the main difficulties encountered in estimates of this type of model is the multicolinearity between the explanatory variables. In addition to the measurement known as the Belsley Index and the tests carried out by the TRIO program, we developed an estimation procedure that brings in variables step by step so as to minimise the risk of multicolinearity.

.gasoline

Figure 2. Monthly development of total mileage.

166 Structural Road Accident Models

Figure 3. Monthly development of average speed.

8

8 I

§

I

Figure 4. Monthly development of the number of injury accidents

0j\mk^^^ 8

S

B

i

Figure 5. Monthly development of the rate of minor injuries per 100 injury accidents.

The TAG-1 Model for France 167

Figure 6. Monthly development of the rate of serious injuries per 100 injury accidents. W9&S!Wmf^9^iWs9?>9iWii0i!

Figure 7. Monthly development of the rate of deaths per 100 injury accidents In our analysis of the resuhs, the impact of different explanatory variables on the seven dependent variables is measured in terms of elasticity. In the case of continuous variables, elasticity is defined as the percentage ratio of variation of two variables measured at a reference point, namely the average of the sample of these two variables. For example, if the explanatory variable X increases by 10 per cent and the dependent variable Y rises by 7 per cent, the elasticity is 0.7. However, in the case where the explanatory variable is an intervention variable, otherwise known as a "dummy" variable, the elasticity is the percentage impact on the dependent variable of the presence of this intervention variable.

6.4. T H E ESTIMATES PRODUCED BY THE TAG MODEL As well as constructing a model of risk exposure and risky behaviour based on total mileage

168 Structural Road Accident Models and average speed, we also developed models of the two measures of accident frequency (nonfatal and fatal injury accidents), the three degrees of gravity (minor, serious and fatal injuries) and the three categories of road casualty (minor injuries, serious injuries and fatalities) to buttress our analysis. 6.4.1. Model of road transport demand In this section, we will estimate the equation of the number of kilometres travelled by road vehicles on the French road network. This model identifies the factors affecting road transport demand and evaluates the direction and intensity of their effect (Annex 1.6). Factors having a positive incidence on the activity of the road transport sector (figure 8). Among the factors that explain the growth in total mileage, the main ones to be taken into account are home-to-workplace journeys, the stock of private cars and commercial vehicles per unit of work, the consumption of wine per adult, temperatures and holiday travel. A 10% increase in, or the advent of, this phenomenon engenders, ceteris paribus, a change in mileage of: Vehicle linspectid Real price of fuel per kilometre Share oi small ca|rs Share ofHGVs Gulf w4r Fog Snow u^ iDp^

Unemployempnt W^ek-ends |3ank holjdays Share ofldiesel-erjgine car^

^ ^

m^

^ r^:

2%

4%

10%

12%

14%

Figure 8. Factors having a negative or positive incidence on total mileage (The unfilled histograms represent factors having an insignificant positive or negative effect on mileage.)

16%

The TAG-1 Model for France 169 While keeping all other factors constant, the reference model as specified reveals that a 10 per cent rise in the employment index (a proxy for home-to-workplace journeys) results in a 15.1 per cent increase in the number of kilometres travelled. Over the past 25 years the average distance of home-to-workplace journeys has almost doubled, and urban development and urban spread have brought a sharp increase in the physical space in which people move in their daily lives (Orfeuil, 1993). Furthermore, according to some studies, the modal split, meaning the share of different modes of transport as a proportion of the number of journeys, tends towards the quickest modes of transport and especially the private car (Orfeuil and Zumkeller, 1991). Taken together, these two phenomena imply an increase in private car use as a means of transport, and hence in total mileage driven. The representative factor of the stock of private cars per unit of work increased throughout the period, resulting in a greater increase in the stock in relation to the employment index. This factor has an elasticity of 0.74 on total mileage and a high degree of certainty. That phenomenon is attributable to an increase in multi-car households engendered by urban spread and the increase in households' purchasing power. As a result, at a constant employment level, the stock increases, and that imply an increase in total mileage. The consumption of alcohol is used in many countries as a powerful indicator of social activity. There is a strong correlation between the frequency of social outings and the frequency of alcohol consumption. In effect, it is found that a 10 per cent increase in wine consumption per adult implies, ceteris paribus, a 3.5 per cent increase in road transport demand for recreational purposes. A 10 per cent rise in average national temperature implies, ceteris paribus, a 3.2 per cent increase in mileage driven. Climate, and particularly temperature, has an influence on the behaviour of road users. Temperature does not directly reflect the amount of sunshine, but there is a degree of correlation. Periods of fme weather encourage people to go out, and conversely, cold spells, and especially freezing temperatures, force them to do things at home. It is interesting to note that the growing proportion of diesel-engine cars has a positive and very significant impact on road transport demand (elasticity of+0.13). These findings are supported by studies showing that, although they already belong to the category of high-mileage drivers, motorists who decide to switch to diesel-engine cars nevertheless increase their mileage by around 25 per cent (Hivert, 1993). Factors having a negative incidence on the activity of the road transport sector (figure 8). Some interesting findings emerge from the reference model, relating in particular to certain laws and regulatory measures (such as the introduction of technical inspections in 1992), the

170 Structural Road Accident Models price of fiiel per kilometre and the proportion of small cars. We find that the introduction of technical inspections in 1992 had a considerable impact on total mileage. This phenomenon is attributable in part to the disappearance of vehicles that did not comply with the new safety standards. Moreover, the intensity of vehicle utilisation reflects the influence of economic factors such as the price of fuel per kilometre, which combines the price of petrol and diesel fuel. An increase in the price of fuel per kilometre has the effect, ceteris paribus, of reducing the number of kilometres driven (-0.23). In Quebec, the same phenomenon was observed as regards the impact of the price of gasoline per kilometre on the kilometres travelled by petrolengine cars (-0.3) (Gaudry et ai, 1994). Madre and Lambert (1989) conclude that "users' sensitivity to the price of fuel is moderate and is greater on motorway link roads (-0,3) than on main roads (-0.2)." Any change in the structure of the automobile stock leads to a change in the activity of the road transport sector (cf the shift to diesel-engine private cars). Hence, we find that, according to the reference model, an increase in the share of small cars (under 5 fiscal horsepower) in the total stock implies a reduction in total mileage (elasticity of-0.19). The growing share of small cars is the counterpart to the growth in multi-car households. In effect, a family's second car is mainly used for short journeys in urban areas rather than for inter-urban journeys over long distances.

6.4.2 Constructing a model of average speed In this section, we will present the equation of the average speed driven on the inter-urban network at the national level. The model identifies the factors affecting average speed and evaluates the direction and intensity of their effect (Tables 2 and 3). Factors having a positive incidence on average speed (figure 9). The four factors identified as having a positive impact on speed are the motor vehicle price index, the rate of safety belt wearing, the proportion of motorway traffic, and the percentage of private motorcars rated at 11 fiscal horsepower or more. Their impact is modest (+0.2%, +0.08%, +0.07%) and +0.06%) respectively), but not negligible. In effect, since estimated interurban speed is an average at national level, it suggests that a majority of drivers have to alter their behaviour in speed terms in order for there to be an incidence at an aggregate level.

The positive impact on average speed of an increase in the motor vehicle price index, the

The TAG-1 Model for France 171 proportion of motorway traffic or the percentage of private motor cars rated at over 11 fiscal HP corresponds to expectations. The outcome as regards the wearing of seat belts is interesting, insofar as it confirms the existence of a retroactive effect on driver behaviour vis-a-vis a safety measure aimed at reducing the severity of accidents.

Factors having a negative incidence on average speed (figure 9). Among the factors identified as having a negative incidence on average speed we find first and foremost the laws relating to speed limits, both in urban areas (-2.9%) and the countryside (-2.5%). Moreover, the results are very significant.

Table 2. Estimations of the elasticities, the parameters with the Student t for the model of average speed in France (simplified version). AVEMGESFEED L I S T FACTORS

A 1 0 % INCREASE « Ceteris paribus

IMPACT

CERTAINTY

Elasticity

« t » of Student

»

1. Risk exposure 1 Total mileage travelled in France

^ 0,2 %

-5,70

710,06%

1,27

71 0,08 % ^ 0,05 %

2,30 -0,89

71 0.07 % ^ 0,2 %

2,22 -4,31

^0,8% 71 0,2 %

-13,55 8,23

:^0,06%

-1,35

^:J 0,3 %

-0,95

^ 2,5 % -0% NJ0,4% ^ 0,9 %

-9,85 0,15 -1,52 -3,47

^^ 2,9 %

-17,68

2. Characteristics of vehicles 1 Number of private cars (> 11 CV) / kilometre (all vehicles - whole France) 3. Behaviour variables of the road users 1 Rate of seat belt wearing 1 Wine consumption taxed in Metropolis per adult 4. Networks 1 Share of motorway traffic 1 Share of main road traffic 5. Prices Real price of fuel per kilometre of private cars Index of price of a car 6. Reason for goods transport 1 Industrial activity (except energy and automotive engineering) per unit of work 1 7. Variable of temporality Number of public holidays and weekends

[|

T H E PRESENCE O F THIS LAW OR THIS EVENT

Generalised speed limit (90/110/130 km/h) (09/73) 1 Introduction of system of no-cl aims bonus-maius (1/78) 10% reduction target (7/82) 1 Tightening of drink-driving offences (07/87) 50 km/h limit in town (12/90)

1

172 Structural Road Accident Models Of the road safety policies we studied using the TAG model, enforcement measures certainly seem to be the most effective. We fond they had an indirect effect on average speed. That was the case in particular of measures introduced to tighten up on drunk-driving offences (-0.9%). Table 3. Goodness of fit statistics of the simplified model applied to average speed. 1

RESULTS OF ESTIMATES

AVERAGE SPEED

1

I. PARAMETERS Box-Cox transformations Lambdaof Y and X

2.455

(« t » of Student: compared to 0)

(5.65)

(« t » of Student: compared to 1)

(3.35)

1

II. GENERAL STATISTICS 324

Number of observations Log-Likelihood

- 232.47

Pseud O-R2

0.853

A 10% increase in, or the advent of, this phenomenon engenders, ceteris paribus, a change in speed of: 50 k m / h ilimit in town G e n e r a l i ed speed limit Tighteniiig of drink- driving

offences

Real pride of fuel per k i l o m e t r e H 10 % r e d u c t i o n

target

Week-en ps b a n k h o l i d a y s Share ofi main r o a d

traffic

Total mii cage Industrial

activity

Wine c o n s u m p t i o n per a d u l t No-claims

bonus

|re of cars |over 1 1 fi Share

of|motorway|

Rate of] seat belt Car pricel -3%

-1,5%

0,5%

Figure 9. Factors having a negative or positive incidence on average speed (The unfilled histograms represent factors having an insignificant positive or negative effect on average speed.)

The TAG-1 Model for France 173 Consumers' sensitivity to the price of fuel per kilometre leads to a reduction in their speed (0.8%) as well as a decline in their kilometric demand. This finding confirms the growing awareness of drivers about the positive relationship between speed and their vehicles' excessive fuel consumption.

6.4.3. Analysis of the results by risk indicator.

The average speed, the number of motorised two-wheelers, total mileage and industrial activity have a significant positive impact on accidents (figure 10). This result, which corresponds to expectations, reveals the preponderant incidence of speed on personal injury and especially fatal accidents. Speed is without any doubt the most important risk factor, therefore. Conversely, few factors engender a reduction in accidents. Some temporary events (such as the Gulf war) and various road safety measures (random breathalyser tests and the rate of seat belt wearing) have had the effect of reducing the fatal and non-fatal injury accident toll. Fatal and non-fatal injury accidents A 10% increase in, or the advent o j thisphenomenon engenders, ceteris paribus, a change in

fatal and non-fatal injury accidents o$

Figure 10 Factors having a negative or positive incidence on injury and fatal accidents (The unfilled histograms represent factors having an insignificant positive or negative effect on average speed.)

174 Structural Road Accident Models The average speed, the number of motorised two-wheelers, total mileage and industrial activity have a significant positive impact on accidents (figure 10). This result, which corresponds to expectations, reveals the preponderant incidence of speed on personal injury and especially fatal accidents. Speed is without any doubt the most important risk factor, therefore. Conversely, few factors engender a reduction in accidents. Some temporary events (such as the Gulf war) and various road safety measures (random breathalyser tests and the rate of seat belt wearing) have had the effect of reducing the fatal and non-fatal injury accident toll. The factors studied are found to have similar effects on fatal and non-fatal injury accidents. The elasticities are nevertheless greater in the case of fatal accidents. The share of small cars is the only factor that simultaneously reduces personal injury accidents and increases fatal accidents. That seems to be attributable to the particular characteristics of these vehicles, which are essentially used in urban areas but whose inferior passive safety relative to that of other vehicles increases the risk of fatal accidents. Casualties. A 10% increase in, or the advent of, this phenomenon engenders, ceteris paribus, a change in casualties of.

Gulf war Random breathalyser tests Share of small cars belt wearing Real price of fuel per kilometre Share of m o t o r w a y traffic Unemployment Share of main r6ad traffic Wine consumpt Weelc-ends bank holidays

Fatalities Serious injuries Lightj injuries

Temperature Industrial activity Share of H G V s i Motorised two-yvheelers fleet Total mileage [Average speed -5%

Figure 11. Factors having a negative or positive incidence on casualties (The unfilled histograms represent factors having an insignificant positive or negative effect on average speed.)

The TAG-1 Model for France 175 The conclusions to be drawn from the model's estimates regarding victims match those relating to accidents (figure 11). Average speed, total mileage, the number of motorised two-wheelers and the share of HGVs increase the number of minor injuries, serious injuries and fatalities. The dominant and very high incidence of average speed also stands out. Only safety measures such as random breathalyser tests or the rate of seat belt wearing bring a significant reduction in risk as regards casualties. The direction and size of each factor's effects are consistent at the level of all equations and in line with our expectations.

6.4.4. Analysis of results by explanatory factor Incidence of road transport demand. Traffic risk depends in the first place on exposure to risk as measured by road transport demand. The construction of this variable required considerable effort and its reliability is essential for the estimations of the TAG model. All traffic risk indicators except gravity rates are closed linked to the development of total mileage (figure 12). This finding assures us, firstly, that our construction of the mileage series is reliable (Jaeger and Lassarre, 1997 and Jaeger, 1998), and, secondly, that it is consistent with analyses of the same type carried out at a world level. As a general rule, referring to the critical review of macro-economic traffic risk models carried out by Hakim et al. (1991), exposure to risk expressed in terms of kilometres travelled bears a positive correlation with the number of accidents and deaths. Hence, the DRAG-2 model concludes that a 10 per cent increase in mileage has an effect of 8 per cent and 7.5 per cent on personal injury and fatal accidents respectively (Gaudry, Fournier and Simard, 1995).

^1^^. <±^^~ Fatal accidents

Minor severity

Serious severity

Figure 12. : Impact of a 10% increase in mileage (Unfilled histograms represent insignificant effects)

176 Structural Road Accident Models It is significant that an increase in mileage increases the risk of fatal accidents more than that of non-fatal injury accidents (figure 13). The same phenomenon is apparent in terms of the number of minor and serious injuries and of deaths. This differential effect on injuries (nonfatal injury accidents) and deaths (fatal accidents) is hard to explain. In any event, it cannot be linked to the nature of journeys, which increase at the aggregate level of mileage, ceteris paribus, according to the type of network, since the distribution of traffic is included in the model. The explanation has to be sought in other variables not included in the model, such as the day/night share of traffic, since a substitution effect may be induced by the differences in the gravity of accidents according to the time of day or night. Risk-taking. With a view to gaining a better understanding and measuring the incidence of the user's behaviour on the accident toll indicators, we are lumping together in this section speed, safety belt wearing and wine consumption. These variables do not permit a global approach to driver behaviour. However, each variable expresses a relationship to a social norm as well as a form of risk-taking in vehicle driving.

;idents

Fatal accidents

Minor severity

Serious severity

Fatal severity

IVIinor injuries

Serious injuries

Fatalities

Figure 13. Direct impact of a 10% increase in average speed (Unfilled histograms represent insignificant effects) Speed, which describes the incidence on traffic risk of an individual's behaviour in terms of his control over his vehicle has very high direct incidence on risk indicators. Speed limits on the country road network have particularly significant effects (Lassarre, 1986) and are due more to a reduction in the dispersion of speeds than to a drop in average speed (figure 14). The results confirm that the elasticity of speed on the number of fatal accidents is close to double that on non-fatal injury accidents (Cohen et al., 1998).

The TAG-1 Model for France 177 The rate of seat belt wearing reflects the user's reaction to risk through self-protective behaviour. In effect, wearing a seat belt demonstrates the driver's determination to reduce his risk of being injured in the event of a collision. The results confirm the effectiveness of safety belts (figure 15). The fact of wearing a seat belt more often reduces traffic risk directly. However, the direct impact is low, averaging 1 to 2 per cent. An analysis of the results also shows that there is a contrary compensating phenomenon, since wearing a seat belt also leads to an increase in speed (figure 9) and hence in the accident toll by encouraging users to modify their behaviour and increase their speed.

Injury accidents

Fatal accidents

Minor severity

Serious severity

Fatal severity

Minor injuries

Serious injuries

Fatalities

Figure 14. Direct and indirect impacts of a 10% increase in the rate of seat belt wearing (Unfilled histograms represent insignificant effects)

injury Fatal accident accidents

Minor severity

Serious severity

Fatal severity

Minor injuries

Seriou injuries

Figure 15. Direct impact of a 10% increase in wine consumption (Unfilled histograms represent insignificant effects)

178 Structural Road Accident Models This is attributable to phenomenon known as driver behaviour retroaction. According to Evans (1991), visible technological changes lead to a certain degree of behavioural retroaction, unlike invisible changes (lateral protection bar). This contrary indirect effect remains marginal and is much weaker than the direct effect, since according to Lund and O'Neil (1986), safety measures whose effect is to reduce the gravity of the damage sustained (which is the case of safety belts) rather than the probability of an accident should result in less compensation on the part of users. The TAG model seeks to assess the incidence on the accident toll of user behaviour vis-a-vis alcohol (figure 16). Driving under the influence of alcohol is a frequent cause of accidents. According to a survey conducted by ONSER^ in 1997, 36.5 per cent of drivers or pedestrians involved in fatal traffic accidents had a blood-alcohol level exceeding the legal limit of 0.8g/l. According to Dally (1985), the probability of being involved in a fatal accident is multiplied by 1.9 for blood-alcohol levels of between 0.50 g/1 and 0.79 g/1 and by 10, 35 and 75 respectively for blood-alcohol levels of between 0.8 g/1 and 1.19 g/1, 1.2 g/1 and 1.99 g/1, and > 2 g/1. In the TAG model, a 10% increase in wine consumption per aduh increases fatal accidents by 1.6 per cent and the numbers of minor injuries, serious injuries and fatalities by 1.2 per cent. 1.4 per cent and 2.1 per cent respectively. Alcohol consumption also has an indirect effect through its impact on speed, but that is more associated with recreational journeys. The DRAG-2 model concludes that there is an increase in personal injury accidents and the number of injured but a decrease in fatal accidents and the number of people killed. But it integrates three categories of alcohol, beer, wine and spirits, and the breakdown of consumption reveals very different results according to the type of drink. In the TAG model, wine consumption provides only an imperfect reflection of total alcohol consumption, and although wine consumption in France forms part of people's habits, the consumption of spirits and especially beer is gaining ground. Effects of economic situation. The economic climate has an influence on the road traffic system. In this section we lump together a set of explanatory variables of the TAG model that are directly or indirectly linked to the economic situation. The structure of the vehicle stock The economic situation determines the structure of the vehicle stock to some extent. The number of heavy goods vehicles is directly linked to the level of economic activity. The increase in the number of small cars is a result partly of the increase in purchasing power, which leads to a rise in the number of households owning two vehicles, and partly of the search

^ Observatoire National de Securite Routiere (Alcool, conduite et insecurite routiere : Cahiers d'etudesn°65, 1985).

The TAG-1 Model for France 179 for savings by consumers wanting to acquire vehicles with low fuel consumption. The increase in the HGV stock increases traffic risk. Furthermore, the number of HGVs mainly affects accidents whose degree of gravity is limited. Hence, a 10 per cent increase in the HGV stock increases the numbers of minor injuries, serious injuries and fatalities by 3.9 per cent. 4.9 per cent and 6.3 per cent respectively. At a given mileage, the share of small cars in the total private motor car stock brings a marked improvement in the accident toll. The direct impact is reinforced by the indirect incidence of the share of small cars on total mileage, which declines when that proportion increases. There is a relatively positive internal and external gravity risk compensation effect, whereas according to Fontaine (1997), the driver and passengers of a small vehicle (< 800 kg) have a higher risk of being killed than the occupants of a heavy vehicle (> 1,000 kg) - 13 deaths per 1,000 vehicles involved as against 8 per 1,000. Evans (1984) attributes the negative incidence on the accident toll of the share of small cars to a retroaction effect. Since the structure of these vehicles increases the probability of fatal or non-fatal injury damage, in the event of an accident users alter their behaviour because they are aware of that increased risk. This retroactive aspect of the behaviour of drivers of small cars could come on top of the effect induced by the fact that small cars have less powerful engines. An increase in the number of motorised two-wheelers results in an increase in the accident toll. It engenders a marked deterioration in the level of road safety, of around 7 per cent as regards fatal and non-fatal injury accidents and the number of casualties. Nevertheless, the effect on gravity rates is considerably weaker, if not negative, which is attributable to the fact that this type of vehicle has fewer occupants. Configuration of the network The level of economic activity also affects the characteristics of the road network. The implementation of infrastructure programmes, road maintenance, the elimination of risk factors associated with the network and the construction of motorways are all linked to the economic situation. The differential risk as between networks is very great. It is of the order of four as between motorways, which had a fatality rate (per 100 million vehicle-kilometres travelled) of 0.78, and main roads, one of 2.85, in 1992. An increase in the proportion of motorway traffic leads to a decrease in personal injury accidents and in the number of serious injuries, but at the same it increases the numbers of fatal accidents and deaths. However, these results have to be treated with caution because of the average significance of the elasticities. An increase in traffic on main roads unquestionably leads to a substantial increase in the accident toll. The particularity of these findings lies in the uniform effect that main road traffic has on all traffic risk indicators. In addition, the compensation effect linked to the lower speed on main roads is

180 Structural Road Accident Models insufficient to reduce significantly the overall traffic risk of this type of network. Economic indicators The real price of fuel per kilometre and the unemployment rate are very representative of economic variables that have a direct impact on traffic risk. A change in the price structure improves the accident toll mainly at the level of fatal and non-fatal injury accidents and the number of deaths, but the low significance of the effect inevitably limits the scope of our analysis. The user seems to equate the cost of driving with fuel consumption. An increase in the real price of petrol leads him to drive in a more flexible manner, which results in a reduction in traffic risk. This phenomenon is amplified by the indirect effects of an increase in the price of petrol on mileage and speed, both of which are reduced. The overall impact of petrol prices thus reduces traffic risk. The overall effect on non-fatal and injury accidents and on the numbers of injured and killed are of the order of-2.8 per cent, -4.5 per cent, -2.0 per cent and -3.4 per cent respectively. The incidence of an increase in the real price of fuel on traffic risk is found to be slightly lower in France than in Quebec, since a large part of the effect is related to speed, which is not taken into account in the DRAG model. Land and McMillen (1980) also report that the price of petrol has a negative effect on the number of accidents. At a given mileage, user behaviour seems to be affected by a set of subjective factors linked to the labour market situation. The results indicate that a rise in unemployment increases traffic risk to a negligible extent, and this effect is strengthened by the indirect effect of unemployment on mileage. An unstable economic environment that creates uncertainty about future income may have a negative influence through psychological mechanisms on driver behaviour (lack of attention, and so on). In the case of the DRAG-2 model and various studies carried out by Wagenaar et al. (1984), Partyka (1984) and Haque (1991), the authors conclude that unemployment has a negative incidence on variations in accidents. These results conflict with our conclusions, leading us to formulate the following hypotheses: either the difference in the construction of the variable distorts the interpretation of the results, or social protection systems, because of their inequality, do not enable us to postulate that people behave homogeneously when faced with unemployment. An increase in journeys related to industrial activity has a positive effect on road safety indicators. Nevertheless, there is evidence of a negative incidence on all measures of gravity. It thus appears that vehicles linked to industrial activity are involved in accidents that are characterised by vehicles with low occupancy rates.

The TAG-1 Model for France 181 6.5. CONCLUSION The structure of the TAG model enabled us to acquire a comprehensive perception of the production of risk within the road transport system. It gives us an estimate of the incidence of the internal and external factors of the road system on the measurement of exposure to risk, behavioural and technological risk, and the different traffic risk indicators. TAG is a long-term explanatory model. It complements the models that exclude seasonal factors, such as GIBOULEE (Le Breton et ai, 1996), and short and medium-term forecasting models, such as RES (Bergel, 1999). The interest of the model's structure of simultaneous, recursive equations lies in its capacity to evaluate the direct and indirect impacts of the main factors on the road accident toll through total mileage and average speed, and hence the substitution effects between the different categories of accident and the different levels of gravity. The first difficulty with this kind of aggregate risk model is to cover the complete field of explanatory risk factors by means of all the explanatory chronological series available. The most one is hoping for is to approach the measurements made within the road transport system, such as the average speed driven, even if limited to the daytime period and to the interurban network, or the wearing of safety belts, even if no observations are made at night. If no measure is available, one uses variables that are assumed to be correlated and accessible, such as wine consumption (that of beer is not), together with the percentage of drivers under the influence of alcohol. The second difficulty, which limits the scope of the model, is that one cannot introduce risk factors which change very slowly, such as the proportions of young and old people in the population. A breakdown of driving licence owners by age and sex, for which there are no monthly figures, would make more sense. The third difficulty arises from technical and human advances in all the components of the transport system, whether it be the infrastructure, the vehicle, the driver, or advances in accident and emergency healthcare, which we find it very hard to appreciate because they spread slowly through the whole system and do not take the form of the wholesale introduction of an innovation. An examination of the results relating to total mileage shows that work-related journeys and the stock of private cars and commercial vehicles are the main factors that have a positive influence on road transport demand. On the other hand, if national and European transport policy-makers wish to limit road traffic in the pursuit of environmental goals or energy savings, encouraging the purchase of small cars or increasing fuel prices would undoubtedly be effective. Evaluating the impact of various factors on the behavioural and technological risk represented by average speed showed us that measures related to speed limits have proved effective. In addition, other enforcement measures, especially as regards drinking and driving, have

182 Structural Road Accident Models persuaded drivers to abide by speed limits. Total mileage travelled, average speed and the proportion of small cars all appeared as major factors in traffic risk. An increase in the first two exacerbates the risk, while an increase in the third diminishes the risk. We observed an effect of speed adjustment under the influence of the wearing of safety belts. A second group of factors composed of variables relating to the composition of the vehicle stock (private cars and motorised two-wheelers), the price of fuel, temperature and taxed wine consumption have a strong average influence on risk. Other factors have lesser but not negligible effects. Road safety measures such as speed limitation or random breathalyser tests have proved to be effective. Since the model runs only to 1993, it is not possible to evaluate recent measures, such as the introduction of mandatory technical inspections and the points system for driving licences. Substitution effects between degrees of gravity (minor, serious and fatal injuries) were found, for example as regards speed. By making obvious the mechanisms of the effects of the risk factors by TAG model, we are able to portray the complexity of the accident process in the road transport system. The two subsequent developments planned are the validation of the model by a projection for the 19941998 period and the updating that will integrate mandatory vehicle testing and the points system for driving licences in order to produce a comprehensive assessment of the change in risk affecting 1999.

6.6 REFERENCES Bergel, R.(1998). Modelisation multivariee du risque d'accident sur le reseau routier national. Actes de la Conference PTRC. Cohen, S., H. Duval, S. Lassarre, J. P. Orfeuil (1998). Limitations de vitesse. Les decisions publiques et leurs effets. Hermes, Paris. Dally, S. (1985). Conduite automobile et alcool. La documentation frangaise, Observatoire National Interministeriel de Securite Routiere. Evans, L. (1984). Involvement rate in two-car crashes vs. driver age and car mass of each involved car. Report GMR-4645, General Motors Research Laboratories. Warren, Michigan. Evans, L. (1991). Traffic safety and the driver. Van Nostrand Reinhold, New York. Fontaine, H. (1997). Gravite differentielle des vehicules-conducteurs. In; L'agressivite des vehicules dans les accidents, Actes INRETS 56 , Arcueil. Fridstrom, L. and S. Ingebrigtsen (1991). An Aggregate Accident Model Based on Pooled, Regional Time-series data. Accident Analysis and Prevention, 23, 5, 363-378. Gallez, C. and J.L. Madre (1993). Demeler les facteurs structures et economiques : la dynamique de I'usage de I'automobile. Dixiemes journees de micro-economie appliquee.

The TAG-1 Model for France 183 Sfax. Gaudry, M. (1984). DRAG, un modele de la demande routiere, des accidents et de leur gravite, applique au Quebec de 1956 a 1982. Centre de Recherche sur les Transports, Universite de Montreal. Gaudry, M., F. Fournier and R. Simard (1994a). DRAG-2, un modele econometrique applique au kilometrage, aux accidents et a leur gravite au Quebec - Partie 2 - Cadre methodologique. Direction etudes et analyses, SAAQ, Quebec. Gaudry, M., F. Fournier and R. Simard (1995). DRAG-2, un modele econometrique applique au kilometrage, aux accidents et a leur gravite au Quebec - Partie 3 - Application du modele aux accidents, a leur gravite et aux victimes de la route. Direction etudes et analyses, SAAQ, Quebec. Gaudry, M., and R. Simard (1994b). DRAG-2, un modele econometrique applique au kilometrage, aux accidents et a leur gravite au Quebec - Partie 3 - Application du modele au kilometrage a I 'essence et au diesel. Direction etudes et analyses, SAAQ, Quebec. Hakim, S., D. Shefer, A.S. Hakkert and L Hocherman (1991). A critical review of macro models for road safety. Accident Analysis and Prevention, 23, 5, 379-400. Haque, D. M., Ohidul (1991). Unemployment and Road Fatalities. Vic Roads, GR 91-10, Danemark. Hivert, L. (1993). Achats de voitures, dieselisation et kilometrages des menages : Essai de quantification dpartir de donnees de panel. Rapport de convention INRETS-ADEME. Jaeger, L. (1998). L'evaluation du risque dans le systeme des transports routiers par le developpement du modele TAG. These de doctorat es Sciences Economiques, Universite Louis Pasteur, Strasbourg. Jaeger, L. and S. Lassarre (1998). Pour une modelisation de revolution de I'insecurite routiere : Estimation du kilometrage mensuel en France de 1957 a 1993 : Methodologie et resultats. Rapport DERA n° 9709, Institut National de Recherche sur les Transports et leur Securite, Arcueil. Jaeger, L. and S. Lassarre (1998). Pour une modelisation de revolution de I'insecurite routiere : Estimation du modele TAG : Methodologie et resultats. Rapport DERA n° 9808, Institut National de Recherche sur les Transports et leur Securite, Arcueil. Land, K.C. and M.M. McMillen (1980). A macro dynamic analysis of changes in mortality indices in the United States, 1946-1975 : some preliminary results. Social Indicators Research 7, 1-46. Lassarre, S. (1986).The introduction of the variables traffic volume, speed and belt-wearing into a predictive model of the severity of accident. Accident Analysis and Prevention, 18, 2, 129-134. Lassarre, S. (1991). Comparaison et evaluation des performances des systemes de recueil de vitesses sur le reseau routier. Rapport INRETS n°136, INRETS, Arcueil, France. Le Breton, P., F. Vervialle, M. Truffier (1996). Utilisation des series desaisonnalisees pour

184 Structural Road Accident Models ranalyse de Vinsecurite routiere. Note d'information n°104, SETRA/CSTR, Bagneux. Loeb, P.D. (1987). The determinants of automobile fatalities. Journal of Transport Economics and Policy, 21, 279-287. Lund, A. and B. O'Neil (1986). Perceived risks and driving behaviour. Accident Analysis and Prevention, 18, 5. Madre, J.L. and T. Lambert (1989). Previsions a long terme du trafic automobile. Rapport d'etude, CREDOC, Paris. OCDE (1997). Road safety principles and models : review of descriptive, predictive risk and accident consequences models. OCDE, Paris. Orfeuil, J.P. (1993). France: a centralized country in between regional and european development. A billion Trips a Day, tradition and transition in European travel patterns. Editions Kluwer. Orfeuil, J.P. and D. Zumkeller (1991). Concevoir et tarifer les transports pour un developpement viable ; reflexions a partir des comparaisons franco-allemandes. Recherche Transports - Securite, 32, 165-171, INRETS, Arcueil. Partika, C. (1984). Simple Models of Fatality Trends Using Employment and Population Data. Accident Analysis and Prevention, 16, 3, 211-222. Sommers, P.M. (1985). Drinking age and the 55 mph Speed limit. Atlantic Economic Journal 13, 43-48. Wagenaar, A.C. and R.G. Maybee (1986). The legal minimum drinking age in Texas :effects of increase from 18 to 19. Journal of safety research, 17, 165-178. Zlatoper, T.J. (1984). Regression analysis of time series data on motor vehicle deaths in the United States. Journal of Transport Economics and Policy 18, 263-274.

The TRACS-CA Model for California 185

THE

TRACS-CA

MODEL FOR CALIFORNIA

Patrick McCarthy

7.1. INTRODUCTION

Starting in the 1980s and continuing on into the 1990s, individual states in the United States (US) passed various pieces of legislation aimed at furthering improvements in traffic safety. Two widely debated policies were motor vehicle occupant restraints, commonly referred to as mandatory use laws (MUL), and speed limits on rural interstate highways which states were permitted to relax in the 1987 Federal Highway Bill. Although several studies (Crandall et al, 1986; Garber and Graham, 1990; Godwin, 1992; Graham and Garber, 1984; Lave, 1985; Lave, 1992; McCarthy, 1993, 1994) have evaluated, either separately or jointly, the highway safety effects of these laws, different conclusions have been reached, particularly in regards to the higher speed limits. Not surprisingly, the research to date characterizes a variety of research methods, different time periods and geographical areas, and alternative definitions of highway safety. In addition to speed limit and mandatory seat belt use, policies related to traffic enforcement, alcohol availability, and alcohol consumption have gained widespread attention. Although traffic enforcement is widely thought to enhance highway safety, there is relatively little information Kenkel, 1993; McCarthy, 1999a; McCarthy and Oesterle, 1987), often due to the unavailability of traffic arrest data, on the extent to which enforcement is beneficial. And several recent studies have analyzed alcohol availability, generally finding a positive relationship between availability and highway crashes (Brown et ah, 1996; Chaloupka et aL, 1991; Ruhm, 1995). Various studies have studied the effect of alcohol consumption, measured by total or per capita gallonage consumed, typically finding that increased consumption has a deteriorating effect on highway safety American Medical Association, 1986; Borkenstein et al., 1964; Laixuthai and Chaloupka, 1993).

186 Structural Road Accident Models This paper constitutes an exploratory analysis at developing and estimating a structural aggregate model of highway safety, termed TRACS-CA, based upon historical time series data from California, that is consistent with Gaudry's (Gaudry, 1993; Gaudry et ai, 1993a, b) multiequation approach, termed DRAG, initially developed in the mid-1980s for Quebec, Canada. TRACS-CA generalizes a previous version of the model by refming empirical specifications in traffic exposure and crash frequency models and by including additional models for crash mortality and morbidity. Monthly observations for California during a nine-year period, January 1981 through December 1989, constitute the sample for this study. The 'age' of the data is an indication of the exploratory nature of the analysis and the focus upon the methodological approach. However, as is seen in the following sections, the analysis also sheds light upon important aspects of highway safety and provides a strong justification for devoting additional resources to update the existing data series and develop richer models. In addition to the study's longer term goal of furthering our understanding of the complex relationships that exist between highway risk, crash frequency, and crash severity, a more immediate goal of the study is to provide policy makers with additional information on role that traffic enforcement, alcohol-related, and highway safety policies have upon highway safety.

7.2. TRACS-CA MODEL STRUCTURE This section summarizes the empirical structure of TRACS-CA. For each component in the structure, the model's dependent variable is defined and the variable's behavior over the sample period is identified and briefly discussed. In addition, this section identifies and comments on the set of explanatory variables included in each sub-model.

7.2.1. Exposure and crash losses Diagram 7.1 depicts the model structure for TRACS-CA that is used in the present paper. The right hand side of the diagram identifies three components of the structure: Risk Exposure, Crash Frequency, and Crash Severity. The Risk Exposure component comprises one model whereas the Crash Frequency Component comprises three sub-models, corresponding respectively to Fatal, Non-Fatal Injury, and Materials Only crashes. The Crash Severity component has a separate sub-model for Crash Mortality and Crash Morbidity. The right hand side of the diagram also identifies for each sub-model the measure used to define the dependent variable in the empirical specification. For this analysis. Total Vehicle Miles Traveled (VMT) on State Highways is an index of risk exposure associated with travel on all roads in California. Although annual VMT for California is available, statewide monthly VMT on all roads is not available. However, since total VMT on state highways is highly

The TRACS-CA Model for California 187 correlated with statewide total VMT, vehicle miles traveled on state highways provides a good instrument for traffic risk exposure in California.

Diagram 7.1. Model structure for TRACS-CA For this study, Crash Frequency is disaggregated by severity and this leads to three submodels: Fatal Crash Frequency, Non-Fatal Injury Crash Frequency, and Material Damage Only Crash Frequency. Crash Severity refers to whether any individual in the crash experienced an injury and is disaggregated according to whether the injury was fatal or non-fatal. In particular, the Crash Severity module comprises two sub-models: Mortality Rate, defined as the number of fatalities per fatal crash; and Morbidity Rate, defined as the number of non-fatal injuries per non-fatal injury crash. In sum, the relationships among the sub-models in the three structural modules are: == f(xi) Exposure Crash Frequencyj ^ g(Exposure, X2j) j = fatal, non-fatal injury, materials only Crash Severityj = h(Exposure, X3j) j = fatal, non-fatal injury where Xjj (i = 1,2, 3) corresponds to a vector of (not mutually exclusive) explanatory variables associated with module i and crash type j Q = fatal, non-fatal injury, materials only). Notice also that two additional outcomes, the number of fatalities and the number of injuries, can be

188 Structural Road Accident Models derived from information on Crash Frequency and Crash Severity. In particular, Number of Fatalities = Fatal Crash Frequency x Mortality Number of Injuries = Non-Fatal Injury Crash Frequency x Morbidity The left hand side of Diagram 1 provides sets of explanatory variables that are hypothesized to influence or determine the structure's dependent variables. In general, three sets of factors are important to highway safety: socioeconomic factors, transport system factors, and environmental factors. Diagram 7.1 also identifies the specific variables used in this analysis. As will be seen in Section 7.2.3, not all variables are included in all sub-models and the definition and specification of a particular variable may differ from one sub-model to another. 7.2.2. Historical trends Graph 7.1 through Graph 7.6 graphically depict the model structure's dependent variables and how these highway safety measures varied over the period of study, January 1981 through December 1989. Graph 7.2 depicts total VMT on state highways in California and indicates an upward trend with seasonal troughs in January/February and peaks during mid-summer. Over the nine-year period, VMT averaged 9.13 billion with a standard deviation of 1.36 billion. Throughout the 108-month period, VMT on state highways increased 39.8 million miles per month. ^ Graphs 7.2, 7.3, and 7.4 provide information on the frequency of fatal, non-fatal injury, and materials only crashes in California during the sample period.^ Similar to the peaks and troughs associated with VMT, we see in Graph 7.2 that the fewest fatal crashes per year tend to occur during the January/February period and the most crashes during the summer months. On average, there were 377 fatal crashes per month with a 48.3 standard deviation. In addition, the series exhibited a slight positive trend over the sample period, increasing fatal crashes by .62 per month. Graph 7.3 depicts the incidence of non-fatal injury crashes during the nine-year period and here we see a stronger positive trend, although relative to the total number of crashes, the trend is still small. In comparison with an average of 17,861 non-fatal injury crashes per month (1,954 ' The reported trends were estimated from a simple time series regression that included month as the only explanatory variable. With the exception of mortality, all trend coefficients were statistically significant (.01 level). ^ The California data are police reported crashes. Since fatal crashes and crashes involving an injury must be reported, these data will be more accurate than materials only crashes which are generally underreported.

The TRACS-CA Model for California 189 standard deviation), the overall trend rose by 52.1 crashes per month during the period. Also, in contrast to total VMT on state highways and fatal crashes, we do not observe sharp peaking in Figure 7.3. The incidence of non-fatal injury crashes seasonally fall during the winter months but thereafter remain relatively constant. Graph 7.1 Vehicle Miles Traveled on State Highways 14000 12000 10000-1 8000 6000 4000 2000

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G r a p h 7.2 Fatal C rashes

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Graph 7.3 Injury Crashes

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190 Structural Road Accident Models The graph for materials only crashes, displayed in Graph 7.4, is a bit more erratic than the prior figures but does exhibit peaking tendencies during the winter months. For the sample period, there were 25,491 materials only crashes per month, with a 2,760 standard deviation and positive trend of 68.1 crashes per month. The last two graphs, Graph 7.5 and Graph 7.6, depict the historical trend of mortality and morbidity rates during the nine-year period. For the 108-month period, there were 1.11 fatalities per fatal crash (.02 standard deviation) but the series exhibited neither an upward nor downward trend. In contrast, there were 1.47 injuries per non-fatal injury crash (.02 standard deviation) and similar to all but the mortality series, there was a slight upward trend, amounting to a monthly increase of 6.07e-04. Graph 7.4 Materials Only Crashes 35000 30000 25000 20000 15000-t 10000 5000 Jan-86

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Graph 7.5 M ortality Rate 1.22 -, 1.2 1.18 1.16 1.14 A

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Although the graphs for mortality and morbidity appear more variable than those for crash frequency, it is important to note that this depends upon the graph's scale. Normalizing the standard deviation by the mean gives the coefficient of variation and provides a more accurate picture of a variable's variation for the sample period. For VMT, Fatal Crashes, Non-Fatal Injury Crashes, and Materials Only Crashes, the coefficient of variation is, respectively, .149, .128, .109, and .112, signifying that each of these series is generally similar in its variation to

The TRACS-CA Model for California 191 the other series. On the other hand, the coefficient of variation for Mortality and Morbidity is an order of magnitude lower, at .019 and .016 respectively. Relative to the mean, there is much less variation in fatality and injury rates than in crash frequencies. 7.2.3. Determining variables included in the TRACS-CA structure Table 7.1 summarizes the explanatory variables included in each of TRACS-CA sub-models. The columns in the table represent one of the six dependent variables in the model structure and the rows identify the set of explanatory variables. A V indicates that a variable has a direct effect on the dependent variable and is explicitly included in the model. From Column (a) in the table, we see that the Demand for Road Space depends upon three economic factors, Real Gasoline Price, an Index of Per Capita Expenditures, and Unemployed per 1000 Drivers. In preliminary model estimations. Real Per Capita Income was included as an explanatory variable but this led to significant multicollinearity problems. Table 7.1. TRACS-CA Explanatory variables according to the dependent variable

INDEPENDENT VARIABLE

Demand for

Fatal

Injury

Morbidity

Crashes

Crashes

Materials Crashes

Mortality

Road Space (a)

(b)

(c)

(d)

(e)

(0

V V V

V

V

V V V

Socio-Economic Factors Index of Per Capita Expenditures Real Gasoline Price (1982-84 $) Unemployed per 1000 Drivers Unemployed per VMT

V V V

V

V V

V

V V V

V

V V V V V

V

V

Alcohol Licenses per Capita Alcohol Consumption per Capita Monthly Population Weekends

V

^l V

V

V

V

V

V V

V V

V V V V V

V V

V V V V V

V

V

V

V

Transport System Factors Exposure Traffic Arrests per Capita Motorcycle Density Seat Belt Law Relaxed Speed Limit Law

A/

Environmental Factors Average Rainfall

The Index of Per Capita Expenditures is strongly correlated with the income measure but led to lower collinearity problems thus providing a good instrument for income in the empirical model. All else constant, it is expected that the demand for travel and the real price of fuel be inversely related. Further, assuming that travel is a normal good and that downturns in the

192 Structural Road Accident Models economy lower household and firm demands for travel, it is expected that the expenditure index will have a positive sign whereas the unemployment variable will have a negative sign. In addition to these economic factors, Monthly Population is included in the Demand for Road Space model. However, as seen in the following section, this variable only enters the heteroskedasticity portion of the model. To accommodate differences in driver preferences for travel and risk taking, two variables are included in the model. Alcohol Consumption per Capita and Weekend Travel. For the exposure equation. Alcohol Consumption per Capita is defined as Statewide Beer Consumption per Capita and reflects potential risk taking behavior on the part of those who drink and drive. Weekend Travel is defined as the number of weekend (including Friday, Saturday, and Sunday) days per month and reflects a myriad of alternative travel behaviors as well as alternative preferences, including that for no travel, that occur on the weekend relative to more work oriented travel during a typical weekday. All else constant, it is expected that the demand for travel fall during weekend periods. With respect to the transport system and regulations, the exposure sub-model is sparse, only accounting for the effect of increased speed limits on rural interstate highways. A priori, the effect of higher speed limits on the demand for travel is uncertain. Similarly, there is only one environmental variable in the model, Average Rainfall, defined as the number of inches of rain per month. It is expected that heavier rainfalls will have a non-increasing effect upon highway travel. Columns (b) through (d) identify the explanatory variables in the crash frequency equations. Here, with the exception of Alcohol Licenses per Capita, which is absent from the Fatal Crash equation, and the Alcohol Consumption per Capita variable, which is defined differently across the equations, the crash frequency models include the same set of explanatory variables. For the frequency models, it is not known whether an increase in fuel prices will produce more or fewer crashes, once we control for exposure. Rising fuel prices affect various aspects of one's travel behavior. In the short run, for example, rising fuel prices have a greater effect on the relative money price of trips on higher speed relative to lower speed roads and on congested relative to uncongested networks. Increasing fuel prices also induce modal shifting, for example from single occupancy automobiles to carpooling or public transit. In the long run, rising fuel prices lead to shifts in the size and fuel efficiency of vehicle fleets. Depending upon the magnitude of these effects as well as upon their safety implications (e.g. high speed roads are relatively safer, congested traffic produces more property damage but fewer injury crashes), rising fuel prices could improve or deteriorate highway safety. A priori, the sign on this variable is uncertain.

The TRACS-CA Model for California 193 Increases in both the number of Alcohol Licenses per Capita, reflecting the availability of alcohol, and the apparent consumption of alcohol, which reflects risk taking behavior, are expected to increase the number of fatal, non-fatal injury, and materials only crashes.^ It should also be noted that the definition of alcohol consumption generally differs across models. In the Fatal Crash model, alcohol consumption is defined as Statewide Consumption of Distilled Spirits per Capita whereas for the Non-Fatal Injury and Materials Only models, consumption is based upon wine gallonage per capita."^ The number of weekend days is a surrogate for changes in the distribution of trip purposes and trip timing that often occurs on weekends, fewer work related trips and more social trips as well as more evening trips for entertainment and social purposes. A priori, there is no reason why one might expect crash frequencies on weekends to be higher or lower, all else constant. The crash frequency models also include a number of transport related variables that are expected to affect highway safety. Most important among these is exposure to risk. All else constant, greater risk exposure is expected to increase the frequency of fatal, non-fatal injury, and property damage crashes. In addition, the enforcement of speeding, drinking and driving, and other traffic laws is expected to be safety enhancing. Potentially complementing traffic enforcement was California's enactment in 1986 of a mandatory seat belt use law. To the extent that they increase the number of drivers using seat belts, MULs are expected to reduce the incidence of fatal crashes. However, the use of seat belts may induce some offsetting behavior by providing individuals with a safety-related incentive for taking more risks. Moreover, to the extent that these laws reduce fatal crashes there may well be an increase in non-fatal injury or materials only crashes. Also, in May 1987 California reduced speed limits on rural interstate highways and although higher speed limits are expected to increase crash severity in the event of a crash, there is less agreement on whether the higher speed limit led to an increase in the frequency of crashes. A final transport related variable in the crash frequency models is Motorcycle Density, defined as the proportion of registered vehicles that are motorcycles. It is expected that an increase in this variable will significantly increase the incidence of crashes. As in the travel demand equation, Average Rain is the only environmental variable in the crash Apparent alcohol consumption is expected to produce a stronger deterioration of highway safety than alcohol licenses. In addition to reducing the time cost associated with purchasing alcohol, increasing the number of licenses also reduces traffic exposure since one has to travel a shorter distance to purchase alcohol if there are many alcohol outlets (McCarthy, 1999b). ^ The alternative definitions of the alcohol consumption in the exposure, crash frequency, and severity models depended upon which variable provided the best overall fit.

194 Structural Road Accident Models frequency equations. Since individual drivers compensate in their driving for poorer weather conditions, it is expected that an increase in average rainfall will reduce the frequency of fatal crashes but increase the frequencies of non-fatal injury and materials only crashes. The last three columns in Table 7.1 identify the variables in the severity sub-models. The Morbidity model includes the same set of explanatory variables as the Fatal Crash equation, including the definition of the alcohol consumption variable. The Mortality equation, on the other hand, differs from the Morbidity equation and the frequency equations in excluding the Real Price of Gasoline and Motorcycle Density but including Unemployed per Vehicle Miles Traveled. Among the six equations, the Mortality equation was most sensitive to collinearity problems, particularly when fuel price and motorcycle density were included. Moreover, there was a significant improvement in the model's fit when Unemployed per Vehicle Miles Traveled replaced Unemployed per 1000 Drivers as an explanatory variable.

7.3. ESTIMATION RESULTS

7.3.1. Statistical summary Table 7.2 is a statistical summary of the estimation results and provides for each model information on the degree of statistical significance for the included variables, the heteroskedasticity and autocorrelation structure, Box-Cox parameters, and the log-likelihood function. With the exception of the Mortality equation, all models fit the data well. For the optimum model, the pseudo-R^ ranged from a high of .996 for the exposure equation to a low of .326 for the Mortality equation. However, in the Mortality equation six of the eight variables were highly significant, suggesting that, despite considerable randomness, the model is identifying significant determinants of mortality rates. With respect to model forms, we see from the table that multiple serial correlation characterized all models. For the exposure equation, there were adjustments for 11 autocorrelation coefficients, ranging from the 1^^ to the 13* order. For crash frequencies, the number of adjustments for serial correlation increased from Fatal to Materials Only crashes but other than a common adjustment for 13* order serial correlation, there was little similarity in the pattern of adjustment. All models were tested for alternative heteroskedasticity structures but only for the exposure equation. Column (a), was there any identifiable structure. Here, the variance was significantly related to Monthly Population. Box-Cox specifications for the dependent variable and the independent variables, in common and separately, were tested in order to explore the non-linearity of a model's empirical specification. With the exception of Injury Crashes (Column (c) in Table 7.2), whose optimum specification was log-linear, each of the models accepted the alternative hypothesis that some

The TRACS-CA Model for California 195 parameter was Box-Cox transformed with a Box-Cox parameter value other than 1 or 0. For the Demand for Road Space, Materials Only Crashes, and Mortality equations, there occurred Box-Cox transformations on both sides of the equation. In addition, the Fatal Crash and Materials Only Crash equations included quadratic specifications for the alcohol related variables. More general specifications accepted the null hypothesis that these quadratic forms were symmetric. Table 7.2. Statistical summary of models (a) Demand for Road Space

(b) Fatal Crashes

(c) Injury Crashes

(d) Materials Crashes

(e) Mortality

(f) Morbidity

Explanatory Variables: (2 < t)

2

4

5

6

6

7

Number of t-statistics (1 < t < 2)

3

2

2

2

0

0

Number of t-statistics (0 < t < 1)

2

4

1

3

2

2

Heteroskedasticity (no. of parameters)

1

0

0

0

0

0

Autocorrelation

11

4

6

8

7

6

Number of t-statistics

(no. of rhos)

Stochastic Form of Model 1.58

-

Fixed at 0.0

-0.775

-7.55

-

-4.03

-1.41

2.73

3.17

HX2)

-

X(Z)

Fixed at 1.0

-

-

0.581

-

-

-

^(y) X{X,)

Log-likelihood at optimum form

-462.5

-365.3

-625.0

-660.5

241.0

290.1

Log-likelihood at >. = 1 (linear)*

-475.5

-368.7

-626.6

-671.0

237.1

288.4

Log-likelihood at ?u = 0 (logarithmic)*

-476.4

-367.0

-628.7

-674.4

232.1

277.0

Pseudo-R^ (adjusted for d.f.)

0.996

0.994

0.931

0.924

0.326

0.802

83

83

83

83

88

84

Sample Size

* Includes quadratic terms. We also see in Table 7.2 log-likelihood values for the optimum model in comparison with two common restrictive specifications, the linear model and the double-log model. Not surprisingly, there is little difference between these values for the Injury Crash equation whose optimum form is a linear in parameters specification. However, we see larger differences in the loglikelihood functions for those models whose optimum forms are non-linear in parameters, particularly when both sides of the equation have Box-Cox transformations. 7.3.2. Common variable results Figure 7.1 provides elasticity measures for vehicle miles traveled, real price of gasoline, and alcohol related determinants, variables which are commonly found in other DRAG-type structures. From the graph, we can make several observations. First, risk exposure is an

196 Structural Road Accident Models important determinant of highway safety in each sub-model. Notwithstanding the lack of information on statewide total vehicle miles traveled, the size and strength of elasticity measures in Figure 7.1 imply that total vmt on state highways is a good proxy. For each submodel, risk exposure has a t-statistic greater than 2. Further, we see in the graph that fatal crashes and materials only crashes have relatively high vmt elasticities, respectively equal to .89% and .75%. With an elasticity equal to .53, non-fatal injury crashes are less sensitive to risk exposure, all else constant. Not only does risk exposure increase the frequency of crashes, we also see that there is an increase in mortality and morbidity. A P/o increase in state highway vmt increases the fatality rate (number of fatalities per fatal crash) and non-fatal injury rate (number of non-fatal injuries per non-fatal injury crash) by .08% and .10% respectively. Somewhat surprisingly, increases in Real Gasoline Price had no impact upon the demand for travel. However, this may be less surprising when one recognizes that the opportunity cost of highway travel is a generalized cost that includes the monetary and time cost of travel. Although changes in real fuel prices capture the monetary component of highway travel, it does not account for the time cost of travel. All else constant, an increase in the real price of gasoline increases the relative price of trips whose monetary price is a larger component of generalized cost. Thus, without explicitly controlling for the travel time, firms and households may increase the demand for relatively time intensive trips so that, on balance, there is little impact upon total vmt. From Figure 7.1, the primary effect of real gasoline prices is upon non-fatal crashes and injury morbidity.^ A Wo increase in Real Gasoline Price reduced the incidence of Non-Fatal Injury and Materials Only crashes by .30% and .27% respectively and decreased Morbidity by .04%. For each of the sub-models, some measure of alcohol consumption or alcohol availability significantly affected the highway safety measure. However, from these results it is not possible to say that a particular category of (apparent) alcohol consumption affected all measures of highway safety. In the risk exposure equation, a 1% increase in per capita beer consumption increased the demand for travel, suggesting some increased risk taking activities on the part of beer drinkers. Increased risk exposure effects were not identified for increased wine and distilled spirits consumption. However, increased consumption of distilled spirits did have a significant impact upon the fatality related equations. A 1% increase in the (apparent) consumption of distilled spirits per capita increased the incidence of fatal crashes by .05% and mortality by .03%. In other words, higher consumption of distilled spirits produced more fatal crashes and more fatalities per fatal ^ Real gasoline price had little effect upon fatal crashes and was excluded from the Mortality equation due to collinearity problems.

The TRACS-CA Model for California 197 crash. Although the best fit of the morbidity equation included the consumption of distilled spirits per capita, this variable was not statistically significant. Elasticities - Common Variables Statewide Alcohol Licenses Per Capita (10% of actual value)

Statewide Consumption of Wine Per Capita

^^^0118**

2 < I t-Stat I 1 <|t-stat|<2

g 0.047*^ f0.036*

0.003

Statewide Consumption of Distilled Spirits Per Capita

E] 0035*^

a Morbidity m Mortality Statewide Consumption of Beer Per Capita

S Materials Crashes H Injury Crashes D Fatal Crashes • Exposure

Real Gasoline Price

Vehicle Miles Traveled on State Highways

Figure 7.1. Elasticities for the common variables Interestingly, the consumption of total wine per capita rather than beer or distilled spirits had the greatest impact upon injury and materials only crashes. From Figure 7.8, we see that a 1% increase in (apparent) wine consumption per capita increases the frequency of non-fatal injury and materials crashes by .03% and .04% respectively. Also important to the frequency of nonfatal crashes is the availability of alcohol. The frequency of non-fatal injury and materials only crashes is elastic with respect to the number of alcohol licenses per month. A 1% increase in monthly licenses generates a 2.1%) and 1.1% increase in non-fatal injury and materials only crashes, respectively.

198 Structural Road Accident Models 7.3.3. Specific variable results Figure 7.2 reports the calculated elasticities for four explanatory variables that are specific to the TRANS-CA model structure, an environmental variable and three highway safety variables. Starting with Average Rainfall, we see in the figure that, with the exception of the risk exposure sub-model, the amount of rain significantly affected, albeit differently, each of the other equations. A common result from wealth maximizing models of uncertainty is risk compensation. If an individual finds herself in a more (less) risky environment, she will compensate by behaving more (less) safely. The results for Average Rainfall are consistent with risk compensating behavior. Average Rainfall significantly affects the frequency and severity of crashes but the direction of the effect is negative for fatal crashes and mortality and positive for non-fatal crashes and morbidity. That is, if drivers in more inclement weather compensate for the poorer weather, for example, by traveling at lower speeds, increasing headways, and switching lanes less frequently, then we would expect fewer fatal crashes and more non-fatal crashes. The results in Figure 7.9 are consistent with this. A 1% increase in Average Rainfall reduces fatal crashes by .001% and increases non-fatal injury and materials only crashes by .008% and .037%), respectively. With respect to traffic enforcement. Figure 7.2 illustrates that there are significant benefits of increased enforcement for fatal crashes. A 1%) increase in the number of traffic arrests reduces the frequency of fatal crashes by .19%). Enforcement also reduced non-fatal injury crashes but the effect was not significant. However, Figure 7.9 also reveals that increased enforcement led to an unexpected rise in the frequency of property damage only crashes. Whether this is due to endogeneity bias, model misspecification, or risk substitution effects that shift crashes away from more serious and towards less serious crashes is not known at this point. The most puzzling results in Figure 7.2 relate to the mandatory seat belt use and relaxed rural interstate speed limit laws. Increased speed limits are seen to slightly reduce risk exposure by an estimated .007%). Higher speeds had no effect on fatal crashes but a strong positive impact on the frequency of non-fatal injury crashes, raising these by .04%). Interestingly, and contrary to expectations, passage of the law significantly reduced mortality and morbidity by .009%) and .002%o respectively. All else constant, there were fewer fatalities per fatal crash effects after the law as well as fewer injuries per non-fatal injury crash. Although the highway safety effects of higher speed limits is controversial, less contentious is the notion that higher speeds result in more serious injuries given that a crash occurs. The mortality and morbidity results call for further research in order to confirm or negate these findings. A similar question arises regarding the seat belt results. In Figure 7.2, we see that, with two exceptions, Seat Belt Use decreased highway safety. First, there was no effect on the frequency of fatal crashes. Although Seat Belt Use reduced such crashes by .003%), the effect was not statistically significant. Second, Seat Belt Use significantly increased the incidence of non-fatal

The TRACS-CA Model for California 199 Injury Crashes and Materials Only Crashes. These results are mildly consistent with the notion of risk substitution, whereby seat belt use reduces the most serious crashes and increases the less serious crashes. However, as noted above, the belt use law did not produce any beneficial effect with respect to fatal crashes. Third, the findings indicate that Seat Belt Use increased Mortality but decreased Morbidity. Thus, although there is no effect on the incidence of fatal crashes, when a fatal crash occurs, more persons are killed. In addition, the law significantly increased non-fatal injury crashes but, when an injury crash occurs, fewer individuals suffer an injury. The findings for the morbidity rate are consistent with expectations. But the results for the mortality rate are not consistent, unless drivers are engaging in significant risk compensation which seems unlikely.

Elasticities - Specific Variables -0.009**! -0.020** r j -0.018 [T|

Speed Limit Law

** 2<|t-stat| T—n 0.044** * 1< I t-stat I < 2

-0.019*' Seat Belt Law

m Morbidity • Mortality • Materials Crashes • Injury Crashes • Fatal Crashes

Average Rainfall

I Exposure

Statewide Traffic Arrests Per Capita

0.116* -0.199* r

-0.300

-0.200

-0.100

0.000

0.100

Figure 7.2. Elasticites for the specific variables

0.200

200 Structural Road Accident Models 7.3.4. Further results In two model estimations, fatal crashes and materials crashes, a quadratic specification for alcohol consumption significantly improved the overall model fit. For fatal crashes, Statewide Consumption of Distilled Spirits per Capita and its squared term were statistically significant.^ The coefficient for the linear term was positive and that for the quadratic term was negative, implying that fatal crashes initially increase with consumption, reach a maximum, and then begin to fall with additional consumption. Solving the quadratic expression for the turning point gives a value of .1846, which is .019 greater than sample mean consumption at .1656. An examination of the data, presented in Graph 7.7, indicates that the highest consumption levels tend to occur in the traditional Christmas season, with per capita rates that may exceed .30, although the data also reveal a general downward trend during the sample period. The estimation results indicate that these high consumption levels decrease fatal crashes thus producing a safer environment. However, it must be remembered that the measure of alcohol consumption in this paper is 'apparent' consumption, which is based upon tax receipts. Although alcohol purchases and actual consumption rise during the Christmas season, it is likely that purchases rise considerably more (i.e. purchases and actual consumption are less correlated during this period) as firms and households often give distilled spirits as gifts. Thus, without further information, it is reasonable to conclude that only the upward portion of the curve is relevant for policy purposes. A similar phenomenon occurred for materials only crashes and Statewide Consumption of Wine per Capita. The linear term was positive, the quadratic term was negative, and the turning point occurred at .399, .025 greater than consumption at the sample mean of .374. As with distilled spirits, apparent wine consumption tends to peak in the holiday season; in addition, however, consumption during the summer period also rises. Although purchases may significantly exceed actual consumption during the holidays (the 'gift giving' phenomenon), this is unlikely to be the case during the summer months so that the relationship between apparent consumption, actual consumption, and materials crashes is less clear.

7.4. DISCUSSION AND FUTURE DIRECTIONS

The elasticity results presented in Figures 7.1 and 7.2 have a number of implications for public policy but also raise questions that call for further research. Less controversial are the findings that aggregate alcohol availability and the apparent consumption of alcohol compromise highway safety. The availability of alcohol increases the frequency of non-fatal crashes. The *" More general specifications rejected a non-symmetric relationship between fatal crashes and distilled spirits consumption.

The TRACS-CA Model for California 201 apparent consumption of distilled spirits increases the incidence and severity of fatal crashes whereas the apparent consumption of wine increases the incidence of non-fatal crashes. From these results, pricing and other policies aimed at lowering the consumption of distilled spirits will reduce the most serious crashes on California's highways. Similarly, increases in gasoline prices were found to benefit highway safety by reducing the incidence of non-fatal crashes as well as reducing crash morbidity. However, such a policy lacks focus and the additional costs it imposes on other travelers and trip making could very well offset any highway safety effects. An implication from the analysis is that more research is needed to identify the mechanisms through which raised fuel prices affect highway safety. Graph 7.7 Statewide Apparent Consumption of Distilled Spirits per Capita

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janv-90

Of particular importance to state and federal governments is the finding that increased enforcement has a beneficial effect for the most serious crashes, those involving a fatality. At the same time, enforcement was found to increase property damage crashes, a result which could reflect risk substitution on the part of drivers or endogeneity problems as greater enforcement responds to increased materials crashes. The findings in this study on the effects of relaxed speed limits in rural interstate highways and mandatory seat belt use are less intuitive. The Seat Belt Use law increased the incidence of non-fatal crashes as well as mortality rates but decreased morbidity rates. There was no effect on fatal crashes. The Relaxed Speed Limit Law, on the other hand, increased the incidence of injury crashes but decreased mortality and morbidity rates. These findings are contrary to commonly expected effects whereby seat belt use laws decrease (increase) the incidence and severity of the most (least) serious crashes; and relaxed speed limits increase rather than decrease mortality and morbidity rates. A possible culprit for these findings is that passage of the two laws was sufficiently close in

202 Structural Road Accident Models time (15 months apart) that it is not possible to disentangle the separate effects, particularly when average speeds were increasing prior to passage of the relaxed speed limit law. To investigate this, the five safety sub-models (fatal, non-fatal injury, materials only, mortality, and morbidity) were re-estimated with only one of the law variables (Seat Belt Law, Speed Limit Law). The hypothesis being tested is whether one variable is capturing the net effect of both laws, starting from the time that the belt law was passed or from the time that the speed limit law was passed. Table 7.3 provides selected information from these models. Column (a) reports the coefficient estimate and t-statistic for a model in which only Seat Belt Law is included; Column (b) reports similar results when only Speed Limit Law is included; and Column (c) reflects the model identified in Tables 7.1 and 7.2. The most important finding in Table 7.3 is the robustness of the signs and, almost uniformly, the significance of the law variables. Despite the fact that the laws were passed within 15 months of one another and that average speeds were creeping upwards prior to passage of California's relaxing rural interstate speed limits, the two law variables appear to be capturing different phenomena. Only in the case of Materials Only Crashes, and to some degree Mortality, is there a large change in the magnitude or significance of Seat Belt Law (Column (a) versus (cl)) and Speed Limit Law (Column (b) versus (c2)). This suggests that we will have to look elsewhere for an explanation of the signs and significance of Seat Belt Law and Speed Limit Law variables. Certainly, as suggested previously, this warrants additional research on the extent to which risk compensation and risk substitution occurs on our nation's highways. Table 7.3. Sensitivity of resuhs to law variables*

MODEL

Fatal Crashes Non-Fatal Injury Crashes

Seat Belt Law

Speed Limit Law

(a) Estimate t-stat 12.2 .77

(b) Estimate t-stat -7.05

-.42

Seat Belt Law (cl)

Speed Limit Law (c2)

Estimate

t-stat

Estimate

t-stat

12.3

.75

-7.32

-.41

.025

1.53

.045

2.76

.022

1.43

.05E-04

1.50

-.00002

1.57

.00004

1.08

-.07e-04

Mortality

.0092

3.24

-.0099

-5.08

.0059

3.83

-.0088

-5.42

Morbidity

-.029

5.39

-.010

-1.08

-.028

-5.26

-.012

-2.11

Materials Only Crashes

.044

2.70 -.74

*Other variables in the models are the same as those identified in Table 7.1. As a last point regarding these findings, we can explore the total effect that the relaxed speed limit law and mandatory seat belt use law had upon injuries. We focus upon injuries for this illustration because neither law had a uniform effect on frequency and severity. In each case, the law increased the frequency of non-fatal injury crashes but decreased morbidity so that the net effect is uncertain. In the case of the mandatory seat belt use law, passage of the law increased injuries .003% which is simply the sum of the direct elasticities reported in Figure 7.2. The more interesting case is the relaxed speed limit law because passage of the law

The TRACS-CA Model for California 203 directly affected exposure, crash frequency, and morbidity. At the same time, the law had an indirect effect upon frequency and morbidity through its impact upon exposure. Taking these effects into account gives a total elasticity with respect to relaxed speed limits equal to -.009. In other words, the total effect of the law reduced non-fatal injuries .009%.^ There is little question that the models and findings reported in this analysis are exploratory. There are relatively few explanatory variables in the estimated models and the time series data are over a decade old. Despite these shortcomings, however, the multi-equation structural methodology has provided additional insights into highway safety that are generally absent from single equation analyses. The ability to model non-linear in parameter specifications as well as to control for multiple serial correlation and flexible patterns of heteroskedasticity enable one to more accurately explain observed behavior and to predict the often complex effects of alternative public policies on the incidence and severity of highway crashes. As discussed throughout the paper, this analysis has implications for public policy and perhaps more importantly, the analysis provides a solid foundation upon which to build more complex yet realistic models of highway safety in the US and for developing a richer set of variables over a longer time span. 7.5 REFERENCES American Medical Association, Council Report (1986). Alcohol and the driver. J. Am. Med. Assoc, 25, 522-527. Borkenstein, R. F. et al. (1964). The role of the drinking driver in traffic accidents (the grand rapids study). Report to the Division of Accident Prevention, Bureau of State Services, U.S. Public Health Service. Brown, R. W. et al. (1996). Endogenous alcohol prohibition and drunk driving. Southern Econ. J., 62, 1043-53. Chaloupka, F. et al. (1991). Alcohol control policies and motor vehicle fatalities. National Bureau of Economic Research, Inc. Working Paper No. 3831. Crandall, R. et al. (1986). Regulating the Automobile. The Brookings Institution, Washington Let V refer to vehicle miles traveled, F to non-fatal crash frequency, M to morbidity (injuries per non-fatal injury crash), I to injuries, and L to passage of a law (i.e. relaxed speed limits or mandatory belt use). Then, in elasticity form, Ei, L = (EF, L + Ep, V EV, L) + (EM, L + EM, V EV, L) where the first term in parentheses gives the direct and indirect impact of the law on crash frequency and the second term in parentheses gives the direct and indirect effect of the law on morbidity. In the case of the mandatory belt use law, the indirect effects are absent since belt use has no effect upon exposure so that the elasticity is the simple sum of the direct elasticities. However, for relaxed speed limits the total effect (from Figures 7.8 and 7.9) is [.044 + (.007)(.532) + (-.009) + (-.007)(.101) = -.009.

204 Structural Road Accident Models DC. Garber, S. and Graham, J. D. (1990). The effects of the new 65 mile-per-hour speed limit on rural highway fatalities: a state-by-state analysis. Ace. Anal. andPrev., 22, 137-149. Gaudry, M. (1993). Le modele DRAG: elements pertinents au monde du travail—une analyse exploratoire. Publication CRT-948, Centre de recherche sur les transports, Universite de Montreal. Gaudry, M., F. Fournier et R. Simard. (1993a). Application of econometric model DRAG-2 to the frequency of accidents in Quebec according to different levels of severity. Proc. VIII Canadian Multidisciplinary Road Safety Conf. June 14-16, Saskatoon, Saskatchewan. Gaudry, M. et al n993b). Cur cum TRIO? Publication CRT-901, Centre de recherche sur les transports, Universite de Montreal. Godwin, S. (1992). Effect of the 65 M.P.H. speed limit on highway safety in the U.S.A (with comments and reply to comments). Transp. Rev., 12, 1-11, 13-14. Graham, J. and Garber, S. (1984). Evaluating the effects of automobile safety regulation. J. of Policy Anal, and Management, 3, 206-21. Kenkel, D. S. (1993). Drinking, driving, and deterrence: the effectiveness and social costs of alternative policies. J. Law andEcon., XXXVI, 877-913. Lave, C. (1985). Speeding, coordination, and the 55 mph speed limit. Amer. Econ. Rev., 75, 1159-64. Laixuthai, A. and F. Chaloupka. (1993). Youth alcohol use and public policy. Contemporary Policy Issues, XI, 70-81. Lave, C. (1992). Comment on Stephen R. Godwin's: effect of the 65 m.p.h. speed limit on highway safety in the U.S.A. Transp. Rev., 12, 11-12. McCarthy, P. and Oesterle, W. (1987). The deterrent effects of stiffer DUI laws: an empirical study. Logistics and Transp. Rev., 23, 353-71. McCarthy, P. S. (1993). The effect of higher rural interstate speed limits in alcohol-related accidents. J. Health Econ., 11, 281-299. McCarthy, P. S. (1994). An empirical analysis of the direct and indirect effects of relaxed interstate speed limits on highway safety. /. Urban Econ., 36, 353-364. McCarthy, P. S. (1999a). Public policy and highway safety: a citywide perspective. Reg. and Urban Econ., 29, 231-244. McCarthy, P. S. (1999b/ Alcohol availability and motor vehicle highway safety in small American cities. Working Paper, Purdue University. Ruhm, C. (1995). Alcohol policies and highway vehicle fatalities. Working Paper No. 5195, National Bureau of Economic Research.

Comparing six DRAG-Type models 205

8

COMPARING SIX D R A G - T Y P E MODELS

Nicolas Chambron

Over the past few years, different transport research estabhshments have adopted a common method for modeUing road safety indicators. The Road Demand, Accidents and Their Severity (DRAG model: modele de la Demande Routiere des Accidents et de leur Gravite) model, conceived by Marc Gaudry in Quebec in the mid-1980s, is the reference. Germany, France, Norway, Sweden and the State of California have recently taken up this approach albeit adapting it to their statistical particularities, on the one hand, and to their socio-economic and political environments, on the other (cf. Table 1). The findings are expressed in terms of the elasticities of risk indicators as a function of explanatory variables that are common to the different models in the DRAG family. The risk factors retained here are those, which appear to be the most important, and which are shared between several models, so that a comparison can be made of their effects. These factors are classified by theme: Risk exposure (section 1), driver behaviour (section 2) and economic variables, divided between households' economic situation and the factors that influence the frequency and distance of people's journeys (section 3). The other groups of variables relating to the characteristics of the road traffic system (defined as the road network, the vehicle stock and users), to climatic conditions, to time-ofyear variables and to regulatory measures are either limited to a single country, or too specific to one of the countries to bring any added value through a comparative analysis. Interested readers can refer to a report (Chambron, 1999).

206 Structural Road Accident Models Table 1. Models belonging to the DRAG family Ref.

1 A

2F

Country

—

II

3N

•!•«

4S

m

5Q

6C

m\wm

^

• JIHI 11 B D D

\m.

Authors

Monthly period

Model

Particularities

Network Whole West Germany

Blum, Gaudry

68-89

SNUS

Material damage accidents included. Light and severe material damages distinguished. Two separate models for fuel and diesel vehicles.

Jaeger

57-93

TAG

Modelling of the average Estimation of monthly mileage.

Fridstrem

73-94

TRULS

Tegner

70-95

DRAGStockholm

57-93

DRAG

Gaudry Foumier, Simard,

McCarthy

81-89

speed.

Model estimating the traffic from the fuel sales including the evolution of the fuel consumption of vehicle according to different categories, meteorological conditions ...Two econometric equations explaining the seat-belt rate and the car fleet. The results are disaggregated by road users categories (pedestrians, cyclists, motorcyclists, car drivers). Many U-shaped effects

Whole France

Whole Norway

Stockholm

Material damage accidents included. Impact of the traffic congestion. Forecasts up to 2004.

Whole Province

Material damage accidents included. For the severity of accident, use of the death TRACS-CA rate (number of fatalities / accident) and the injury rate. Elasticities have been recalculated.

State Highways

8.1. R I S K E X P O S U R E

Total mileage driven on the national road network is the principal component of risk. This explanatory variable can be measured simply by referring to the findings of investigations or surveys, whose results are published periodically. In that case, the information obtained is often incomplete, since it only relates to part of a network and disregards movements across frontiers. One can try to correct that bias, but it requires meticulous and often very timeconsuming work (Jaeger, 1998). Other models give priority to simpler approaches. Take the case of West Germany, where road demand is expressed as the total mileage driven by both petrol and diesel engine vehicles. Each variable is obtained by dividing the monthly consumption of petrol or diesel oil by the average rate of consumption of these two types of vehicle. In the case of the TAG model, there is an extremely strong relationship between total mileage driven and accidents (cf Table 2; model 2F). In effect, the short-term elasticity is 0.65 for

Comparing six DRAG-Type models 207 personal injury accidents and 0.9 for fatal accidents. In other words, a 10% increase in the number of kilometres driven would result in a 6.5% rise in personal injury accidents and a almost proportionate increase in fatal accidents. Table 2. Elasticities of traffic risk indicators as a function of mileage Ref. 1A 2F

Origin

Variable Total mileage

^

II

Total mileage

Traffic share on national roads'^

SlSiiii

Total mileage

Traffic density (mileage / network length)

^^

• ^H

injury

fatal

Total mileage

Traffic density (mileage / network length)

Victims

severe

injured

killed

NS**

1,4%

NS**

5,6 %

7,1%

6,5 %

light

2,0 %

Gasoline vehicle share

Traffic share on motorways'^

^ -^

Accidents

6,5 % -0,3% 0,7%

8,9 % MS"** 3,2%

9,1 % -4,1%

9,2 %

^^**

A/S***

-0,5%

7,2 %

2,7%

3,6%

10,3 %

6,8 %

7,6 %

-2,4%

7,7%

2,3%

11,5% -0,2%

5Q

^3 Bi

Total mileage

7,8 %

4,6 %

7,2 %

6,0 %

6C

J^

Total mileage

5,3 %

9,0 %

6,3 %

9,8 %

* wzY/z a constant total mileage. ** NS : (Highly) Non Significant, that is to say the Student t associated to the estimated is lower than 1

The results indicate that an increase in mileage driven leads to an appreciable increase in the risk of fatal accidents relative to the risk of personal injury accidents. The same phenomenon is found to a lesser degree in the level of the severity indicator (minor injuries, serious injuries, fatalities). It is hard to explain this different impact. It could be attributable to the differing development in the share of traffic for each type of network. But this factor is integrated in the model as an explanatory variable. For that reason, since the elasticity of the number of accidents (and of victims) with respect to mileage is calculated on a ceteris paribus basis, this argument is not valid. The explanation could lie at the level of other variables that are absent from the model, such as the breakdown of traffic between day and night. Nevertheless, it is still very instructive to study the development of accidents as a function of relative traffic density on different road netw^orks. The TAG model looks at the consequences on road risk of an increase in the proportion of traffic on main roads (motorways), assuming the proportion of traffic on motorways (main roads) is constant. For that reason, any relative increase in traffic on main roads takes place at the expense of other components of the road network (secondary roads, rural roads and others). The positive correlation between the share of traffic on main roads and the number of personal injury, and especially fatal, accidents is

208 Structural Road Accident Models attributable in part to a reduction in inter-urban journeys, which are less risky than longer journeys. Conversely, the increase in the share of motorway traffic has a far smaller impact on the accident rate, since it helps to divert some traffic away from secondary roads, which are more dangerous. At an international level, the Califomian model (6C) generates similar results despite considerable differences at the level of the reference period and the construction of the reference variable. As regards this last point, it is noteworthy that, on the one hand, P. McCarthy measures the exposure variable in miles and not in kilometres (1 mile =1.6 km) and, on the other hand, that he focuses on only part of the road network. In fact, monthly data are only available for state highways'. But the risk factor (vehicle-miles) on this type of road is strongly correlated with that of the whole network, which appreciably reduces the bias in estimating these parameters. The German, Quebec and Norwegian models (respectively SNUS, DRAG and TRULS) lead to slightly different results from the last two models. But any comparative analysis has to be undertaken with caution since each model may: (i) refer to very particular contexts; (ii) bring different variables into play, whose presence or absence changes the significance of the estimated parameters. In West Germany, the elasticity of personal injury accidents with respect to mileage is lower than in France (0.2). That difference may be attributable to greater traffic congestion in West Germany. Over a comparable period, S. Oppe ^, with the help of a logistic function, showed that growth potential in the volume of traffic was between 10% and 15% in West Germany, which is comparable with the situation in the Netherlands, whose road network is the densest in Europe, along with Belgium's. By comparison, growth potential in the United Kingdom is thought to be 20%. Let us consider the work of L. Fridstr0m. The frequency of personal injury accidents has an elasticity of 0.9 in relation to the total mileage of motorised vehicles. For that reason, injury accidents increase in almost the same proportion as the volume of traffic, ceteris paribus. However, that elasticity applies to a traffic density
Comparing six DRAG-Type models 209 that d does not vary. Conversely, there is the more realistic scenario in which the network does not change. In this case, the elasticity of personal injury accidents with respect to mileage is approximately 0.5 for 1994. For the other indicators, the elasticities with respect to mileage given a constant network are as follows: minor injuries: e = 8%; serious injuries: e = 9%; fatalities: e •= 10%. These are more or less the results of the French TAG model. Lastly, the Norwegian model measures the phenomenon of the clogging of the road network by introducing a variable for traffic density. The latter has a negative impact on personal injury accidents and minor injuries and a positive one on the numbers of more serious injuries (cf. Table 2). These results could be attributable in particular to lower traffic speeds and reduced compliance with safety headways. Gaudry, Fournier and Simard suggest that traffic congestion be transcribed on to the development of road risk by introducing first a symmetric, and then an asymmetric, form of mileage. The mileage per square then takes the form of a new explanatory variable. Concretely, the numbers of accidents and victims increase to a maximum, then traffic congestion leads to a progressive reduction, despite an increase in mileage. This approach was used in the TAG model and yielded the following elasticities. The addition of elasticities in a quadratic form does not necessarily signify the same incidence for mileage as in the initial model, but it does enable account to be taken of the effect of risk exposure on accidents. Table 3. Impact of traffic congestion (TAG model) Accidents injury (a) Mileage

Victims severe

light

fatal

fatal

-2,3 %

-4,3 %

-2,7 %

11,5%

-2,4 %

(b) Squared mileage

4,8 %

7,8 %

5,9 %

-2,0 %

6,4 %

(c)-(a) + (b)

2,5 %

3,5 %

3,2 %

9,5 %

4,0 %

8.2. DRIVER BEHAVIOUR 8.2.1. Speed The relationship between speed and road risk is relatively intuitive: "The more the user increases his speed, the less he can take effective action against any disruption in his environment" (Jaeger, 1998). The first consequence of excessive speed is thus the loss of

210 Structural Road Accident Models control of the vehicle. In addition, it increases the degree of severity of the accident. This predominant factor is nevertheless absent from most econometric models, except for the TAG model, which reveals a high incidence of average speed on the different indicators of road risk (cf Table 4). Table 4. Elasticities of road risk indicators with respect to average speed (TAG model) Victims Accidents Ref. Origin Kind of network severe light fatal fatal injury 2F

MM

All

11,1%

18,1%

11,4%

20,8 %

16,7%

A 10% increase in average speed would lead to a direct increase of 11.1% in the number of personal injury accidents. The impact is even greater on fatal accidents (an increase of 18.4%)), which confirms that speed is an exacerbating factor. This aspect is also underscored by the very high values of the elasticities calculated for the victims. The above conclusions enable us to gauge the importance of speed limits. In that regard, Table 5 shows that the regulatory measures introduced in Germany and Quebec in the 1970s resulted in a sharp drop in the numbers of personal injury and fatal accidents. Oddly enough, though, the results obtained in France, Sweden and the State of California are far less convincing. However, the elasticities given in Table 5 may be biased to greater or lesser extent for two reasons: (i) Any speed limit acts at two other levels of risk: mileage driven and, above all, speed, (ii) The indicative variables measure the partial effect of the measures and not the overall effect. While the first does not seem to have a major impact on accidents, the second tends to prove that speed limits have an indirect effect and operate through the reductions in speed they bring about. In effect, the different speed limit experiments carried out on French rural road networks (1973) resulted in a 2.5%) reduction in average speed, and for that reason, in a 3% to 5%) reduction in all risk indicators, depending on the degree of severity. This outcome appears modest, but has to be seen in context, as the following point suggests. For example, the dummy"^ variable used in the TAG model was applied to the month of September 1973, whereas the intervention period lasted for several months. Once again, the estimations appear far lower than the reality.

A dummy variable is a binary variable (0, 1) reflecting the introduction of a regulatory measure or the occurrence of an exceptional event.

Comparing six DRAG-Type models 211 Table 5. Impact of mandatory speed limits Ref.

Origin

1A

"^M

Impact TOTAL

Victims

Accidents injury

fatal

light

severe

injured

-8,9 %

-9,3 %

Direct impact (dummy Oct. 1973-March 1974)'*

-8,5%

^5"

NS

-0,8%

Indirect impact (through mileage)

-0,8%

NS

-0,5%

NS

2F

killed

-6,7% -2,2 Yo

• 1

Direct impact (dummy September 1973)^

Small probability for a significantly impact different from zero.

Indirect impact (through mileage)

Small probability for a significantly impact different from zero.

Indirect impact (through average speed)

-2,8%

110/90km/h

4S

• ••

5Q

11 B mandatory seat Q 11 belt (08/76)

-4,5%

-2,8%

-4,2 Yo

-5,0%

1,2%

-0,4 %

Speed limit +

6C

;.m

Indirect impact (dummy May 1987)^ Indirect impact (trough mileage)

TOTAL

-9,4 %

-14,7%

-9,1 %

-17,8%

0,4 %

-0,3 %

0,3 %

-0,5 %

0,4% £

-0,2% -0,1%

0,3 %o

-0,4 Yo

£

-0,1%

The elasticities in the SNUS (lA) model seem closer to reality, because the dummies were "active" (equal to 1) during the whole regulatory phase, which is to say from October 1973 to March 1974.

During this period, which corresponded to the first oil shock, traffic was restricted on Sundays and a speed limit of 100 kph was imposed on motorways. ^ Decree of 9 November, 1974: speed limits were imposed on all roads: (i) 90 kph on single carriageway roads; (ii) 110 kph on four-lane, dual carriageway expressways; (iii) 130 kph on motorways. This decree followed various experimental measures and temporary speed limits (decrees of June and December). ^ In May 1987, the State of California reduced speed limits on rural interstate highways.

212 Structural Road Accident Models S. Lassarre (1986) had already studied the impact of road safety measures introduced in France in 1973: - July 1973 (i) A speed limit of 100 kph was imposed over the whole network, except on motorways and certain other types of road; (ii) Wearing of seatbelts was made compulsory for front-seat passengers of vehicles equipped with safety belts. Between July and November 1973 the overall effect was a 9% drop in the number of fatalities. This gain broke down as follows: (i) 11% due to the speed limit, (ii) 89% due to the wearing of seatbelts: In December 1973 a speed limit of 120 kph was introduced on motorways. The speed limit was reduced to 90 kph for other roads. The overall effect (including the July 1973 measures) was a 14% reduction in the accident risk and a 25.8% drop in the number of fatalities. As regards the latter, the gain broke down as follows for the period from December 1973 to April 1974: 60% due to speed limits, 40%) due to the wearing of seatbelts. Altogether, these different (and, in some cases, temporary) speed limits led to an appreciable fall in the number of deaths on the rural road (which represents 2/3 of the total number), in the region of 15%).

8.2.2. Seatbelt wearing The resuhs of the SNUS (lA), TAG (2F) and TRULS (3N) models, all of which are concerned with this factor, are outlined in table 6. As regards this explanatory variable, the three models cover comparable estimation periods. While it is true that the TAG model starts in January 1957, the rate of seatbelt wearing in France was virtually zero up to the early 1970s, because most vehicles were not equipped with safety belts. On the other hand, the methods for calculating this statistic differ. For example, the surveys carried out in Norway and West Germany are confined to drivers, while the O.N.I.S.R. includes front-seat passengers. Moreover, the Norwegian surveys relate to the "rate of nonwearing" of safety belts, whereas the others concentrate on the rate of wearing. The principal singularity of the French approach is that it distinguishes between the direct and indirect effects of speed. Seatbelt wearing, like any other apparent technological advance, enhances the driver's safety to the point where he changes his behaviour and increases his speed. This indirect effect remains much weaker than the direct effect, however. L. Fridstr0m studies another indirect effect on accidents: the size of the fines imposed for any breach of the regulations governing seatbelt wearing. A 10% increase in the fine would engender a 1.3% rise in the rate of seatbelt wearing. Combining this coefficient with the elasticities in Table 6, one can calculate the estimated effect of an increase in the real value of the fine. It was raised to 500 Norwegian kroner (about 62 Euros) in 1994.

Comparing six DRAG-Type models 213 Table 6. Elasticities of road risk indicators with respect to seatbelt wearing Ref. 1A 2F

mjury

Total

Drivers''

-1,1 %

Total

Front seat occupants

-0,88 %

W^ •1

Direct impact Indirect impact (through average speed) 3N

^inH

m\wm

Accidents

Estimation of the rate

Impact

Origin

Total

Drivers "Non-user rate"

Victims hght

severe

fatal

NS

-0,5 %

-1,1 %

-1,88 %

-1,48 %

-1,88 %

-2,08 %

-0,9%

-1,9%

-1,5%

-1,9%

-2,1%

0,02%

0,02%

0,02%

0,02%

0,02%

1,9 %

1,2 %

0,6 %

2,3 %

fatal

In 1994 a 10% increase in the fme corresponded to a 1.3% rise in the rate of seatbelt wearing, taking it up from 88% to 89.2%. That is equivalent to a fall of nearly 10% in the "rate of nonwearing" (from 12% to 10.8%), which results in a reduction of: 22% in the number of personal injury accidents; 1.8%) in the number of slightly injured; 1.1% in the number of seriously injured; 0.6%) in the number killed. These results have to be put into perspective by taking account of the gradual fall in the real value of the fme, due to inflation. This phenomenon probably has the opposite effect. As far as the global impact is concerned, it can be seen that an increase in seatbelt wearing has a strongly negative incidence on personal injury accidents in Norway, West Germany and France. As regards severity, TRULS (3N) models lead to a different conclusion: safety belts are more effective for preventing minor, and even serious, injuries than for saving lives. It has to be realised in that regard that in most fatal accidents, wearing seatbelts cannot diminish the consequences, either because belts do not exist for those involved (pedestrians, two-wheelers, public transport), or because the shock is too great for them to have any effect (substantial deformation of the body). In West Germany and France, on the other hand, growing use of safety belts would result in a greater reduction in the number of fatalities (of 1.1%) and 2.1%). There even appears to be a transfer phenomenon in terms of severity from fatal to serious and minor injuries.

^ The series for the rate of seatbelt wearing (as a percentage) is corrected, because it only exists since 1975 (when the rate was close to 40%). The monthly data for the 1968-1975 period were thus obtained by extrapolation.

214 Structural Road Accident Models 8.2.3. Consumption of alcohol An increase in alcohol intake has a direct impact on road risk indicators (cf. Table 7). The variables used to take account of this factor are not generally expressed in terms of a level. In effect, to offset the problems of multicolinearity, consumption can be treated as a reference variable. For example, the DRAG (6Q) model looks at sales of alcohol/?^r adult, because there is a positive and virtually systematic relationship between these two variables. In the TAG (2F) model, a 10% rise in wine consumption per 1,000 kilometres driven increases the numbers of injury, fatal accidents, minor injuries, serious injuries and fatalities by 1.2% to 2.1%), depending on the circumstances. It would thus seem that this variable has more of an effect on the severity of injury accidents than on their number. Wine consumption per adult also has an indirect impact through its effect on speed, but that remains very limited (0.04%)). These results have to be treated with caution since, by introducing wine consumption as an explanatory variable, the author makes the strong assumption that variations in this factor correspond to those in drink-driving. However, while wine accounted for 75%o of the alcohol consumed (in litres of pure alcohol) in 1970, the proportion was down to two-thirds by 1994. The consumption of beer and spirits increased during that period and accounted for 15%) and 17%) of the total respectively in 1994. Furthermore, studies aimed at producing a typology of drivers who have consumed alcohol highlight the predominant share of aperitifs and beer. It can be seen, finally, that these proportions differ according both to geographical area (consumption of wine in the south of France and of beer in the north) and to age (the young mainly drink beer and spirits). In terms of the numbers of personal injury and fatal accidents, the DRAG {6Q) model generates coefficients that are close to zero and not at all significant. But given the available data, the variable describing alcohol consumption per adult does have some limitations. For one thing, since this variable was constructed on the basis of sales of alcohol, there may be a lag between the purchase and the effective consumption of alcoholic drinks, especially when it comes to purchases for the "festive season". For another, this variable reflects the state of the whole adult population. There is an even greater bias in the case of the TRACS-CA (6C) model, since the consumption of wine and spirits is related to the total population. However, it shows that effects differ according to the variable considered: wine consumption has a positive impact on personal injury accidents, whereas that of spirits, which are stronger, plays a significant role in fatal accidents. The SNUS (lA) model focuses on the production of beer, which is closer to the Germans' actual consumption than the production of alcohol in general. But there is still a bias, which is linked, on the one hand, to the import and export of beer and, on the other hand, to the lag

Comparing six DRAG-Type models 215 between production and consumption. In Germany, going to a bar is a leisure activity in itself. The particular cultural context enables beer consumption to be used as an indicator of mobility, which is to say as a social activity variable. In effect, a 10% increase in beer consumption would lead to a 2% rise in the number of kilometres travelled by petrol engine vehicles. Table 7. Elasticities of road safety indicators with respect to alcohol consumption Ref. Origin 1A

Variable

!•"

TOTAL

Victims

Accidents injury

fatal

light

severe

fatal

1,4%

Direct impact (I eer production per worker and per mileage)

1,0%

-0,7%

NS

NS

Indirect impact (beer production per worker)

0,4%

NS

0,3%

1,1 %

1,2%

1,4%

2,1 %

2F

%W

Wine consumption / 1 000 driven km^

1,2%

o 3N

4S

mmm

m ^m

1

Shops per 1 000 inhabitants

0,8 %

0,5 %

2,0 %

2,0 %

% selling strong beers'^

0,2 %

0,4 %

0,5 %

0,2 %

% selling liquors / wine

0,4 %

0,4 %

0,4 %

With licence to serve alcohol/1 000 inhab. -0,06 %

mm Kx

6C

;m

8

0,6 % -1,8% -1,0%

% Serving liquor / wine

-2,5 %

- 0 , 1 % -3,5 % -2,9 %

% Serving strong liquor

0,2 %

-0,4 % -0,5 % -0,6 %

Sma 1 alcohol consumption level High alcohol consumption level

5Q

1,6%

Alcohol total sales / adult Wine consumption per inhabitant

-6,7 % -20,0 % 3,9 %

11,7%

NS

NS

iV5' - 1 , 3 %

0,4 %

NS

7V5

iV5

NS

0,5 %

NS

7^5-

Of course, this explanatory variable also highlights the impact of drunkenness on road risk. For that, the authors based themselves on beer consumption per employee by total mileage travelled by petrol engine vehicles. It reveals a slightly positive effect on personal injury accidents, but one little removed from zero on the victims (except the slightly injured). The TRULS (3N) model provides more information by making a dual distinction between: (1) different categories of alcoholic drinks; (2) shops licensed to sell alcohol and restaurants. ^ The only long series available is taxed wine consumption in mainland France expressed in hectolitres, from the Monthly Statistical Bulletin of the French National Economic Statistics Institute (INSEE). > 4.5 % of alcohol by volume.

216 Structural Road Accident Models In Norway, the choice of variables is linked to the legislative context governing the distribution of alcohol. Access to alcohol is more tightly controlled in Norway than in most other industrialised countries in the west. Wine and spirits are sold exclusively by government stores, mostly located in big towns. Moreover, the sale of beer requires a licence granted by the local authority. Similarly, restaurants need a licence from central or local government to serve alcoholic drinks. Finally, some Norwegian town councils prohibit the sale of alcohol altogether. The first variable (the total number of stores authorised to sell alcohol per 1,000 inhabitants) seems to have coherent effects, since it results in positive elasticities for each degree of severity. It can be seen that these elasticities are higher for the seriously injured than for those who die. Judging from these estimations, Norway's restrictive policy towards alcohol has prevented a number of road accidents and above all reduced their severity. The other two variables, relating to stores authorised to sell particular types of alcoholic drink (strong beer, wine and spirits), still have a positive impact on accidents, but it is less marked. On the other hand, the density of restaurants authorised to serve alcohol has a negative effect. That is especially true for restaurants authorised to sell wine (as opposed to beer gardens). This relationship is certainly linked to the fact that alcohol consumption in restaurants is more moderate. In any event, these results show that there is a complex relationship between alcohol consumption and accidents. The DRAG-Stockholm (4S) model proposes a non-linear relationship between these two variables, distinguishing low levels of alcohol consumption from higher levels. The first have a negative impact on accidents, while the second have a positive one. M. Gaudry tried to explain this relationship: "When aggregate consumption of alcohol increases from relatively low per capita levels, the overwhelming majority [of drivers] drink a little, which reduces their risk below that prevailing when they do not drink, perhaps because they compensate or are less aggressive; this reduction may more than offset the increase in risk for the minority [of drivers] who drink more heavily, depending on their relative proportion on the road."

8.2.4. Consumption of medicines Only the German, Quebec and Swedish models incorporate a variable relating to the consumption of medicines (cf Table 8). Overall, there is a strongly positive impact on personal injury accidents (except in Germany) and especially on fatal accidents. This result is the result of behavioural problems (drowsiness, giddiness, dizzy spells, etc.) caused by some medicines, whether used properly or misused. In particular, the accident risk rises when there are problems of overdoses, dependence or combination with alcohol.

Comparing six DRAG-Type models 217 Table 8. Elasticities of road risk indicators as a function of the consumption of medicines Victims Accidents Ref. Origin Variable injury fatal light severe injured killed 1A

•r

4S

•

H

Consumption of medecines / adult Prescriptions of medecines

5Q Ei ma Consumption of medecines / adult

NS

0,7 %

NS

2,6 %

2,4 % 7,8 % 2,2 % 3,1 %

2,2 % 3,6 %

We must guard against any erroneous interpretation here. An increase in the consumption of medicines is not the cause of a deterioration in road safety. These two variables are only linked through a correlation that essentially reflects the attitude of users who are not worried about the side effects of medicines, especially on their ability to drive. Given the importance of this phenomenon, France decided to require any new medicine likely to have an effect on people's ability to drive to carry a warning sign on the pack consisting of a little white triangle with a red border and a black car in the middle. In view of the results obtained in Quebec and Sweden, this measure should effectively contribute to reducing the number of road accidents and their severity.

8.3. E C O N O M I C V A R I A B L E S

8.3.1. Households* economic and financial situation A motorist's driving can be altered by his or her own economic and financial circumstances. At a more general level, road risk moves in tandem with household morale, and the economic environment plays an important role in increasing or reducing it. This argument is supported by the fact that only the variables related directly to households are significant. Those associated with national output (GNP, GDP) or corporate income provide very little information. To understand the incidence (or incidences, since the effect can vary from country to country) of the economic environment on road risk, we will thus focus on the following variables: unemployment, household income and final consumption (cf Table 9). Unemployment. In the case of this variable, the French model yields diametrically opposed results to those of the Norwegian, Quebec and Swedish models. In effect, unlike the other approaches considered in this paper, the TAG (2F) model is characterised by positive and globally non-significant elasticities. There is a straightforward theoretical explanation for the positive impact of unemployment on accidents: any period of economic uncertainty may have the effect of reducing the attention of

218 Structural Road Accident Models the driver concerned, since he is probably worried about the consequences on his financial situation. The difficult situation in which the user finds himself results in an appreciable increase in his accident risk. Once this connection has been made, how is that we may find negative correlations between unemployment and road accidents in other countries? Table 9. Elasticities of road risk indicators as a function of economic factors Ref.

Origin

Accidents

Variable

injury

fatal

light

severe

injured

Victims

UNEMPLOYMENT 2F U

•

unemployed / 100 adults

Direct impact

NS

Indirect impact (through mileage) 3N

miMi HlHH

NS

NS

0,4%

Small probability for a significantly impact differentfirom zero.

unemployed

-0,3 %

-0,8 %

-0,8 %

Direct impact

-0,2%

-0,6%

-0,6%

Indirect impact (through mileage)

-0,1%

-0,2%

-0,2%

4S

4,1 %

•r'iBi

5Q H

B

NS

employed

IB unemployed /driving licenses B

-1,2 %

-1,1 %

-0,2 % £

-0,2%

-1,5 %

-1,1 %

HOUSEHOLD INCOME 3N

m\mm mXmm

household income

Direct impact

£

Indirect impact (through mileage)

1 A

"••r

8,0 %

8,0%

13,4 %

16,2 %

16,6 %

0,6%

2,5%

0,7%

12,8%

13,7%

15,9%

FINAL CONSUMPTION food & clothes

Direct impact Indirect impact (through

mileage)

0,1 %

1, 5 % 1,1 %

NS

0,1 %

0,4%

NS

0,3%

0,2%

-1,1%

NS

1,2%

A number of hypotheses can be advanced: Historically, unemployment rates in Norway and Sweden are considerably lower than in France (cf. Chart below^^). For that reason, the consequences of an increase in this indicator do not have as marked an impact on household morale in those countries, where the labour market is more open. An increase could even engender greater mobility, that is to say a higher exposure to risk. This argument has to be put into perspective, of course, taking account of the results of the Quebec study, which suggest

Unemployment rate in France, Sweden and Norway - source: OECD.

Comparing six DRAG-Type models 219 that unemployment has a negative impact, whereas the labour market situation is comparable to that observed in France. The inequality of social welfare systems means we cannot assume that individuals behave in a homogeneous fashion towards unemployment. The higher share of social security spending in total public expenditure in France could partly explain the fact that the TAG model yields nonsignificant results. The variables under consideration vary from one country to another, depending on the way they are constructed and calculated. Household income and final consumption. These two variables highlight the importance of the effect on road accidents of households' financial situation in relation to their economic situation. In the TRULS model, total elasticities with respect to the number of jobless does not exceed 1% in absolute terms, whereas they amount to between 8% and 17% when income is taken into account. It can be seen that these elasticities are calculated on a ceteris paribus basis, which applies to, among other things, the total length of the road network. Household income has an essentially indirect incidence on road risk indicators, via the mileage driven. At this stage, one should note that the Norwegian model uses the stock of private vehicles as a relative variable. This enters the mileage equation both as an endogenous and then as an exogenous variable. For that reason, the elasticities shown in Table 16 are aggregate numbers and correspond to long-term elasticities. In fact, the effect induced by a variation in the vehicle stock is included in the elasticity of mileage, which is itself incorporated in the elasticity of personal injury accidents and victims.

198 1 ^Quebec

1 986 •France

^Suede

1 993 •^orvege

Lastly, an appreciable rise in household income implies an increase in sales of private motorcars and hence in the number of kilometres driven. That chain of causation, which

220 Structural Road Accident Models ultimately leads to an increase in road risk, is evidence of the recursive structure of the TRULS model. An improvement in households' financial situation has a more marked incidence on the severity of injury accidents than on their number. That could be due in part to the arrival of more powerful vehicles, which would induce an increase in average speed. The elasticities in the SNUS model seem to conflict with this result. However, the authors integrate a variable for private household consumption relative to their daily consumption (food) or at least to their regular consumption (clothing). But this generally implies short trips in an urban environment. That is why its incidence is greater on material damage accidents (elasticity=0.41) than on personal injury accidents (0.11). Furthermore, this variable was incorporated more to gauge the impact of changes in households' travel habits than to estimate the effect of variations in their economic and financial circumstances. In any case, there is another reason why it would not be able to do the latter: the major part of this variable is incompressible (necessities, consumption habits).

8.3.2. Fuel prices The German, French, Norwegian and Quebec models come to the same conclusion: an increase in fuel prices improves road safety outcomes, mainly at the level of fatal accidents and fatalities (cf. Table 10). The TAG model highlights three effects: (i) direct: an increase in the real price of gasoline encourages users to drive in a more flexible fashion, which engenders a reduction in road risk; (ii) indirect through mileage driven, i.e. users compensate for the rise in price by a consequent reduction in mileage. L. Jaeger and L. Fridstrom conclude that this is the predominant effect in the short and the long term respectively. In West Germany, on the other hand, the effect is found to be marginal. That is essentially due to the low elasticities of road safety indicators with respect to mileage driven, since the influence of fuel prices on road demand is much the same from one country to another. In Germany, for instance, a 10% rise in gasoline (or diesel oil) prices would result in a 1.8% (2.6%)) reduction in mileage driven by petrol engine (diesel engine) vehicles. By comparison, the French and Norwegian models yield corresponding elasticities of-3.3% and -2.6%. The Californian model produces even more surprising results, concluding that the real price of petrol has no effect on road demand. The author explains this phenomenon by starting from the definition of the opportunity cost of road journeys. The latter incorporates two components relating to monetary expenditure, on the one hand, and journey time, on the other. However, any change in the real price of gasoline has no effect on this last factor, whose relative weight

Comparing six DRAG-Type models 221 in the total cost is low. For that reason, P. McCarthy puts forward the hypothesis that companies and households react to an increase in this factor not by limiting their journey time but by increasing their demand for short trips. At the end of the day, the impact on total mileage is marginal. Table 10. Elasticities of road safety indicators as a function of motor vehicle fuel prices Accidents Victims Ref Origin Variable light severe injured killed injury fatal 1A H^H

Real fuel price^^

-1,9 %

Direct impact

A^^

-0,6%

NS -0,4% -0,9%

NS -0,3%

NS -1,0%

2 F 1H H I Real fuel price^^ / kilometre -2,9 % -4,8 % -2,5 % -3,2 %

-5,5 %

Direct impact Indirect impact (through mileage) Indirect impact (through average speed)

-0,5% -1,3% NS A^^* -1,5% -2,0% -1,6 %\ -1,5% -0,9% -1,5% -0,9% -1,7%

-1,7% -2,1% -1,7%

3 N 1 jjjjl"^ 1 Fuel price

-1,3 %

-2,1 % -2,5 %

-2,6 %

-0,1%

-0,3%

-0,1%

-2,0% -2,2%

-2,5%

Indirect impact (through mileage)

Direct impact

£

Indirect impact (through mileage) 5Q

11 O

-0,4%

Real fuel price / km

6C

Real fuel price ' ^ Direct impact Indirect corrected impact (through mileage)

-1,3% -3,9 % -4,4 %

-3,9 % -4,9 %

-3,0 %

-3,5 %

-3,0%

NS

-3,5%

NS

NS

NS

NS

NS

However, in the context of an international comparison, other elements may have to be taken into account. To start with, the price of petrol in the United States is appreciably lower than in Europe because of much lower taxes, and state highways are not subject to tolls. Added to that are the characteristics of American vehicles (automatic gearboxes are the norm) and the low speeds they are driven at, which makes for lower fuel consumption. For that reason, the cost of road travel is not a very important component of the household consumption function. Hence, an increase in petrol prices has little effect on their mobility and travel habits. Furthermore, public transport is less developed in Californian towns than in large European cities, which ' ^ The authors take the real price of petrol per kilometre driven, so as to correct the effect related to the fuel's performance. '^ It is important to extract the effects linked to inflation by expressing prices in constant values so as to isolate the incidence of a real increase in fuel prices on road demand.

222 Structural Road Accident Models does not enable a true judgement to be made. (iii) Indirect via average speed, i.e. at the same time, users reduce their speed in order to lower their fuel consumption. The presence of a speed-related factor in the TAG model explains why the incidence of an increase in the real price of fuel on road risk is slightly higher in France than in Quebec. The Norwegian approach measures the impact of a rise in fuel prices on the development of the stock of private motorcars. This turns out to be substantially negative, at -1.5% for a 10% increase in the price of petrol. Taking account of this phenomenon makes it possible to calculate long-term elasticities. In the very long run, the petrol price effect translates almost totally into a commensurate variation in mileage driven.

8.3.3. Competing supply from public transport According to the TRULS model, the cost of public transport, and especially underground (subway) and tram fares, has relatively modest effects on road accidents in the long term (cf. Table 11). Nevertheless, these effects are very significant and virtually equivalent to the indirect effects conveyed through the mileage variable. To be more precise, an appreciable increase in transport fares would encourage households to decide in favour of purchasing private motorcars. Of course, their decision would not be taken immediately, so the consequences on mileage and then on traffic accidents are not felt until some time after the announcement of a new pricing policy. Table 11. Elasticities of road safety indicators with respect to transit fare. Victimes Accidents Ref Origin Variable injury fatal light severe killed

3N

as

Fare price for metro or tramway Direct impact Indirect impact (through mileage)

0,4 %

0,7 %

0,9 % 0,8 %

£

0,1%

0,2%

0,4%

0,6%

0,7%

£

0,8%

8.4. CONCLUSION The main contribution of DRAG-type models is to identify the variables that have a preponderant impact on road safety outcomes and on which regulatory measures should thus be focused.

Comparing six DRAG-Type models 223 Punishing motorists who drive too fast remains an inescapable means for achieving a significant reduction in the number of victims. In France, a 1% increase in average speed would bring a commensurate rise in the number of road deaths. The other variables related to user behaviour have less of an incidence on personal injury accidents and their severity. They include, firstly, the wearing of safety belts by drivers and front-seat passengers, since that is strongly embedded in our habits. Secondly, there is the consumption of alcohol, a variable whose limited effect is essentially attributable to its general nature. In effect, the series generally used is national alcohol consumption. However, most drivers, aware of the effects of alcohol on the organism, compensate for the increase in risk by driving more cautiously. Only a small proportion of users do not practise this kind of retroaction. In addition, the French model concentrates on wine consumption, whereas a large number of personal injury accidents is associated with the consumption of beer or spirits, especially by the young. A more interesting estimation could be to separate alcohol served in restaurants from that sold in other licensed establishments (such as bars and discotheques). For that, however, long series would have to be available, as in Norway. In any event, the beneficial effects of the very strict regulatory regime in force in this country (close to prohibition in some municipalities) deserve to be emphasised. Other variables yield interesting results. For example, periods of economic prosperity for households are associated with an increase in accidents, since risk exposure is greater. The actions of the authorities and the forces of law and order could be stepped up at such times. The models studied here are destined to be regularly updated, at the level of the estimation period, on the one hand, and of the integration of new risk factors, on the other. Even if the authors face obvious measurement problems, new explanatory variables could be "dreamed up" to build in the effects linked to the consumption of drugs such as cannabis, which is reported to be a factor in around 16% of personal injury accidents occurring in France'"^, or to the growing use of mobile telephones at the wheel, which is estimated to increase the accident risk by a factor of four^"^. It would be equally interesting to know the impact on road risk of the active and passive safety features of vehicles.

The French Toxicology Society. ^"^ The 14-month study involved 699 drivers in the Toronto region (Canada) and was published in the February 1999 edition of the New England Journal of Medicine. It pointed out that the accident risk when the driver is on the telephone is similar to that resulting from consuming the legal limit of alcohol. Unfortunately, no distinction was reported among severity levels, as risk compensation could increase the total but reduce the severity.

224 Structural Road Accident Models 8.5. REFERENCES Blum, U., M. Gaudry (1999). SNUS-2.5, a multimoment analysis of road demand, accidents and their severity in Germany, 1968-1989. Centre de recherche sur les transports, Universite de Montreal. Carre, J.R., S. Lassarre et M. Ramos eds. (1993). Modelisation de I'insecurite routiere. INRETS, Actes du seminaire, Arcueil. Chambron, N. (1999). La modelisation de I'insecurite routiere par une approche commune: Allemagne de I'Ouest - France - Norvege - Quebec - Suede. Federation fran9aise des societes d'assurances, Paris. Cohen S., H. Duval, S. Lassarre, J. P. Orfeuil (1998). Limitations de vitesse. Les decisions publiques et leurs ejfets. Hermes, Paris. Dally S., (1985). Conduite automobile et alcool. La documentation fran9aise, Observatoire national interministeriel de securite routiere. COST 329 (1999). Models for traffic and safety development and interventions. European cooperation in the field of Scientific and Technical Research. Fridstrom, L. (1998). TRULS: an econometric model of car ownership, road use and accidents in NORWAY. Institute of Transport Economics. Gaudry, M., Foumier, F. et R. Simard (1995). DRAG-2, un modele econometrique applique au kilometrage, aux accidents et a leur gravite au Quebec. Synthese des resultats, Societe de I'assurance automobile du Quebec. Jaeger, L. (1998). L'evaluation du risque dans les systemes de transports routiers par le developpement du modele TAG. These de doctorat de sciences economiques, Universite Louis Pasteur, Strasbourg. Jaeger, L. et S. Lassarre (1997). Pour une modelisation de I'insecurite routiere : estimation du kilometrage mensuel en France de 1957 a 1993. Rapport DERA n°9709, INRETS, Arcueil. Lassarre, S. (1997). Analyse des progres en securite routiere pour dix pays europeens. INRETS-DERA, Arcueil. Lassarre, S. (1994). Cadrage methodologique d'une modelisation pour un suivi de I'insecurite routiere. Synthese INRETS n°26, Arcueil. Tegner, G. (1998). Human behaviour and road traffic safety-a regional, Swedish long-term perspective-The DRAG Stockholm-2 Model. TRANSEK Consultants.

The Road, Risk, Uncertainty and Speed

225

THE ROAD, RISK, UNCERTAINTY AND SPEED Marc Gaudry Karine Vernier

9.1 RISK, UNCERTAINTY AND OBSERVED ROAD ACCIDENT OUTCOMES

This paper, adapted from Gaudry and Vernier (2000) and Vernier (1999), demonstrates how one can identify the impact of fine road infrastructure features, both geometric and surface, on road speed and safety, taking due account of the simukaneous nature of the speed-safety relationship. This is done within a DRAG-type multi-level structure where accident frequency or severity and speed equations are estimated with Box-Cox transforms. Our explanation of accident outcomes distinguishes, beyond pure randomness, between two (subjective) components of observed (objective) risk, namely calculated risk linked to speed, and uncertainty or "dangerousness", in the manner of Frank Knight (1921). Both components are part and parcel of a new empirical measure of "perceived risk", expected maximum insecurity EMI, that is demonstrably separable from other determinants of speed choice. In our structure, the first two groups of equations explain accident frequency and severity with discrete choice Logit models admitting of non linearity and choice-based sampling—^permitting future completion of the database at low cost. The third and last group of equations also consists of non-linear flexible-form models and explains the mean and the variance of speeds. The tentative results, briefly outlined here, of tests done with more than 80 road infrastructure trace and pavement characteristics, show clear speed-safety trade-offs and validate the view that drivers do not fully respond to perceived risk through speed adjustments. Results provide a clear rationale for interventions on the part of road safety authorities, e.g. through refined road design and signage improvements. Such improvements should demonstrably reduce the role of the "systematic surprise" element (uncertainty), to the benefit of the chosen risk element (speed) and of pure randomness, in the explanation of observed accident outcomes. Speed, as indicator of voluntary risk-taking, would then fully account for observed non-random risk, and observed risk itself imply a perceived risk EMI level precisely sustaining the chosen speed.

226 Structural Road Accident Models 9.2 MODEL STRUCTURE: SIMULTANEITY AND PERCEIVED RISK Framework. Although the original DRAG-1 structure (Gaudry, 1984) was specified as simultaneous, the absence of speed data forced the estimation of reduced form equations (with the speed variable replaced by its determinants) of a recursive nature (with accidents a function of road demand, but not the reverse). Despite general agreement on the simultaneous nature of the speed-safety phenomenon, authors estimating both accident equations with a speed variable and a speed equation proper also end up with recursive systems (with accidents dependent on speed but not the reverse) in recent two-level (Cardoso, 1997; Aljani et al., 1998) and threelevel (Jaeger, 1999) cases. The difficulty of a strict simultaneous formulation arises from the need to specify an adequate representation of the safety risk within the speed equation. Hence our approach, summarized in Figure 9.1, within the D-P (Demand-Performance) framework. Figure 9. 1. Multilevel simultaneous DRAG-type framework Demand road use -, ETC.] (DR)

VICTIMS

Accidents <-{Calculated Uncertainty of Driver, ETC.} and their risk, [traffic+road+vehicle], severity (A), (G) ^ {

Travel <Time

(f(V)), (u[(DR,N-I),M-C]),

(h(Y-G))

}

Realized ^-{Perceived risk, Comfort, Vehicle, DWver, ETC.} Speed (V)

~]

Exposure Risk

d

Frequency and severity Risks

p

Calculated Risk L^l

In that figure, consumer self-protection and self-insurance activities are reduced to the choice of speed V because the characteristics of vehicles M-C and safety belt use are assumed given. Speed is then assumed to be determined by maximum expected insecurity EMI = u(A,G) and by k(N-I), the subset of road characteristics N-I that determine driving comfort. Driving comfort itself, chosen jointly with speed, is not observed, and the traffic level DR is also given. We assume that, for a driver with characteristics Y-G, chosen speed V determines a "calculated risk" f(V) but leaves a latent residual uncertainty u[(DR, N-I), M-C] associated with exogenous traffic DR, road characteristics N-I and his own vehicle characteristics M-C. The calculated risk corresponds to Knight's (1921) notion of "risk" (the object of probability calculations) and the uncertainty corresponds to his notion of globally apprehended intuitive "uncertainty" where the weight of each element is not calculable but the whole dangerousness is, in his language, "directly" perceived. Dangerousness of the road environment is not pure randomness but covers notions like "legibility" of the road and systematic "surprises" due to incorrect

The Road, Risk, Uncertainty and Speed

227

perception of situations or conditions. Taken together, these factors determine objective accident frequency and severity risks A and G. Theoretical model. Formally, our theoretical structure is given by the following equations: (9.1) Frequency:

y], = p^^.R^, + P2^^,YG^, + Pa^.MC,, + p4,,y, + TI,,

(9.2) Severity:

y; = p,,R,. + p , J G , + P,,,MC, + p , , y , + TI,

(9.3) Speed:

y, = p , , R , + p^^.YG, + p , , M C , + p^^.y^ + p,,y^ + TI, ,

with: (i) yl et y^: vectors of latent non observable variables representing risk for a given individual. To each vector corresponds a discrete variable taking different values, according to the levels of yl and y*^ ; (ii) rj^ and r/^ : residuals with given distribution; (iii) R, Y-G, M-C: vectors of explanatory variables for the road (R = (DR, N-I)), driver (Y-G) and vehicle (M-C); (iy)y,,: continuous variable describing chosen speed; (v)/;^,: speed equation residual of zero mean and standard error a^I; (vi) y^^ and y^ : variables describing the frequency and severity risks perceived by the road user; (vii) /?. ^, /?. ^ and y^, ^ are vectors of parameters to be estimated. More precisely, we define for these two variables inclusive values: (9.4) yi; = ISR, YG, MC, y^,) =

J]^xp{v;)

(9.5) y'^ = I^(R, YG, MC, yj=[

Z exp{Vt)

the natural logarithm of which yields the expected maximum insecurity measure EMI, an analog of the Williams-McFadden (Williams, 1977) expected maximum utility measure EMU. These measures give the maximum insecurity expectation of the road user over all feasible alternatives. Estimated model. Because of the absence of data on individual drivers Y-G and their vehicle characteristics M-C, the estimable model is a simplified version of (9.1)-(9.5) with right-hand side endogenous variables derived from two previous steps: (9.6) Frequency:

y* = Pi^R^ + p2,ayv + 'Ha

(9.7) Severity:

y; = p,^,R^. + ^,^1

(9.8) Speed:

y, = p , , R , + p^^.y^ + P^vYg + ^v

+ T]^

228 Structural Road Accident Models with y^^nd y^ drawn from first-step models for inclusive values: (9.9) perceived frequency risk:

y^ =I^(R) =

(9.10) perceived severity risk:

y^=I^(R) =

\^exp(V")

and expected speed y'^ is calculated from (9.8): (9.11)

y:=E{y^,)

because estimates of speed equation (9.8) are obtained from a sample disjoint from that used for the accident frequency and severity equations, thereby insuring independence (Arellano and Meghir, 1992) between errors of (9.8) and those of (9.6) and (9.7); and finally: (9.12)

E[l^(R)/fj,,] = E[I^(R)/?^]

= 0, and E[y:/?],] = ^ X / ^ / , ] = 0.

Note that inclusive measures ya and y^ are obtained from (9.8) and (9.9) without including in these equations a first-step estimate of chosen speed y^, as desired, because of severe limitations of our sample of speeds, taken on only 17 links. These limitations also affect the efficiency of our unbiased estimates of (9.8) and consequently the efficiency of y^^ in (9.11). Econometric specification. The accident frequency and severity equations (9.6) and (9.7) are Logit models given by:

(9.13) P(Yi = 6) = i;^

—-

j=i

and

P{Z. = a/Y. = S} =

^—^-^, /=i

where Yj denotes the probability of occurrence of an accident of any severity, Zj the conditional probability of occurrence of three severity levels (light bodily damage; severe bodily damage; fatal) and Vjg the representative ufility function of 6 for individual i, with X^. the Box-Cox transformation (1.4) of variable X^, in conformity with perspective PF-1 in Chapter 1: linearity is not a priori credible in any utility function and the response curve shown on Figure 1.7 is not more likely to be symmetric here than anywhere else in economics or transportation, to say nothing about the impact of form on signs in Logit models (Gaudry, 1999): (9.14)

V,,=

Zp,X.

The Road, Risk, Uncertainty and Speed

229

Estimates are obtained by the Weighted Endogeneous Sample Maximum Likelihood (WESML) procedure in TRIO (Liem and Gaudry, 1993) with 2541 observations for the accident frequency equation and 1225 observations for the severity equation. The sample of 2541 observations contains those 1225 accident cases and 1316 non-accident cases drawn at random from the remaining 48758 links (of the total sample of 50000 homogeneous non urban links of an average length of 70 m. from 14 French departements) where there was no accident over the sample period (1991-1995). To express results in a convenient way, we use as elasticity the "probability points" measure: (9.15)

7r(pjij,X3) =

-X,,

that expresses in probability points the effect on the probability P(yyi=i) of a percentage change in the explanatory variable Xj. The conventional elasticity measure is clearly inadequate for probability (and share) models because their dependent variables are already restricted between 0 and 1 (or 0 and 100 %): results on "the elasticity of the probability" are not directly interpretable without the mean sample values in mind. By contrast, the probability point measure is interpretable and clearly satisfies perspective PR-1 of Chapter 1. Also in conformity with perspective PF-1 in Chapter 1, the speed equations explaining the mean and standard deviation of speeds on individual road links are formulated as (1.1) and estimated with the LEVEL-1.5 algorithm (Liem et al, 2000) found in Chapter 12. The sample points for the average speed and the standard error of speed on 17 road links are constructed from 60000 instantaneous speed measurements, but we did not correct Student's /^-statistics to reflect this efficiency and consequently understate their true values. Results are expressed with the conventional elasticity measure (1.9). In all cases, extensive y^ tests were carried out to find which of the variables were individually significant. The 80 variables belonged to 17 groups, labeled G 1 to G 17 (these labels are retained in tables below), distributed among 5 classes of factors: behavioural (chosen speed, expected risk), traffic environment (quantity and type), horizontal and vertical dimensions of road sections (flat or not, turning or not, etc.), width and length dimensions (number of lanes, width, etc.) of road sections and pavement surface indicators (smoothness, cracking, type of pavement and chemical composition of surface treatment, etc.) of road sections. We already mentioned above that data on the characteristics of vehicles and drivers were not available: we do not know whether these factors are independent from road and traffic characteristics, but assume that they should be, thereby insuring that our results are not too biased.

230 Structural Road Accident Models 9.3 SELECTED RESULTS: ACCIDENT FREQUENCY AND SEVERITY Results for 3 factors. The full paper presents results for the impact on accident frequency and severity of traffic levels and composition (heavy truck mix), slopes as one approaches the segment and other (vertical) forms of the segment (e.g. top of a hill), initial (horizontal) angle of the segment, and its geographic orientation, segment width, and various surface indicators such as gravel stone emergence (hauteur au sable), repairs of various importance and diverse pavement deformations. But we will only select a few factors other than those to demonstrate our method. Some factors influence both frequency and severity but others influence only frequency and are neutral with respect to the severity mix: 32 factors were retained in the frequency model and 68 in the severity model. The former had four Box-Cox transformations on various variables and the latter only one. Also, to interpret ^statistics correctly, one has to remember that, in Logit models, they test whether the difference between the coefficient of the chosen state and that of the reference state differs from say zero (and that they do not constitute an exact test as compared with the results of the y^ tests that previously decided the inclusion of the variables): in the frequency model, the reference state is the absence of accident, in the severity model, it is the state oi lightly severe bodily damage (L.S.), indicated by (r.s.). The effect of expected speed variables. Consider first Table 9.1, where the expected speed and standard error of speed are shown to influence both frequency and severity. In probability terms, higher speeds reduce the probability of having an accident, which is as expected to the extent that speed and driving attention are complementary and that relatively few accidents happen at high speeds. The Box-Cox transform suggests a rapidly decreasing probability, as exhibited in Figure 9.2. However, in terms of severity, higher mean speeds increase the relative probability of severe bodily damage or fatal accidents. By contrast, the standard error of individual speeds (retained by the x^ tests) has, judging by the (albeit imprecise) r-statistic, a marginal impact on accident frequency and but reduces the relative probability of fatal bodily damage, perhaps due to the fact that drivers are more careful in heterogeneous flows than in more homogeneous ones. Overall, these results appear consistent with those of Solomon (1964) who had found a positive correlation between accident frequency (without distinguishing among severity levels) and the standard error of speed individual speeds. The effect of turns. An interesting question is whether many stylized facts concerning the difficulty of turning left are correct. Generally speaking, models pertaining to infrastructure characteristics do not use many variables. In models of accident frequency, the geometric

The Road, Risk, Uncertainty and Speed Table 9.1. Effects of behaviour: average speed and standard error of speed 1 Probability of accident (a) Model prob: 2 ^=0 t=l \ I. Probability points; Students t of variable 7i(a) G 1. Speed 1 Expectation of average speed MV -.202 (-1.63) 1 Expectation of stand, error of speeds (0.03) SV .001 [1.73] [1.32] Box-Cox Transformation of VM 4.233 ^4 Probability of severity (g): Light Severity = (L. S.) Heavy Severity = (H.S.) Fatal ^(F.) \L Probability points; Student's t of variable G 1. Speed 1 Expectation of average speed MV 1

Expect, of stand, error of speeds Box-Cox Transformation of VM

SV

^4

231

\ |

1

Model grav: 2

Mg) (L.S.) -24 (H.S.) 138 (F) . 108 (L.S.) .029 (H.S.) .023 (F.) -.052 1.802

r= o

t=l

1

|

(r.s.) (1.37) (1.54) (r.s.) (0.01) (-1.31) [0.29] [0.13]

Figure 9.2. Effect of mean speed on accident probability

configurations of curves appear: Barber et al. (1998), for instance, use a model of frequency with 5 variables, 4 of which defme curvature, after Zeeger et al. (1991); in probabilistic models of severity, we could find only two (Shankar et al., 1995, 1996) using road geometry variables. Others (Ato Eguakun and Wilson, 1995; Weiss, 1992 and O'Donnel and Connor, 1996) had no such variables. What do we find? We find that, although the direction of turning has no effect on the severity of accident, it does indeed affect the probability of accidents, as indicated in

232 Structural Road Accident Models Table 9.2 where the distinction between the two definitions of turning depends on the turning arc's radius and the results are to be interpreted relative to going straight. Interestingly, slow (long radius) left turns are more dangerous than slow right turns. And both are more accident inducing than straight sections. But, again, strong (short radius) left turns are more dangerous (-16.8 % on the probability of accident, relative to -6.8 %) than strong right turns, although both are safer than straight sections. Turning left is therefore always more dangerous than turning right, irrespective of the radius. This may be because the slope of the pavement reduces centrifugal forces more when the vehicle turns right than when it turns left, or because loss of control on right turns benefits more from the accotement than on the left. Table 9.2. Effects of the road: geometric characteristics in the horizontal plane Model prob:2 1 Probability of accident (a) \L Probability points and Student's t of variables t=0 t=l n(a) |G 8. Type of section in the horizontal plane (r.s.: straight section) 1 circle to the right (long radius) .004 (0.12) cerdro 1 circle to the left (long radius) .031 (1.05) cergau (-3.60) 1 connection to the right (short radius) racdro -.168 -.068 1 connection to the left (short radius) racgau |(-l-75)

^ ] 1

1

The effect of pavement smoothness. Interestingly, smoothness (the absence of undulations) does not appear to influence accident severity: it impacts only on frequency, as indicated in Table 9.3. In the first model, the less plane the pavement, the smaller the accident probability. In the second model, a model that does not include speed as an explanatory variable (that is the only difference between models 1 and 2), this is not the case any more: if proper account is not taken of speed, the effect of smoothness is estimated to be of some import, embodying the effect of speed. In reality, smoothness leads drivers to a speed adjustment and has no real impact on accident probability once this speed adjustment is accounted for. Table 9.3. Effects of the road: surface smoothness |G 14. Smoothness, friction, tracks, surface coating loss... Model prob:l 1 Probability of accident (a) \L Probability points and Student's t of variables ^=0 ^=1 71 (a) 1 index of pavement planeness valuni -.070 (-1.42) bvaluni .074 associated binary variable (1.23) Box-Cox transformation of planeness .535 [2.77]['-2.46] X, 1 Probability of accident (a) \I. Probability points and Student's t of variables 1 index of pavement planeness valuni associated binary variable bvaluni Box-Cox transformation of planeness

h

Model prob:2 f=0 / = ! n(a) -.035 (-.60) (1.12) .073 .525 [2.56]["-231]

^ \

1 \

1

The Road, Risk, Uncertainty and Speed

233

9.4 SELECTED RESULTS: SPEED Models that explain speed choice tend to focus on average speed (MV), as in Badeau et al. (1997) or Jaeger (1999), and on a measure linked to the standard error of individual speeds, the 85"' percentile of speeds (V85), or its complement the 15* percentile (VI5), as in Godlewski (1985), Lamm et al. (1987, 1990), Collins and Krammes (1996) and Cardoso (1997). We studied both MV, the average speed, and SV, the standard error on links of our 60000 individual speed readings. Table 9.4 presents in three sections: (I) elasticities and ^statistics (uncorrected for efficiency—see above) of variables retained by the y^ tests; (II) functional forms used and corresponding /^-statistics (with respect to linear and logarithmic cases of the Box-Cox parameters); and (III) general statistics. We could not use excellent information showing a clear tomahawk-like pattern [ 0 = = ] between speed on the y-axis and vehicle length on the X-axis (see Gaudry and Vernier, 2000, or Vernier, 1999) because this information was not available for links used in the accident models. Statistical comments: (i) in both equations, the variables "perceived severity risk", "planeness", and "severe transversal cracks" make negligible contributions to the log-likelihood unless BoxCox transformations are used on them; (ii) in both equations, the logarithm of the inclusive value of insecurity, or log-sum, is obtained, but the standard error equation also admits of the linear form; (iii) all infrastructure variables retained have to do with driving comfort, as stated by expression (k(N-I)) in Figure 9.1; (iv) in the standard error equation both the presence (the dummy dftgrav) and amount of severe cracking (the variable//^grav) matter (see Section III). Mean speed equation (MV): (i) perceived severity risks lead to more significant drops in speed than "^QxcQiNQd frequency risks, but the estimated Box-Cox form (-0,33) implies diminishing marginal responses with higher perceived risk, as if the drivers did not easily want to reduce speed to zero; (ii) the degradation of smoothness leads to speed reductions, a result found elsewhere (Gambard, 1986); (iii) the weak impact of traffic may be due to lack of congestion. Standard error of speed sequation (SV). The standard error of speeds is influenced principally by geometric characteristics of the road (the presence of turns and the number of lanes), by the presence and quantity of severe structural deformations (cracks), and by surface features (the kind of bonding used for small gravel on the top pavement layer). In particular: (i) the standard error is 45 % higher on 2 by 2-lane highways than on 2-lane roads; (ii) surface cracks increase the standard error by 50 %; (iii) untreated gravel reduces the standard error by 25 %, as compared to bitumen-bonded gravel surface rolling layers.

234 Structural Road Accident Models

Table 9.4. Speed equation results Model of MV

Model of SV

1 /. Elasticities and t-stat of variables | ^ 1 li^) n 1G 2. Perceived risk 1 perceived probability risk Ml -.207 (-.68) EMUp 1 Box-Cox tr. of EMUp LAM2 [^2] 1 perceived severity risk EMUgr -.124 (-1.67) .043 1 Box-Cox tr. of EMUgr LAM2 [^2] 1G 3. Traffic volume 1 aver, annual daily traffic |MJA | -.071 (-.56) IG 6. Approaching section horizontal feature (r.s.: straight section) -.226 1 Turn 1 virage | 1G 11. Hierarchical classification within the network (r.s.: route nationale) | LACRA | .243 (1.39) .450 1 highway section 1G 14. Smoothness, friction, tracks, surface coating loss... 1 index of planeness valuni .242 (1.50) 1 Box-Cox tr. of planeness LAMl [^1] 1G 15. Structural: various cracking patterns -.002 1 severe transversal cracks ftgrav 1 1 Box-Cox tr. of ftgrav LAMl [^1] .535 1 associated binary variable dftgrav 1 1G 17. Bonding used for surface layer: bitumen,...(r.s.: grave bitumen) .230 1 hydraulic grave GH -.248 1 untreated small gravel NT [//. Values of A.; Student's t 1 Transformation X^ 1 Transformation X2

^m -.135 -.327

\IIL General statistics Log-likelihood with estimated X parameters Log-likelihood with X=\ 1 Log-lik. with A. = 1 and variable dftgrav included 1 Pseudo-R-2 |Pseudo-R-2 adjusted for degrees of freedom 1 Number of observations 1 Number of estimated parameters - Betas - variables - constants - associated binary variables 1 - Box-Cox transformations

t=0; t=l -.04 ; -.34 -.27;-1.08

{%] -9.99

W^)

\\

||

(.07)

1

(.52)

1 | |

(-1.76) (3.02)

.•

1 1 1 )

f (-2.01) (6.85) j

(0.76) 1 (-1-59) t=0; t=l -.01;-.01

1 1 1 1

\ -57.185 -59.842

-37.431 -4"5".864 -42.856

.746 .549 17

.861 .683 17

5 1 0 2

7 1 1

1

J 1 1

1

The Road, Risk, Uncertainty and Speed 9.5

235

CONCLUSION

We have formulated and estimated a simultaneous system showing interactions between speed behaviour and perceived risk, defining for the latter a new measure, expected maximum insecurity (EMI), that we have decomposed between globally apprehended uncertainty and calculable risk components along the famous lines proposed by Knight. We have not found other studies using any of these three dimensions. Our results, obtained with a limited but improvable database, were derived by combining y^ tests to determine the relevance of fine infrastructure characteristics and Box-Cox tests to determine the actual form of their effects. In particular, the log-sum special case was found optimal to represent the effect of perceived risk on speed choice and, yet again, the Box-Cox transformation yielded both better fits and more reasonable results than those obtained with predetermined forms, linear in particular.

9.6

REFERENCES

Ato Eguakum, G. and F. Wilson (1995). Modelling risk as a choice process. Compte rendu de la 9^*"^ conference canadienne multidisciplinaire sur la securite routiere, Montreal, pp. 173184, 23-25 mai. Aljani, A. A. M., Rhodes, A. H. and A. V. Metcalfe (1998), Speed, speed limits and road traffic accidents under free flow conditions. Accident Analysis and Prevention, 31, 161-168. Arellano, M. and C. Meghir (1992). Female labour supply and on-the-job search: an empirical model estimated using complementary data sets. Review of Economic Studies, 59, 537-557. Badeau, N., Baass, K. et C. Poirier (1997). Stabilite des vehicules dans les courbes: experimentations, analyse et interpretation des resultats. Non Public, Centre de recherche sur les transports, Universite de Montreal, 75 p., Janvier. Barber, P., Badeau, N. et K. Baass (1998). L'effet des profils horizontal et vertical sur la securite routiere. Tome I. Non Public, Laboratoire de circulation et de securite routiere, Ecole Polytechnique, Montreal, 73 p., decembre 1998. Cardoso, J. L. (1997). Accident rates and speed consistency on horizontal curves in carriageway rural roads. International Conference Traffic Safety on two Continents, 22-24 Dec. 1997,Lisboa. Collins, K. M. and R. A. Krammes (1996). Preliminary validation of a speed-profile model for design consistency evaluation. Transportation Research Board, Preprint 960590. Gambard, J. M. (1986). Approche du comportement cinematique du vehicule libre sur un axe routier. These de docteur ingenieur, Ecole Nationale des Fonts et Chaussees, Paris. Gaudry, M. (1984). DRAG, un modele de la Demande Routiere, des Accidents et de leur Gravite, applique au Quebec de 1956 a 1982. Publication CRT 359, Centre de recherche sur les transports, Universite de Montreal, 216 p. Gaudry, M. (1999). Are fixed-form regression models ever credible? Some evidence from transportation studies. In Gaudry, M. and R. R. Mayes, eds.. Taking Stock of Air Liberalization, Ch. 12, 171-188, Kluwer Academic Press. Gaudry, M. et K. Vernier (2000). Effets du trace et de I'etat des routes sur la vitesse et la securite: une premiere analyse par equations simultanees non lineaires distinguant entre

236 Structural Road Accident Models risque et incertitude. Rapport INRETS rf 224, Les Collections de I 'INRETS, Institut national de recherche sur les transports et leur securite, Arcueil, France, 68 p. Godlewski, D. (1985). Optimisation de la gestion routiere: utilisation de I'uni longitudinal. These de docteur ingenieur, Ecole Nationale des Fonts et Chaussees, Faris. Jaeger, L. (1999). Developpement d'un modele explicatif des accidents de la route en France sur une base mensuelle de 1956 a 1993. Publication CRT-99-11, Centre de recherche sur les transports, Universite de Montreal, 348 p. Knight, F. H. (1921). Risk, Uncertainty and Profit. Houghton, Mifflin. Lamm, R. and E. M. Choueiri (1987). Rural roads speed inconsistency design method. Part 1. Operating speed and accident rates on two-lane rural highway curved sections: investigation about consistency and inconsistency in horizontal aligment. Clarkson University, Report 87026, Postdam. Lamm, R., Choueiri, E. M. and T. Mailaender (1990). Comparison of operating speeds on dry and wet pavements of two-lane rural highways. Transportation Research Record, 1280, pp. 199-207. Liem, T. and M. Gaudry (1993). PROBABILITY: The P-2 program for the Standard and Generalised Box-Cox Logit models with disagreggate data. Publication CRT 527, Centre de recherche sur les transports, Universite de Montreal, 34 p.. Liem, T.C., Gaudry, M., Dagenais, M. and U. Blum (2000). The TRIO L-1.5 algorithm for BCGAUHESEQ regression. In Gaudry, M. and S. Lassarre, eds, Structural Road Accident Models: The International DRAG Family, Ch. 12, 263-325, Elsevier Science, Oxford. O'Donnel, C. J. and D. H. Connor (1996). Predicting the severity of motor vehicle accident injuries using models of ordered multiple choice. Accident Analysis and Prevention, Vol. 28, pp. 739-753. Shankar, V., Mannering, F. and W. Barfield (1995). Effect of roadway geometries and environmental factors on rural freeway accident frequency. Accident Analysis and Prevention, Vol. 27, pp. 371-389. Shankar, V., Mannering, F. and W. Barfield (1996). Statistical analysis of accident severity on rural freeways. Transportation Research Board, Preprint 960112. Solomon, D. (1964). Accidents on main rural highways related to speed, driver and vehicle. U.S. Department of Transportation, Federal Highway Administration, July. Vernier, K. (1999). Les incidences du trace routier et des etats de chaussee sur le comportement des usagers vis-a-vis de la vitesse et de la securite. Publication CRT-99-16, Centre de recherche sur les transports, Universite de Montreal, 319 p. Weiss, A. A. (1992). The effects of helmet use on the severity of head injuries motorcycle accidents. Journal of The American Statistical Association, 87, pp. 48-56. Williams, H. C. W. L. (1977). On the formation of travel demand models and economic evaluation measures of user benefit. Environment and Planning, 9a(3), 285-344. Zeeger, C.V. et al. (1991). Cost-effectiveness geometric improvements for safety upgrading of horizontal curves. FHWA-RD-90-021, Federal Highway Administration, McLean, Virginia, 237 p.

The RES Model by road category for France 237

10

THE R E S MODEL BY ROAD TYPE IN FRANCE Ruth Bergel Bernard Girard

10.1. INTRODUCTION

After the focus of attention was shifted at the beginning of the 1990's in France to monitoring short-term indicators in order to isolate a trend for local fluctuations (Bergel and al., 1995), a need is now felt for an "explanatory" monitoring of risk indicators at a disaggregate level in relation to their principal determinants, for anticipation/prediction purposes. The choice of monthly periodicity, which is the product of a compromise between the variability and availability of existing data and that of the determinants-weather and time-ofyear conditions, traffic flow or its substitutes (economic activity, prices, and network development), behavioural variables and road safety measures-position this approach mid-way between the modelling of daily fluctuations and the modelling of long-term trends. Taking the recommendations of the COST 329 group as a basis, which point out the need of disaggregate models, we sought to develop a model for road risk data that focused on a breakdown by type of road throughout France. It complements the TAG model that was constructed on French data at national level (Jaeger, Lassarre) and then utilised for international comparison purposes. We extended that approach to a vectorial framework and limited the number of determinants to about 10 for reasons of robustness, so the model is both explanatory and predictive. To give an example, we will present a version of the model limited to two types of network.

238 Structural Road Accident Models main roads and toll motorways^*\ We will first describe the structure of the model as a whole, and then go on to outline the main results: the outcome of tests of hypothesis related to econometric specification, the values for the indicators' elasticities to their determinants, showing their development over the period, and the simulations carried out. Lastly, we will discuss these findings and suggest avenues of enquiry for continuing this research. 10.2. S T R U C T U R E O F T H E M O D E L

10.2.L General outline The approach adopted consists of constructing a model for the risk exposure-risk-severity triangle, taking account of explanatory factors and the search for an optimal functional form (Gaudry, 1984; Gaudry et ai, 1997).

Table 10.1. The structure of the model Levels:

Risk exposure

Risk

Dependent variables

Traffic

Number of accidents Number of victims (personal injury. (deaths, serious injuries. minor injuries) fatal)

Explanatory variables

Traffic Economic activity Network length Fuel prices Weather conditions

Economic activity Network length Fuel prices Weather conditions Speed Use of seatbelt

Severity

Traffic Number of accidents Economic activity Network length Fuel prices Weather conditions Speed Use of seatbelt

The dependent variables of the three levels risk exposure/risk/severity, selected in this paper. ^*^In the first instance, we confined ourselves to the national road network (main roads and motorways), comprising 24,000 km of main roads and 7,000 km of motorways, which account for over one-third of France's entire road traffic. The rest of the French network is composed of departmental roads, on the one hand, where the degree of severity is the greatest, and urban roads, on the other, where the accident rate is high. We thus have four distinct fields covering the whole of France and which present very different risk profiles within the overall road safety picture.

The RES Model by road category for France 239 are the traffic, the number of accidents and the number of deaths. We first constructed a model independently on main roads and on toll motorways, using an univariate form, and we also extended this approach to the three pairs of indicators simultaneously defined on main roads and toll motorways, using a vectorial form.

10.2.2. The data base The database, compiled from monthly data for the 1975-1993 period, was created as part of the activities of an inter-ministerial working party comprising representatives from SES, SETRA and INRETS. It is particularly informative as regards the main road and motorway networks, for which monthly mileage travelled is available, recorded at different points along the national road network. It consists of: - the numbers of personal injury accidents (fatal and non fatal) and victims (killed, seriously injured, slightly injured), broken down between four fields: main roads, motorways, departmental roads and urban roads, in built-up areas of aver 5,000 inhabitants; - the data for kilometres travelled per month, on both main roads and toll motorways; - data for transport supply (the road network length, fuel prices and motorway tolls) and for transport demand (gross domestic product, industrial production and household consumption); - climatic variables relating to temperature, the occurrence of frost and the height of rain, which measure the average meteorological effect, for the whole month and the whole country; - and behavioural variables: speed and seatbelt use on the basis of surveys, broken down by network. Figure 10.1. Traffic on main roads (in hundreds of millions of vehicle-kms)

240 Structural Road Accident Models

Figure 10.2. Traffic on toll motorways (in hundreds of millions of vehicle-kms)

Figure 10.3. Number of accidents on main roads

CD

03

The RES Model by road category for France 241

Figure 10.4. Number of accidents on toll motorways

O)

O

O)

Figure 10.5. Number of deaths on main roads

O

242 Structural Road Accident Models

Figure 10.6. Number of deaths on toll motorways

Table 10.2. The data in 1992 National roads Toll highways rrattic(10 8 vehkm) Month, mean Accidents Month, mean % of total network Deaths Month, mean % of total network Network length (km) Average speed (km/h) Speed excess (%) Seat belt use (%)

786,95 78,92 14 070 1 172 9,8 2 241 187 24,7 24 000 87 45 90

454,38 37,86 2 443 204 1,7 338 28 3,7 5 728 117 24 94

Public highways Vational network Total network 268,76 22,40

1 510,09 125,84

3 275 273 2,3 228 19 2,5 1 381 105 39 83

19 778 1 648 13,8 2 807 234 30,9 31 109

143 362 11 947 9 083 757

10.2.3. Economic formulation All the explanatory determinants of traffic volume, accident risk and accident severity are taken into account, including weather/time-of-year effects and those of an economic and behavioural nature, whether their impact is immediate or delayed for a few months. The variables are retained in the definitive model if they are regarded as statistically significant on the basis of a Student test.

The RES Model by road category for France 243 We asked ourselves whether there might be non-constant elasticities and whether there might be interactions between the endogenous demand variables (mileage) on the different networks, especially on main roads and motorways, which would show up on the risk and severity variables. These two questions are at the root of the research described here. The econometric specification that we are now going to outline is thus decisive for trying to give an answer. 10.2.4. Econometric specification 10.2.4.1. Univariate specification. The econometric formulation retained in the univariate approach is a multiple regression of a dependant variable y on K explanatory variables xk, with a functional form and a generalised structure of heteroskedasticity and autocorrelation of the residuals, described in Equations (1.1) - (1.3). The econometric formulation retained at the outset corresponds to the constraints Xy = 0 and X^^^ = 0 for the primary explanatory variables. We also looked for a more general functional form, while maintaining the constraint on the variable to be explained and freeing those relating to the primary explanatories. The test we conducted (cf 10.3.1) does not enable us to reject the nullity constraint of the X^^, from a strict statistical point of view. This is the reason why, in the vectorial specification we are now going to outline, the endogenous variables are in a first approach logarithms of the traffic, of the number of accidents and of the number of fatalities, and the exogenous variables are also logarithms when they are primary explanatories. 10.2.4.2. Vectorial specification. A more general form of the earlier auto-regressive univariate equation is:

(10.1)


+ 0 ( 5 ) w,

where: y is the endogenous variable to be modelled (possibly filtered by an F(B) filter) zi are the exogenous variables (possibly filtered by Fi (B) filters) w is a white noise not correlated to the past of Y and the Zi, and O, ^ and 0 are polynomials in B, and B is the backward shift operator. When the Wt residuals corresponding to two equations on the same level, for example the numbers of accidents on main roads and on motorways, are correlated (whether immediate or delayed correlation), this correlation can effectively be taken into account by using a vectorial specification of two components for the vector of the numbers of accidents on the two types of road. In the general case, the auto-regressive vectorial equation retained for an endogenous vector consisting of p components is

244 Structural Road Accident Models

(10.2)

0 ^ (B) cD(B) Yt = 0 ^ (B) ^(B) Zt + Wt

where: Y and W are the endogenous vectors and white noise consisting with p components, Z the exogenous vector consisting with q components, and O, ^ and 0 are matrices of which each term is a polynomial in B. One can also use a markovian representation, which entails introducing an unobservable "state" vector X that takes over the dynamics of the system in the form of an auto-regressive vectorial of the order of 1. The Y vector observed is thus broken down into two parts: an exogenous part, and its own dynamics expressed as a function of X : (10.3)

X t + l = F X t +KWt

(10.4)

Yt -D(B)Zt + HXt + Wt

where: D is a polynomial matrix in B, and F, K and H are three matrices of real coefficients. 10.2.5. Algorithm The numerical results have all been obtained using the SAS software. In the case of the vectorial modelling, we used a specific algorithm for modelling multivariate time series with exogeneous variables (Azencott et al., 1997). 10.2.5.1. Univariate modelling. We maximise the likelihood for a grid of X values. This likelihood is calculated for each value using a regression of the endogenous variable on the exogenous ones with auto-regressive residuals whose coefficients for the auto-regressive part are obtained stepwise (the SAS autoreg procedure, option ml). The likelihood is calculated using the Kalman filter and is optimised using the Gauss-Newton method. 10.2.5.2. Vectorial modelling. We could estimate the parameters of the vectorial entry (10.2), but we prefer to estimate those of the markovian representation (10.3)-(10.4), which enables the vector's dynamics to be distinguished from the exogenous effects. Determining the orders of the auto-regressive polynomial matrices and mobile averages O and ^ , or of the system's rank (the minimum dimension of the State vector), for the markovian representation is an essential element in identifying the model's dynamics. For that, we used a method making it possible to determine automatically the rank of the markovian representation, and an algorithm enabling the system (10.3) - (10.4) to be constructed. This is carried out in three successive steps. The first step involves correcting the endogenous vector for the exogenous effects in such a way as to isolate the dynamics. This is done through the intermediary of a long auto-regressive vectorial model with delayed exogenous variables.

The RES Model by road category for France 245 We thus obtain the corrected vector YCt - Y t - D ( B ) Z t The second step concerns the modeUing of the YC vectorial process : X t + l = F X t +KWt YCt = H Xt + Wt The system's rank is determined by several tests constructed on the basis of a canonical analysis of the Hankel matrix of the YC process. The F, K and H matrices are then estimated using the Desai and Pal algorithm. Lastly, we use the values obtained for the F, K and H matrices from the markovian representation to re-estimate the exogenous effects, which is to say D (B), through a maximum of likelihood on the initial Y vector. 10.3. T H E R E S U L T S

10.3.1. Tests of functional form Here we look for the optimal Box-Cox forms for a small number of exogenous variables, which are the primary explanatory variables. In the example of the model for the two networks, main roads and motorways, for which we have six equations, these exogenous variables are the two types of traffic on main roads and motorways for the equations of the two accident/severity levels and the determinants of traffic (final household consumption, fuel prices and the length of the motorway network) for the equations of the volume of traffic. The object is to test whether, for these variables, there was a statistically significant difference between the estimated value of the vector of parameters X(X\ ,X2 ,... , A.^) - the vector thus has between one and three components, depending on the case - and the zero value X=0 initially retained (the logarithmic transformation retained in the RES), or the value X=\ (the linear form is sometimes retained). The results obtained are detailed in Table 10.3. In any event, the test concluded that there was a statistically insignificant difference between the model obtained and the initial model. Conversely, the difference between the model obtained and a linear form model may be significant, and that is the case three times out of four for risk and severity equations. Hence, there is a risk of error if one accepts the linear form for the main exogenous variables in the risk and severity equations, but conversely one cannot reject the linear forms in equations for road demand. At a statistical level, there is thus no significant difference between the model with Box-Cox transformations on the primary explanatory variables and the initial model. We can then turn to other criteria that might lead us to prefer one or other of the two cases in question. If the coefficients linked to the dynamics and impact of the exogenous variables do not vary much, there is a difference to be seen in the form of the endogenous variable's curve of elasticity to

246 Structural Road Accident Models the exogenous variables, which is more general when a Box-Cox transformation is performed on the exogenous variables, and in the functional form itself that links the endogenous variable to the exogenous one. To give an example, if one takes a value close to -2 for the Box-Cox parameter associated with household consumption (an explanatory factor behind motorway traffic), then the elasticity of household income to traffic is assumed to decline over time, falling from 0.5 in 1975 to 0.2 in 1992, whereas it is assumed to be constant and close to 0.4 if parameter X is zero. As for the functional form linking traffic to household consumption, there is a long-term saturation effect in the first case that does not exist in the second. 10.3.2. Measuring elasticities The elasticities of explained variables are assumed to be constant in the initial model, whereas in the optimal model they are assumed to develop monotonously, growing or decreasing depending on whether the Box-Cox parameter has a positive or negative value, and as an example we detail the annual elasticity values obtained for 1975, 1984 and 1992 (see Table 10.3: The univariate models in estimated fiinctional form, 1975-1993). The elasticity values are consistent with those found in the literature on the subject. On French data, Madre and Lambert (1989) have estimated traffic elasticities values on main roads and motorways, using on econometric approach on an annual basis. Recent results have been obtained with quarterly time series (Bresson and al., 1997) and other estimations on a monthly basis as well as their development over time have been proposed (Bergel, Nespoux, 1997). As for risk and gravity, elasticities values to their determinants have been estimated alltogether at the national, level (Jaeger, Lassarre, 1998) The elasticities of traffic, of the number of accidents and of the number of fatalities to economic activity as measured by household consumption are positive and less than 1, the average being 0.3 over the period , but it can be seen that the separation of the two effects of consumption and the extension of the network, using a model based on two strongly correlated variables, can over-estimate the effect of extending the network, which is seen to average 0.7 over the period. The elasticity of traffic to fuel prices is more important on motorways than on main roads (the average being -0.2 and -0.4). Fuel prices and the development of the motorway network appear to have an indirect effect on the numbers of accidents and fatalities through the intermediary of traffic. The elasticities of the numbers of accidents and fatalities with respect to traffic are positive and less than 1, with an average value on the period that vary from 0.4 to 0.8 depending on the dependant variable and on the network. Climatic variables generally have a lesser effect, except for the impact of temperature on the numbers of accidents and fatalities on the motorway network, which are twice as high in

The RES Model by road category for France 247 Table 10.3. Univariate models with estimated functional form on the main exogenous variables, 1975-1993 ACCIDENTS Toll motorways Main roads

TRAFFIC Main roads Toll motorways Dynamics A(l)

-0,265

***

DEATHS Toll motorways Main roads

-0,471

-0,344

-0,209

-0,331

A(2)

-0,518

A(12)

-0,969

A(14)

0,503

0,369

0,242

0,237

***

***

***

***

*** ***

Exogeneous variables (*) Traffic on toll motorways

-0,381

***

-0,976

***

-0,379

***

Fuel price

Toll motorways' network length Rain

***

-2,283 0,51/0,28/0,18

***

*** 1,424 0,ll/-0,13/-0,08

*** **

**

***

**

***

-0,364 0,59/0,45/0,36

**

***

***

-200 0,05/0,06/0,06 0,04/0,05/0,05

***

***

-0,01

0,06

***

***

0,05 0,33/0,32/0,34

***

-1,333 0,96/0,81/0,62

***

***

*

0,01

*

***

0 0,05/0,06/0,06

**

0,06

*

***

0,02 0,43/0,42/0,45

**

0,03

*

0,04

0,03 0,23/0,24/0,23

**

***

0,06

170,166 2,935 0,686

286,009 1,053 0,331

-65,111 23,544 0,455

170,851 2,917 0,27

accepted 0,13 accepted 0,99 accepted 0,56

rejected 0,01 accepted 0,54 accepted 0,61

accepted 0,56 rejected 0,0038 accepted 0,14

accepted 0,12 accepted 0,92 accepted 0,64

rejected 0,04 accepted 0,34 accepted 0,1

209 LL0=340,468 HO accepted

209 LL0=484,233 HO accepted

209 LL0=170,118 HO accepted

209 LL0=285,942 HO accepted

209 209 LL0=-65,416 LLO= 169,615 HO accepted HO accepted

LL0=341,047 HO accepted

LL0=484,515 HO accepted

LLO-161,642 HO rejected

LLO-284,857 HO accepted

LL0=-69,242 HO rejected

*

0,02

0,16

342,208 0,661 0,405

484,603 0,182 0,349

accepted 0,32 accepted 0,76 accepted 0,13

Comparative tests Sample (01/75-12/93) HO:lambda=0 vsHl:lambdaoO H0:lambda-1 vsHl:lambdaol

White noise

***

0,102

***

-0,087 -0,24

-0,217 0,82/0,68/0,64

***

-0,403

0,808 0,24/0,29/0,34

-0,01 Goodness of fit Log likelihood SSE R2 Tests on residuals Sampling P-value Normality

***

***

0,06 Winter temperature

***

-0,283 0,61/0,59/0,56

-0,01 Summer temperature

-0,188

***

-0,01 Frost

***

-0,552

-0,04 0,44/0,42/0,40

Traffic on main roads Final household consumption

***

***

***

LL0=167,475 HO rejected

(*)The successive results are: for the dynamics, the parameter and its significance (* 1 and <2; *** for >2) and for the exogenous variables: the significance, the lambda value, and the elasticity value (annualmeanfor 1975, 1984, 1992)

248 Structural Road Accident Models summer as in winter, the highest average value being 0.4 in the summer, for the number of deaths on motorways. The vectorial approach confirms the existence of common interaction effects and common external effects that are not explicitly modelled; these effects are found to be greater for the numbers of accidents than for the numbers of fatalities, and still greater for traffic. The parameters of the model's exogenous variables may differ, and the differences mainly relate to the effect of economic growth, which would be more important on motorway traffic than on main roads traffic (0.7 and 0.3 respectively), and to the effect of traffic, which would also be more important on the number of accidents and deaths on motorways than on the number of accidents and deaths on main roads (0.7 and 0.5, and 0.6 and 0.4 respectively).

10.3.3. Short and medium term simulations Simulations for the development of the number of accidents and fatalities were carried out on periods of several years and using known exogenous variables. This exercise reveals the errors attributable to a single model and deliberately ignores the errors made on exogenous variables, which could be significant in a real forecasting exercise. It can be seen from these examples that the model's dynamics do take account of the trend over the coming years, which makes it potent for the medium term. The monthly development profiles are more or less well reproduced: the exogenous variables available are not sufficient to explain the development of these profiles from one year to the next (the residual integrates, for example, the time-of-year effect, the effect of random events or the introduction of a road safety measure). 10.4. C O N C L U S I O N

We will retain three types of conclusion. The first is the result thrown up by tests on the Box-Cox transformation: the logarithmic form cannot be rejected (X=0 is accepted for the six equations), but nor can the linear form be accepted, at least not for the two levels of risk and severity {X=\ is rejected in three out of the four equations of risk and severity). Given comparable quality, the model with Box-Cox transformation differs from the model with the logarithmic form in terms of the shape of the elasticity curves of the endogenous variable with respect to the main exogenous variables and in terms of the functional form itself, which may reveal saturation effects. The second is the fact that the vectorial approach enables account to be taken of interaction effects between the endogenous variables for risk exposure, risk and severity on main roads and toll motorways; these effects appear greater for the numbers of accidents than for the numbers of fatalities, and greater still for traffic. The division of the effects is slightly different, and one

The RES Model by road category for France 249 is also able to take account of the effect of seatbelt wearing by front-seat passengers. Finally, we find that the vectorial model makes it possible to break the link with the trend effects taken into account in the dynamics, and hence to isolate the short-term effects. It is in that respect that it constitutes a potent tool for the short and medium term, to the extent that models are constructed of these two types of effect. The focus of this research is now on finding an interpretation of the models' functional forms, and this would mean the model-builder could choose between the different values for the BoxCox parameters (X=0, X estimated, ^=1), when they do not differ at a statistical level alone. The time-of-year effect remains to be integrated as well, and work to that end is under way. Above all, the way the behavioural variables, relating to speed excess and to seatbelt wearing, can also be taken into account, needs to be appreciated. At the moment, the database has been extended to the 1975-1998 period, and the model to the whole of the national road network, including the complete motorway network, and to all categories of accident (fatal/non fatal) and casualty (killed, seriously injured, slightly injured). It is being applied to the recent past so an analysis can be made of road safety performance in 1998, when there was an exceptional increase in the severity of accidents throughout France, for different reasons of a temporary nature, but probably also because of a slippage in driving behaviour resulting in an increase in speed over the past two years. 10.5 REFERENCES Azencott, R., B. Durand, B. Girard, Y. Girard and C. Vernier (1997). Un logiciel de modelisation de series temporelles. DIAM-Recherche/SAMOS/SCIPRE, Universite Paris 1Pantheon-Sorbonne. Bergel, R. (2000). Modelisation multivariee des indicateurs d'insecurite routiere. Rapport sur convention INRETS/DSCR. Arcueil. Bergel, R. (1997). Multivariate modeling of accident risk on the national road network. In: Proceedings of the Conference Traffic Safety on two Continents, Lisbon. Bergel, R., B. Girard, S. Lassarre and P. Le Breton. (1995). A model of seasonally corrected indicators of road safety. In: Proceedings of the Conference Road Safety in Europe and Strategic Highway Research Program, Prague. Bergel, R., V. Nespoux (1997). Modelisation des trafics terrestres de voyageurs et de marchandises. In: Actes du Seminaire de VINRETS sur la Modelisation du trafic, INRETS, Arcueil. Bresson, G., J.L. Madre et A. Pirotte (1997). Prevision du trafic automobile sur differents types de reseaux aux niveaux national et regional. Rapport sur convention INRETS/SETRA, Arcueil. COST 329 (1999). Models for Traffic and Safety Development and Interventions. Final Report of the Action, Directorate General for Transport, European Commission. Brussels. Gaudry, M. (1984). DRAG, Un modele de la demande routiere, des accidents et de leur gravite, applique au Quebec de 1965 a 1982. Centre de recherche sur les transports, Universite de Montreal. Gaudry, M., Fournier, F. et R. Simard (1993-1995 et 1997). DRAG-2, Un modele

250 Structural Road Accident Models econometrique applique au kilometrage, aux accidents et a leur gravite entre 1957 et 1989 au Quebec. Societe d'assurance automobile au Quebec, Quebec. Jaeger, L. et S. Lassarre (1998). Pour une modelisation de revolution de Vinsecurite routiere Estimation du modele TAG. INRETS, Arcueil. Madre, J.L. and T. Lambert (1989). Previsions a long terme du trafic automobile. CREDOC, Paris.

Postscript and Prospects 251

11 POSTFACE AND PERSPECTIVES Sylvain Lassarre The purpose of this chapter is to evaluate the eight models that were described in detail in the preceding chapters, with the aim of drawing lessons for their improvement. We will examine these models from the standpoint of road safety research to assess their relevance as regards the quantification of risk factors. Does the use of flexible functional forms and the three-tier "exposure-accident-severity" structure yield satisfactory results? Thanks undoubtedly to a systemic approach; models have been constructed of retroactive effects that give a better account of the complexity of the factors at work. Has this increased our knowledge of the influence of risk factors? To fmd out more, we will review the risk factors one by one, broken down by category-mobility, economics, demography, regulatory measures, and so on-in order to determine the contribution made by models belonging to the DRAG family to explaining the effects of these factors on the risk triangle. From this starting-point, we will go on to identify the means for improving the way these factors are taken into account for the analysis of road risk and advocate that action be taken at the level of the data (quality and comprehensiveness), of the model's structure (comprising three or more than three levels) and of the breakdown of variables by type of road and user, indeed their desegregation by individual road location or vehicle x driver. We will then go back to our assessment of the relevance of these models from the viewpoint of risk management. Is the transfer from research to the operational level an easy one? On what conditions? Knowing there is a great need in this respect, whether at the level of countries or blocs (Europe), these models are generally suitable tools available to government departments and private companies for performing three functions identified by the COST 329 group (1999): (i) establishing an explanatory assessment for interpreting the annual development of the trend; (ii) evaluating the effectiveness of safety measures, and (iii) setting quantified objectives and making predictions with the help of scenarios. Lastly, after casting a critical eye on the new zero risk spectators, we will end by looking ahead

252 Structural Road Accident Models at the contribution that DRAG models could make to solving the classic problems of speed and drink-driving, and the more recent problems resulting from the introduction of new technology or new methods of driving instruction.

11.1. RELEVANCE OF MODELS FOR UNDERSTANDING THE INFLUENCE OF RISK FACTORS The models belonging to the DRAG family are designed to convert into equation form the direct and indirect effects of risk factors on the toll of road accident victims. Their three-tier "exposure-accident-severity" structure makes it possible to separate out the effect of one factor on the number of victims via the accident level and the exposure level. Propagation and retroaction effects can thus be taken into account by the form of the model through combining elasticities. While the model performs satisfactorily as regards the treatment of systemic interaction phenomena, as, for example, between the average speed driven and the rate of seatbelt wearing, and their impact on risk, it is important to evaluate the relevance of the results obtained in constructing models of the effects of risk factors. Applying the metaphor of the street lamp, that entails identifying zones that are lit and those left in shadow in terms of a factor, indeed a field, of road risk, and determining the reasons that lead to some risk factors being better treated than others. An immediate criterion is the frequency with which this factor turns up in the six models described in part I. The factors can then be classified in three groups according to a scale of incidence: high, average, low (Table 1). Table 1. Breakdown of the models' variables taken as a factor of risk according to frequency of use High

Average

Low

- total traffic

- composition of motor vehicle stock,

- speed.

- weather conditions

- traffic breakdown by type of road,

- detailed characteristics of vehicles, users

- fuel prices

- rate of seatbelt wearing,

and the infrastructure.

- economic activity

-sales of alcohol,

- police monitoring activities,

- regulatory measures

- rate of vehicle occupation, - supply of hospital services for rescuing and caring for the wounded

The second criterion relates to the degree of proximity between the variable selected in the model and the risk factor it is supposed to represent. A third criterion has to do with the quality of the statistical information. While the risk factor "rain" can be measured in terms of the duration of the rain or its height, there are many other factors that cannot be measured directly. That calls for the implementation of an assumed chain of causality to link the risk factor to a

Postscript and Prospects 253 measurable proxy variable. For instance, driving experience can be measured indirectly in terms of the proportion of novice drivers in the whole driver population or, better still, in total traffic. If that information is lacking, one can use instead the proportion of 18 to 20-year-olds in the population. These last two criteria will be evaluated only superficially. A fuller evaluation would require us to go back to the detailed documents on the establishment of time series found in the models. Nevertheless, we will try to gauge the distance between the risk factor and the indicator selected by the model-builder. Let us start by what is well covered in terms of variables in the models: total motor vehicle traffic measured by the number of vehicles x kilometres; weather conditions (temperature, rain, snow); fuel prices; and indices of economic activity. The first two variables correspond to well identified risk factors, since mileage driven is a measure of risk exposure while icy roads, rain and snow falls create more dangerous driving situations. It is important to understand that the traffic taken into consideration includes motorised four-wheel vehicles and sometimes-motorised two-wheelers as well, but not journeys by bicycle or on foot. Traffic is sometime replaced by sales of gasoline and diesel, which introduces a bias, especially for countries that share borders with a number of neighbouring countries. The last two variables cannot be regarded as risk factors. The price of fuel has an influence on speed, which is a purveyor of road accidents, and on driving style. In the models it plays the role of an intermediate variable for speed, provided traffic is in the equation. The indices of economic activity used are those for household consumption by type of product and industrial output by sector of activity. The first are designed to provide a breakdown of the purpose of the journey (shopping, leisure, travel between home and work) as a proportion of all journeys by people. Assuming there is a differential risk according to the purpose, a journey generated by a given reason takes places in a specific environment and in more or less dangerous driving conditions. The second set of indices is used to distinguish the shares in total traffic of different categories of heavy goods vehicle (HGV), reflecting the respective outputs of the industrial and agricultural sectors. For example, the timber industry in Quebec generates long-distance journeys by articulated lorries, while the construction and public works industry generates trips by smaller trucks over short distances. The availability of economic statistics encourages us to use these indicators even though they do not have a direct relationship with risk factors. Risk factors in this case are linked rather to different sizes of HGV traffic or to private motorcar traffic in different conditions (day/night). Here is what the variables cover to an average extent: composition of the vehicle stock; breakdown of traffic by type of road; rate of seatbelt wearing in the front seats of private cars; sales of alcohol; and road safety regulations.

254 Structural Road Accident Models All these variables may be directly considered as risk factors or as being linked to risk factors. The first two serve to distinguish risk according to type of user and type of road. Not all vehicles on the road present the same degree of aggressiveness or the same protection in a collision. The heterogeneity of the motor vehicle stock (two-wheelers, private cars, HGVs and buses) is a source of danger. In fact, the vehicle stock (updated on the basis of new registrations) is used in place of the mileage driven, which is not available at monthly intervals. It should be noted that the models do not contain measures for the risk exposure of the most vulnerable users, namely pedestrians and cyclists. The share of motorised traffic by type of road (urban, motorway, main roads, secondary roads) is a relative measure of risk exposure used to convey the risk differential induced by the particular infrastructure characteristics of each type of road. The other two variables, rate of seatbelt wearing and sales of alcohol, reflect the degree of risk in driver behaviour (especially drivers of private cars). The protective measures taken by the user are aimed at reducing the risk, whether it be the wearing of seatbelts in the front and back seats of private cars and light commercial vehicles', the wearing of crash helmets by motor cyclists, moped riders or cyclists, or the use of protective devices for children transported in vehicles (special seats). Of all these protection activities, the only one included in the models is the rate of seatbelt wearing in front seats, as observed during the day (not at night). In that regard, it is interesting to compare estimates of benefit in terms of fatalities when the rate of seatbelt wearing in front seats increases from 80% to 100% as between the TAG model and an in-depth analysis of fatal accidents involving cars. In the TAG model, the average elasticity is 0.2, meaning that for a 25% increase in seatbelt wearing (from 80% to 100%)) there is a 5%) reduction in the number of fatalities, equivalent to some 450 lives in 1990. An in-depth analysis of the circumstances of fatal car accidents that year, in which 50%) of those killed out of a total of 6,000 were not wearing a seatbelt, estimated that there would be a gain of 1,800 lives from a 100%) rate of seatbelt wearing (Foret-Bruno, Le Coz, 1999), of which about half would come from fataf accidents where the vehicle rolled over (somersault). Taking account solely of front seat occupants, one arrives at a gain of 1,500 lives, which is three times greater than that estimated by the TAG model. The size of that difference raises a problem, and calls for thought about the value and form of the elasticities. Estimates derived from epidemiological surveys are made on the basis of samples of collisions that date from the 1980s, and even the 1970s, which reflect the conditions of the vehicle stock and drivers' practices at that time. Changes have appeared in the

' The wearing of seatbelts is not yet mandatory for all drivers of HGVs in some countries, nor for bus passengers on inter-urban journeys.

Postscript and Prospects 255 composition of the vehicle stock and the driver population, which may alter the values for relative risk estimated at the time. That is why a longitudinal approach may lead to different evaluations from those in the cross-sectional approach, which is dated. For certain risk factors, such as seatbelt wearing, we used exposed/not exposed epidemiological surveys to estimate their intrinsic effectiveness equal to the ratio of the probabilities of being killed in a collision with and without a safety belt for a driver of a private car. Wearing a seatbelt in the event of a collision reduces the risk of being killed by 50% (Hartemann, 1985); in the case of somersaults, the effectiveness is close to 90%; and in side shocks, it is nearly 20%) (Evans, 1991). It then becomes possible to calculate a theoretical elasticity of the number of killed to the wearing of seatbelts on certain assumptions. All things being equal, from a multiplicative model of the number of killed, where T is the proportion of drivers wearing seatbelts, we have (11.1) n = A kilom" {er + \-T) and its derivative (11.2) dn = A kilom" (e - \)dT. One deduces (11.3)

dn _ (e- \)dT _ (e- l)r

dr

n

T

l + {e-V)T

l + (e-V)T

The elasticity of the number of fatalities among drivers of light vehicles is a decreasing monotonous function of the proportion of drivers wearing seatbelts (11.4) el =

--^^-'^' l-(\-e)x

The elasticity of seatbelt wearing is quite low when it first becomes mandatory. It grows in absolute terms when safety belts give greater protection (Figure 1). For example, for a rate of belted drivers of 70%) (estimated from fatal accidents in 1990), the elasticity is equal to '• '-— = -0.53 . If the proportion of belted drivers rises from 70%) to 100%), the gain in 1-0.5x0.7 30 terms of fatalities is equal to 100x 0.53 x — = 23.1%o. This quantity is also called the attributable risk (Kleinbaum et aL, 1982), which is the proportion of fatalities attributable to the non-wearing of seatbelts, namely 30%. If one applies the gain to all front seat occupants of private cars, one arrives at a gross gain of 5,000 x 0.231 = 1,153, which is less than the 1,500 estimated from the analysis of fatal accidents but more than double the figure generated by the

256 Structural Road Accident Models TAG model. 0

-0,2

.

0,1

0,2

0,3

0,5

0,6

0,7

0,8

0,9

=*

-0,4

^^^^^^^^^^^ --^.___ I ^ ^ ^ ^ ^ ^ T C T ^ ' ^ ^,^

"^^z^7^^

-0,8

e=0,6 ^^•

_X---Tr\a

1.0

e=0,5 •

e=0,4

\ .

-1,2 -1,4

\.

-1 .6

Rate

of

seatbelt wearing

Figure 1. Elasticity of the number of LV drivers killed to the rate of safety belt wearing according to intrinsic effectiveness One way of improving the construction of the model so as to remedy this under-estimate of the elasticity is to continue developing the preceding simple model with the aim of predicting a theoretical form of the elasticity of the total number of accidents to the rate of seatbelt wearing. If we now add the internal risk for front-seat and rear-seat passengers to that for drivers, as well as an external risk vis-a-vis vulnerable users, tav and tar, the rates of occupation of front and rear seats are of the order of 1.2 and 0.2. We assume that the risk for rear-seat passengers is equal to that for unbelted front-seat passengers. The relationship of the internal over the external risk is estimated by the ratio G^i of the number of fatalities among vulnerable users hit by a vehicle and that of occupants of vehicles killed. We assume it to be constant and equal to 0.2 and to be independent of seatbelt wearing without homeostasis of the risk. The total number of fatalities involved in a collision with at least one LV is given by (11.5) n = Akilom"' (f,,,(er+ l - r ) + ^ ^ , + G ; ) . The elasticity of the total number of fatalities to the wearing of seatbelts, which is equal to (11.6) el-=

1 +1^^ + G. - (1 - e)x

is even lower and levels off for high rates of belt wearing (Figure 2).This function may be approximated by a function of the second degree or of the first degree if one considers the denominator close to a (11.7)

el = -r^„(l - ^)^(l - C - G- + (1 - e)x)

el = -t^X^-e)x

Postscript and Prospects 257 and the total number of fatalities involved in accidents with an LV for an effectiveness of 0.5 It follows from this that in a model of the number of fatalities; the theoretical form of the elasticity to the rate of seatbelt wearing should be of the quadratic type (11.8) Logn = log A + alogkilom + ar + br^ Taking the logarithm of the proportion of belted drivers, the elasticity become constant and does not reflect the quadratic form in the second approximation of the theoretical elasticity nor the linear form in the first approximation. The regulatory measures introduced into the models are often in the form of 0-1, such as the introduction of a speed limit, mandatory wearing of crash helmets or a special driving licence. However, many other measures are brought progressively into effect, such as technical vehicle inspections or the introduction of a points system for driving licences, and giving an account of their effects calls for a less rudimentary approach. At present, only measures that have an immediate and national impact are included in the models. Sales of alcohol are used as intermediate variables for measures of the exposure of drivers taking the wheel under the influence of alcohol. There is assumed to be a relationship between sales, consumption and traffic. In some models, the quantities of wine, beer and spirits sold are aggregated, taking account of the alcohol content of the drinks. In other models, series by category of drink are introduced.

total 0,5 e=0,5

Figure 2. Elasticity to the rate of seatbelt wearing of the number of LV drivers killed The following are all the variables that the models take only minor account of: statistics for speeds driven (average, V85, standard deviation); detailed characteristics of vehicles, users and the infrastructure; police monitoring activities; vehicle occupancy rates; and supply of hospital services for rescuing and caring for people injured on the road. There is only one model (France) in which average speed is explicitly built in as a risk factor, and furthermore it emerges as a major contributor to the development of the trend with the greatest elasticity. In

258 Structural Road Accident Models the other models, the fuel price variable replaces it as a risk factor. The detailed characteristics of vehicles, users and the infrastructure are rarely introduced. A breakdown of private cars by age group, size or horsepower is introduced in one or two models, but no indicator reflecting the progress made in the shock resistance of new vehicles has been produced. The same is true of the infrastructure, where capital investment measures and the installation of safety devices (such as roundabouts) seem difficult to grasp. The characteristics of users, and above all drivers, are sometimes taken into account by indicators, such as the proportions of novice drivers, young drivers and older drivers. Nothing is to be found about women drivers, apart from a series on pregnancy. A series on the sale of medicines (in Quebec) is used to estimate the influence of taking medicines on drivers' psycho-physiological state (vigilance). The intensity of police monitoring activities is the only indicator found in models that seek to give an account of checks and enforcement activities carried out by police forces, and is measured in terms of the number of hours of police presence on the road. One could try to draw a distinction by the type of offence targeted (exceeding the speed limit, for example). Vehicle occupancy rates are not included in the models at all, nor is the supply of hospital services for treating those injured on the road. Moreover, although it is crucial to explaining the number of victims, there is no series on vehicle occupancy rates, which is seasonal and may be linked to temperature in current models. Lastly, the progress made in rescuing those injured in road accidents and treating them in hospitals is not integrated in the models.

11.2. OUTLOOK FOR RESEARCH IN CONSTRUCTING RISK MODELS The previous observation showed us that some risk factors were given precedence more because of their availability in an existing statistical information system than for their relevance in risk terms. To have them included in the construction of models, we are going to suggest solutions we hope model developers will test. To make models more comprehensive as regards their coverage of risk and the accuracy of their quantification, we think there is need for more work in the following areas: data extraction; the addition of levels to the structure; a breakdown of indicators by type of road and type of user; and a further desegregation by location or vehicle x driver. 11.2.1. Data extraction The choice of additional risk factors to be introduced into the models should be guided more by the strong influence of the risk than by ease of access to statistical information. The major risk factors still are speed and drink driving, followed by driving experience, the heterogeneity of the vehicle stock and the quality of the infrastructure. The risks associated with these risk

Postscript and Prospects 259 factors have been estimated in epidemiological surveys, such as, for example, the relative risk of being killed as a function of the quantity of alcohol in the blood or the physical mass of the vehicles in the event of a collision. Variables conveying these risk factors have to be given priority in the models. Comparing the results of aggregate models with those of desegregated models as regards the major risk factors is a fount of knowledge, as we saw in the example of the effect of wearing a safety belt. Evaluations made on cross-sections of particular samples have to be complemented by longitudinal analyses across broader fields in order to judge the factor's aggregate effect. The same rationale applies to the factor of the heterogeneity of the vehicle stock, where there is a difficult balancing act to perform between the internal and external risk differentials between "large and small" vehicles. Similarly^ safety measures have to be built into the model, whether they are measures relating to the protection of the occupants of an LV or other users, or regulations governing speed limits and breathalyser tests. One of the objectives of constructing models is to evaluate the effectiveness of safety measures, taking account of confusion factors. Measurements of risk exposure do not exist for vulnerable users-pedestrians and cyclists-whereas urban transport policies encourage these modes of transport. Vehicle occupancy rates are known only vaguely, whereas they should complement measurements of the number of vehicles x kilometres to obtain the number of passengers x kilometres. It can clearly be seen that measurements of risk exposure by type of user (pedestrians, cyclists, moped riders, motor cyclists, motorists, occupants of heavy goods vehicles and buses) are extremely rare and that a special effort needs to be made to develop them, as the COST 329 group recommends. 11.2.2. Adding levels to the structure Two models demonstrate the merit of filling out the risk triangle by adding more levels: the Norwegian model with the vehicle stock and the French model with average speed driven. In the first model, the short and medium term elasticities of fuel prices were separated out, and in the second model retroactive effects between seatbelt wearing and speed were tested. One might consider including a separate level for the numbers of offences recorded by the police, which would be explained by variables relating to user behaviour on the road and the intensity of police checks on drivers. Another avenue for research is to use a system of simultaneous equations to test the retroactions of the levels of risk on risk exposure, the behaviour of users and the activity of police forces, as K. Vernier and M. Gaudry did by means of a desegregated analysis of risk on country roads. Lastly, U. Blum and M. Gaudry opened up a path for constructing a model of the moments of second order of the risk indicators. The average risk and the variance are built into the model and interpreted jointly.

260 Structural Road Accident Models 11.2.3. Breakdown of indicators by user and road types Breaking down risk indicators by type of user and type of road is a rewarding strategy when one is seeking to record detailed effects that would be drowned in an aggregate model at the level of the entire road system. The impact of the stock of motorcycles and the wearing of crash helmets shows up better in a model of risk indicators relating to accidents involving motor cycles than in a general model, because they are a small number in the overall statistics despite their very high risk. The same is true of statistical information by type of road (urban, motorway, countryside); a breakdown by road type is a must, knowing that the levels of risk and their development are very different from one type to another. Two forms of model can be selected to handle these multidimensional series of the numbers of accidents and victims by type of user or road. A distributive model (probit, multinomial logistic) yields a model of the proportions of accidents and victims divided among these types. A VAR-type multivariate model may be used to produce a joint model of risks on different types of road, as in the book on the multivariate model by R. Bergel and B. Girard. Lastly, a combination of temporal and spatial dimensions in the Norwegian model yields a more solid estimate by providing more information. 11.2.4. Disaggregation by location or vehicle x driver Aggregate models cannot be used to study certain detailed factors relating to the characteristics of vehicles, users or the infrastructure. One has to switch to a desegregated model of the Poisson, probit or logistic type, while retaining the three or four level "exposure-behaviouraccident-severity" structure and the use of flexible forms to search for non-linear relationships. M. Gaudry and K. Vernier give a good example in their analysis of the risk generated by the characteristics of the road surface. 11.3. RELEVANCE OF THE MODELS FOR MANAGING ROAD SAFETY It is strange that the zero fatality risk option is now being taken up in the field of road transport after being launched by the Swedish transport authority, although in other fields of transport it is admitted that zero risk does not exist. For more than 20 years people have advocated using the ALARP (As Low As Reasonably Possible) principle for managing the risk of complex systems, such as nuclear power stations and petrochemical plants, in a bid to reach an acceptable level of risk at a reasonable cost. Even though the road transport system is very badly designed from the safety standpoint and kills thousands of people every year, the zero risk policy seems to be a step backwards in the direction of denying the risk and not wanting to accept a residual risk. The role of risk models is fundamental, since thanks to them the risk is

Postscript and Prospects 261 quantified. It is then possible to estimate the potential benefits that can be obtained through measures targeted at risk factors: reducing speed, increasing seatbelt wearing, improving road surfaces .... Even if one were to open the floodgates wide, meaning if one were to give limited values to the models' explanatory variables-which is in itself a risky venture, since one is likely to stray out of the model's field of validity-the risk of an accident and of being killed on the road is still there, because, as we have said, the existing system was not designed to ensure optimal safety. But we think aggregate or desegregated models provide some help to managers and decision-makers responsible for road safety, since they can fulfil the three functions identified by the COST 329 group: producing an explanatory assessment of the annual development of road risk; evaluating the efficacy of road safety measures; and setting quantitative targets with the help of scenarios. Few government agencies or private companies have developed an information system for processing chronological series on road risk for the purpose of formulating a road safety policy. The SAAQ in Quebec is an exception, but it is difficult to transfer this information processing technology by means of models even though there is a great need for decision support systems. Studying the obstacles to transposing the models developed by research on to an operational level would teach us more. The assessment of the annual development of risk by risk factor provided in the Swedish model is a good example of the explanatory power of models for highlighting the influence of different factors and judging the effectiveness of a policy. There is a need for dialogue between researchers and managers in order to initiate a reciprocal learning process enabling models to be used as a partial decision-support tool. Evaluating an intervention and analysing the reasons for its success or failure should be the main element in the decision-making process if policy is to be better directed at its objectives. One sometimes wonders whether there is any real political will to implement this evaluation process. It might be necessary to create an independent agency to take responsibility for evaluating the effectiveness of policy, but it would also need to have adequate human resources, since we have seen that the safety measures taken at the moment are complex processes which cannot be represented by all or nothing variables and necessitate a major data extraction effort. In 1994 an OECD group of experts advocated the setting of quantified objectives to rationalise the decision-making process and the implementation of road safety policies. Applications are beginning to emerge in Scandinavian countries and in the Netherlands. There is a crucial need at a European level. A European model, CEDRAG, has been proposed following a tender issued by the European Commission, but it did not attract any offers. Objectives cannot be set for Europe as a whole without first considering the development of risk in each country, and then integrating the data in a multivariate model. Some countries, such as Spain, are improving their road safety performance, while others, such as the Netherlands, are standing still. Each country has its own rate of progress in the area of

262 Structural Road Accident Models road safety, even though the influence of the main risk factors is the same everywhere. A comparative analysis would inform us about the effectiveness of the global or specific solutions applied in different countries. As far as speed and drunk driving are concerned, a comparative analysis of the results of models belonging to the DRAG family could teach us a lot. Would it confirm the 1,2,4 scale of risk for material damage, personal injury and fatal accidents as a function of the average speed? Are the estimated effects of speed limits consistent? In the case of drinking and driving, the effects of reducing the legal blood-alcohol limit to 0.8, 0.7, 0.5, 0.2 or 0 g/1 could be amalgamated by using the experience of different countries. The impact on risk of the type of drink consumed (wine, beer or spirits) could be studied. It is clear that we are only at the beginnings of a joint analysis of models by country and that by exploiting the particular conditions of each as variations in an experimental field, it is possible to extract fresh knowledge about the effects of risk factors or safety measures. Another possibility to explore in future research is prediction supported by scenarios. Models in the DRAG family are long-term models that are particularly suited for that exercise. At the moment, demographic and economic scenarios can be used for making forecasts, as we saw in Quebec. The problem is that one has to predict the effects of new factors or new measures that have no equivalent in terms of variables in the model. For instance, introducing a probationary driving licence for young drivers or imposing restrictions on night driving by young drivers would call for the construction of new models. There again, it is the role of research to anticipate demand and develop models capable of evaluating the effects of policies targeted either at driver training, or at vehicle design, or at traffic regulation. New risks become more pressing as those linked to the intake of medicines or illicit drugs in combination with alcohol, new driving aids appear on the market, and new types of vehicle and road use emerge, which all present challenges to model-builders, which must push them towards more original models. 11.4 REFERENCES COST 329 (2000 forthcoming). Models for traffic and safety development and interventions. Final report, European Commission, Directorate General for Transport, Brussels L. Evans (1991). Traffic safety and the driver. Van Nostrand Reinhold, New York. J.-Y. Foret-Bruno, J.-Y. Le Coz (1999). Les gains potentiels de victimes avec les nouveaux systemes de retenue en choc frontal. Notes de cours de la Formation continue de I'Ecole Nationale des Fonts et Chaussees, Cycle de securite routiere, Paris. F. Hartemann (1985). La ceinture de securite. Le concours medical, 107-30, 2841-2847. Kleinbaum, D. G., L . L. Kupper, H. Morgenstern (1982). Epidemiological research Principles and quantitative methods. Van Nostrand Reinhold, New York. OECD (1994). Road Transport Research. Targeted road safety programmes. OECD, Paris.

The L-1.5 Program for BC-GAUHESEQ Regression 263

12 THE T R I O L E V E L - 1 . 5 ALGORITHM FOR BC-GAUHESEQ REGRESSION Tran Liem Marc Gaudry Marcel Dagenais Ulrich Blum

12.1

I N T R O D U C T I O N A N D STATISTICAL M O D E L

12.1.1 Introduction In applied regression analysis, the most important aspect in model specification is the choice of the functional forms of the dependent and independent variables. As Zarembka (1968) has noted, economic theory rarely indicates the appropriate forms under which the variables should appear, except for the signs of the regression coefficients which are expected to be positive or negative according to the assumptions made on the economic behavior of the dependent variable with respect to the changes of the explanatory variables. The three classical forms often encountered in econometric studies are the linear, semilog and log-linear forms due to their computational ease with a standard regression computer package. One way of letting the data detemiine the most appropriate functional form is the use of a class of power transformations considered by Box and Cox (1964). The main advantage of this approach is ^ Over the past 20 years, the five earlier versions of this algorithm documented in English (Liem, 1979, 1980; Liem et al, 1983, 1987, 1991 and 1993) or German (Liem et al, 1986), implemented and maintained since 1987 in the TRIO program (Gaudry et al, 1993 etc.), were supported at various stages by Transport Canada, by the M.E.S.S.T. and F.C.A.R. programs (iointly with the M.T.Q. and the S.A.A.Q. from 1990 until 1996) of Quebec, by the S.S.H.R.C. and the N.S.E.R.C. of Canada, and by the Alexander von Humboldt-Stiftung of Germany. This sixth version was directly supported by the SAAQ {Societe de Vassurance automobile du Quebec) contribution to the development of the international DRAG network (1994-1999), by the National Sciences and Engineering Research Council of Canada (N.S.E.R.C.C.) and by the Deutsche Forschung-Gemeinschaft (DFG) of Germany. The work also benefitted from Marc Gaudry's tenure as a 1998 Centre National de la Recherche Scientifique (C.N.R.S.) researcher at BETA, Universite Louis Pasteur and UMR CNRS 7522 and from Ulrich Blum's 1999 guest professorship at Universite de Montreal. Richard Laferriere recently suggested inclusion of one of the B? measures computed in this version.

264

Structural Road Accident Models

that statistical tests can be performed on the Box-Cox parameters to discriminate the estimated functional forms against the classical forms which all appear as special cases of the BoxCox transformation. Early applications of this transformation can be found in various fields of economic analysis: monetary economics [Zarembka (1968), White (1972), Spitzer (1976, 1977)], income analysis [Heckman and Polachek (1974), Welland (1976)], production theory [Appelbaum (1979), Berndt and Khaled (1979)], and transportation [Kau and Sirmans (1976), Hollyer et al (1979)]. The assumption originally made by Box and Cox (1964) that the transformation, in addition to its main puipose of choosing the appropriate functional form, also renders the distribution of the transformed dependent variable nearly nomial and homoskedastic may be unrealistic, since in the case where the residuals are heteroskedastic, Zarembka (1974) has shown that the estimated Box-Cox parameter on the dependent variable will be biased due to the effect needed to make the transformed dependent variable more nearly homoskedastic. To deal with this problem, models which estimate the flexible functional form by allowing for the simultaneous correction of heteroskedasticity have been proposed by Gaudry and Dagenais (1979) and Egy and Lahiri (1979). Another important case is the problem of autocorrelation of residuals which usually exists with time-series data: Savin and White (1978) have considered the simultaneous estimation of the functional form and the first order autocorrelation, and Gaudry and Wills (1978) have extended the approach to multiple order of autocorrelation. Generahzing the procedure for a first autoconelation order outhned in Gaudiy and Dagenais (1979) to incorporate a higher autocoiTelation order simultaneously in the estimation of the functional form and heteroskedasticity, this computer program performs for a single regression equation the maximum likelihood estimation of the functional forms of the dependent and independent variables with the Box-Cox Transfoimation (BCT), and the functional form of heteroskedasticity with the Inverse Box-Cox Transfomiation (IBCT) in which the variables used to explain the error variance are themselves subject to BCT. Further details of the general procedure for a higher autocorrelation order are also given in a working paper by Gaudry and Dagenais (1978). The program gives a number of useful outputs such as the elasticities of the first three moments of the dependent variable (expected value, standard error and skewness), the marginal rates of substitution and elasticities of substitution among the moments, and among the independent variables, as well as forecasts made by simulation and maximum likelihood techniques. An application of the analysis of marginal rates of substitution and elasticities of substitution among moments of the dependent variable is found in Blum and Gaudry (1999).

The L-1.5 Program for BC-GAUHESEQ Regression 265

12.1.2 Log-likelihood function A. Model Following the approach considered above, for a sample of n observations the regression equation with a flexible functional form of the dependent and independent variables and a generalized structure of heteroskedasticity and autocorrelation of the residuals can be written: K

(12.1)

F/'^U^^,4J-^^)+U,,

(t = l,...,n)

k=i

(12.2)

ut

(12.3) where

vt

^vtf{Ztf'\ = ^ Pt'^i-t + wt, ^=i

(i) the dependent variable Yi and the independent variables X^i's are subject to the BCT which is defined as a power transformation with a parameter A on any positive real variable Vt: (12 4)

ifA^O, if A = 0,

y{X)^[{V,^-l)l\ ^ \lnT4

with the corresponding "Box-Cox-Gaudry" BCG or inverse function IBCT:

(12.5)

^/'^"-

p r

+ lj

Uxp(FO

if A 7^0,

ifA = 0,

where the expression in square brackets must be positive. Note that in (12.4), if A = 0, then y/^) = (y^A _ i)/A|;^^o = 0/0, which is an indetemiinate form. Using L'Hospital's rule gives lim V} ' = lim ^ ^wo(— = km V^lnVt = In 14; (ii) the first-stage vector of residuals u = {ut} is assumed to be heteroskedastic with mean E{u) — 0 and covariance matrix E\uu\ E(vf)f{Zt)

= 0 = diag(cjn, ...,cj„,„,) where LOU — E{uf) —

and f{Zt) is a function of a vector of M variables, Zt = (Zi^,..., ZMI), used

to explain the variance of ut. Note that these variables can be chosen from the set of independent variables Xkt's or totally exogenous; (iii)the second-stage vector of residuals v = {vt} is assumed to follow a stationary autoregressive process of order r, with mean E{v) = 0 and covariance matrix E{vv') — a^;^, where dlf^ — E{wf) and ^ is a symmetric matrix of order n, whose element can be expressed in terms of the pis\ (iv) and the third-stage vector of residuals w = {wt} is assumed to have a mean E{w) = 0 and a covariance matrix E{ww') = af^In where a'^^ is the variance of wt and In is an identity matrix of order n.

266

Structural Road Accident Models

In (12.1) it is clearly understood that some of the X^t's, such as the regression constant, the dummies and the ordinary variables not strictly positive, cannot be transformed by BCT. In (12.2) the functional form of heteroskedasticity for f{Zt) is assumed to be an IBCT with a parameter A^^ applied to a linear combination of the variables Zmt's which are themselves subject to BCT: 1/A.

M

(12.6)

f{Zt) = A

{X.m)

^0 + X ] ^rn,^m.t

+1

Hence the variance of Uf associated with a typical diagonal element of H can be expressed as:

E{ui)^uu=^''f(Zt) 1/A.

(12.7)

= i>HK So + ^SmZ',mi

+1

where the expression in curly brackets must be positive since it is associated with a variance. The constants c^o and ij) are necessary to preserve the invariance with respect to changes in measurement units of the Z ^ / s [Schlesselman (1971)]. The form (12.7) is quite general since it includes a great number of special cases encountered in empirical studies and fully discussed in Judge et al

(1985):

(i) Setting A^^ equal to zero yields a first form of heteroskedasticity:

(12.8)

LOtt — i^^ exp <^o + y ^6m,Z^vat

•• %l)^ e x p

Y^^mZ\ mt

= i^'fiZt)

where ip'^ = '0''exp(^o) and f{Zt) = exp This form is commonly called "multiplicative" heteroskedasticity since f[Zt) can also be expressed as a product of M exponential functions, or equivalently the logarithm of the variance UJU is a linear combination of the Zl^^ (12.9)

(12.10)

=

s\

r^'']le^^[8mZ^^r^]

^ In^tt = In ^2 + Xl^m^i^r^ = 5) + X^^mZ, -"mi

The L-1.5 Program for BC-GAUHESEQ Regression 267

where ^o — In V^^ = In ^^ + S^ Harvey (1976) considered a special case of (12.9) - (12.10) where every X^m is set to one: (12.11)

ujtt = i^^Y[Gw[^m{Zmt-'^)]=i^

(12.12)

^ Inujtt = ^0-^/2

^^^^^^

JJexp(^^Z^t)

~ •^) ^ ^0 ^" X / ^ ^ ^ ^ *

m

m

where ij) — wY[ exp {—8m) and 8Q = 8Q — ^^ 8.^.. Another special case of (12.9) - (12.10) m

m

can be obtained if every X^m is set equal to zero: (12.13)

(12.14)

^u - if^Hexp

8m

{8mlnZmi) =

^^]\Z\rni

< ^ l n S « = lnV^2 + ^ ^ m l n Z ^ < = ^o + ^ m

<^m In ^mt

m

which was considered by Dagum and Dagum (1974). The univariate case (M=l) has been often used in practice, for example in Geary (1966), Park (1966), Kmenta (1971) or Glejser (1969) who has proposed in his test for heteroskedasticity specific values of ^i such as 2, 1, 1/2, -1/2 and - 1 to give what he called "pure" heteroskedasticity. (ii) A second form of heteroskedasticity corresponds to the case where A„ ^ 0. An important case found in the literature [Hildreth and Houck (1968), Theil (1971), Goldfeld and Quandt (1972), Froehlich (1973), Harvey (1974), Amemiya (1977)] corresponds to the special case where Xu and every Xzm are set equal to one, i.e. where the variance UJU is assumed to be a linear combination of the Z ^ / s :

UU = ^ ^ k o + X ] ^rn{Zrat \_ rn

(12-15)

-

J

=«^^[.5o-E*- + l ) + E ^ ' * - ^ - ' \

m

/

= <^0 + 2_^ ^rn Zmt

where 80 = ip^ [8Q -^8m

l) + 1

+ I ) and ^.^ = ^|;^8,

m

268

Structural Road Accident Models

For the univariate case (M=l), Glejser (1969) considered the cases where A^^ = 1 and 2 with A^i = 1, and also the subcases which he called "mixed" heteroskedasticity corresponding to AM = 1 and 1/2 with Xzi = Si = 1. Due to the positivity constraint on the diagonal elements of n in the special form (12.15) as well as in the more general form (12.7) with A^.^ ^ 0, it is hard to estimate the ^-coefficients and A-parameters without violating the constraint, as experienced in preliminary tests with an earlier version of our computer program. Therefore, in this program version, we will only consider the first form of heteroskedasticity (12.8) with \i = 0 which still includes a great number of interesting cases. B. Likelihood function Before considering the likehhood function for the observed Vi's, it is convenient to rewrite the model (12.1)-(12.2)-(12.3) into a more compact form where various expressions which will be frequently used throughout the manual will be defined. The equation (12.3) for the residuals Vf's can be expressed as a function of the residuals u / s given in (12.2): no 1^^

^^

Y^

where /(Z,) = /(Z,) - exp E^mZ^r from (12.1) as Y} ""^ - Yl^kX^^

^t-i

Y

fiz^y^'

.

fiZi following (12.8). Replacing Uf which is derived

and ut_i by an analogous expression int - I yields:

k ^r{\y)

(12.17)

y{\.u)

iiZ^f''

k

j^(Axfc)

yi^y)

Y

fiZtf'

r

fiZf-tf

t

fiZt-ef'

Y; - Y. ^*-^**< = E «^'-^ - E /'t E p^^li-t+""* k

e

k

e

where F / = Y}^'"^/f{Ztf^^^ and X^^ = xl^""'^//(Ztf^^^ The corresponding expressions for t — ^ are obtained by replacing thy t — L The resulting form (12.17) can be more compactly rewritten as:

i

(12.18)

k

\

i • Wi

k

where F/* = F / - Y^P^ytU and A-* = X^ i

i

-Y^Pt^.-t

Assuming that the residuals wt's are independently and normally distributed N(0,cr'^y) and dropping the first r observations to simplify the procedure for higher order autocorrelation, i.e.

The L-1.5 Program for BC-GAUHESEQ Regression 269

assuming that the first observations on Yf are given, the likeHhood function associated with the last n — r observations on Yt can be written as follows:

(12.19)

C=

T\

—±==exp--^

dwt dYt ^^^ ^^^ jacobian of the trans-

where the residual Wf is given by (12.18) as Y** — J^/^k^lh it

formation from Wf to the observed Yt is: \dwt/dYt\ —Y^^

jf{Zt) ' . The corresponding

log-likelihood function is:

(12.20)

i = - y In (2^^™) - ^

E '^i' - 1 E 1 " •^(^') + ^^y - 1) E 1 " ^'

where N — n — r and the index t for the summation runs from 1 + r to n. Note that this function depends on all the parameters of the model: 11= f ^ , aj;, A^,, A^, A^, ^ , /> j where (j^y and \y are scalars, and l3^\^\z^S

and p are the column vectors associated with the /^j^'s,

Ax it's, A^rn's, <^m's and /9£'s respectively. C. Concentrated log-likelihood function Since the model (12.18) rewritten in terms of the transformed variables y;** and X^'s

is just

linear in the /3f-coefficients, we can concentrate the log-likelihood function on the h'^ and (T1^ by setting the first derivatives of the function with respect to these parameters to zero and solving for their values which will be replaced in (12.20). In matrix notation, the compact form (12.18) can be expressed as:

(12.21)

F** = X**P + w

where in view of (12.19) and (12.20), F** is a column vector containing the last A^ observations, X** is an (A^ x A") matrix, ^ is a (A x 1) vector of coefficients and if; is an (A^ x 1) vector of residuals. Using (12.21) to replace '^.w'l in the log-likelihood function (12.20) by w'w, the first

r

derivatives of the function with respect to /? and crj, are given by: dL_ (12.22)

^^

I dw'w _ 2.7^ a/3

ld{Y**-X**Pi al

dfi

= l.x**'(Y** - X**/9) = \(X**'Y**

^

^^ - X**'x**/3)

270

Structural Road Accident Models

dL _ _A^J_ (12.23)

_]_ ,

^"^'if^'^^r \ 2
< )

By equating these two derivatives to zero and solving for fi and al^^, we obtain:

(12.24)

(5= (^X**'X**V X**V**

(12.25)

al = ^w'w = ^ ( r * * - X**/?)'(r** - X**^)

Replacing ^ by ^ in (12.25) gives a value of crJJ, in function of ^:

(12.26)

al = ^ ( ^ * * - A"**/^) (>"** - A^**/^) •

Substitution of 13 and a"^^ in L yields the concentrated log-likelihood function:

(12.27)

L = -!^[l+\n{27:)]-j\nal-^-Y,\nf{Zt) + {Xy-l)J^\nYt

which now depends only onfl= f A^, A^., A^, ^ , /> ) . 12.1.3 Computational aspects A. Maximization procedure The Davidon-Fletcher-Powell (DFP) algorithm [Fletcher and Powell (1963)] is used to maxi/

/

/

/ A'

mize the concentrated log-likelihood function L with respect to IT = f Ay, A^, A_^, ^ , /? j . The gradient of L which is used in DFP can be written as: (12.28)

where

=^

^y,^

^

^

+ ^\nYt

The L-1.5 Program for BC-GAUHESEQ Regression 271

(12.29)

(12.30)

(12.31)

(12.32)

(12.33)

dwt ^ dXy

1

dY,(A,)

-E

1

dxl';"''>

= -A- fiZtf'

dXrk

(A.)

v^

dKk

ut dztr^ fiZ,f'^ 5A.„

'Jwt

dX.

dSm

f{Zt

dwt

Pe_

^/{Zt^ef

E^fiZ,.eY'' t

dxt-l dX,k

dX

f{Zt-i)

Ut_

^—

fiZt-ef if IIj ^ Xzm. and 8^,

(12.34)

dlnfiZ,)

_\

— \

^I[^

dZ^ 771 dX,

f^m.

0 1

(12.35)

The derivatives dY^^'^/dXy

n 111 — y^zmj

if^^^Ay, i f n , = Ay.

in (12.29), dxl^^^'^/dX^k

in (12.30) and dZ^^^^^/dX,m,

in (12.31)

and (12.34) as well as their lagged expressions are computed by the generic formula:

(12.36)

QV}^^ Ji[^/lnT4-y/'^] dX

k^n'V,

ifA^O, if A = 0.

272

Structural Road Accident Models

B. Asymptotic covariance matrix of the parameter estimates At the maximum point of L, hence L, the asymptotic covariance matrix of all the parameter estimates 11 is evaluated by the method of Berndt et al. (1974). The log-likelihood function L in (12.20) can be rewritten as a sum of N individual log-likelihood functions associated with each observation t\

(12.37)

1 2a-

L = Y.^^ = Y.

.

In {2^CJI) - -^wi

1

--In

f(Zt)

+ (A, - l ) l n F ,

where Lf is the log-likelihood function corresponding to the observation t and is equal to the expression in square brackets. The asymptotic covariance matrix of n is given by:

Var,(B)

(12.38)

dLt dLt

E dUdW

where the column vector dLf/dU has the following form:

^ ^^^

du~

2aidii

2

du

2

du

'^

an

'

with

(12.40)

(12.41)

(12.42)

2wt dwt

-rn

/ 2

-4

ifn,=<x:i,

d{wf /o

dn^

duH dn.

I T^rT

= <^ dn, -x»

It n. = n., ifn. = /'4.

Note that the derivatives dwf/dlli in (12.42) for the typical elements of 11 are already given in (12.29)-(12.30)-(12.31)-(12.32)-(12.33). Since f{Zt) is a function of the A^^/s and Sm's only,

The L-1.5 Program for BC-GAUHESEQ Regression 273

the derivative dlnf{Zt)/dI[i

is also given in (12.34) for 11, = 11^ = X^rn. and Sm- Finally the

derivative d{Xy — l)/dUi is equal to zero if 11^ j^ \y and one if 11^ = A^ as in (12.35). C. Scaling of the variables Although at the initial and final steps of the maximization of L, all the outputs are given in terms of the original units of the variables Yt, X^t and Zmt, an automatic scaling of these variables is performed during the iterations to avoid numerical problems which can slow down or inhibit the convergence process. Each variable Vt is scaled as follows:

(12.43)

Vt=s,Vt

where Vt represents 1/, Xkt or Zmt in their original units, and s^ is the scaling factor of the -

; LaxiytAj

log

form 10 L V * / -I in which the square brackets indicate that the greatest integer is being taken for the expression inside. In general, the 13 and ^-coefficients and their unconditional t-values computed from (12.38) are not invariant with respect to the scaling of Yt, Xkt and Zmt, due to the fact that the Pcoefficients depend on the measurement units of Yt and Xj^/s, and the ^-coefficients depend on those of Zmt'^- In contrast, the A^, A^^, A^ and /9-parameters as well as their unconditional t-values remain invariant, since these parameters are pure numbers, i.e. do not depend on the measurement units of the variables. Note that the concentrated log-likelihood function Z, hence L, and the error variance cr'^^^ are affected by the scaling of Yt only. Therefore the concentrated log-likelihood values which are listed in terms of the scaled units of Yt during the iterations would not be the same as if they were in the original units of Yt, unless the scaling factor of Yt happens to be equal to one for a particular data set. When a test for stability of the ;6f-coefficients over various data sets is performed in tenns of the log-likelihood values or the error variance estimates, these quantities must be in the original units of Yt, or more generally in a system of units which is common to all data sets used. Table 12.1 summarizes the relationships between the original ^ and 8 and the scaled /? and 8, as well as the original and scaled log-likelihood functions L and L due to the scaling of the variables Yt, X^t's and Zmt's, when each variable is transformed by Box-Cox. For the dependent variable, due to the Box-Cox transformation, all the /^-coefficients including the regression constant, as well as the log-likelihood function, will change. For the independent variables, only a strictly positive variable X^t or a quasi-dummy Qt, which is defined as a nonnegative variable in Section 2.6, can be transformed by Box-Cox. An associated real dummy Dgt is always created for each quasi-dummy Qt, since the null observations of Qt cannot be transformed by Box-Cox. The main difference between Xkt and Qt, when they are

274

Structural Road Accident Models

scaled, is that due to the Box-Cox transformation, there will be a shift in the regression constant (3o for X^t and in the ^5-coefficient of the associated real dummy, [3DQ, for Qt, in addition to the usual changes in the /^-coefficients of X^f and Qt themselves. The log-likelihood function will not be affected by the scaling of Xkt or Qt. For the heteroskedasticity variables, only the (^-coefficient associated with each Zm,t will change. There will be no effect on the regression constant and the log-likelihood function. TABLE 12.1 Relationships between the original (P^S^L) and scaled {l3^6^L) for Box-Cox transformed variables. 1

Variable

Regression constant

P or (5-coefficient

Log-likelihood |

Dependent

?/ = s,Y,,

A=v"/5o + 4^''*

Th = Sy'Pk for all k's

L=

^o = /Jo-4'r'/5;t/4r

~Pk = hi sit

L=L

L-N\nsy\

Independent • Strictly positive ^kt

=

Sxk^kt

(invariant) • Quasi-dummy Qt = sqQt

/?o = A) (invariant)

r^Q=f^QhQ'

L=L

T^D., =

(invariant)

f^Do-^o'^fiQls'o' Heteroskedasticity ^mt

^^

^zm.^mi

A = /9o (invariant)

^rn — ^ml ^zm

L=L (invariant)

D. Constraints on the /9-paranieters Strictly speaking, the autocoiTelation parameters pi's in (12.3) must satisfy a very large number of conditions to ensure stationarity in an autoregressive process of order r. For simplicity, a constraint —l
every pi is implicitly introduced by using the Fisher's z-

transformation:

(12.44)

- In -;

I

(-00


\ - Pi

with the corresponding inverse transformation pi — tanh^^. This parameter change from Pi to zi implies that the maximization of L is performed with respect to the ^-parameters.

The L-1.5 Program for BC-GAUHESEQ Regression 275

Therefore, during the iterations the ^-values are listed instead of the /?-values which are only given at the convergence of L when all the estimated parameters of the model IT are printed with their standard-errors and t-statistics. The derivative of L with respect to each z-parameter is computed as follows:

(12.45)

1^ = |i£« dze

dpi dzi

where dL/dp£ is already given in (12.28) and (12.33), and dpi/dz£ = sech'^^^ = 1 — tanh Z£ — 1 — pj. 12.1 A Model types Four types of Box-Cox regression models can be specified using the different options available: BC. This model type only includes the Box-Cox (BC) transformation on the dependent and independent variables in the regression equation (12.1) without considering the problems of heteroskedasticity and autocorrelation for the residuals M/S and Vf's respectively. The parameters involved are /S, o•f^^, Xy and A^ where the variances of Ui,vt and wt are all identical; BC-HE. This model type extends the previous one to the case where the functional form of HEteroskedasticity (HE) of the first-stage residuals wt's, f{Zt) = exp X^^m^^/"" L {_ m

J

is specified simultaneously with the BC transformation on the dependent and independent variables. The additional parameters to be considered are the A^ and (^-vectors which are involved in f{Zt). Since the problem of autocorrelation of the residuals vt'sis not considered, the variance of wt remains identical to the variance of vt, but the latter is different from the variance of uf, BC-GAU. This model type allows for a GeneraHzed AUtoregressive (GAU) structure of the second-stage residuals v^s, to be estimated jointly with the BC transformation on the dependent and independent variables, while the first-stage residuals w/s are assumed to be homoskedastic. Only the /^-parameters are added to the set of parameters already included in the BC model. Hence the variance of Wf, is different from the variance of vt, but the latter is identical to the variance of ut; BC-GAUHE. This model type corresponds to the general case where the first-stage residuals Uf's are assumed to be heteroskedastic with a functional form f{Zt) and the second-stage residuals vt's follow a stationary autoregressive process of order r. All the parameters of the model, 11 = i/S ,a^^^,Xy,X^,X^,S ,p j , are simultaneously estimated. Therefore the variances of Uf, vt and Wf are all different. The BC-GAUHE model type includes all the first three as special cases provided that: (i) The same set of dependent and independent variables, Y and X, is used for the estimation of the parameters, with the same set of A^, and A,^-parameters being estimated or fixed

276

Structural Road Accident Models

across the four model types. For example, if A^ or some of the A^;-parameters are fixed at zero, then the same constraints should be retained across all the model types; (ii) The same set of variables Z is specified in the functional form of heteroskedasticity for both BC-HE and BC-GAUHE, with the same set of A^ and ^-parameters being estimated or fixed in both model types. For instance, if every \zm is set to one, then the same constraints must be used in both; (iii) The same structure of the autoregressive parameters is estimated for both BC-GAU and BC-GAUHE so that the simpler model type is nested in the general model type. For example, if BC-GAU includes three estimated /^-parameters of orders 1, 4 and 12, then the same structure should be preserved in BC-GAUHE. Note that the BC-HE and BC-GAU model types are completely different since they are based on different assumptions on the structure of the residuals, hence they cannot be compared between each other, although the simpler BC model type is nested in both provided that the condition (i) is satisfied. 12.1.5 Model estimation Since the four model types presented above are by increasing degrees of nonlinearity in the parameters as the specification of a model under study is allowed to be more and more general by relaxing the constraints on the parameters of heteroskedasticity or autocorrelation or both with respect to the BC model type, a normal procedure to succeed in estimating all the four model types can be suggested as follows: [Step l.| Start the estimation of the model with the BC type. If the number of independent variables is too large to allow for a distinct BC transformation on each variable, regrouping all the variables which can be transformed by Box-Cox into a few main categories and specifying one A^-parameter for each category of variables will reduce the number of estimated A.^parameters. The program includes two types of constraints on the A^ and A3;-parameters: (i) A^ and any A^;-parameter can be fixed at a constant value instead of being estimated. This type of constraint permits specially the estimation of the classical functional forms such as the linear, semilog and loglinear forms: for example, setting A^, equal to zero and every A3;-parameter to one yields the semilog form; (ii) Xy can be set equal to one of the estimated A^-parameter. This type of constraint is useful in particular for the case where the model is specified with only one estimated A-parameter common to all the dependent and independent variables. This form can then be compared to the linear and loglinear forms considered above. To improve the specification of the model, three special options are available:

The L-1.5 Program for BC-GAUHESEQ Regression 277

(iii) Detection of multicollinearity among the independent variables X with the correlation matrix and specially with the regression coefficient variance decomposition proportions based on the spectral decomposition of X' X in terais of the original and Box-Cox transformed variables (Section 12.3.1); (iv) Graphical analysis of the estimated residuals Ui which are plotted by increasing values against a variable Zm that is thought to explain the variance of ut in the case of heteroskedasticity which can arise not only with the cross-section data but can also be induced by the BC transformation on the dependent variable even with the time-series data (Section 12.3.2); (v) Statistical tests of the estimates of the autocorrelation function and the partial autocorrelation function at different lags of the residuals «<'s for large time-series samples in order to detect the most significant orders of autocorrelation (Section 12.3.3). [Step 2,1 Proceed with estimation according to the results of the previous analyses: (i) If the problem of severe multicollinearity among a group of independent variables exists, then it should be first treated before considering the problems of heteroskedasticity and autocorrelation: some of the variables must be dropped from the regression equation (12.1) or replaced by their proxies, i.e. the variables which are used to explain essentially the same phenomena as the original variables, but which are less collinear among themselves. The model with a new specification of the variables is reestimated following Step 1. The whole process is repeated until a satisfying set of independent variables is obtained. (ii) If only the problem of heteroskedasticity is the most serious, then different types of the "multiplicative" form of heteroskedasticity in (12.8) can be tried using the BC-HE model type since the program allows the A^-parameters and the ^-coefficients in j{Zt) to be estimated or fixed at a constant value. At the end of each estimation, the first special option can be used again to detect multicollinearity among the Box-Cox transformed independent variables corrected for heteroskedasticity X|p and for time-series data, the third special option can also be used to analyze the structure of autocorrelation of the estimated residuals v/s. This will allow us to proceed further with the estimation of the model using the BC-GAUHE type if autocorrelation is present. (iii) If only the problem of autocorrelation is important, then using the BC-GAU model type is sufficient to improve the specification of the model. Note that at the end of each estimation with the BC-GAU or BC-GAUHE model types where the correction for autocorrelation has been applied with a chosen set of estimated /9-parameters, a further check on the estimates of the autocorrelation and partial autocorrelation functions will indicate whether the estimated residuals wt behave approximately as a white noise or other autocorrelation orders remain to be corrected for.

278

Structural Road Accident Models

12.2

ESTIMATION RESULTS

12.2.1 Definitions of moments of the dependent variable In a standard linear regression model where the dependent variable Yt is not subject to a BoxCox transformation, the calculated value of Yt is equal to the expected value of 1^. In constrast, in a Box-Cox regression model where Yt is transformed by Box-Cox, this property no longer holds. In this case, the expected value of Yf is more relevant than the calculated value of Yf. Moreover, we know that the use of the Box-Cox transformation on the dependent variable will affect the standard error and the skewness of the distribution of Yt. In this section, we will define these four elements, namely the calculated and expected values of Yt, the standard error and the skewness of Yt. A. Calculated value of the dependent variable VJ Using the equation (12.18), the calculated value of the transformed dependent variable Y** is obtained by replacing all the parameters of the model by their maximum likelihood estimates:

(12.46)

Yt** =

^^kXrt k

where

v: = y.^'-^imr . (\ \

^kt

= ^kt

~ /

1/2

^Pi^k,t-i

The L-1.5 Program for BC-GAUHESEQ Regression 279

xit = xf^"'^lfXz^)" ^k,t-e

- ^k/-i

f{Zt)

= exp

(12.47)

f{Zt-i) Replacing Yt

If{^t~i)

Y^'^^zg'

= exp

in the left-hand side of (12.46) by its expression given in (12.47), we solve the

resulting equation for the calculated value of the original Yt: l/\y if Ay 7^ 0

(12.48)

k

Ytt = \ exp

V i

f{Zt-i)

k

B. Expected value of the dependent variable Yt Following our approach in which the dependent variable Yt is assumed to be censored both downwards and upwards, e < Yt < v, where e and u are respectively the lower and upper censoring points common to all observations in the sample, the generalization of Tobin's model (1958) to a doubly censored dependent variable yields the following expression for the expected value of Yt: lutiv)

ivt{e)

(12.49)

/ (f{w)dw+

E{Yt) = e

-oo

/

Yt{wt)ip{w)dw + ly

wt(e)

/

(f{w)dw

totii')

where (f{w) is the normal density function of w with zero mean and variance a'^^,:

^("^ = ;;^^^n-^]'

(12.50) Yt(w) is a function of w:

1/Xy

1 + XyfiZtf' (12.51)

(Epiy;-e l^Ht-^t-i'

\ i

Yt{w) = { exp

InYt-i

+ E^kXt;

+1

ifA, ^ 0 ,

k

if Ay == 0 ,

280

Structural Road Accident Models

and the generic formula for wt(e) and wt(i/) is

Depending on the sign of the BC parameter A^, distinct cases can be derived for E{Yt): 1. \\y > 0.| If e and u tend towards 0 and oo respectively, following (12.52) the limits of wt{e) and wt(iy) can be obtained as:

(12.53)

1/2 " J2 ^^^'i-^ ~ Yl f^'^^^kt

iv; =: \\m wt(e) = -

and lim wtM = 00. Hence the first and last ternis in (12.49) disappear and the expected value of Yt reduces to the second temi which can be written as: 00

(12.54)

E(Yt) = I Yt{ivMw)dv

2. |A^ = 0.| If e and i/ tend towards 0 and 00 respectively, then lim Wf{e) == — 00 and lim wt(iy) = 00. Hence the first and last teims in (12.49) disappear and the expected value of Yt reduces to the second teiTn which can be written as:

-I

(12.55)

E{Yt) = / Yt{w)ip{w)dw

3. |A^ < 0.| If e tends towards 0, then lim Wi{£) — —00, and the first terai disappears. Before taking the limit of wt{iy) as i^ tends towards 00 in the last two ternis of (12.49), the expected value of Yt can be expressed as:

(12.56)

E{Yt) =

/ Yt{w)(f{w)dw-\-ly

/ (p{w)dw.

If p tends towards 00, the limit of wt,{iy) has a finite value:

The L-1.5 Program for BC-GAUHESEQ Regression 281

(12.57)

lim w^{p) =

^—-

- Y, PlYt*-i - E ^XH .

oo

Hence lim ly f ip(w)dw does not exist and EiYA does not have a finite value. In this case, it is still possible to obtain an approximation of E{Yt) for every observation t provided that the user selects a large value of v, say z/, such that the probability that Yt exceeds this upper limit V will be negligible, i.e. given the values of Y^.Xl* and Zt in wt{T'), the value of E{Yi) will be much smaller than V. The value of v can be chosen very approximately without affecting the numerical precision of E{Yt). By default, the program will generate a value of z/ determined as follows:

(12.58)

V - min [max Yt + lOa^, (-0.8o-,^A.^)^/^^|

where o-y is the sample standard error of Yf and cr^^ and A^ are evaluated at their estimated values. Note that if A^ is fixed at a certain value in an estimation run, then this value will be used. For the three cases of A^, the integrals corresponding to the second terai in the general expression of E{Yt) are numerically computed using the Gauss-Legendre 32-point quadrature. As a sample measure of the degree of numerical approximation of E{Yt) for the two cases Ay > 0 and A^ < 0, the mean probability of Yt in the sample to be at the lower limit e if Ay > 0 or at the upper limit v if A^ < 0 is also computed:

(12.59)

Pr{Yt <e)^

=^?/ —y

/ ip{w)dw

and oo

(12.60)

-K{Yt>u) = ^Yl

I "^H* tOt{v)

where the integrals involving Just the nonnal density function Lp{w) are computed with the formula 26.2.17 in Abramowitz and Stegun (1965). A value of Pr{.) lower than 1% indicates that the sample contains practically no limit observations.

282

Structural Road Accident Models

C. Standard error of the dependent variable Yt The standard error of Yt is defined as the square root of the variance of Yt which is also known as the second moment of Yt centered about the mean E{Yt): (12.61)

a{Yt) = / M ^ = ^jE[Yt-E{Yt)f = ^JE{Y^) - [E{Yt)t

where the first and second noncentral moments E{Y^), r — 1,2, are given by: oo

(12.62)

E(Yl) =

j Y;{w)if{w)dw

if Ay > o

J Y;(w)ip{w)dw

a Xy = 0

— oc

/

Y{ {w)(p(w)dw-i-ly'' J ip{w)dw if Ay < 0

-oo

iut{i')

D. Skewness of the dependent variable Yt Skewness is a measure of asymmetry of a distribution. It indicates how the data are distributed in a particular distribution relatively to a perfectly symmetric one. Several types of skewness can be defined, but the most usual one is the Fisher Skewness defined as: /^3 3/2

(12.63)

f^h

/^3 rr-^

where /i2 and /is are respectively the second and third moments centered about the mean of 1/2

the distribution, and firj = <J is the standard error of the distribution. Note that the third moment fis is expressed in cubic units. In order to compare the results from one distribution to another, the skewness is expressed in standard units, i.e. as the third moment divided by a^. It does not depend on the units of measurement of the variable considered in the distribution. For example, the skewness of a normal distribution is zero because it is symmetric about its mean, whereas a lognormal distribution has positive skewness, i.e. the right tail is longer than the left one. Conversely a negative skewness indicates that the left tail is longer than the right one. Usually a distribution is considered to be: Slightly asymmetric, if I7I < 0.5 . Moderately asymmetric, if 0.5 < I7I < 1 . Highly asymmetric, if I7I > 1 .

The L-1.5 Program for BC-GAUHESEQ Regression 283

Using the definition of the skewness in (12.63) applied to the distribution of the dependent variable Yf, 7(1^) can be written as:

(12.64)

,(K,) = 0 M

where fi3{Yt) = E[Yt ~ E{Yt)f is the third moment of Yt centered about the mean E{Yt) and (T{Yi) is the standard error of Yt defined in (12.61). The third central moment fJ^siYt) can be expressed as a function of the first three noncentral moments of Yt: (12.65)

/*3(Vi) = E{Yt') - EiYt)[3E{Y,^)

-

2[E{Y^)f]

where the first three noncentral moments E{Y^)^ r — 1,3, have the following forms: (

00

if Ay > 0

j Y;[w)^{w)dw

(12.66)

E{Y;)\

II ^^ * 00 I yti^WH^w

ifAj^-O

I —00

/

Yl(w)ip{w)dw-\-jy''

-00

.

J ip[w)dw if Aj, < 0 %ut{v)

12.2.2 Derivatives and elasticities of the sample and expected values of the dependent variable Two types of elasticity are computed in the program: The first type of elasticity, denoted as 7/^ , is defined in terais of the sample value of Yt'. it measures the percent variation of the dependent variable 1^ due to a percent variation of an independent variable Xjt, given all the other independent variables X^t^ as well as all the variables Zmt^ {Zmt ^ Xjtym)

in f{Zt) fixed at their observed values.

The second type of elasticity, denoted as r/^^ , which is more relevant since the model is nonlinear in 1^, is defined in temis of the expected value of Yt derived in Section 2.1. When heteroskedasticity is present, two types of elasticity, namely T/J^^ and ?/^^^, can also be computed with respect to a heteroskedasticity variable Zmt specified in the function f{Zt). Two specifications of the variable Zmt should be considered: The variable Zmt specified in f(Zt) is also used as an independent variable Xjt. It is only specified in f{Zt), not elsewhere. A. Derivatives and Elasticities of the Sample Value of Yt The sample value of Yt has the same form as Yt{w) in (12.51), but with w replaced by wt. The derivative and the elasticity of the sample value of Yt with respect to an independent variable Xjt can be written as:

284

Structural Road Accident Models

(12.67) _ dYt X,t __ 1 (f.^\,,,.,

\

where (12.68)

if f{Zi) is not a function of Xjt^

Gx, = 2^rn^rnt

(A.X.)

nzir'At-zihxi't

if f{Zt) contains one Zmt = ^jt-

and At — T^^p(Y*_^ + Y^kl^k^kt- Similarly, the derivative and the elasticity of the sample value of Yt with respect to a heteroskedasticity variable Zmt which appears only in f{Zt) are: r.s '-'

(12.69)

_ dYt

_Z-J,Gz^, y/^-i

dZmt dYt Zmt dZmt Yt

where Gz^, =

^mZir

f(ZtY^'At-Y:ihxl

Gz^, K/V

^A^.k)

B. Derivatives and Elasticities of the Expected Value of Yt Using E{Yt) in (12.54)-(12.55)-(12.56) depending on the value of A^, the derivative and the elasticity of the expected value of Yt with respect to an independent variable Xjt can be written as:

^z, = ^ § ^

= J [Yf.Ht'^

[t^jXJr-'

+ X-/Hx,A^)]

^{w)du

(12.70)

Rw

where Rio is the integration domain of w : [w;J^, oo], [—oo, oo] and [—oo, wt{j^)] if ^y > O^Xy = 0 and A^ < 0 respectively, and

The L-1.5 Program for BC-GAUHESEQ Regression 285

1

0

if/(-^t) is not a function of X^^,

X r 1/2 (X ^1 ^SmZ^T \f(Ztr (At + w) - E/^fc4t""M

if/(^O contains Zmt = Xjt.

Likewise, the derivative and the elasticity of the expected value of Yt with respect to a heteroskedasticity variable Zmt which appears only in f{Zt) are as follows: _ dEjYt) _ R.

(12.72) "^Zrr.

dZmt

where Hz^,{w) = \SmZ^T

E{Yt)

E{Yt)

R„

f{Ztfl\A,-Vw)-Y.PkXt

C. Derivatives and Elasticities for the Linear and Logarithmic Cases of Yt The most usual forms of the dependent variable Yf encountered in pratice are the linear (A^, = 1) and logarithmic (A^ — 0) cases: For these two cases, the explicit forms of the sample value of Yt and its derivative and elasticity are given in Table 12.2, depending on the presence or absence of heteroskedasticity. To be completely general, autocorrelation is considered in these forms which can be further reduced in the absence of autocorrelation by setting all the autoregressive coefficients /?/'s included in At equal to zero. Similarly, for both cases, explicit forms of the expected value of Yt and its derivative and elasticity — whenever possible due to the integrals which cannot be reduced to closed forms — are summarized in Table 12.3.

286

Structural Road Accident Models

TABLE 12.2 Explicit forms of the sample value of Yt and its derivatives and elasticities for the linear and logarithmic cases of Yt. STATISTIC

HETEROSKEDASTICITY

HOMOSKEDASTICITY

CASE

(No f{Zt)

involved)

7^ ^ m t

•^jt

^3t

—

'^mt

|

1 SAMPLE VALUE A, = 1

l + At

\ +

f(Ztf'''At

Yt A, = 0

exp(At)

exp

[f(Zt)"''At\

1 DERIVATIVE \y = 1

^j^jt'

-^

^miGz^t

^Xu

(/?,x;\--^ + z-jGz„,)yJ

Ay-0

A, = 1

Not applicable

z;n\Gz,^,

Same as above

A, = 0

Not applicable

Zmi Gz,r,.t Yt

Same as above

^k..

1 ELASTICITY A, = 1

{piX^r +Gz,^)lYt

P^X^t'/Yt

"^x,, A, = 0

P^xfr+Gz,^,

nk,, =vx,, =PjX}r

\y = 1

Not applicable

Gz^^jYt

Same as above

A, = 0

Not applicable

Gz^,

Same as above

^L* where At = Y^ t

k

Z * ; andGz^, = \SrnZi%- \^f(Zt)"''At

- E/^fc^''^'^] •

T h e L-1.5 P r o g r a m for B C - G A U H E S E Q R e g r e s s i o n 287

TABLE 12.3 Explicit forms of the expected value of Yt and its derivative and elasticity for the linear and logarithmic cases of Yt. STATISTIC

HOMOSKEDASTICITY

CASE

(No f(Zt)

HETEROSKEDASTICITY

involved)

Xjt = Zmt

Xjt / Zm

EXPECTED VALUE K = 1

E{Yt)=

f^Yt{tu)^(w)div

where Ytiiv) = 1-{-A, + iv |LiMiTCASE| lim E(Yt) = 1 + At

E{Yi)

^^^^^ Yt(w) = 1 + f(Ztf^\At

'

ILIMITCASEI

lim

+ iv)

E(Yt) = 1 + / ( Z ^ ' ^ ' ^ t

oo

A, = 0

E(Y^) = f Yt(lD)ip{w)dlV Where Y,(w) = exp (A^ + iv)

^^^^

Yt(iv) = exp \f(Z,f\At

+ w)]

DERIVATIVE \

A^ = 1

Same as left column

1

+ D | „ J A , = 1) I LIMIT CASEI

A^ = 0

[LIMIT CASEI

A.,.,-1

Jim

Dx .^ - /?j X-

Jim

D^.^

=

Same as left column

D'x,,=f3^xfr~'E{Y,)

A. = 1

Not applicable

DkJ\

= ^)

Same as above

A„ = 0

Not applicable

D'z^A\

= 0)

Same as above

ELASTICITY Q-X

A^ = 1 ^x.t

^ CASE!

Ay

Jnn

^^

same as left column

OO

r]\^^ =

ILIMIT CASE]

/3,^,,

-jj^

./™oo^^^-^" 1+kz.y^^A. Same as left column

= 0

ix,,

^^x„

=f^jX.;'

+vz,.A\=^)

A, = 1

Not applicable

^ k , ( A y = 1)

Same as above

Ay = 0

Not applicable

VZ,„, (Ay = Q)

Same as above

'IZr.t

where A, = ^PJY;_, +EPkXl* I k oo

, Gz^, = \SmZ^„:r \f{Zif'''At L

^l,.,(Ay = 1) = ^ - ; / Hz^A^v)^{w)drv,D%^^{\y

- 0) = Z " ! /

I Hz,^,{w)(p{iv)dw,

E'z^^^iXy = 0) = ^ ^

f — oo

lu*

and Hz,„A^v) = ^S^Z^T

Yt{w)Hz^X^)^{w)dw,

GO

io

^z,„,(Aj/ = 1) = ^^)

- E/^^^'^'^^^ fc J

oo

\f(Ztf'\At

+ w) - E / ? f c 4 t " ^ ' '

Yt{w)Hz,^,{w)cp(iu)dw,

288

Structural Road Accident Models

12.2.3 Derivatives and elasticities of the standard error of the dependent variable The concept of elasticity can be extended to the standard error of the dependent variable Yt, (j{Yt), to give a measure of the percent variation of the standard error of Yt due to a percent variation of an independent variable Xjt or a heteroskedasticity variable Zm.t specified in the function f{Zt).

Like for the derivatives and elasticities of the sample and expected values of Yt,

two cases of heteroskedasticity can be distinguished for the derivatives and elasticities of cr{Yt): The heteroskedasticity variable Zm.t is also used as an independent variable Xjt. It is specified only in the function f{Zt),

not elsewhere.

A. Derivative and elasticity of cr(Ft) with respect to an independent variable The derivative and the elasticity of the standard error of Yt with respect to an independent variable Xjt are given by: _ cMYt) c)Xjt

-^^'

(12.73)

1X,,

^

1 ''^'^ 2a(Yt) dXjt

da{Yt)

Xjt

dXjt

cr{Yt)

2EiY^)^^^ dXr,

where the derivatives of the first and second noncentral moments

1,2, with

E{Yl)

respect to Xjt are: dE{Y;)

(12.74)

dXjt

Note that Yt{w),Ru,

and Hx,,(w)

are already defined in (12.51), (12.70) and (12.71) respec-

tively, and that Hx t(w) includes the first case of heteroskedasticity where Xjt appears also as a heteroskedasticity variable Z^t used in f{Zt).

After some algebraic manipulations, the

derivative and the elasticity of the standard error of Yt with respect to Xjt can be expressed as: ^ ^{Yt

/

[Yt{w)t'nytH

- E{Yt)] [iSjXf--'

+

X~,'HX,,H]^{w)du

(12.75)

ih.

= y ^ ) J [ytHf-'nYtH

- E{Yt)] [fJjX^r +

Hx,A^)]^Hdw

B. Derivative and elasticity of cr(l^) with respect to a heteroskedasticity variable Since the first case of heteroskedasticity has been previously treated, only the second case where Zmt appears only in f{Zt) is considered here. The formulas for D^^^^^ and T/J^^ are analogous to D^^ and ryJ^ without the terais f3jX:^^~

and PjX^^^ respectively:

The L-1.5 Program for BC-GAUHESEQ Regression 289

^L. ^ ^

/ [ytH]'-'nyiH -

mt^z

^{w)(f{w)dw,

(12.76) il-A. [Yt{w)Y-''y[Yt{w) -

rz^^ =Var(F,)

E{Yt)]Hz^,{w)ip{w)dw.

where Hz^,{w) is already defined in (12.72). C. Standard error of Yt for the linear and logarithmic cases of Yt In Table 12.4, explicit forms of cr(l^) are given for the linear and logarithmic cases of Yt depending on the presence or absence of heteroskedasticity. As in Tables 12.2 and 12.3 for the sample value Yt and the expected value E{Yt) respectively, autocorrelation is considered in these forms which can be further reduced in the absence of autocorrelation by setting all the autoregressive coefficients pi's included in At equal to zero. Linear Case (A^ = 1) Homoskedasticity: cr{Yt) varies slightly depending on the value of the lower bound of integration w^ except if w'l tends towards — oo, cr{Yt) is constant and equal to the standard error of the residuals wt's : (12.77) I

lim

a{Yt) =

lim

\

oo

(Y^{w)ip{ w)dvi [V*

[l + At

/ (p{w)dw-\- / w'^ip{w)dw+ 2(1-j-At) -co

/ wip(w)dw

—oo

oo

oo

(1+A( I / (p[w)dw + / wLp[w)du oo

/

w (f[w)dw

where Yt{w) = I + At + w, J Lp{w)dw = 1, / w(p{w)dw = 0 and / w'^Lp{w)dw — crj^. — oo

—oo

—oo

Heteroskedasticity: in contrast, cr{Yt) varies for each observation t even if w;]f tends towards

290

Structural Road Accident Models

-oo. In this case, o-{Yt) changes proportionally to

f{Zt)^:

(12.78) v^

Jim

w'—>• —oo

(7{Yt)=

hm

tu; ^>- — oo

oo

{ j Y^^{w)i^{w)dw -

I Yt{w)ip(w)dw

oo

oo

[l + / ( Z O ^ A , ] ' / ip{w)dw-^ i\Zi)

I

— oo

—oo

w\{w)dw

oo

+ 2[l+ f(Zt)'^At^^f(ZtY I w^{w)dw — oo OO

+•fiZiMi] .

1

OO

I
I

wif{w)dv

CJO

fiz>

/ w'^(p{w)dii

where Yt{w) = 1 + f{Zt)^{At

f{Zt)^^...

+ w) .

Logarithmic Case (A^ = 0) Homoskedasticity: o-{Yt) varies proportionally to exp (At): oo

(12.79)

a{Yt)

=exp{At

/ e x p (2w)ip(w)dw

oo

I exp

—

H e t e r o s k e d a s t i c i t y : cr(y;) is a nonlinear function of At and

{w)ip{w)dw

f{Zt)^:

(12.80) oo

!

/ exp [2f{Zt)^w]ip{w)dw

r oo

-

y exp

[f{Zt)'^w'\ip{w)d'.

D. Derivatives and elasticities of a(Yt) for the linear and logarithmic cases of Yt In Table 12.4, explicit forms of the derivatives and the elasticities of o-(Yt) with respect to Xjt and Zmt for the linear and logarithmic cases of Yf are given in tenns of an observation t to show how both statistics vary with each observation. a.

Derivative and Elasticity with respect to Xjt 1.

Linear Case (A^ — 1)

The L-1.5 Program for BC-GAUHESEQ Regression 291

•

Homoskedasticity: the derivative D^ and the elasticity r/J vary slightly depending on the value of the lower bound of integration w^, but are both equal to zero as w^l tends towards — oo, since

•

lim

CO

/ [Yi{w) — E(Yt)](p{w)dw is equal to zero.

Heteroskedasticity {Xjt ^ Zmt)' this case is identical to the previous one since the heteroskedasticity function f{Zt) does not include any Zmt which is used also as an independent variable Xjt.

•

Heteroskedasticity {Xjt = Zm.t): both D^^ and 7/J^ include two components: one resulting from the variation of Xjt alone and the other from the variation of Zm,tIn the limit case where w^ tends towards —00, only the second component remains.

2. Logarithmic Case (A^ = 0) Homoskedasticity: the derivative D J

is a function of Xjt and u{Yt), but the

elasticity 7^J ^ is a function of Xjt only and it is identical to both elasticities TJJ^ and rj'x^^. •

Heteroskedasticity {Xjt ^ Zmt)- this case is identical to the previous one since the heteroskedasticity function f{Zt) does not include any Zmt which is used also as an independent variable Xjt. Heteroskedasticity {Xjt = Zmt)'' the derivative DJ^ and the elasticity 77J^ depend on Xjt,Zmt.c7{Yt)M^).E{Yt)

and Hz^,.

b. Derivative and Elasticity with respect to Zmt 1. Linear Case (A^, = 1) •

Homoskedasticity: since f{Zt) is not involved in this case, the derivative D^^^ and the elasticity T/J^^ do not apply.

•

Heteroskedasticity {Xjt ^ Zmt)' w^,ZmtTcr{Yt),Yt{w),E{Yt)

the derivative and the elasticity depend on

and Hz^t.

If w^ tends towards —cx), both statistics

have the same limits as in the case A.l with heteroskedasticity {Xjt — Zm,t) since only the heteroskedasticity component remains. Heteroskedasticity {Xjt = Zmt): this case is identical to the case A.l with heteroskedasticity {Xjt = Zmt)2. Logarithmic case (A^ = 0) •

Homoskedasticity: since f{Zt) is not involved in this case, the derivative D^ and the elasticity 7/J ^ do not apply. Heteroskedasticity {Xjt 7^ Zm.t)'' the derivative and the elasticity depend on Zmt.cT{Yt),Yt{w),E{Yt) and Hz^,. Heteroskedasticity {Xjt — Zmt)'- this case is identical to the case a.2 with heteroskedasticity {Xjt — Zmt)-

292

Structural Road Accident Models

TABLE 12.4 Explicit forms of the standard error of Yt and its derivatives and elasticities for the linear and logarithmic cases of Yf. 1

STATISTIC

CASE

HETEROS KEDASTICITY

HOMOSKEDASTICITY (No f{Zt)

involved)

X,t

^ Zmt

1

Xjt — Zmt

1

2>|l/2

1

1 STANDARD ERROR A, = 1

a{Yt) = I

jY,^{tu)(f{tu)diu-

jYtixi

)ip{xo)dxu

) = i + fiZty^\At ^{Yt)

ILIMUCASE!

^lim

a{Yt)

^

[LIMITCASEI

(Tru

\mi^aiYt)

1

+ xo)

|

= fiZty^^a,,

1

n2>|l/2

\y = 0

JY{

(r{Yt) = I 7 Yt^{to)
v)(p{xo)dxu

^^^^^ Yt{to) = exp \f{Zty^^{At

+ to)]

\

1 DERIVATIVE Same as left column

Xy = 1

1

^X. = ^ ^ ^ 7 D ^ . M - £ {Yt)Mxo)dxu

^h.

[LIMIT CASEI

lim

^x,,

= 0

ILIMIT CASEI

^lim

^x,^

=

1

^6mZ^'r~'fiZt)'^^cr^ \y = 0

Dx,t=l^jX'r~'^iYt)

Same as left column

1

Xy = l

Not applicable

^z,.,(A,-l)

Same as above

1

Xy=:0

Not applicable

^l,„,(A,-o)

Same as above

1

sam e as left column

1

DZrr..

1 ELASTICITY Xy = l

+

''^,«

JLiMFT CASE!

lim

/?x,,

=

0

'

[LIMIT CASEI

1

ifx^t = ¥rnZ^r

1

Same as left column

1

;im

Xy = 0

VXjt = VXjt = VXjt =

A, = . 1

Not applicable

riz,^^i^y

= ^)

Same as above

1

Not applicable

Vz,^A^y

= 0)

Same as above

1

PjXjt'

^fZmi A^ = 0 1 where At = T^PiY^U 1 i

+

T.l^kXH, k

Dz^r^Mv

= 1) = ^

\{Yt{w)

Dz^My

= 0) = ^

7 Yt{xo)[Yt{iu) -

'n'Zrr.Mv

=

1) =

US™t(^^ = o) = and Hz^^H

-

1 7 [Ytiw) Var (Yt) J^ 1

Var(Yi) J

= ^mZ^T

/

E{Yt)]Hz,^AtoMtu)dxu, E{Yt)]Hz^AwM^o)dto, EiYt)]Hz,r.tM(p{xv)diu,

Yt{xv)[Yt{xu) -

E{Yt)]Hz,r.i{io)(p{xu)dxu

[ / ( ^ t ) ' / ' ( A t + to) - E ^ f c ^ t ^ ' ^ j •

The L-1.5 Program for BG-GAUHESEQ Regression 293

12.2.4 Derivatives and elasticities of the skewness of the dependent variable A. General case Since an independent variable Xjt can also appear as an heteroskedasticity variable Zmt in the heteroskedasticity function f{Zt), two cases for the derivatives and elasticities of 7(1^) can be considered: 1. The independent variable Xjt also appears in /(Z^). The derivative and the elasticity of 7(It) with respect to an independent variable Xjt can be obtained as: 1

d^{Yt)

dX.j«

dXu

(12.81)

ih.

dl(Yt) X,t dXjt 7(y,)

J

•

Using (12.65), the derivative of /^3(1<) with respect to X^t can be written as: (12.82)

3M^-4E(rO^^(^') dX,t dx,t

dx,t The derivative of o^(Yt) with respect to Xji is:

''''^''*K,.^(Y)'<^*)

(12.83)

dX„

dX.t

where 1

(12.84)

dX,,

2a{Y,)

'Jjm^2EiY/J^dX, dX,,

In (12.82) and (12.84), the derivatives of the first three noncentral moments of Yt with respect to Xjt are given by: (12.85)

dE{Y;) dXjt

• J lYiw)]^-^' [l^.X^r'

+ X-t'Hx„M]^Hdw,

(r = 1,3)

where Yt{w) is defined in (12.51) and the heteroskedasticity terni Hx,,(w) in (12.71). Note that for the special case where Xjt is not specified in f(Zt), then the heteroskedasticity term Hx ,(w) is null. 2. The heteroskedasticity variable Z^t appears only in f{Zt).

The derivative and the

elasticity of 7(yi) with respect to Zmt are analogous to (12.81) with Xjt replaced by Zmf-

(12.86)

Dl„

di{Y)

1

dZmt (T^Yt) d-f(Yt) Z„,,t

1z„.= dZmi ^{Yt)

•

dZmt

9Zm.t

294

Structural Road Accident Models

The derivatives of the first three noncentral moments of Yt with respect to Zmt are given by: ^^^•^^^

" '^'"* J i^*Mr''Hz„.M'p{w)dw , (r = 1,3)

^§Sr

Rw

where Hz^^{w) is defined in (12.72). B. Derivatives and elasticities of ^{Yt) for the linear and logarithmic cases of Yt For the linear case of the dependent variable, the skewness of Yt is zero, since the third central moment of Yt is equal to zero: -I 3

(12.88)

iX.k)

//3(K,) = E[Yt - E{Yt)f = E

E^)=0

k

where for simplicity, we suppose that Yt == ^fh^lt''''

'^ ^^ without heteroskedasticity and

k

autocorrelation. Since the residuals ut's are nonnally and independently distributed, all the odd-order moments of ut are zero, in particular the first and third-order moments. Hence the derivative d'j{Yt)/dXjt

as well as the elasticity T/J are all zero in the linear case of Yt. For

the logarithmic case of the dependent variable, there are no closed forms of the skewness j{Yt), the derivatives D^

and DJ , and the elasticities 7/J and ?/J ^. They should be computed

numerically using (12.64) and (12.81)-(12.87). All these cases are summarized in Table 12.5.

The L-1.5 Program for BC-GAUHESEQ Regression 295

TABLE 12.5 Explicit forms of the skewness of Yt and its derivatives and elasticities for the linear and logarithmic cases of F^. 1 STATISTIC

CASE

HETEROSKEDASTICITY

HOMOSKEDASTICITY (No f{Zt)

involved)

7^ ^mt

^jt

Xjt

= ^mt

\

SKEWNESS 0

\y=l

liYt) ^(Y^

A, = 0

H^iM

E{Y,-)-E{Y,){3E{Y-)-2[E{Y,)f]

where E{Y{), r = 1,3, is given in (66). DERIVATIVE A, = l

0

0

A^-0

^h

^x. + ^L

A, = l

Not applicable

0

A, = 0

Not applicable

D^ ^z^.

0 ^l.+^L

1 ELASTICITY 0

0

A,=:l

^x. A, = 0

'^Xji -

^x,t

A.-1

Not applicable

0

0

Ay = 0

Not applicable

^L-^L|^

^x.. + ^L.

^L. where ^J. = W and

^x.. + ^L,

^7(1^1

=^ [ ' f B -

^I, = l e ^ = ^

^ ( ^ - 0 ^ ] - given in (81)

[ ^ e ^ - T ( y < ) % ^ ] as given in (86) .

1

296

Structural Road Accident Models

12.2.5 Ratios of derivatives of the moments of the dependent variable In the three preceding sections, we have considered the derivatives of the three moments of Yt with respect to an independent variable Xjt, namely dE{Yt)/dXjt (12.73) and dj{Yt)/dXjt

in (12.70), da{Yi)/dXjt

in

in (12.81). Using these derivatives, three different relations can be

obtained for the ratios of two derivatives of the moments of Yt. Consider them in turn: The ratio of the derivatives of a same moment with respect to two different independent variables Xit and X-jt gives the Marginal Rate of Substitution (MRS) between these two variables: • '

'''•'''

dX,t

dE{Yt)/dX,t

d
d^(Yt)ldX,t

'

This MRS can be shown below to be independent from any moment chosen for the derivatives. The ratio of the derivatives of two different moments m^ and m^ with respect to a same independent variable Xjt gives the Marginal Rate of Substitution (MRS) between those two moments: (12.90)

Mi?<,~ = ^ arris

= 1 ^ ^ ^ ^ , (r / s = 1,3 and J = 1, A') . oms/oXjt

where mi, 7722 and ms represent respectively E(Yt), cr{Yt) and j(Yt).

It will be shown

below that this MRS does not depend on any Xjt with respect to which the derivatives of the moments are taken. For two different moments rrir and rrig, and two different independent variables Xit and Xjt, we have the following relation for the ratio of derivatives: (12.91)

MRS^-^^ = ^ ^ ^ = ^ ^ = MRS--'^^.MRSx.,x, • ^''^•^ drris/dXjt drris dXrt ' It is a combination of the two preceding cases, i.e. a product of a MRS of moments and a MRS of variables. But the economic interpretation of this case remains to be found. We therefore focus on the first two ratios and on the elasticity of the second one. A. Marginal Rate of Substitution between two independent variables The marginal rate of substitution between two independent variables Xa and Xjt can be computed from the ratio of the derivatives of the first moment E{Yt) with respect to Xit and Xjt. It will be shown below that this MRS does not depend on any moment chosen for the derivatives in the ratio. Since the independent variables Xit and Xjt may also appear

The L-1.5 Program for BC-GAUHESEQ Regression 297

in the heteroskedasticity function f{Zt), two cases of the MRS between Xa and Xjt can be considered: f{Zt) is not a function of Xu and Xjt. Using (12.85) for the expressions of the derivatives of E{Yt) with respect to Xa and Xjt, where the heteroskedasticity terms Hx,X^) ^^^ ^XjtM '^^^ i^uU, the MRS can be written as: (12.92)

f [YAw)f~^'fJ^X^/^-^if(w)dw ^^'^^

ax,t aE(y,)/^^;^

, ,

/ [Yt{w)]^~^'f3jX^r'~^ip(w)dw ft-x^;^"^ '

For example, in the context of travel demand, the marginal rate of substitution between travel time (T) and cost (C), also known as the value of time, can be computed as: ^^^ - ^ - ^ ' ^ "

(12.93)

f{Zt) is a function of both Xa and Xjt- Using (12.85) again for the expressions of the derivatives of E{Yi) with respect to Xit and Xjt, where the heteroskedasticity terais Hx.ti'^) ^^d Hxjti'^)'^^^nonnull, the MRS cannot be obtained in closed form, but should be computed numerically: (12.94) MRSx

^^ ^^,^,^^,, / [Yi[w)t^y b,Xt-'^X-'Hx,{^)Uw)dw X = " - "^^(^^^/^^^'^ - ^dXu dE{Y,)ldXj, J [y,(^)]i-A.Ux^^-^ +X-ii/;,^^(^^)lc^(^^)^.

Two special subcases of (12.94) can be noted: a. If only Xit appears in f{Zt), then Hx^tM is mill. b. If only Xjt appears in f{Zt), then Hx^^^w) is null. Now we turn to the proof for the double equality of the ratios of the derivatives of the moments of Yt: (J2 95)

dE(Yt)/dX,t dE{Yt)/dX,t

_ da{Yt)/dXH da(Yt)/dXjt

and (12.96)

da{Yt}/dXH _ d'i(Yi)ldX,t da(Y^)idx,t d^{Xt)ldXjt

Structural Road Accident Models

Using (12.73) for the expressions of the derivatives of a(Yt) with respect to Xa and Xjt, the ratio of these derivatives can be written as: (12.97) da{Yt)ldX,t da[Yt)ldX,t

dEjYt) dX,t

2a{Yt)

_

since

dEjYt) dX,t

^ ^"(^-)

dE(Y,^) dx^t dXu dE{Yt) dX,t

• 2E{Yt)

dE{Yt)

^^V-^^

dE(Yt)ldX,t dE{Yt)/dXjt

^ ^ ( ^ t ^ ) 9X^i dXu dE(Yt)

_ ^Hy') dXjt

dX,, dE(Yt)

_

dE{Y;') dE{Yt) '

Using (12.81) for the expressions of the derivatives of 7(1^) with respect to Xa and Xjt, the ratio of these derivatives can be expressed as:

(12.98)

d^(Yt)/dXa

^HTT)

dj{Yt)/dX,t

1

^l^W

dXu

dXa

J

dX,t

[ dX,t

dcr^Yt)

l\^t)\

dXjt

By noting that (12.99)

3a'{Yt)da{Yt)/dX,t 3a'{Yt)da{Yt)/dXjt

dcj\Yt)ldX,t da\Yt)ldX,t

da{Yt)/dXu da{Yt)/dXjt

and (12.100)

dii^{Yt) ^X^t dX,t d<j\Yt)

dXjt

dcj\Yi)

_ d^3(Yt) dcj\Yt) '

we obtain the equality of the ratios of the derivatives: .,2 i o n

d^{Y,)ldXu di{Yi)ldXji

_

da{Yt)ldX,t da(Yi)ldX,t

•

B. Marginal Rate of Substitution between two moments Since there are three moments of Yt, we have six possible combinations of two different moments for the ratio of the derivatives of two moments of Yt with respect to a same independent variable Xjt.

For each combination, we will show that this ratio is independent

from any Xjt with respect to which the derivatives of the moments are taken: 1. MRS between E[Yt)

and cr(lt).

da{Yt)/dXjt,

we can write:

(12.102) dEjYt) da{Yt)

dE{Yt)/dXjt da{Yt)/dXjt

Using (12.73) for the expression of the derivative

dE{Yt)/dXjt [dE{Y^)/dXjt-2E{Yt)dE{Yt)/dXjt]/[2a{Yt)]

'

The L-1.5 Program for BC-GAUHESEQ Regression 299

Dividing both the numerator and the denominator by dE(Yt)/dXji,

(n.m)

MRS

we obtain:

=^-^^^—^-^^——--—-.

Clearly, this expression does not depend on any Xjt with respect to which the derivatives of the two moments are taken initially. 2. MRS between E{Yt) and 7(1^). Using (12.81) for the expression of the derivative dy{Yt)/dXjt, no 104)

we can write:

myj^^dEiYtydX^^ dy(Yt) di{Yt)ldX,t

dE{Yt)/dX,t [dix,{Y^)ldX,t-i{Y^)da\Yt)ldX„]la^{Yi)-

Dividing both the numerator and the denominator by dE{Yt)/dXjt,

we obtain:

(U 1051

MR^'--^ = ^-^(^') = '^^(^0 d^(Yt) dMYt)/dE{Yt)-^iYt)da^{Yt)/dE{Yt)Clearly, this expression does nor depend on any Xjt with respect to which the derivatives of the two moments are taken initially. 3. MRS between a{Yt) and 'y{Yt). Using (12.73) and (12.81) for the expressions of dcr{Yt)/dXjt and dj{Yt)/dXjt

respectively, we can write:

ri2 106> M ^ = aa{yt)/dX,t df{Yt) dt(Yt)/dX,t

^ [dE(Y^)/dX,t - 2E(Yt)dE(Y)/dX,t]/l2
Dividing both the numerator and the denominator by dE{Yt)/dXjt, (12.107)

we obtain:

,,,, _ daijtl _ a'{Y)[dE{Yt^)/dE(Y)-2E(Yt)] MRS"-'' = d-/{Yt) 2[dtiz(Yt)ldE{Yt) - ^{Yt)da\Yt)ldE{Yt)]

'

Clearly, this expression does not depend on any Xjt with respect to which the derivatives of the two moments are taken initially. We have shown that the ratio is independent from any Xjt with respect to which the derivatives of two moments are taken for the three cases dE{Yt)/dcT{Yt), dE{Yt)/d'y(Yt) and dcr{Yt)/dj{Yt). The three remaining cases (9o-(yt)/<9£;(F0, d-f{Yt)/dE{Yt) mdd-f{Yt)/da{Yt) are just the inverse ratios of the three preceding cases. Furthemiore it is interesting to note that the six MRS's are interrelated. Using the chain rule, the product of (12.103) and (12.107) gives (12.105): n 2 108^ ' ^

9Eiyt)da(Yt) ^ da{Yt) d^{Yt)

dEjYt) d^{Yt) •

Each MRS on the left hand side can then be deduced: ri2 1091

g-E(y,) ^ dE(Yt) da(Y) d
300

Structural Road Accident Models

and da{Yt) _ dE{Yt,) dj{Yt) dj{Yt)'

(12.110) The inverse cases d-f{Yt)/dE{Yt),

da{Yt)/dE{Yt)

dE{Yt) da{Yt) • and dj{Yt)/da{Yt)

can be obtained by tak-

ing the inverse of every component in (12.108), (12.109) and (12.110) respectively. Although the sign of each MRS taken separately can be: Positive, if the derivatives of the two moments with respect to a same independent variable Xjt have the same sign, Negative, if they have opposite signs, the equation (12.108) implies that once the signs of any two MRS's are known, the sign of the remaining one is detemiined. Therefore, only four out of eight (2^) combinations of signs of the three MRS's are feasible. Table 12.6 gives these four feasible combinations. TABLE 12.6 Feasible combinations of signs of the MRS's among moments.

Feasible 1

Signs of MRS's

|

combinations

MRS'^''

MRS'''''

MRS'''''

1

A

-

-

+ +

1 1 1 1

1

1 1

+

B

c p

+ +

-

+

C. Elasticities of substitution between two moments In the preceding section, the MRS's among the three moments were defined. Since the first two moments E{Yt) and cr(Yt) are measured in the same units as the dependent variable Yf, whereas the skewness 7(i^) is a pure number, the MRS's where the first two moments are involved, namely MRS^''^

and MRS'^'^,

are dimensionless in contrast to the other four MRS's where

the skewness ^{Yt) is involved with one of the first two moments. Therefore, the elasticity of substitution between two moments rrir and m^ should be also computed to obtain a statistic which does not depend on the measurement units of 1^: (12.111)

^rn,^,n^.^^J^^m.,m.^^^m^!!^

rrir

urris rrir

Since the first moment E{Yt) and the skewness j{Yt) can be positive or negative —

E{Yt)

can be negative if the dependent variable Yt is not transformed by Box-Cox, hence Yt does not need to be strictly positive — whereas the standard error cr{Yt) is always positive, four cases for the signs of the elasticities of substitution among moments are given in Table 12.7, for each feasible combination of signs of the MRS's among moments reported in Table 12.6. Note that the first row associated with each feasible combination of signs of the MRS's corresponds to

The L-1.5 Program for BC-GAUHESEQ Regression 301

the case where all the moments are positive, hence the signs of the elasticities of substitution are identical to those of the MRS's. TABLE 12.7 Feasible combinations of signs of the elasticities of substitution among moments. Feasible

Cases

of signs of

Signs of moments e

Signs of the elasticities of substitution

(7

7

^e,cT

^-,7

7/^'^

+ + + + + + + + + + + + + + + +

+ +

-

-

+ -

MRS's

A

B

C

D

A.l

+

A.2

-

A.3

+

A.4

-

B.l

+

B.2

-

B.3

+

B.4

-

C.l

+

C.2

-

C.3

+

C.4

-

D.l

+

D.2

-

D.3

+

D.4

-

-

+ + -

+ + -

+ + -

+ -

+ + -

+

+ + -

-

+ + +

+

4-

-

-

+ + -

+ -

+ + -

H-

+ + + + + +

1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

12.2.6 Evaluation of moments, their derivatives, rates of substitution and elasticities A. Evaluation of the moments of Yt In Section 2.1, the expected values of the dependent variable Yt, as well as its standard error and skewness are computed: For each observation t in the estimation period. For a subset of observations included in the estimation period. Note that this option applies only to the expected values of Yt.

302

Structural Road Accident Models

Note that the calculated value of Yt given in (12.48) is not computed due to the fact that the expression in the square brackets can be negative and hence cannot be raised to the power l/\y,

if A, ^ 0.

B. Evaluation of the derivatives and elasticities of Yf In Sections 2.2, 2.3 and 2.4, the derivatives and elasticities of the sample and expected values of Yt, and those of the standard error and skewness of Yt are computed: At the sample means of the dependent and independent variables, and the heteroskedasticity variables if any, for the estimation period. At the sample means of these variables, for a subset of observations included in the estimation period. Note that this option applies only to the sample and expected values of Yt. C. Corrected elasticities for a quasi-dummy and a real dummy The concept of a derivative, hence an elasticity, strictly implies that an independent variable is of continuous type. Since all the independent variables specified in a model may not be necessarily continuous, a classification of the variables into four categories is needed to allow for the correction of the elasticities for the positive observations of two categories of variables, namely the "quasi-dummies" and the real dummies: Category 0: Continuous variable which is strictly positive, hence can be transformed by Box-Cox. Category 1: Any continuous variable which is not strictly positive, hence cannot be transformed by Box-Cox. Category 2: Quasi-dummy, i.e. a continuous variable containing only positive and null values, for example the level of snowfall per month, which is positive in winter, but null in summer. Strictly speaking, this type of variable cannot be transformed by Box-Cox due to the null values, but if a Box-Cox transformation is absolutely necessary, then only the positive observations of the quasi-dummy are transformed. Since the null observations cannot be transformed, an associated real dummy, which has a value of 1 for the positive observations of the quasi-dummy and a value of 0 otherwise, must also be created and specified in the regression equation. Category 3: Real dummy which has only two values: 0 and a positive constant, for example a binary ( 0 - 1 ) variable such as sex (0 for male and 1 for female), or a dummy associated with a quasi-dummy defined above. Unlike a quasi-dummy, a real dummy cannot be transformed by Box-Cox since all the values of 1 will be reduced to zero, if they are transformed with any value of the Box-Cox A-parameter, so that the transformed real dummy will contain only null observations.

The L-1.5 Program for BC-GAUHESEQ Regression 303

Two cases of the quasi-dummy should be considered: 1. Quasi-dummy not transformed by Box-Cox. The correction of the elasticities for the positive observations is based on the fact that one is interested in the effect on the dependent variable only when an activity or a phenomenon represented by the quasi-dummy or the real dummy really occurs, i.e. when the observations of the dummy are only positive. The following correction formula considered in Dagenais et al (1987) is used:

x+ V^i^t

- ,^. , ^ - ^., ^* . ^ ^ ^ ^ , , ,J+ = ^=

(12.112)

N

t

where rjj represents T/^^, rj^.^, Vx t^^ ^x evaluated at the sample means, Xj is the sample mean of a quasi-dummy or real dummy Xj computed from the total set of observations used in estimation, and X^ is the sample mean computed only from the positive observations of Xj whose number is N^. Note that for a subset of observations from the total set, the correction formula is analogous. 2. Quasi-dummy transformed by Box-Cox. If a quasi-dummy, say Qt, is transformed by Box-Cox, the correction will change slightly for TJQ^ since the expected value of Yt evaluated at the sample means of X and Z, say ^(^t)lx,Z' should be computed in fact from the sample mean of a strictly positive variable Qf instead of Qt so that the wellknown properties, namely E(Yt)\x z — ^ ^^^ ^ojx z ~ '^Qt\x,z i^ ^^^ standard linear regression case, may be preserved. Transfomiing only the positive observations of Qt by Box-Cox while leaving the null observations of the variable untouched is exactly equivalent to applying the Box-Cox transformation on a strictly positive variable Q| such that Ql is identical to Qt for the portion of positive observations Qf and Ql is equal to 1 for the portion of null observations of Qt, since Q^ ^ = 0, if Ql = 1 for any value of \Q. Thus, the correction formula in this case has the following form:

(12.113)

.Q. = VQ.^ = r,^'^;Q^

= ^Q'^[Nl

+ ^,Qt

where T/Q* is evaluated at the sample means of X and Z, with the mean of Q replaced by the mean of Q*, and NQ and NQ are respectively the numbers of positive and null observations of Qt.

304

Structural Road Accident Models TABLE 12.8 Correction of the elasticities 77^^^, ri^.^, Vx t^^^ ^x„ ^^ ^^ sample means, for the positive observations of a quasi-dumy or a real dummy. 1

ELASTICITY

CATEGORY

^h'^'^^h

"^h

^h

(2) Quasi-dummy N

• No Box-Cox

N

3

N(

• With Box-Cox

Vt. \

N(

V^

'"''Nl\N^,+Y.Xti)

\

iv/ ? ^

1

"-iv+l^yv^ + E x + J

(3) Real dummy N

• No Box-Cox

N

N

J

3

where rjj represents 77^^, rj^^ ^, TJ^ ^ or 7/J evaluated at the sample means, TJQ* represents 7/Q*, TJQ*, TJQ^ or r/g* evaluated at the sample means, N^' is the number of positive observations Xt's,

N^ is the number of null observations of X^t and A^ is the total number

of observations (N = NJ-\- N^ j used for estimation.

D. Evaluation of the ratios of derivatives of the moments of Yt and elasticities of substitution among moments In Section 2.5, three different relations were obtained for the ratios of two derivatives of the moments of Yt.

The first relation gives the Marginal Rate of Substitution between

two independent variables, the second the Marginal Rate of Substitution (and elasticity of substitution) between two moments, and the third the product of a MRS of moments and a MRS of variables. Since the third relation cannot be interpreted in economic temis, only the first two relations are computed at the sample means of the dependent and independent variables, and the heteroskedasticity variables if any, for the estimation period. Table 12.9 summarizes the evaluations provided for the derivatives and elasticities of the moments, and for the ratios of the derivatives of the moments and their elasticities.

The L-1.5 Program for BC-GAUHESEQ Regression 305

TABLE 12.9 Evaluations provided for the derivatives and elasticities of the moments, and for the ratios of the derivatives of the moments and their elasticities. Evaluation provided Statistics

Equation # reference

At the sample means Full set

Subset

|

Corrections for Quasi-dummies and Real Dummies

1

1

\

|

Derivative

|

Dx

(67)

/

/

^x.

(70)

/

/

Dx,.

(73)

/

(81)

/

^it

Elasticity

|

1x„

(67)

/

/

/

^h.

(70)

/

/

>r

^x,.

(73)

/

/

^l.

(81)

/

/

1 MRS between variables

1 MRSx.,x,

| (89)

/

\ MRS between moments MRS"^^-"^^

| (90)

/

1 Elasticity of substitution between moments m.r,ms

(111)

/

12.2.7 Student's t-statistics The t-statistics for the estimated parameters can be computed as follows: At the initial step of the maximization procedure, only the t-statistics for the estimated pcoefficients conditional upon the initial or fixed values of A^, A,^, A^, ^ and p can be given since they are based on the results of a standard regression of the transformed dependent variable F** on the transformed independent variables X**. Hence these tests are invariant to changes of measurement units in the original variables F, X and Z. At the convergence of the log-likelihood function, two types of t-statistics can be obtained:

|

306

Structural Road Accident Models

a. Unconditional t-tests for the estimates of ^6^, A^, A^;, A^,^,/> and cr^^ using the standard errors derived from the covariance matrix (12.38) are computed under the null hypothesis that the parameter is equal to zero. These tests are invariant to changes of measurement units in the original variables F, X and Z, except for ^ and (^-coefficients due to the application of the Box-Cox transformations on these variables. For the A-parameters, apart from the value 0 which corresponds to the logarithmic form of the variable, another interesting value to test against is the value 1 since it corresponds to a linear specification of the variable, hence unconditional t-tests for the estimates of Xy, A^; and Xz are also computed under the null hypothesis that the parameter is equal to one. b. To obtain invariant t-tests for the /? and ^-estimates, which are more reliable in hypothesis testing than the unconditional ones which are not invariant, hence can be boosted at will by a proper choice of measurement units of the variables, a simple procedure is to compute the t-tests for 0, 8, p and cr^^^ conditionally upon the estimates of Xy.Xj^ and Xz by using the conditional covariance matrix of /?, 8, p and alj which is dLtdLt

Etf

in (12.38) J n=n by deleting the rows and columns associated with A^, Ax and Xz (Dagenais et ai, 1987).

equal to the inverse of a submatrix derived from the matrix

When the program fails to converge to a maximum point, the covariance matrix of the estimated parameters cannot be obtained. In this case, the standard errors will be set equal to unity so that the final table can be printed to give information about the gradient. 12.2.8 Goodness-of-fit measures Two goodness-of-fit measures which indicate the accuracy with which a specified model adjusts to the observed data are given in the program: the i?|; measure is computed with the expected value of Yt, E{Yt), defined in section 2.1 and the i ? | measure is based on the likelihood ratio test statistic A. Since both measures are just nonlinear extensions of the standard linear regression case, they are called pseudo—R} measures. A. Pseudo-(E)-i?2 Instead of computing the usual li^ measure defined for a standard linear regression model:

(12.114)

R^ = l - -

rr

Y.{yt-Y)' t

which can give a negative value in the nonlinear case where the dependent variable Yt is transformed by a Box-Cox, since the sum of squares of residuals can be greater than the total sum of squares of Yt in deviation around its sample mean Y, we compute the square of the

The L-1.5 Program for BC-GAUHESEQ Regression 307

simple correlation coefficient between Yt and E{Yt) like Laferridre (1999): i2

[E{yt-y){E{Yf)-E{Y))

(12.115)

i:{yi-yfj:(E{Yt)-E{Y)) where Y and E{Y) are respectively the means of Yt and E{Yt): Y = J^Yf/N and E{Y) = t

^E(yi)/A^. This R?^ measure will always give values within the range 0 - 1 , and will be identical to the usual B? measure if the dependent variable Yt is linear (Johnston, 1984). B. Pseudo-(L)-i?2 — unadjusted:

(12.116)

Rl =

l-A^/^

— adjusted:

^i =

(12-117)

i - ] ^ ( i - ^ i )

where A is the ratio of the likelihood function C when maximized with respect to the regression constant /3o only, to the one with respect to U* as defined above:

A = max£/max£.

(12.118)

^0

n*

The Rl measure always remains inside the interval 0 - 1 since the maximum of C associated with /3o — which corresponds to the most restrictive model where no independent variables except for the constant term are specified — is necessarily smaller than the maximum associated with n* which includes less restricted parameters. Note that for the standard linear regression case (A^ = X^k = 1, VA:) without heteroskedasticity (^ = 0,VA^) and autocorrelation {p = 0), the two measures, R\ and R^^ coincide since reduces to the ratio of the unexplained sum of squares to the total sum of squares of Yt in deviation form:

(12.119)

A"^/^

-2/7V

E[Yr-Yt t

EiYt-yf

[t

fiZt)

1/2 t

308

Structural Road Accident Models

where Yt

is the calculated value of Y^** given in (12.46).

C. Moments of Yt observed and estimated, and of E{Yi) estimated In a regression model where a Box-Cox transformation is used on the dependent variable Yt, one is interested to see how much the transformation will make the distribution of the expected values of Yt close to the distribution of the observed values of Yt. The first three moments (mean, standard error and skewness) computed for the observed and estimated Yt's and also for the expected values of Yt given by E{Yt) constitute a set of goodness-of-fit measures. Note that in column B of Table 12.10, the three moments of the estimated Yt's are evaluated at the sample means of the variables. TABLE 12.10 Moments of Yt observed and estimated, and of E{Yt) estimated. MOMENT

(A) MOMENT OF K<

(B) MOMENT OF Yt

OBSERVED

ESTIMATED AT

(C) MOMENT OF E{Yt) ESTIMATED

THE SAMPLE MEANS

Mean

EiYi)

t

E(Y) =

t

^E{Y)

Standard error ">

V

Skewness

where "^i

v^——^—

liYt)

lfE{Y) =

lY,[EiYt)-W)Y

where Af-l

=


7y = ^sZ-Sy

j^^E{Yt)

^ 3

-

^t/^EiY)

^ J2[E{Yt)-E{Y)y 7V-1

1

The L-1.5 Program for BC-GAUHESEQ Regression 309

12.3

SPECIAL OPTIONS

12.3.1 Correlation matrix and table of variance-decomposition proportions A correlation matrix for the independent variables (excluding the constant) and the dependent variable in terms of the original variables {X and Y) is always given before the maximization procedure begins. Another correlation matrix in teims of the transformed variables (X** and y**) is also computed at the maximum of the log-likelihood function. The matrices are stored and output in a lower triangular form where the last row represents the pairwise correlations between the dependent variable and each of the independent variables. To detect the presence of multiple linear dependencies among the original independent variables X, the spectral decomposition of X X is used (Judge et al, 1985). This method is similar to the singular value decomposition of the matrix X given in Belsley et al (1980). The analysis is also performed for the transformed variables X** at the maximum of the log-likelihood function. The spectral decomposition of X X is defined as: k

(12.120)

X'X

= Y^^rVrP^, 1=1

where pi is the (A" x 1) eigenvector associated with the i-th eigenvalue ji of X X and the columns of X are scaled to unit length but not centered around their sample means, because centering obscures any linear dependency that involves the constant teim. Belsley et al use a set of condition indexes which is a generalization of the concept of the condition number of a matrix to detect the presence of near dependencies among the columns of X: — Condition number of X: K,{X) = (7^,ax/7mn7y^. where 7^,0.^ and 7mm are respectively the greatest and smallest eigenvalues of the 7i's. This number measures the sensitivity of b to changes in X X or F F in linear systems represented by the noraial equations X'Xb

=

X'Y;

— Condition indexes: r}i = (7max/70^ , ^ = 1,..., A". Note that if 7^ is equal to 7mm, then rii has a maximum value which corresponds to the condition number of X: ^max = t^{X). To determine which variables are involved in each near dependency, a decomposition of the variance of bk(k = 1,...,A") is performed: K

(12.121)

Var(6,) ^ a^ J ] (p^,/7,)

310

Structural Road Accident Models

The proportion of Var{bk) associated with any 7^ is then computed:

(12.122) These results can be summarized in a table of variance-decomposition proportions where the elements in each column are reordered according to the increasing values of the 7i's. TABLE 12.11 Eigenvalues of X'X, Condition Indexes of X and Proportions of Var(6^). EIGENVALUE

CONDraON

VARIANCE-DECOMPOSITION PROPORTIONS

INDEX

Var(bi)

Var(b2)

71 (7min)

m = «(X)

nil

Yin

HiK

72

V2

n2i

n22

n2K

7K (7max)

7/K = 1

HKI

nK2

HKK

Var(bK) 1

The sum of the proportions flip's in each column associated with Var(6jt) is equal to one. The following rules of thumb can be used to detect the presence of near dependencies: 1. High values of the condition indexes {r]i > 30) signal the existence of near dependencies, while high associated liik's in excess of 0.5 indicate which variable Xk is involved in the collinear relations. 2. When a given variable X^ is involved in several collinear relations, its proportions n^j^'s can be individually small across the high T/^'S. In this case, the sum of these proportions which is in excess of 0.5 also diagnose variable involvement. 12.3.2 Analysis of heteroskedasticity of the residuals At the initial and final steps of the maximization of the log-likelihood function, the estimated residuals of ut, vt and wt are plotted around their means, against the observations ordered by increasing values of one selected independent variable to provide a graphical analysis of the functional form of heteroskedasticity related to this variable. The residuals in standard units are plotted to scale on the horizontal axis, whereas on the vertical axis, the independent variable is not, due to the large number of observations: one line represents an observation, irrespective of the real distance between two consecutive observations.

The L-1.5 Program for BC-GAUHESEQ Regression 311

12.3.3 Analysis of autocorrelation of the residuals At the initial and final steps of the maximization of L, autocorrelation and partial autocorrelation functions are estimated for a time-series wt that is produced by a differencing operation applied to the estimated residuals e/'s which can be the Uf/s, Vf's or wt's depending on which model type is specified:

(12.123)

wt = {l- B)\l

- B'fet

, (t =: 1 + r,..., n)

where B is the backward shift operator defined as B^t = et_i, hence B'^et = ^t-m, d is the degree of consecutive differencing (0 < (i < 3), Z) is the degree of seasonal differencing (0 < D < 3) and s is the period of seasonal differencing (1 < 5 < 31). An additional constraint for
^i(''^*'~^'w)('^^+k-f'^)]

= 0,1,...,A^)

^ Cov{wuWt-^k) =lK(k

Y ^ [(^* - f^wf] E [{^t+k -1^where the mean and variance of wt are respectively fi;^ = (T'L = E\ {wt — z^;^)

E{wt)

=

E{wt+k) and

— ^ {^t-\-k — /^.^) » and the autocovariance at lag A: is 7^; =

Cov{wt,wt_^.k) = Cov{wt+rn,^t+m+k) foi" ^uy t, k and m. Note that 'po = 1 and 70 = cr^^. The autocorrelation function {p^} is dimensionless, i.e. independent of the scale of measurement of the time-series. Since 7 > \'^k\ {k = 1,2,...), all the ^^'s lie between - 1 and 1. It is symmetric about zero, that is 'pk = ^-k, hence considering the positive half of the function {k > 0) is sufficient to analyze the time-series. Since 7^ = ^A;7O = ?A;^^» a normal

312

Structural Road Accident Models

stationary process wt is completely characterized by its mean fi- and autocovariance function {jk}, or equivalently by its mean fi-, variance <J~ and autocorrelation function {^k}An estimate of the autocorrelation function, called the sample autocorrelation function, can be computed as:

(12.125)

rk = — Jk = 0,l,...,K Co

V

where Ck is the estimate of the autocovariance function at lag k:

(12.126)

Ck = ^

{wt - w) {wt+k - w)/{n

w is the sample mean of wt computed as

J^ wt/{n — r) and CQ is the sample variance t=l+r

of Wt computed as

^

(^wt — w) /{n — r).

Note that for long time-series, the sample

t=l+r

autocorrelation function will closely approximate the true population autocorrelation function, but for small samples, e.g. less than 50 observations, it will be biased downward from the latter. B. Standard error of the autocorrelation estimate To check whether the theoretical autocorrelation 'pk is effectively zero beyond a certain lag q, Bartlett's (1946) approximation for the variance of the estimated autocorrelation coefficient of a nornial stationary process can be used: .

(12.127)

Var(r;t) - jr

+00

^

{^t + pv+k?v-k - ^h^v'Pv-k + 2^?^)-

? ; = —cxo

If all the autocorrelations ^y are zero for v > q, all ternis except the first between the parentheses are zero when k > q. Thus at lags k greater than q, the variance of the sample autocorrelations nt can be expressed as:

(12.128)

Var{h)c^Ui.

+ 2^]^A

, {k >l) •

To obtain an estimate of Var(r'jt), say Var{r^k)^ the sample autocorrelations r^v (^ = 1:2,...,^) are substituted for 'py. The square root of this estimate is referred as the large-lag standard error. For example, if ^ = 0, i.e. the series wt is assumed to be completely random, then for all lags, the large-lag standard error reduces to:

The L-1.5 Program for BC-GAUHESEQ Regression 313

\jVar{n) '.

(12.129)

which is printed next to each row of the sample autocorrelations r^^ and can be used in a first step to test the null hypothesis that 'pk = 0, (A; = 1,2,...). A second step would be to select the first lag q at which an autocorrelation coefficient was significant and then use (12.128) to test for significant autocorrelations at lags k longer than q. For example, if n were significant, r2,r3,... could be tested using the standard error [(l + 2r^)/N]

. For moderate N, the distribution of an estimated autocorrelation coefficient,

whose theoretical value is zero, is approximately Noraial (Anderson, 1942): on the hypothesis that 'pk is zero, the estimate r^k divided by its standard error will follow approximately a unit Nomial distribution. C. Estimate of the partial autocorrelation function Whereas the sample autocorrelation function is used as a first guess which is certainly not conclusive for the significance of each autocorrelation coefficient, the partial autocorrelation function <j)££ which is based on the fact that for an autoregressive process of order p which has an autocorrelation function infinite in extent, (j)u is nonzero for l < p and zero for l > p, i.e. it has a cutoff after lag p, provides a means for choosing which order of autoregressive process has to be fit to an observed time-series.

Let (l)£j be the j t h coefficient in an autoregressive process of order £ so that

is the last

coefficient. The <^£y's can be shown to satisfy the set of equations:

(12.130)

pj = (j)nP3-\ +h2pj-2

+ ... + (l)uPj-e , (j = 1,2,...,^)

leading to the Yule-Walker equations:

(12.131)

1 PI

PI 1

P2 Pi

lpi-1

pi-2

Pes

'Pi-l'

'hi'

?£-2

4>t2

1

.(t^u.

7i' =

h .Pi,

The coefficient (j)££, regarded as a function of the lag £, is called the partial autocorrelation function. It measures the correlation between wt and Wf-i given m-i,...,

w;t-(£-i)- The

estimated partial autocorrelations ^^^'s can be obtained by substituting the sample autocorrelations F / s for the 'pj's in (12.130) to yield:

314

Structural Road Accident Models

(12.132)

rj = ^nrj-i

+ fcf^--2 + • • • + fer^-/ , {j =

l,2,...J)

and solving the resulting Yule-Walker equations. A simple recursive method for estimating the (/>£-,'s, due to Durbin (1960), may be used if the values of the parameters are not too close to the nonstationary boundaries: i f ^ = 1, ^-1 ^

(12.133)

(l>a = {

if^ = 2,3,...,X. 1 - Yl 't>i-i,jr.j

where <^^j = ^i-i,j - hi4>e-i,i-j (i = 1 , 2 , . . . , ^ - 1) and L is the maximum lag of the partial autocorrelation function. Note that L should be less than or equal to K which is the maximum lag of the autocorrelation function used in (12.124). D. Standard error of the partial autocorrelation estimate If the process is autoregressive of order p, the estimated partial autocorrelations of order p-|-1, and higher, are approximately independently distributed (Quenouille, 1949), and the standard error for these autocorrelations is:

(12.134)

^ ^ ( 4 ) ^ - L , (£ > p)

which can be used to check whether the partial autocorrelations are effectively zero after some specific lag p. Note that on the hypothesis that (f)a is zero, the estimate (!)££ divided by its standard error is approximately distributed as a unit Normal deviate. 12.3.4 Forecasting: Maximum likelihood and simulation forecasts A. Maximum likelihood forecast To obtain the predicted values of the dependent variable over the p periods for t = n + 1 , . . . , n+ p,, the following approach is adopted: for the first period of forecast n + I, the maximum likelihood predicted value of F„,+i is given by the first order condition dLn+i/dYn+i — 0, since maximizing the total log-likelihood function

77-+1

n

J2 ^t — S

t=l+r

t=l+r

^* + ^77+1 with respect

to Yn+1 is equivalent to maximizing the log-likelihood function Ln+i associated with the

The L-1.5 Program for BC-GAUHESEQ Regression 315

(n + l)-th period only. The first derivative of the log-likelihood function Ln-\-i with respect to K+i can be written as: (12.135) 1 dLn+i

_

dYn+1

"

Yrn+l /2

Yn+l

1

- 1 -

Yr, n+1

(A.

where U.+1 - C i

1

/

Un+1

2^^^f(y I

^l/2 JK^n-¥l-i) '^n+l-i .1/2

1^ pr ~t f{Zn+l-i

1/2

alf{Zn^^f'\f{Zn+^^'^'

if A^ 7^ 0 if A„ =: 0

.(A.,)

" E/^^^rnti k

\\y ^ 0-1 Two subcases must be distinguished: (i) If Aj, y^ 1, setting the derivative equal to zero and solving for Y^^^, yield an equation of the second degree in F^^j:

Y^':i\+BY^'l,-^C

(12.136)

=0

where

(12.137)

Un+l-i

B•

t

nZn+l-l)

and

C=-\y{Xy-l)aif{Z^^^).

(12.138)

To ensure that the predicted value of F^+i, say Yn+i, is real and positive, both roots of (12.136), namely Y^^^ = (-B ± VB'^-iC]/2,

must be real, i.e. if ^^ - 4C > 0, and

the greater root will be chosen since it will be always positive as shown in the next section. Finally, the chosen root must also satisfy the second order condition d'^Ln+i IdY^j^^ < 0. (ii) If A J, = 1, i.e. the dependent variable is specified in a linear form, the first derivative reduces to:

(12.139)

Un+l

dYn+1

^li\Zn

xl/2

f{Zn +1)a / 2

E

Pi-

'^n+l-i

f{Z^ +1-1)

a/2

316

Structural Road Accident Models

which gives a unique maximum likelihood predicted value of Yn+i by setting the derivative equal to zero and solving for Yn+i:

K,+i = i + Yl ^kxj^:i + m,,+if' Yl pi-

(12.140)

k

£

'J'n-i-l-i

a/2 f{Zn+l-l)

\Xy — 0-1 Setting the first derivative equal to zero and solving for Yn+i also give a unique maximum likelihood predicted value of Yn+i:

(12.141)

Yn+i

=exp

pr E^^<:i + n^n.^r'j: k i

^n+l-t f{Zn+l-i

1/2

• ^tuf{Zn-\-l)

Note that for all cases, the predicted value F„,+iis computed at 11 = 11 Clearly, the approach can be easily extended to the next forecast periods, since in each single period, only the log-likelihood function associated with that period is needed for the maximization, given the sample observed values of the dependent variable in the estimation period and its predicted values in the preceding forecast periods. Thus for the i-\h period of forecast, the predicted value of Yn+i is also given by the first order condition dLn+i/dYn+i

= 0,

subject to the second order condition d^^Ln+i/dY^^_^^ < 0. The formulas are analogous to (12.135) - (12.141) with n + 1 replaced by n + z. B. Variance of the forecast error The variance of the forecast error for the first period of forecast, cr^^^^, is computed from the covariance matrix of the parameters 11 and Yn+i, which is equal to minus the inverse matrix of the second derivatives of the total log-likelihood function i>(i) = S";jj^_,_,^Lt with respect to n and Yn+i, evaluated at FI = 11 and y„+i = K„+i:

(12.142)

COV^d) =

COV^o)

Cn (1)

-^(1)

where COV(o) is the covariance matrix of 11 which is already estimated in (12.38) and is used instead of —[d'^L(^-^^/dIldW]

which is too complex to be evaluated,

(7(1) is the column vector of covariances between the elements of fl and Yn+i:

(12.143)

^(1) = -COV(^Q^

dHn(1) dUdYn+i

d'k^) dY^^,

The L-1.5 Program for BC-GAUHESEQ Regression 317

and al_^-^ is the variance of the forecast error for F^+i •

9'hi)

-.2

(12.144)

dYn'+i

+ a;,,c70V(-/qi)

In general, the variance of the forecast error for the z-th period of forecast, cr^^^, can be recursively computed from:

(12.145) COV^,) where COVi-i

is the covariance matrix of ^{i-i)

=

(n',K,i+i,... ,l^j,4.j_i j

which

is already estimated in the preceding period of forecast, and is used instead of — ^^L(^>)/5n(i_i)5n/_jx

which is too complex to be evaluated,

(7(i) is the column vector of covariances between the elements of Tl[i-.\) and Yn+i'.

(12.146)

a. =

-GOV.,

a^L i^)

ff'L {^) dIli^,_i^dYn+t

and cr^^_^i is the variance of the forecast error for Yn+i'-

(12.147)

~n+i

d'L (0 ^y2 .

^C[^COV^,\fi^-

Note that a2L(,)/OT(,_i)aK+, = ^X,,+,/OT(,_i)ay;+, and dH^^jdY^i^^ - d^'Ln^.jdY^,^^, since Y^+i appears only in Xn+z-

The second derivatives d^^L^+iIdIl(^i_i^^dYn^i and

d'^ Ln+i /dY^_^^ are in the form Y^~l/\alf{Zn+if^^^\

times a component which is given

in Table 12.12 for each element of II^^) = f n|-_-^N,Fn+z) crossed with Yn+i.

318

Structural Road Accident Models

TABLE 12.12 Components of the second derivatives of Ln+i with respect to !!(,) and Yn+i Yr.^.

A^.k)

S^(A.fe)

/ ( W / ^ ?'^>(^.^._,)^/^ "n+i/l^™

1 nz„,.r'

• tVn-\-i In y^

1

dyl^' QK

f(Zr,.,ir^

ff^-EV / ( 2 n + . - 0 " '

tl'n+j-C

9X,

5^-^

9^^^

...1/2

2 ^ ^ % / 7 ..

t

.1/2

fiZn+,.-iy

dZ

Y7^V;^i-_.!^

(A.™)

m _

7 1/2 / ^ m , n

Yn+^

Pi

8X1

f{Z„^0"'

/(^n+,;-.)

-E ^nz„^<.e)"'

2^

~Z^^^

/(Z„+,_,)^/'

./7

(s = 1,... ,miii(i - 1, r) and i > 2)

-(2yi^^^ + B)/[yn+,/(^n+0''

^^-

.l/2^m„n+i-

The L-1.5 Program for BC-GAUHESEQ Regression 319

In Table 12.12, the different derivatives dY^^^} jdXy , ^^i;^;+V^A^^, and dZ^^^^^JOX^m as well as their lagged expressions are computed by the generic formula given in (12.36). In evaluating the second derivatives of Ln+i at the maximum point of the total log-likelihood function L^^) = T>^^j_^^Lt, the predicted values of (Yn+i,..., Yn+i) obtained by the approach above are used just as when the predicted value of Yn-^i is computed, the predicted values of {Yn+i,..., Yji-^i-i) are also used. The error of forecast is computed in percentage whenever the observed value of the dependent variable is available in the forecast periods t = rz+1,..., n+p:

(12.148)

e,::. ( 1 - ^ | 1 0 0

where Yf and Y^^ are respectively the predicted and observed values of y^ The 95% confidence interval for Yt is also computed:

(12.149)

Yt - 1.96a,
+ 1.96a,

where at is estimated from (12.147). C. Simulation forecast In addition to the maximum likeHhood predicted value Yt (t = n + 1 , . . . , n + p), the simulated value of the dependent variable Yt is computed at 11 = 11 as:

(12.150)

[i + Xyf{Ztf\j:ePiytU Yt= ' ^

+ ^kf^kXi;)f

^'

if A, ^ o

^

exp [f{Ztf' {^^^p^-Mip + ^khXi:)]

if

Xy^^

where Yt-t is replaced by Yt-i if Yt-i is not observed, i.e. if t — ^ > n. Note that for the case Xy ^ 0, if the expression between the squared brackets happens to be negative, then it cannot be raised to the power IjXy and the program will be stopped at that point of the run. |A^ 7^ 0.| Two subcases should be considered: [Subcase c A,, ^ T | : If both Yt and Yt are positive, the bias Yt — Yt, ov equivalently Y^^ — Y^/^, can be shown to be negative if A^ < 0 or A^ > 1 and positive if 0 < A^, < 1. For the first period of forecast t = n + 1, if the two roots of (12.136) are denoted by Y^' = (-B + VB'^ -4C)/2

and ? / = (-B - \/B'^ - AG\ /2, then replacing -B by

its value which is equal to Y^ ^ in the greater root 1^ ^ yields:

320

Structural Road Accident Models

Y^y = Y^y M + y^i - 4a/F/^M /2 ,

(12.151)

If Ay < 0 or A^ > 1, then C is negative, \Jl - 467?/^^ > 1 and the bias Y^' - F / ^ is negative. If 0 < A^ < 1, then C is positive, A/I - 4(7/?/^' < 1 and the bias F / ^ - F / ^ "^A

^ A

'^'^^

'^A

^ A

is positive. Furthermore, since Y^^ =Y^^ -\-Y^ , the bias Y^^ —Y^^ will be given by the smaller root Yi . These results can be illustrated by the following diagrams: iNegative bias| : f / '

< F / ^ ( A ^ < 0 Or A^ > 1)

Y

Y y

-B/2

Y y

-B

iPositivebiasI : F / ' > F / ^ (0 < A^ < 1) Yf'

^/'

Yi^"

-B/2

-B

Choosing the greater root Y^ ^ will ensure that the maximum likelihood predicted value of Yt is always positive since the smaller root F^ can be negative. These results will hold for every period of forecast t = n + l , . . . , n + p. I Subcase A^ z^ i.| I In this casc, the bias Yf — Yt is null, i.e. the simulated value of the dependent variable Yt is identical to the maximum likehhood predicted value Yt given in (12.140):

Yt = 1 + fiZtf (12.152)

fc

PiYt-i + E PkXS k

=i + fi^Ztf' E Pi Yt-i - E/^kXi.^i k

= Yt

J

+ E f^kXtt k

The L-1.5 Program for BC-GAUHESEQ Regression 321

|A^ = 0.| In this case, the bias Yt - Yt is positive:

Yf

(12.153)

=

exp

-

exip{f(Zt)

— exp

t

\J[^t-i)

k

I k

\)

j:^,xi>;"^+mf'j: Pi f{Zt

"^'/' '-^t-i)

> % since Yt contains in the argument of the exponential (12.141) an additional negative term equal to -alf{Zt) 12.4

.

REFERENCES

Abramowitz, M. and LA. Stegun (1965). Handbook of Mathematical Functions. Dover Publications, New York. Amemiya, T. (1977). A Note on a Heteroscedastic Model. Journal of Econometrics, 6, 365-370; and "Corrigenda". Journal of Econometrics, 8, 275. Anderson, R.L. (1942). Distribution of the Serial Correlation Coefficient. Annals of Mathematical Statistics, 13, 1. Appelbaum, E. (1979). On the Choice of Functional Forms. International Economic Review, 20, 449-458. Bartlett, M.S. (1946). On the Theoretical Specification of Sampling Properties of Autocorrelated Time Series. Journal of the Royal Statistical Society, Series B, 8, 27. Belsley, D., E. Kuh and R.E. Welsh (1980). Regression Diagnostics. John Wiley, New York. Berndt, E.K., B.H. Hall, R.E. Hall and J.A. Hausman (1974). Estimation and Inference in Nonlinear Structural Models. Annals of Economic and Social Measurement, 3, 653-665. Berndt, E.K. and M.S. Khaled (1979). Parametric Productivity Measurement and Choice among Flexible Functional Form. Journal of Political Economy, 87, 1220-1245. Blum, U. and M. Gaudry (1999). "SNUS-2.5, a Multimoment Analysis of Road Demand, Accidents and their Severity in Germany, 1968-1989". Publication CRT-99-07, Centre de recherche sur les transports, Universite de Montreal; Working Paper No. 99-xx, Bureau d'Economie Theorique et Appliquee, Universite Louis Pasteur, Strasbourg; Dresdner Beitrage zur Volkswirtschaftiehre Nr. 4/99, Dresden University of Technology, 50 p.. May 1999. Gaudry, M. and S. Lassarre, (eds), Structural Road Accident Models: The International DRAG Family, Elsevier Science, Oxford, Ch.3.

322

Structural Road Accident Models

Box, G.P. and D.R. Cox (1964).

An Analysis of Transformations.

Journal of the Royal

Statistical Society, Series B, 26, 211-243. Box, G.P. and G.M. Jenkins (1976). Time Series Analysis: Forecasting and Control, HoldenDay, San Francisco, Revised Edition. Dagenais, M.G., M.J.L Gaudry and T.C. Liem (1980). Multiple Regression Analysis with Box-Cox Transformation and Nonspherical Residual Errors: A Transportation Application. Publication no. 166, Centre de recherche sur les transports, Universite de Montreal. Dagenais, M.G., M.J.I. Gaudry and T.C. Liem (1987). Urban Travel Demand: The Impact of Box-Cox Transformations with Nonspherical Residual Errors. Transportation

Research,

21 B, 6, 443-477. Dagum, C. and E.B. Dagum (1974). Construction de Modeles et Analyse Econometrique. Cahiers de ITnstitut des Sciences Mathematiques et Economiques Appliquees, 8, no. 11-12. Egy, D. and K. Lahiri (1979). On Maximum Likelihood Estimation of Functional Forai and Heteroscedasticity. Economics Letters, 2, 155-159. Fletcher, R. and M.J.D. Powell (1963). A Rapidly Convergent Descent Method for Minimization. Computer Journal, 6, 163-168. Froehlich, B.R. (1973). Some Estimators for a Random Coefficient Regression Model. Journal of the American Statistical Association, Gaudry, M. et al

68, 329-334.

(1993, 1994, 1995, 1996, 1997). Cur cum TRIO? Publication CRT-901.

Centre de recherche sur les transports, Universite de Montreal, 20 p. Gaudry, M.J.I, and M.G. Dagenais (1978). The Use of Box-Cox Transfoimations in Regression Models with Heteroskedastic Autoregressive Residual. Cahier no. 7814, Departement de sciences economiques, Universite de Montreal. Gaudry, M.J.I, and M.G. Dagenais (1979).

Heteroscedasticity and the Use of Box-Cox

Transfonnations. Economic Letters, 2, 225-229. Gaudry, M.J.I, and J.J. Wills (1978). Estimating the Functional Fomi of Travel Demand Models. Transportation Research, 12, 253-289. Geary, R.C. (1966). A Note on Residual Heterovariance and Estimation Efficiency in Regression. American Statistician, 20, (4), 30-31. Glejser, H. (1969). A New Test for Heteroscedasticity. Journal of the American Association,

Statistical

64, 316-323.

Goldfeld, S.M. and R.E. Quandt (1972). Nonlinear Methods in Econometrics,

North-Holland,

Amsterdam. Harvey, A.C. (1974).

Estimation of Parameters in a Heteroscedastic Regression Model.

European Meeting of the Econometric Society, Grenoble, France.

The L-1.5 Program for BC-GAUHESEQ Regression 323

Harvey, A.C. (1976). Estimating Regression Models with Multiplicative Heteroscedasticity. Econometric a, 44, 461-465. Heckman, J. and S. Polachek (1974). Empirical Evidence on the Functional Forni of the Earnings-Schooling Relationship. Journal of the American Statistical Association, 69, 350-354. Hildreth, C. and J.P. Houck (1968). Some Estimators for a Linear Model with Random Coefficients. Journal of the American Statistical Association, 63, 584-595. Hollyer, M., W. Maling and G. Wang (1979). Domestic and International Air Cargo Activity. Report no. FAA-AVP-79-10, U.S. Department of Transportation. Johnston, J. (1984). Econometric Methods, Third Edition, McGraw-Hill, New York. Judge, G.G., W.E. Griffiths, R.C. Hill, H. Llitkepohl and T.C. Lee (1985). The Theory and Practice of Econometrics, Second Edition, John Wiley, New York. Kau, J.B. and C.F. Sirmans (1976). The Functional Forai of the Gravity Model: A New Technique with Empirical Results. Working paper. Department of Finance, University of Illinois. Presented at the North American Meeting of the Regional Science Association in Toronto. Kmenta, J. (1971). Elements of Econometrics, Macmillan, New York. Laferriere, R. (1999). Apply the Pearson Coefficient as a Measure of Jl'^ in Nonlinear Models. Publication CRT-99-25, Centre de recherche sur les transports, Universite de Montreal. Liem, T.C. (1979). A Program for Box-Cox Transfomiations in Regression Models with Heteroskedastic and Autoregressive Residuals. Publication CRT-134, Centre de recherche sur les transports, and Cahier #7917, Departement de sciences economiques, Universite de Montreal, 60 p.. Refereed in The American Statistician, Vol. 34, no. 2, p. 121, May 1980. Liem, T.C, Dagenais, M. and M. Gaudry (1983). L-1.1: A Program for Box-Cox Transfonnations in Regression Models with Heteroskedastic and Autoregressive Residuals. Publication CRT-301, Centre de recherche sur les transports, and Cahier #8314, Departement de sciences economiques, Universite de Montreal, 70 p.. Liem, T.C, Dagenais, M. and M. Gaudiy (1987, 1990, 1993). LEVEL: The L-1.4 program for BC-GAUHESEQ regression —^Box-Cox Generalized Autoregressive HEteroskedastic S^ingle EQuation models. Publication CRT-510, Centre de recherche sur les transports, Universite de Montreal, 41 p. Liem, T.C, Dagenais, M., Gaudry, M. und R. Koblo (1986). Ein Programm fiir Box-Cox Transformationen in Regressionsmodellen mit heteroskedastischen und autoregressiven Residuen. Publication CRT-301-D, Centre de recherche sur les transports, Universite de Montreal, and Discussion paper 4/86, Institut fur Wirtschaftspolitik und Wirtschaftsforschung, Universitat Karlsruhe, 44 p.

324

Structural Road Accident Models

Park, R.E. (1966). Estimation with Heteroskedastic Error Terms. Econometrica, 34, 888. Quenouille, M.H. (1949). Approximate Tests of Correlation in Time Series. Journal of the Royal Statistical Society, Series B, 11, 68. Savin, N.E. and K.J. White (1978). Estimation and Testing for Functional Forai and Autocorrelation: A Simultaneous Approach. Journal of Econometrics, 8, 1-12. Schlesselman, J. (1971). Power Families: A Note on the Box and Cox Transfoniiation. Journal of the Royal Statistical Society, Series B, 33, 307-311. Spitzer, J.J. (1976). The Demand for Money, the Liquidity Trap, and Functional Forms. International Economic Review, 17, 220-227. Spitzer, J.J. (1977). A Simultaneous Equations System of Money Demand and Supply Using Generalized Functional Fomis. Journal of Econometrics, 5, 117-128. Theil, H. (1971). Principles of Econometrics, John Wiley, New York. Tobin, J. (1958). Estimation of Relationships for Limited Dependent Variables. Econometrica, 25, 24-36. Welland, J.D. (1976). Cognitive Abilities, Schooling and Earnings: the Question of Functional Form. Working paper 76-14, Department of Economics, McMaster University, Hamilton. White, K.J. (1972). Estimation of the Liquidity Trap with a Generahzed Functional Fomi. Econometrica, 40, 193-199. Zarembka, P. (1968), Functional Form in the Demand for Money.

Journal of the American

Statistical Association, 63, 502-511. Zarembka, P. (1974). Transformation of Variables in Econometrics. P. Zarembka (ed.). Frontiers in Econometrics. Academic Press, New York.

The IRPOSKML Procedure of Estimation 325

13 THE IRPOSKML PROCEDURE OF ESTIMATION Lasse Fridstrom

13.1. A C C I D E N T F R E Q U E N C Y M O D E L S P E C I F I C A T I O N

In the TRULS-1 model for Norway (Chapter 4), the general form of the accident frequency and casualty count equations is this: (13.1) ln{y„+a) = Y.P,^'rr'+»,rHere, y,^ denotes the number of accidents or victims (of some kind) occurring in county r during month t. x^^. are independent variables, with Box-Cox parameters /L^^, and regression coefficients P^, the w,^ 's are random disturbances, and a is the so-called Box-Tukey constant^ In general, we set ^ = 0.1. Thus, the dependent variable is Box-Tukey transformed, although with a Box-Cox parameter set to zero, yielding a logarithmic functional form. The independent variables may, in principle, all be Box-Cox-transformed, although the Box-Cox parameters need not all be different from each other. As argued by Fridstr0m (1999), casualty counts may be assumed to follow a (generalized) Poisson distribution. This means that the model (13.1) is heteroskedastic, and in a quite particular way, with y,^ -by assumption-Poisson distributed: (13.2) var(u,^) - var^n{y^^ + a)].

* See Box and Cox (1964), Tukey (1957), or Gaudry and Wills (1978).

326 Structural Road Accident Models

var[ln(y+a)]

r""""^"-'

1

\

0.1

^

a=0.0-K^

0.01 ^

a=0.1'

y ^ a=0.5

nr>i

a=1

y ^

^ \

< ^

Graph 13.1. The variance of ln{y + a), where y is Poisson distributed with parameter co. We therefore need to evaluate the variance of the log of a Poisson variable with a small (BoxTukey) constant added. For large Poisson variates, one can invoke the Taylor approximation formula (13.3) var[ln(y,^+a)]^--^^^-^^

when4;;J»«.

For smaller accident counts, however, var[ln{y,^ + a)] is not a linear function of the reciprocal of ^^//•J- It is not even monotonic (see Graph 13.1). Since - to our knowledge - there exists no exact, closed-formed formula linking var[ln{y,^ + a)] to E\y,^ J, we proceeded to construct a numerical approximation. The results of this exercise are summarized in Graph 13.2 and Table 13.1. Curve A shows the exact relationship between var[ln(y + a)] and = ^[3;], for CD values given by (13.4) cD = e-'^''''''

where/= 1,2,3,...,5016,

i.e. for CD values ranging from e~^ = 0.00248 to e^'^"^ = 692, in equal logarithmic steps^.

' A small GAUSS program was written to compute the exact points defining curve A.

The IRPOSKML Procedure of Estimation 327

Table 13.1. Approximations to var[ln{y + 0. l)], when y~ P (&)). I.

BETA (COND.

T-STATISTIC)

VARIANT = VERSION = DEP.VAR. =

qplsOl 12 qpvarOl

qplsOl qpvarOl

Artificial data describing Poisson distribution betaO In(omega) (i e, logarithm of Poisson parameter) Box-Cox betal transformed Poisson parameter Exponential gammal of Poisson parameter divided by 100 Box-Cox transformed gamma2 exponential of omega/100 Cubic inverse of alpha-3 Poisson parameter

.137595E+01 (301.99)

. 900662E+01 (-210.74)

-.998650E+02 • - " .07) (-261.07) LAM ^ AM .146792E+01 (-16.10)

.212181E+02 .64' ^ ' ' ' • 1LA1 ^ .786153E+00 (-81.69)

.111817E+03 (142.96) LAM .542707E-07 (25.51)

.547438E+02 .95' LA .130388E-01 (66.32)

<"^-g|^

Squared inverse of Poisson parameter

alpha-2

.407530E-04 (-29.95)

.438190E+00 (-86.60)

Inverse of Poisson parameter

alpha-1

.108554E-01 (40.71)

.138182E+02 (129.52)

REGRESSION CONSTANT alphaO (-221.44) (-119.67)

.963134E+02 -.126669E+02

Poisson parameter

alphal

.972405E+02 (262.69)

.174562E+00 (219.46)

Squared Poisson parameter

alpha2

.402538E-02 (-351.77)

.616178E-03 (-212.39)

Cube of Poisson parameter

alpha3

.476936E-05 (62.28)

.138938E-05 (137.96)

II.

(omega)

PARAMETERS (COND. T-STATISTIC)

BOX-COX TRANSFORMATIONS: UNCOND: [T-STATISTIC=0] / [T-STATISTIC=l] LAMBDA(Y)

zero

LAMBDA(X)

lambda1

LAMBDA(X)

lambda2

.000 FIXED -.663 [-21.321 [-53.47] -6.526 [-231.141 -266.56

.000 FIXED .993 [68.861

[-.50J

III^^™5^L:_^I^!EI^II^^

-3.667 -104.281 -132.72 _CURVE_=

LOG-LIKELIHOOD PSEUD0-R2 : - (E) - E ADJUSTED FOR D.F. - (L) ADJUSTED FOR D.F. SAMPLE : - NUMBER OF OBSERVATIONS - FIRST OBSERVATION - LAST OBSERVATION NUMBER OF ESTIMATED PARAMETERS : . BETAS . BOX-COX

C

__________ .994 1.000 .994 1.000 4700 1 4700

13585.346 .999 1.000 .999 1.000 3201 1500 4700

11 2

11 2

328 Structural Road Accident Models

var [ LnCy + O . 1 ] ] and approximations, u/here y is Po i sson with parameter omega

Exact var[tnCy+0.1D] omega/(omega + 0. 1 )^2 TRIO PROJECT: DATE: 99 04 13 USER:toi

TRULS - an econometric model or road use, accidents and their severity

trio

Graph 13.2. Alternative approximations to var[ln(y + a)], when y~P (co). Curve B is the approximation given by the first equality sign of (13.3). One notes that this "approximation" is very bad - indeed, outright misleading - for any value of co less than 10 (corresponding to an index / < 3321). C and D are numerical approximations estimated by fitting functions to a sample of observations (given by 13.4) on the exact relationship (curve A). Curve C uses the entire sample up to &> = 314; as one can see, this approximation is not too accurate over the middle range of co values. Curve D, however, is estimated without using the very smallest observations, and provides a quite satisfactory fit in the middle range. Both approximations have the form

(13.5) f{co) = exp\fiMo^hfi,o^^'^^^ry''''^rM''T

+ 2^a,co *=-3

i.e. there are 13 parameters estimated, including the constant a^ and the two Box-Cox parameters Xy and Aj-^^ estimate a casualty equation like (13.1), we therefore proceed as follows. A first-round set of estimates j8l and X[. are computed based on a homoskedasticity assumption within the general BC-GAUHESEQ estimation procedure of Liem et al (1993).

The IRPOSKML Procedure of Estimation 329 Fitted values yl are calculated by the formula yl = exp\Y.i^^^'' ~ a. These estimates are then plugged into the variance approximation C or D^ (of Graph 13.2), to form a set of variance estimates dl (say) for the Box-Tukey transformations ln(y,^ + a), valid under the Poisson assumption. A second round of estimation is run, this time with heteroskedastic disturbances, obtained by specifying

A^, =0,^,= I, z,^, = al, and z,„. = 0 Vz > 1

heteroskedasticity formula: u,^ =

in

the

BC-GAUHESEQ

ul, where the u',^ are homoskedastic. A

new set of fitted values yl is thus obtained, and a new set of variance estimates calculated. Etc. The process is repeated until convergence. It turns out that three to four iterations are usually sufficient for convergence. We refer to this procedure as the Iterative Reweighted POisson-SKedastic Maximum Likelihood (IRPOSKML) method"^. In Graphs 13.3 and 13.4, we show-as an illustration-fmal round sample values for the disturbance variance in the TRULS-1 equations explaining injury accidents and fatalities, respectively. One notes that in the injury accident model, the variance varies by a factor of at least 10 across the sample, a clear indication that our weighting procedure is worthwhile. In the fatalities equation, the variance varies non-monotonically, owing to the rather small expected values found. These values are such as to question the appropriateness of a model with (approximately) normal disturbance terms. For this and other reasons, we prefer to estimate the number of fatalities in a two-step fashion, combining the injury accident equation with an equation explaining the number of fatalities per injury accident (severity).

^ For casualty counts with expected values ranging below 0.1, we use the more robust approximation C. In other cases, we use the (locally) more accurate function D. "^ The "ML" part of the acronym is due to the fact that the basic BC-GAUHESEQ algorithm is a maximum likelihood method based on normally distributed disturbances. The main principle of the IRPOSKML procedure could, however, be applied to any program capable of weighted least-squares regression analysis.

330 Structural Road Accident Models

CaLcuLated

ident

Est imated TRIO PROJECT: DATE:97 12 10 USER:toi

TRULS - an econometric model of n

expected

modeL

number

oF

!, accidents and their

.no

Graph 13.3 Calculated disturbance variance in the TRULS-1 injury accident equation, plotted against the estimated expected number of injury accidents.

C a l c u t a t e d

v o r i a n

e

in

f a t a l

1t y

model

c

> O D • ^

:;

I

o •Q

i

O O c o

*

D

TRIO PROJECT: DATE:971210 USER:toi

Est , • n o t e d TRULS - in econ ometric model of road u se, accident s and their

e x p e c t e d

n um b e r

oF

Fatal

i t i e s

trio

Graph 13.4. Calculated disturbance variance in the TRULS-1 equation explaining fatalities, plotted against the estimated expected number of fatalities.

The IRPOSKML Procedure of Estimation 331 13.2. SEVERITY MODEL SPECIFICATION The general form of the TRULS-1 severity equations is this:

(13.6)

k+a y,r + ^

= Z>3,x;:^"^+".

Here, y,^ denotes the number of injury accidents in county r during month t, while h,^ is the number of victims of a certain severity (road user killed, dangerously injured, or severely injured, respectively). Note that in this case, the dependent variable Box-Cox parameter (//) is unconstrained, and estimated along with all the other model parameters. Severity ratios are subject to heteroskedastic random disturbances, as are single casualty counts. In this case, however, the issue is somewhat more complex, in that we are dealing with a ratio of two random variables, transformed by a general Box-Cox function. Sverdrup (1973, p. 147)

shows how the variance of an arbitrary differentiable function

F = g(xp ^2) of two random variables x, and Xj can be approximated by means a first order Taylor expansion (Y', say) around the means (X\ and Xi ^ say):

(13.7)

Y^r=g{xvX2hi{h-x)^g{XvX2) i=\

^Xi

u (13.8) var{Y)^var{Y') = J^

var{x,) + 2 ^Xi)

cov{x,,x,), ^Xi^Xi

where we use —^ as shorthand for — ;?(x,, x^), ^ „ dXi dx,^ ' ^¥rx:^.

Letting

g(x^,x^^

(/^)

, X2 = 7/^, and Xj = \^, we have

(13.9) - ^ = (77,, + aY\(D,^ + a)~' drj,^

(13.10) f^ = -(;7. + «)''K + «r""

332 Structural Road Accident Models where we have defined 77,^ = E[h,^] and co,^ = E[y,^]. Substituting these expressions into (13.8), we have:

(13.11)v«r

(/O'

k+a y,r + ^

-2{rj,+ar-\c,,-^aY^-'pJvar[K)var[y,)

+ k,+af'-\co,+ay'

var{\)+{rj, +af'{co,^+a)-''-' var{y,;)

+ {?], + af"-' {(D, + ay-' ri, + {rj, + af {co, + a)-''-' co,.,

where /7,,^ equality ^,r=E{yJ

^{K'y,r) ^var{h,^)var{y,^) sign

follows

is the correlation coefficient between h,^ and y^^, and the last

from

the

Poisson

assumptions

r/,^ =E{h,^) = var{h^^) and

= var{y,^).

To estimate a severity equation like (13.6), we proceed as follows. A set of separate casualty equations for y,^ and /z,^are estimated (by the IRPOSKML - a, and Y.1^^^'

procedure above), fitted vales j),^ are calculated by the formula y,^ = exp\ similarly for h,^.

The residuals w,^ = y,^ - y,^ and vt",^ = h,^. - h,^ are calculated, and an estimate of the probabilistic correlation between the two variables is calculated as:

^ _

XI.(";:-""k-"')

where u ^ and u ^ are the means of the respective residuals. Substituting h,^ for rj,^, y,^ for co,^, and p^,y for yO,,^ in (13.11), and choosing a starting value "^ We are interested in the random part of the covariation only, corresponding, in principle, to the residuals. The systematic sample covariation is very much larger than the random covariation, since the independent factors affect victims and accidents in much the same way.

The IRPOSKML Procedure of Estimation 333 jr (= 0, say) for // , we calculate a first-round set of disturbance variance estimates

for the severity ratio.

Using this variance estimate, we estimate a first-round heteroskedastic severity model on equation (13.6), deriving, inter alia, a first-round estimate of jU (ju\ say). Step 4 is repeated, with ju^ instead of / / plugged into (13.11), and so on until convergence. Note that in this procedure, only the dependent variable Box-Cox parameter // is iterated upon. The statistics h^^, y^^, and p^^y are calculated only once. Again, three to four iterations turn out to be sufficient for convergence. In the TRULS-1 data set, the residual correlation coefficients p^^^ come out as follows: 0.1613 between injury accidents and fatalities, 0.2132 between injury accidents and dangerous injuries, and 0.4388 between injury accidents and severe injuries.

CaLcuLated

> F m o r t a l i ty

*^^^%V:<.^^^^*..-v. Est imated TRIO PROJECT: DATE: 97 12 10 USER: toi

TRULS - an econometric model of road use, accidents and their severity

expected

number

of

injury

ace idents

trio

Graph 13.5. Calculated disturbance variance in the TRULS-1 mortality equation, plotted against estimated expected number of injury accidents.

334 Structural Road Accident Models In Graph 13.5, we show the calculated disturbance variance of the mortality model (fatalities per injury accidents), plotted against the expected number of injury accidents. Clearly, the variance is a decreasing function of the "base" number of accidents, by which the fatality count is divided. The larger the base, the smaller the relative scope for purely random variation. Since, however, there are two random variables at work, the relationship is not exact. Here, too, the variance is seen to vary across the TRULS-1 sample by a factor of more than 10.

13.3.

REFERENCES

Box, G. E. P. and D. R. Cox (1964). An analysis of transformations. Journal of the Royal Statistical Society B, 26, 211-243. Gaudry, M. and M. I. Wills (1978). Estimating the functional form of travel demand models. Transportation Research, 12, 257-289. Liem, T., M. Dagenais and M. Gaudry (1993). LEVEL: the L-L 4 program for BC-GAUHESEQ regression - Box-Cox Generalized AUtoregressive HEteroskedastic Single EQuation models. Publication 510, Centre de recherche sur les transports, Universite de Montreal. Sverdrup, E. (1973): Lov og tilfeldighet. Vol 1, 2"^ edition. Universitetsforlaget, Oslo. Tukey, J. W. (1957): On the comparative anatomy of transformations. Annals of Mathematical Statistics, 28, 602-632.

Turning Box-Cox Including Quadratic Forms in Regression 335

14 TURNING BOX-COX INCLUDING QUADRATIC FORMS IN REGRESSION Marc Gaudry Ulrich Blum Tran Liem

14.1 MODEL WITH TWO BOX-COX TRANSFORMATIONS ON A SAME INDEPENDENT VARIABLE Our problem^, prompted by the increasing search for flexible turning functions, for instance in the explanation of how some transport costs might vary with distance, or of how fatal road accident frequency and severity fall might rise and then fall^ with increased traffic (Tegner and Loncar-Lucassi, 1997), or of how the impact of alcohol on fatal road accident frequency might be U-shaped^, is to fmd the conditions under which the function y{x) has a maximum or minimum over the positive region of X in the following model: (14.1)

j;*^^' = y^o+A^^^'^+A^^''^+••• + " '

where the positive independent variable X is transformed by two different Box-Cox parameters (/Ij 7t ;i2) so that the model is identified in terms of the transformed variables. 14.1.1. Solution The first derivative of y{x) with respect to Xis:

^ This paper is the final version of, and cancels, a manuscript, widely circulated in Canada, Germany, Sweden and the United States, usually called: Gaudry, M., "FIQ: Fractional and Integer Quadratic Forms Estimated with the LEVEL algorithm in TRIO", 5 pages, November 5, 1996, augmented on March 9, 1997, and on October 28, 1997. A fourth version, produced with Ulrich Blum on March 20, 1999, was not circulated. ^ The S A A Q in Quebec City has established such relationships since December 1992 with symmetric forms. Since 1997, asymmetric forms are used: they are now part of the official model for Quebec (Fournier et Simard, 1999) and imply that traffic reaches a point where additional vehicles have no impact on accident frequency or even reduce accident frequency, i.e. confer a positive externality. ^ For a summary, see in particular Chapter 1 in Gaudry and Lassarre (2000).

336 Structural Road Accident Models

(14.2)

dX

ay*'.' dX

y .

Equating this derivative to zero and solving for a critical point of X give:

(14.3)

X

I A

What conditions must hold for this critical point to be a turning point? 14.1.2. First-order conditions 1. Since the independent variable X is always positive, the critical value of X should also be positive. Hence the coefficients p^ and pj should have opposite signs for any values of the /I's (>l, "^ ^i)2. Conversely, if the coefficients have the same sign, then X* does not exist, i.e. there is no maximum or minimum. To determine that the critical point X* corresponds to a maximum or a minimum, we have to analyze the sign of the second derivative of y{x) at this point:

/?,(A,-1)X

dX^

^p,{X,-\)X

. ^,-2

_X

A(A,-i) + A a 2 - i ) ^

X

(14.4)

» ^1-2

*

*

M-2

-^2-^1

^1-2

',(^-l) + A(A,-l)

y

- ^

* '^1-2

y'

y'

Since the terms X *

* '^\ ^

A -1

and y '

are always positive, the sign of the second derivative at

X depends only on the sign of the product p^ (/I, - /I2) • By factoring out X *

X

^1-2

like above, an equivalent property can be derived:

* ^2-2

instead of

Turning Box-Cox Including Quadratic Forms in Regression 337 _

ax'

1

A(A,-i)x • ' " ' + y 3 , ( 2 , - l ) X *

A2-2"

A (^ - i)jr •'•"''+/?, (A,-1) y' * ^2-2

(14.5)

A(^-i)

y'

K Pi J

J

* ^2-2

X

» ^2-2

Since the terms X

-2

A (A,-A, )

•

^ _i

and jv '

are always positive, the sign of the second derivative at

X* depends only on the sign of the product fij (^^2 ~ ^1) • 14.1.3. Second-order conditions 1. X* corresponds to a maximum if the second derivative at X*, i.e. the product y^j (A^ - /I2) or A (^^2 ~ ^ ) ? is negative. 2. X* corresponds to a minimum if the second derivative at X*, i.e. the product y?i (/Ij - /I2) or y^2 (^2 ~ A)»is positive. Table 1 combines the first and second-order conditions to obtain a maximum or minimum at X*. Due to the first-order condition that the twoy^ 's should have opposite signs, the secondorder conditions that the product y^, (/Lj - /Ij) is negative for a maximum and positive for a minimum are equivalent to the ones with the product /^j(^2 ~^]) - Moreover, for a given set of specific values of (y^i,/lj,/?2'^2)' Maximum 1 and Maximum2 are equivalent, and so are Minimum 1 and Minimum2, due to the interchangeability of the values of the pairs (y^j ,A,) and

Table 1. Conditions for a maximum or minimum in the model with Box-Cox transformations

[CASE

A

A

Maximum 1

+

-

-

Minimum 1

+

-

+

Maximum2

-

+

+

Minimum2

-

+

-

/ii — /I2

M^-^2)orM^2-^) +

+

1 1

338 Structural Road Accident Models Figures 1 and 2 illustrate respectively the first two cases where the dependent variable y is not transformed by a Box-Cox for simplicity reasons: (1) in Figure 1, we have a maximum, Maximuml: ;^ = 5 + 3X^'^^-2X^^^\ that is equivalent to Maximum2: y = 5-2X^^-^^ +3X^^-^\ when the values of the pairs (/?p^i) and (A'^2) are permuted; (2) in Figure 2, we have a minimum, Minimuml: y = 3 + 2X^^'^^ -4X^~^-^\ that is equivalent to Minimum2: >^ = 3-4X^"^*^+2X^^^\ when the values of the pairs (y^i,/l,) and (y?2''^2) are permuted. Figure 1. Graph of y

5 + 3X^"^ - 2X^''^, with Maximuml at (1.3359, 5.173)

9

I

j

I

i

I

!

i

^

0.5

1.5

2

2.5

3.5

X

Figure 2. Graph of y = 3 + 2X^"^ - 4X^-"^, with Minimuml at (1.206, 2.8174) 3

8

1

i

^

7 6 X

\

>^

3 2 1 n 0.5

1.5

2.5

3.5

Turning Box-Cox Including Quadratic Forms in Regression 339

14.1.4. Special case: quadratic form The quadratic form can be obtained by setting Aj = 1 and ^2 = 2 in model (1):

(14.6)

y^'^^ = p,+l3,X^'Kp,X^'K... + u .

From Table 1, we can only have two cases where ^-X^ is negative: 1. Maximum 1 if y^, > 0 and J32 <0, that is equivalent to Maximum2 when the values of the pairs (J3^,/1^) and (y^2'^2) are permuted. 2. Minimum2 if y^j < 0 and >^2 > 0' that is equivalent to Minimum 1 when the values of the pairs (P^,X^) and (y^j'^i) are permuted. If we consider only the portion y(X) = PQ+ PiX^^'^ + >^2-^^^^ i^ C^)' it is a quadratic function of X which is symmetric with respect to a maximum or minimum point X* Af ^2 differs from 2, then y{X) is a nonlinear function which is no longer symmetric. Figure 3 illustrates the symmetry/asymmetry property of the function y(X) for Aj = 1.5, 2, 2.5. Figure 3. Symmetric (Xj =2 ) and Asymmetric (/I2 ^ 2 ) Forms 30 20 10

-10

y^

j

y >/^

/I 1 ^

^

0 g

1

X

1

/

1

^

1

!

i

1

i

1

'" " r

•

!

1

1

\

i

r

1

\

1

1 i

1

' ;

1

1

!

1

X.

!

1^ LaiTthda2 = I \ IN.

; T

1 ;

-20

r ^ ^ ^ I - J_ambda2 = 1.5

T

1

^

1

-I

^ j

-

\

1 \

Lanlbdf 2 = 2.5

-30

1

1

1

;

1

-40

1

1

1

1

-50

X

I

Y -^

\ \

! C)

1

2

3

4

5

6

7

8

9

1

0

X

14.2. MODEL WITH POWERS X, AND X^ ONLY ON A SAME INDEPENDENT VARIABLE The model (1) can be rewritten in terms of powers X^ and X^ on X as:

340 Structural Road Accident Models

(14.7)

j;*^^' = pl + I5\X^^ + I5\X^' +... + w (2, and /I2 ^ 0),

where the newy^* 's are related to the original /? 's as follows:

1,

^>'i *

•

=

!

•

This model is not equivalent to the model (1) with Box-Cox transformations for two reasons: 1. The simple power transformation X^' or X^'' does not include the logarithmic form of the variable X. 2. The ordering of the data is not preserved when the parameter /I, or /Ij changes its sign. For example, consider two values of X , say 10 and e where 10 > e , then 10"^' <e^' if ;i, < 0, and 10^' > e^' if /I, > 0 , whereas with the Box-Cox transformation, (lO-"' -1)/A, >(e''' -l)/;i, for any value of ;i,. Hence, in practice, we cannot estimate jointly the y^* 's and A 's, but only the J3*'s with fixed values of the A 's. The first derivative of y with respect to X is:

Equating this derivative to zero and solving for a critical value of X

(H.9)

give:

^^^_r_A^i^_r_^i^ AAj

^ A A,

14.2.1 First-order conditions 1. Since the independent variable X is always positive, the critical value of X should also be positive. Hence the terms y^,*!, and jSj/lj should have opposite signs for any values of the /I's (/I, 1^ A^), implying that if the two y^* 's have the same sign, the two /I's should have opposite signs and vice versa. 2. Conversely, if the terms have the same sign, then X* does not exist, i.e. there is no maximum or minimum. To determine that the critical point X** corresponds to a maximum or a minimum, we should analyze the sign of the second derivative of y(X) at this point:

Turning Box-Cox Including Quadratic Forms in Regression 341

A*Aj(A,-i)x**' +p;x,{x,-\)x''

ax' _X*' X

(14.10)

p;x,{x,-\)+p;x,{x,-\)x" p\x,{x,-\) +

p\x,{x,-\i-i^

y' = ^^[A*A,(I,-I)-/?;^(A,-I)] y' **'^'"^ _X P\\{\~K) • Since the terms X** and y ' are always positive, the sign of the second derivative at X** depends only on the sign of the term J5\X^ {X^ - ^2) • By factoring out X

instead of X **^' like above, an equivalent property can be derived:

aV 1 ax' x=x" y ' _X*

p\X, (A, - 1)X" J3*X,(X,-\)X**

+ I5\X^ {X, - \)X* +j3lX,(X^-\)

y'

(14.11)

. X

j3;x,(x,-i)\

^ p\h^ +p;x,{x,-\) p[x.

** ^ ~ = ^rTr[-A*^2(A -1) +A*^2(^2 -i)]

y' Since the terms X** and y ' are always positive, the sign of the second derivative at X** depends only on the sign of the term P^X^ (Xj -X^). 14.2.2. Second-order conditions 1. X** corresponds to a maximum if the second derivative at X* , i.e. the term P^X^(/Lj -X2)or PIX2(X2 - /li), is negative. 2.

X** corresponds to a minimum if the second derivative at X**, i.e. the term P^X^ (/Ij - /I2) or plXj {X2 - /l^), is positive.

342 Structural Road Accident Models Table 2 combines the first and second-order conditions to obtain a maximum or minimum at X**. Due to the first-order condition that the terms y^,*/I, and p\X^ should have opposite signs, the second-order conditions that the tQvmj3*A^(/l^ - Jij) is negative for a maximum and positive for a minimum are equivalent to the ones with the term/?2/l2(^2 ~ A)- The first eight cases correspond to /?*/l, >0 and fi^A^ < 0 , whereas the last eight correspond to y^,*/l, <0 and J3IA2 > 0 . Table 2. Conditions for a maximum or minimum in the model with powers A^ and Aj only CASE

fi:

X,

/?;i,

A*

-

-

+

-

^2

filA,

+

-

/?;;i,(/l,-A2) or 1 ~~

2

-

Maximum 1.1

+

NoMinl.l 1 Maximum 1.2 Minimum 1.2 1 Maximum 1.3 +

-

+

+

-

-

+

+

-

+

-

fi;A,{A,-A,) Since 2, < 0 and A2 >0, the condition /Ij - /I2 > 0 cannot be satisfied for a minimum. |

+

+

-

1 1

+

+

-

Since A^ > 0 and A2 <0, the condition A^ -Aj <0 cannot be satisfied for maximum. |

Minimum 1.4

+

+

Maximum2.1

+

-

Minimum 1.3 NoMaxl.4

+

+

+

-

+

-

+

-

-

-

+

-

-

NoMin2.1 Maximum2.2 Minimum2.2 Maximum2.3 Minimum2.3

+

-

-

-

-

+

-

+

-

+

+

+

-

-

+

+

+ _

+

+

1 1

—

+

u

Since /I, < 0and A2 >0,thQ condition A^ -A2 >0 cannot be satisfied for a maximum. |

+

+ Minimum2.4

+ -

NoMax2.4

Since /L, > 0 and A2 <0, the condition A^ -A2 <0 cannot be satisfied for a minimum. |

!

Turning Box-Cox Including Quadratic Forms in Regression 343 Due to the interchangeability of the values of the pairs (/?,*,A,) and {f5\,X^), the first eight cases are equivalent to the last eight for a given set of specific values of (y^*,/lj,,/?2'^2 )• F^^ example, Maximum 1.1 and Maximum2.1 are equivalent, and so areNoMinl.l andNoMin2.L Figures 4 and 5 illustrate respectively the two cases Maximum 1.1 and Minimum 1.4 where the dependent variable y is not transformed by a Box-Cox for simplicity reasons: 1. Maximuml. 1: ;; = 15 - 4X~^ "* - 2X^ \ that is equivalent to Maximum2.1: 3;=:15-2X^^ -4X"^'* when the values of the pairs {P\,\) permuted.

and {P\,X^) are

2. Minimum 1.4: ;; = -6 + 3X'^ + 5X~^^ , that is equivalent to Minimum2.4: jv = -6 + SX'^^ + 3X^^ when the values of the pairs (/?,*, /I,) and {p\, X^) are permuted. 14.2.3. Special case: Quadratic form The quadratic form can be obtained by setting /I, = 1 and /Ij = 2 in model (7): (14.12)

;;*^^' = /?;+A*X^+y^;x'+... + w .

From Table 2, there are three maxima and three minima that satisfy the condition yl, - /I2 < 0 • 1. Maximum 1.1, Maximum 1.2 and Maximum 1.3, that are respectively equivalent to Maximum2.1, Maximum2.2 and Maximum2.3 when the values of the pairs (/?j*,/lj) and (p\ ,/l2) ^re permuted. 2. Minimum2.2, Minimum2.3 and Minimum2.4, that are respectively equivalent to Minimum 1.2, Minimum 1.3 and Minimum 1.4 when the values of the pairs (y^,*,/l,) and ( p \ ,/i 2) are permuted. 14.3. TWO-STEP TRANSFORMATIONS ON A SAME INDEPENDENT VARIABLE In TRIO estimation procedures (Gaudry et al, 1993), the user frequently adopts a two-step procedure to make transformations on the independent variable X. Having done previous tests with a single monotonic transformation, the user then searches for turning points, e.g.: 1. In the first step, a quadratic form in X is estimated using the equation (12). 2. In the second step, a same Box-Cox transformation on the two independent variables X and X^ involving in the quadratic form of the previous step is estimated: (14.13)

/^^^ =j3l* + Pl* X^'^^ + pl\X^)^^^^ + ... + u .

This model can be reduced to the model (1) as follows:

344 Structural Road Accident Models Figure 4. Graph of j ; = 15 - 4 X - ' ' - 2 X ' ' , with Maximuml.l at (1.0546, 9.0268) 10

8

g

6

I

4

I

2 0

0

0.5

1

1.5

2 X

2.5

3

3.5

4

Figure 5. Graph of y = -6 + 3 X ' ' + SX"'', with Minimuml.4 at (0.9473,1.9892) 10

1 1

8

/

:

/

\ 6

4

2

0

) (

0.5

1

1.5

2 X

2.5

3

3.5

^^

Turning Box-Cox Including Quadratic Forms in Regression 345

/''' = (14.14)

P:^P:X''^'+2P;

= Pl* +pl*X^'^^+2pl*X^^'^^ +... + U = y^o +P,X^^'^ +P^X^^^ +... + U .

where /?« = pl*, p, = p**, p^ = ip^* and A^ = IX,. This is not surprising in view of the property that the combination of a simple power and a Box-Cox transformation gives an equivalent Box-Cox transformation (Gaudry and Laferriere, 1989) with a rescaling effect for the coefficient p*^. In the first step, generalizing the form in X gives: (14.5) y"^'^ =pl^p\X'+p\X''-v...^u where m is a real number.

.

In the second step, the model with the generalized form in X can be rewritten in terms of model (1):

y*^^^^ = Pl'+p;*x^^^^+Pl\x'"y^'^ +...+U n 4 1 ^.

' * " ^ " ' ' •*• " ' ^ " — r - + • • • + "

=^"-'

(14.16)

mX, = Pl* + p;*X^''^ +mp;*X^'"^^U... + u = PQ +P,X^^^^

+... + U .

+P^X^^^

where PQ = pl*, p, = p**, P2 = mp^* and Xj = mX,. Note that when computing the elasticity of ;; with respect to X at the sample means, for the model (13), the program TRIO considers the two independent variables X and X^ as distinct variables not related to each other and gives two distinct elasticities, namely r]{y,X)\-^= Pl*X^^ ly^' and r]{y,X^)\_-=

Pl*X"-^' ly^'. If the second variable X^ is

considered as a function of X as it should be, the total elasticity of y with respect to X at the sample means is given by: (14.17)

ri{y,X)\

yy

=P:-^^2P;-^ y

where the second component of the elasticity can be computed from the elasticity 77(3;, X^)_ — \y,X

given by TRIO as follows: (14.18)

2j}^;

=2r^(^yX\-,^—

.

346 Structural Road Accident Models An example of the two-step procedure comes from the SNUS-2.5 Model (Blum and Gaudry, 2000), where the demand for road use with gasoline cars (y) is explained by the stock of cars per employee (X), among other things. Figure 6 gives the graph of the portion of the equation (13) where only Xis involved, namely y(X) =fi**X^^'^+ fil\X^y^^^ where fi** =-93, J3*2* =4.4 and /I, =-3.3. The first and second derivatives of y(X), y(X) and y\X), are also plotted. Figure 6. Result from SNUS-2.5 Model 1500

E

: 1 ; i

1000

!



1 y"(X) :


'sz 0)

>

500

c 0

>

•o 0)

_c "o

0 y(X)/

(0 CO

O II

g

y'(X)

-500

( -1000

1 ;

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

X = Cars per Employee

14.4. REFERENCES Blum, U. and M. Gaudry, (2000). The SNUS-2.5 model for Germany. In Gaudry, M. Lassarre, eds.. Structural Road Accident Models: The International DRAG Family, Ch. 3, 67-96, Elsevier Science Publishers, Oxford. Foumier, F. et R. Simard, (1999). Previsions sur le kilometrage, le nombre d'accidents et de victimes de la route au Quebec pour la periode 1997-2004, Societe de I'assurance automobile du Quebec, 300 p. Gaudry, M., et al. (1993). Cur cum TRIO? Publication CRT 901, Centre de recherche sur les transports, Universite de Montreal. Gaudry, M. and R. Laferriere, (1989). The Box-Cox transformation: power invariance and a new interpretation. Economics Letters 30, 27-29. Gaudry, M. and S. Lassarre, eds., Structural Road Accident Models: The International DRAG Family, Elsevier Science Publishers, Oxford, 2000. Tegner, G. and Loncar-Lucassi, V. (1997). Demand for road use, accidents and their gravity in Stockholm: measurement and analysis of the Dennis package. Transek AB, Stockholm, 18 p., January.

Appendix 1 347

15 APPENDIX 1. DETAILED MODEL OUTPUTS Marc Gaudry, Sylvain Lassarre Readers wishing to consult and download the regression results obtained for the principal models of this book can find them at http://www.crt.umontreal.ca/crt/AgoraJulesDupuit/ (using A^oraJulesDupuit if the location of the results is changed) and at http://www.inrets.fr. The appendix contain analytical definitions of dependent variables and results for all explanatory variables, each one clearly defined in words, presented in tables numbered according to the model chapter number, as indicated below. All 7 tables or results contain three parts. Part I contains: (i) for models # 1.2 to # 1.7, elasticities calculated according to Equation 12.70 (or 12.112 for dummy variables); (ii) for model # 1.9, probability points calculated according to Equation 9.15 (corrected in the same way as in 12.112 in the case of dummy variables); (iii) for all models, ^statistics (with respect to zero) of the underlying regression p coefficients computed conditionally upon the estimated value of the Box-Cox transformation (BCT) used in the model; (iv)for all models, a flag identifying each variable transformed by a particular BCT. Part II contains: (i) for models # 1.2 to # 1.7, estimates of the BCT associated to variables and of heteroskedasticity or autoregressive coefficients, according to Equations 1.1 to 1.4, as well as ^statistics; (ii) for model # 1.9, estimates of BCT associated to variables according to Equation 9.14, as well as ^statistics (with respect to zero and one). Part III contains, for all models, general statistics, including measures of fit and log likelihood values. Contents of Appendix 1

#

Model considered

1.2 The DRAG-2 model for Quebec

Tablex table of results generated by TRIO 348-354

1.3 The SNUS-2.5 model for Germany

355-360

1.4 The TRULS-1 model for Norway

361-367

1.5 The DRAG-Stockholm-2 model

368-371

1.6 The TAG-1 model for France

372-375

1.7 The TRACS-CA model for California

376-381

1.9 Road, risk, uncertainty and speed

380-385

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