No title

There is also a vital "cost"side to the problem.It is naturalto supposethat if species/library i is boarded,it takes up some space or room on the Ark-let this space coefficientbe denoted Ci. Since overallspace is limited, the amount

1286

MARTIN L. WEITZMAN

of room that a species occupiesbecomes a criticalfactor in the choice problem. For the biblical Ark parable, Ci is measured in units of cubits3. In the real world, Ci would represent the cost of the project that extends an enhanced measureof protectionto species/library i. We can now state the central problem. Noah wishes to select survival probabilitiesthat maximizeexpected diversityplus expected direct utility. The Noah's Ark mathematicalprogrammingproblemis to select values of {Pi} that (13)

maximize{PJ[W({Pi}) + U({Pi})]

subjectto the n individualprobabilityconstraints (14)

Pt < Pt < Pt

Vi,

and subjectto the overallbudget constraint (15)

Ci

=B.

Note that, as formulated,the above programmingproblem is continuousin the probabilityvariable Pi, and thus allows for strictly"interior"values of Pi, which fulfill the condition (16)

P < Pt < iy

An "interior"value of Pi, one that satisfies (16), correspondsto partial protection,or "fractionalboarding"of species/library i on the Ark. "Fractional boarding"might be given a physical interpretationof boarding only some fraction of a reproductivelyviable populationsize. Suppose that, when not a single individualof species/library i is boarded,the survivalprobabilitywould be Pi =P. At the opposite extreme, if population size Ci is boarded, then suppose the survivalprobabilityis enhancedto Pi = Pi. Finally,assume that for "in-between"population size boardings a linear interpolation describes the survivalprobabilities.Under such assumptions,there is a rigorousunderlying basis for sayingthat a survivalprobabilityof Pi for species/library i comes at a budget cost of (17)

ciIP-

Pt

I now seek to characterizethe nature of the solution to the Noah's Ark Problem.The most distinctiveaspectof the mathematicalprogrammingproblem (13)-(15) is the presence of the unusual "expecteddiversityfunction"W(P) in the objective.It mightbe thoughtthat the functionW(P) is so unorthodoxthat it is difficultto say anythinggeneralthat is also interestingabout the solutionto the problem.Fortunately,it turns out that a quite strikingcharacterizationis possible.

1287

NOAH S ARK PROBLEM

The solution of the Noah's Ark Problemis always"extreme"in the following sense. Noah, or the conservationauthorities that he symbolizes,should be concentrating all their resources on maximal protection of some selected species/libraries,even at the expenseof exposingall remainingspecies/libraries to minimalprotection. In an optimal policy, the entire budget is spent on a favored subset of species/libraries that is affordedmaximalprotection.The less favoredcomplementarysubset is sacrificedto a level of minimalprotectionin order to free up to the extreme all possible scarce budget dollars to go into protecting the favoredfew. While a real-worldinterpretationof this result must be properlyqualified, because it is only as strongas the underlyingassumptions,the followingthought leaps to mind.Subjectto the restrictionsof the model, there is some implication that a conservingagency may want to think more in terms of concentrating limited resources,ratherthan spreadingthem out thinly. In order to state the above ideas rigorously,we start with the following definition. DEFINITION

(Extreme Policy Solution): "Almost all" (n

-

1 out of n)

species/librariesare either fullyboarded(Pi = 1-) or not boardedat all (Pi = Pi). At most one species/library j (the "roundoffspecies/library")is "fractionally boarded"with interiorprobabilityPj, where PJ< Pj< Pj. The basic result here is the following: THEOREM

2: The solution of Noah's Ark Problem is an extremepolicy.

PROOF: I present here a concise version,which the reader should be able to follow even thoughit is compact.Not everysingle aspect is spelled out, because to do so requires a lot of algebra and notational detail, which the interested reader should be able to fill in. All the main steps in the underlyinglogic are provided. The proof begins with the observationthat conditioningon the existence of species/libraries 1 and 2 allows the expected diversityfunction W(P) to be rewrittenin the form:

(18)

W(P) =

E

[K(Q)] [P1P2V(Q u

1 u 2) + Pl(1 -P2)V(Q

U 1)

QCS\1\2 +

P2(1 - P1)V(Q U 2) + (1 - P1)(1 - P2)V(Q)]

where K(Q) is definedas the polynomialexpression: (19)

K(Q)- (HP) Q jCE=

(

rI\

kEC S\Q\1\2

(' -Pk))

?0

and set notation such as Q U 1 or Q\ 1 are shorthandfor, respectively,Q u {1} or Q\{1}.

1288

MARTIN

L. WEITZMAN

Mechanicallytaking the second mixed partial derivativeof (18) yields, after some algebraicmanipulation,the formula d2W

(20) 81% P~2

=K(Q)] QcS\1\2

J(Q)]

where 1(Q) is just shorthandnotation for the followingexpression: (21)

J(Q)

[V(Q u 1) + V(Q u 2) - V(Q u 1 u 2) - V(Q)I.

Now the J(Q) formula expressed by (21) actually stands for something. Lookingcarefullyat the appropriateVenn diagramshould convincethe reader that J(Q) stands for the numberof books held jointlyby libraries1 and 2 but not containedin any other librariesof Q. Naturally,J(Q) is nonnegative. Thus,the expression(20) for the second mixedpartialof W with respectto P1 and P2 is nonpositiveand independentof P1 or P2. It follows that the expected diversityfunction W(P) is convex in any two of its variables,holding all other argumentsconstant. The objectivefunction (13) is therefore convex in any two of its variables, holdingall other variablesconstant,while the relevantconstraints(14) and (15) are linear. Because the optimal value of a convex function maximizedover a convex set is alwaysattained at an extreme point of the convex set, it follows that, out of anypair of probabilityvariablesin an optimalpolicy,no more than one wouldbe strictly"interior"in the sense of satisfyingcondition(16). But this means that at mostone species/libraryj, out of the entire set S, is "fracdonally Q.E.D. boarded"with interiorprobabilityPj,wherePj< Pj < Pj. The truly "extreme"nature of this solution can perhapsbest be appreciated by setting Pi = 0, Pi = 1 for all i. Then a diversity-maximizingpolicy makes each

species (except for at most one) either totally extinct or perfectlysafe. It is importantto understandthe intuitivelogic that explainswhy an extreme policy is optimalin the Noah's Ark Problem.Considerthe two-libraryexample depicted in Figure 1. As before, let El be the numberof books distinctiveto library1 and E2 the numberof books distinctiveto library2, while the number of books held jointly in commonbetween libraries1 and 2 is denoted J. Then the relevantexpressionfor expected diversityin this case is given by (5). If library1 alone becomes extinct,then El books are lost. If library2 alone becomes extinct,then E2 books are lost. However-and this is the crucialpoint -if libraries1 and 2 both become extinct, then an additionalJ books are lost. With costs linear,Noah would rathershift probabilitiestowardsavingas fully as possible one of the librariesat the expense of the other in order to "pin down the line" of the J books in common. The strengthof this effect of increased returnsto savingeither one of the librariesis measuredby J, which from (5) is minus the second mixed partial derivativeof expected diversity,corresponding to equation(20).

NOAH S ARK PROBLEM

1289

The idea of "pinning down the line of books in common" seems to me to be an

importantconservationprinciple.In the case of costs that are linearin probabilities, it resultsin policiesof extremeconcentrationof conservationresources.As the incrementalcost of protectionincreasesnonlinearlyin survivalprobability, this extremenessof an optimalpolicyis diluted,or even reversedfor sufficiently strong curvature.Nevertheless,the basic principleremains,and may even give useful insight into the form of an optimal conservationpolicy under some circumstances.When costs are approximatelylinear in probabilitychanges, as might be supposed to be the case for sufficientlysmall APi Pi - Pi, then Theorem2 articulatesa well-definedsense in which a conservingagencyshould be concentratingscarce resources on some librariesfor the sake of "pinning down the line of books in common"that might otherwisebe lost from multiple extinctions. Theorem 2 of this section gives a strong characterizationof the form of a solution to the Noah's Ark Problem.The solution form is an extreme policy where the preservationchanges of some species/libraries are maximallyenhanced by their being boarded completely on the Ark, while others are left completely behind. But this result does not say which species/libraries are favored by being selected for boarding, nor explain fully why they are so favored. As a means towardsthe end of actuallysolvingthe Noah'sArk Problem,there is a need to understandbetter the propertiesof the expecteddiversityfunction and also to incorporate into the analysis some measure of uniqueness or distinctivenessof a species/library. It turns out that these two issues are intimatelyrelated. There exists a strikingconnection between what might be called the "marginaldiversity"of a species/libraryand what mightbe called its "distinctiveness," meaningloosely a distance-basedmeasureof differencefrom other species/libraries. To bring out this importantrelation, some definitions need to be made and some more structuremust be put on the problem.The additionalstructurethat will eventuallybe imposedon species/libraries is that they are "as if" createdby a specificprocessof descentwith modification,which will be called the "EvolutionaryLibraryModel." " DISTINCTIVENESS" IN THE EVOLUTIONARY LIBRARY MODEL

We now wish to define the uniqueness or "distinctiveness"of a library. with the fundamentalmathematiEssentially,we identifyhere "distinctiveness" cal concept of distance.Consequently,there is a need to startwith the relevant measure of distance in this setting. The definition of pairwisespecies/library distanceappropriateto the present context is as follows: DEFINITION: DistanceD(i,j) is the number of different books contained in libraryi that are not containedin libraryj.

For the importantspecial case where libraries i and j contain the same number of different books, distances are symmetricbecause D(i,j) = D(, i)

1290

MARTIN L. WEITZMAN

whenever Mi = Mj. It is importantto note, however,that,distances as defined above are not symmetricin the general case where Mi is not equal to Mi. Now let Q be any assemblage of libraries. In the present context it is appropriateto employthe standardmathematicaldefinitionof the distancefrom a point to a set: (22)

D(i, Q)

minD(i, j).

In the case of Q being the null set, distances are normalized so that D(i, 0) = Mi.

The interpretationof D(i, Q) is that it representsthe distance of library i from its "nearestneighbor"or "closestrelative"in Q. It seems natural to define the distinctivenessof i, denoted Dj, to be the expecteddistanceof i from its nearest neighboror closest relativein S: (23)

(1

Di QsS\i

j)('H ]E Q

(1 -Pk))D(i,Q).

kE S\Q\i

What is the possible connection between distinctivenessand diversity?To answerthis questionsharply,one needs to put more structureon the problem.I now try to impose a "natural"structureon the book collections of the various species/libraries, which corresponds,at a high level of abstraction,to the standardparadigmof evolutionaryrelationshipsamong biological species. The book collections of the various libraries will be modeled "as if' they were acquiredby a process of "descentwith modification." Previously,no restrictionswere placed upon the book collections of the various libraries.Now I want to suppose that the book collections are "as if' they were acquiredby an evolutionarybranchingprocesswith a corresponding evolutionarytree structure.The particularbranchingprocess describedhere is called the evolutionary librarymodel,and it is patternedon the classic paradigm of biologicalspecies evolution. The "evolutionarylibrarymodel" is a branchingprocess that explains the existence of the current library assemblage S as a result of three types of evolutionary/historicalevents. (i) Each existing library acquires new books at any time by independently sampling,at its own rate, out of an infinitelylarge pool of differentbooks. Tfie independentacquisitionof differentnew books by each librarycorrespondsto the evolutionof genetic traits when species are reproductivelyisolated with no gene pool mixingby lateral transfer. (ii) New librariescan be createdby a "speciationevent."A new branchlibrary can be founded by adopting a complete copy of the current collection of an existinglibrary.Henceforththis new librarywill become reproductivelyisolated and acquireits books independently,as describedby (i) above. (iii) Libraries can go extinct. When a library is extinguished, its entire collection of books is lost. Librariesthat have alreadygone extinct in the past

1291

NOAH S ARK PROBLEM

correspondto lost stem taxa, and do not show up in the set S of currently existinglibraries. Figure 2 illustrates the evolutionarylibrary model for the case of three species/libraries.The firsttwo librariescorrespondto the depictionof Figure 1, shown in a previous section of the paper. As was pointed out there, any two librariescan be given an evolutionarytree representation.But for three or more libraries,the evolutionarylibrarymodel mustbe assumedin orderto have a tree representation.Hence, three librariesis the simplestcase to analyzewhere the evolutionarylibrary model is actually imposing additional structure on the librarycollections. The evolutionarylibrarystorytold by Figure2 is somethinglike the following. At the beginningthere was one prototypelibrary,which acquireda collectionof G different books. Then occurred a "speciationevent." Two "reproductively isolated"librarieswere created,each startingoff with identicalcopies of the G different books. Both librariesthen began independentlyto acquire different books. One of the two libraries eventuallybecame, after acquiringindependentlyE3 differentbooks, the currentlyexistinglibrary3. The other librarywas the ancestorof libraries(1, 2). This ancestorlibraryof (1, 2) acquiredindependently J differentbooks before another "speciationevent"occurred-resulting in the fission-likefoundingof two identicallibraries.One of these libraries,after furtherindependentacquisitionof E1 differentbooks,became eventuallylibrary 1. The second library,after furtheracquiringindependentlyE2 differentbooks, evolvedinto library2. In this way did the currentcollection of libraries{1, 2, 3} come into being. I presenthere the "evolutionarylibrarymodel"in a particularlysharpform to emphasize its essential characteristics.As such, the concept is intended to representan "ultimateabstraction"of how entities are createdby an evolutionary branchingprocess. While the model could tolerate some generalizations,

|G

1

1

l

El

~~~~~E3

E2

1

1

2 FIGURE 2.

~~~~~~~3

1292

MARTIN

L. WEITZMAN

such as types of nonindependentsampling,I do not feel it is worth the loss of sharpfocus to pursuehere the most general possible formulation. In thinking about the evolutionarylibrary model, it is almost as if the fundamentalunits are definedas entities that satisfy the three basic axioms. Since the model is at such a high level of generalization,in a way it does not really matter whether "evolutionarylibraries"stand for biological species, or languages,or somethingelse. An "evolutionarylibrary"here is an abstraction standingfor an entity that evolves as a unit independentlyof other such units, and which came into being originallyby splittingoff from anothersuch entity. I think there is some merit in first definingcarefullythe mathematicalessence of an evolutionarybranchingprocess-i.e., the "evolutionarylibrarymodel"-and then, in the biological context, defining species to be units that satisfy, not perfectlybut tolerablywell in practice,such a model. The evolutionarylibrarymodel naturallygeneratesa correspondingevolutionary tree. And when a tree structureis present, it seems to induce naturallyin the human mind a way of visualizingand comprehendingintuitivelyrelationships amongobjectsthat are quite subtle or complicatedto describewithoutthe tree. "Tree thinking"representsa prime example of how one picture may be worth a thousandwords. Take, for example,diversity,which is definedas the total numberof different books.In the evolutionarylibrarymodel, diversityhas a naturalinterpretationas the total (vertical)branchlength of the correspondingtree. For the three-library example of Figure 2, it is readily confirmedthat the diversityof the three librariesis the total branchlength G + J + E1 + E2 + E3. How is "distinctiveness" representedin the contextof an evolutionarySbranching model?Absent uncertainty,the distinctivenessof a libraryis just its distance from its nearest neighbor or closest relative. In the tree correspondingto an evolutionarybranchingmodel, the distinctivenessof a libraryis representedby its (vertical)branchlength off the rest of the evolutionarytree. Thus, in Figure 2, one readilyconfirmsthat the distinctivenessof libraryi is representedby its (vertical)branchlength Ei for all i = 1,2, 3. Note here an importantgeometricrelationship.For an evolutionarylibraryin the deterministiccase, the loss of diversityfrom extinctionis just the length of the (vertical)librarybranchfrom the rest of the evolutionarytree. The precise mathematicalstatementis (24)

V(Q u i)-V(Q)

= D(i, Q).

A vividimage is that the extinctionof a librarycorrespondsto snappingoff its branchfrom the rest of the evolutionarytree. In Figure2, the loss of diversityif library1 goes extinct is E1. The loss of diversityif library2 goes extinct is E2. (The loss of diversityif libraries1 and 2 both go extinctis E1 + E2 + J, which is the total (vertical)branch loss of sequentiallysnappingoff 1 first and then 2 next, or vice versa.) In the deterministiccase, the loss of diversityfrom extinction of a library exactlyequalsthe distinctivenessof the library.Essentially,changeof diversityis

1293

NOAH'S ARK PROBLEM

distinctiveness.This importantrelation generalizes from a deterministicto a probabilisticsituation.The basic idea can be stated formallyas the following result: THEOREM

3: In the case of evolutionarylibraries,

8W (25)

PROOF:

(26)

= Di

for all Pi E [O,1].

Rewritethe diversityfunction(3) as

(H I P) (

W(P)PiE= + (1

-Pi)

H

Ui)

(1 -Pk))V(Q

k c- S\Q\i

jEQ

Qc S\i

E

(

Q sS\i

n Pj) i E-Q

fl

k c- S\Q\i

(

-P)VQ

Takingthe derivativeof (26) with respect to Pi and collectingtermsyields (27)

8W 8Pi

= QcS\i

HPi

JEQ

H f

kcS\Q\i

(1-Pk)

u(Q i)-

Combining(27) with (24) and (23) yields the result(25).

V(Q)].

Q.E.D.

Theorem 3 is a statementthat marginaldiversityand distinctivenessare the same concept in the context of an evolutionarylibrarymodel. The theorem representsan appropriategeneralizationto the uncertaincase of the essential idea from the deterministiccase that loss of diversitycan be visualized as a branch-snappingevent. There is, however, a new wrinkle added by the presence of uncertainty. Distinctivenessnow is associatedwith expecteddistancefrom a nearestneighbor or closest relative, loosely speaking.Under uncertainty,seemingly symmetric librariescan have different degrees of distinctiveness,with consequences for conservationpolicy.It is importantto understandhow this comes about. To see this effect most starkly,return to the simple two-libraryexample of Figure 1. Make the tree picturesymmetricby supposingthat both librarieshave the same number of books so that M1 = M2 = M, and also E1 = E2 = E. Taking the derivativeof expression(5), and usingrelation(25),yields that D1 = M - JP2, while D2 = M - JP1.Thus, in a seeminglysymmetricsituation,the more endangered species is less distinctive. To expose sharplythe underlyinglogic behind this seeminglycounterintuitive result,supposean extremeexamplewhere P1 = .99,while P2 = .01. Then library 1 is practicallysafe, while library2 is practicallyextinct. For such a situation, library2 is much less distinctivethan library 1 in the following sense. The presence of library2 is practicallycontributingonly E different books, since library1 is almost sure to surviveby itself. On the other hand, the presence of

1294

MARTIN

L. WEITZMAN

library1 is practicallycontributingE + J differentbooks, because library2 will almost surely become extinct. In this sense library1 is much more distinctive than library2. The principleof "pinningdown the line" of jointly held books manifestsitself in this exampleby indicatingthe relativelysafe libraryas more distinctive than the relatively endangered library. Pushing even further the extremesymmetryassumptions,if the underlyingcosts of changingprobabilities are the same, then it is optimal to make the safe libraryeven safer at the expense of makingthe endangeredlibraryeven more endangered-because the safe libraryis more distinctiveand has greatermarginaldiversity. At this point all of the necessaryanalyticalapparatushas been developed,and the paperis readyto begin its main theme of developinga rankingcriterionthat solves the Noah's Ark Problem. THE NOAH'S ARK MYOPIC RANKING CRITERION

Suppose Noah wishes to actually solve problem (13)-(15). He wants to maximizeexpecteddiversityplus directutilitysubjectto the relevantconstraints. He does not want,however,to mess aroundwith a complicatedalgorithm.Noah is a practicaloutdoorsman. He needs robustnessand ruggedperformance"in the field."As he stands at the door of the ark, Noah desires to use a simple priorityrankinglist from which he can check off one species at a time for boarding.Noah wishes to have a robust rule in the form of a basic ordinal rankingsystem so that he can board first species #1, then species #2, then species #3, and so forth, until he runs out of space on the ark, whereuponhe battens down the hatches and casts off. Canwe help Noah? Is the conceptof an ordinalrankingsystemsensible?Can there exist such a simple myopicboardingrule, which correctlyprioritizeseach species independentof the budget size? And if so, what is the actual formula that determinesNoah's rankinglist for achievingan optimalark-fullof species? The answerto these questionsis essentiallypositive,and along the following lines. Providedthat (28)

'AP-

Pi

is "relativelysmall"' (for all i) in the usual sense of the prototypicalsmiall project justifyingcost-benefit investmentmethodologylocally, then a priority rankingbased on the criterion (29)

Ri= [Di+ Ui] Ci Ci

Note that the presumed smallness of AP goes somewhat against the spirit of the biblical version of Noah's Ark, for which a fair interpretation might be AP= 1. If so, I plead literary license to justify using the extended metaphor here, because it is so pretty.

NOAH'S ARK PROBLEM

1295

is justifiedin the sense of givingan arbitrarilyclose first order approximationto an optimalpolicy. To intuit why this mightbe so, ask the followingquestion.If we have enough moneyto adjustprobabilitiesa little bit in a particulardirection,whichdirection wouldyou choose? This is askingabout the "gradient"of the objectivefunction. Theorem 4 says that the gradient indicates a derivative of Di + Ui in the directionof Pi, which impliesa policyof the specifiedextremeform that pushes probabilitiesto their maximumor minimumvalue. More formally,we have the followingtheorem. 'THEOREM 4: Maintain the evolutionary-library hypothesis. Suppose one selects a solution of the Noah's Ark Problem to be of the following form: Thereexists a cutoff value R* such that

Ri > R*

Pi = P

(species i is boarded),

Ri < R*

Pi = Pi

(species i is not boarded).

Then the errorintroduced by this proposed solution is of second or higher order in {zlPi}. PROOF: The proof presentedhere is concise.Not everysingle aspect is spelled out, since to do so would requirea lot more algebraand notation.While some of the messier details about the applicationof Taylor-seriesapproximationsare left to the reader,all of the main steps in the underlyinglogic are providedhere. For all {Pi} satisfying(14), n

(31)

W({Pi}) +

W({Pi})

E3W

dpi

i=1 L

-(Pi-P) ~i P=pi

where the symbol = stands for a first order approximation,which is arbitrarily accuratefor sufficientlysmall {APi} in the traditionalsense of a Taylor-series expansionthat omits only terms of order (Ai)2 or higher. But from Theorem3, rSW] (32)

=DPi

Using (32), rewrite(31) as (33)

WQ{Pi})

=

W({Pi})

+

n

5?Di(Pi-P).i i=l

Now substitutethe linearizedexpression(33) into the objectivefunction(13), for which it is an arbitrarilyclose approximation.The reduced form of the

1296

MARTIN L. WEITZMAN

linearizedversionof (13)-(15) is now a programmingproblemthat selects values of {Pi}that (34)

maximizepi,E aicPi+ constant

subjectto the n individualprobabilityconstraints (35)

Pi < Pi < Pi,I

li,

and subjectto the overallbudget constraint (36)

E 83iPi= 'y

In the linearizedproblem(34)-(36), the followingdefinitionsare employed: (37)

ai-Di

+U

and (38)

8i-

C' APi

and C.P. APi

but notationally Using a Taylor-seriesexpansionargument,a straightforward that in function lemma shows the error the introducedby very messy objective as an solution the solution of to the of (13)-(16) using (34)-(36) approximation is of second or higher order in {APi}. Now the problem(34)-(36) is in the form of a classicallinear programming budgetingproblem.The relevantsolutionconcept is to rank"investmentopportunities"by the ratio criterion: (40)

{a}

and alwaysto favorhigher-rankedprojectsover those of lower rank. When applied to the Noah's Ark context, and makinguse of (37), (38), tJ;ie ratio criterion(40) is equivalentto followingpreciselythe statementof Theorem 4. This concludesthe concise proof of Theorem4. Q.E.D. Theorem 4 representsa culminationof the researchstrategymotivatingthis paper. It has been shown that a methodologythat "feels" like a traditional cost-effectivenessapproachcan be constructedto deal with the conservationof diversity.The myopicrankingcriteriondevelopedhere is sufficientlyoperational to be at least useful in suggestingwhat to look at when determiningconservation priorities.At the same time, the formula is rigorouslyderived from an optimizationframework,so that its theoreticalfoundationsare clear.

NOAH S ARK PROBLEM

1297

DISCUSSION

The rankingformula(29) encouragesthe conservationauthoritiesto focus on four fundamentalingredientswhen choosingpriorities: Di = distinctiveness of i = how unique or different is i; Ui = direct utility of i = how much we like or value i per se; APi = by how much can the survivabilityof i actually be improved;

Ci = how much does it cost to improvethe survivabilityof i. I am not intendinghere to argue that it is easy in practice to quantifythe above four variablesand combinethem routinelyinto the rankingformula(29) that defines Ri. The real world is more than a match for any model. The essential worth of this kind of research is to suggest a frameworkor way of thinkingabout biodiversitypreservation,and to indicatehow it mightbe backed by a rigorousunderlyingformulation. The basic hope is that the formulafor Ri could still be used as a roughguide or rule of thumbfor decidingconservationprioritieseven in situationswherewe cannot know Ci, APi, Di, or Ui with any precision. Perhaps one could come away with a sense that when making conservationdecisions in the name of preservingdiversity,it might seem like a "good idea" at least to consider the four factors Di, Ui, APi, and Ci-especially in a policy world so otherwise lackingjustifiableguidelinesfor endangeredspecies protection.One is perhaps further encouraged to think that combining these four ingredients into an overallindex Ri, more or less as indicatedby (29), also seems like a "goodidea" because it is intuitivelyplausibleand backed by a rigoroustheory in a special, but not unreasonable,case. The ultimate hope is that the metaphor of the Noah's Ark Problem, the associated Myopic Ranking Criterion,the underlyingLibraryModel, and the rest of the conceptual apparatushave "stayingpower" as a useful guide to organizedthinkingabout the economicsof biodiversitypreservation. Dept. of Economics, Harvard University,Cambridge,MA 02138, U.S.A. ManuscriptreceivedJuly, 1996; final revision receivedJune, 1997.

REFERENCES H. GOLDSTEIN (1984): A Model for Valuing Endangered Species," Journal of EnvironmentalEconomics and Management, 11, 303-309. CROZIER, Ross H. (1992): "Genetic Diversity and the Agony of Choice," Biological Conservation,61, 11-15. METRICK, ANDREW, AND MARTIN L. WEITZMAN (1994): "Patterns of Behavior in Endangered Species Preservation," Land Economics, 72, 1-16. BROWN, GARDNER M., AND JOHN

REID,

W. V., AND

K. R. MILLER (1989): KeepingOptionsAlive: The ScientificBasisfor Conserving

Biodiversity.Washington D.C.: World Resources Institute.

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L. WEITZMAN

SOLOW,ANDREWR., STEPHAN POLASKY, ANDJAMESM. BROADUS(1993): "On the Measurement of Biological Diversity," Journal of EnvironmentalEconomics and Management, 24, 60-68. (1993): "Searching for Uncertain Benefits and the Conservation of Biological Diversity," Environmental and Resource Economics, 3, 171-181. MARTINL. (1992): "On Diversity," QuarterlyJournal of Economics, 107, 363-406. WEITZMAN, (1993): "What to Preserve? An Application of Diversity Theory to Crane Conservation," QuarterlyJournal of Economics, 108, 157-183.

Econometrica,

Vol. 66, No. 6 (November, 1998), 1299-1325

NEW TOOLS FOR UNDERSTANDING SPURIOUS REGRESSIONS' BY PETER C. B. PHILLIPS Some new tools for analyzingspuriousregressionsare presented.The theoryutilizes the generalrepresentationof a stochasticprocessin termsof an orthonormalsystemand provides an extension of the Weierstrasstheorem to include the approximationof continuousfunctionsand stochasticprocessesby Wienerprocesses.The theoryis applied to two classic examplesof spuriousregressions:regressionof stochastictrends on time polynomials,and regressionsamong independentrandomwalks. It is shown that such regressionsreproducein part and in whole the underlyingorthonormalrepresentations. KEYWORDS: Loeve Karhunenrepresentation,nonsense correlation,orthonormalsystems, spuriousregression,Weierstrasstheorem.

1. INTRODUCTION SPURIOUS REGRESSIONS,OR NONSENSECORRELATIONS as

they were originally called, have a long history in statistics, dating back at least to Yule (1926). Textbooksand the literatureof statisticsand econometricsaboundwith interesting examples, many of them quite humorous. One is the high correlation between the numberof ordainedministersand the rate of alcoholismin Britain in the nineteenth century.Another is that of Yule (1926), reportinga correlation of 0.95 between the proportionof Church of England marriagesto all marriagesand the mortalityrate over the period 1866-1911. Yet anotheris the econometricexampleof alchemyreportedby Hendry(1980) between the price level and cumulativerainfallin the U.K. The latter "relation"provedresilientto many econometricdiagnostictests and was humorouslyadvancedby its author as a new "theory"of inflation.With so manywell known exampleslike these, the pitfalls of regressionand correlationalstudies are now commonknowledge, even to nonspecialists.The situation is especially difficultin cases where the data are trending-as indeed they are in the examplesabove-because "third" factors that drive the trends come into play in the behaviorof the regression, althoughthese factors may not be at all evident in the data. Moreover,as we have come to understandin recent years (althoughthe essence of the problem was evidentlyunderstoodby Yule in his originalarticle),it is the commonalityof trendingmechanismsin data that often leads to spuriousregressionrelations. The original version of this paper, entitled, "Spurious Regression Unmasked," was delivered as an Invited Lecture at the XIV Latin American Meetings of the Econometric Society, Rio de Janeiro, August 5-9, 1996. That version of the paper is available as Cowles Foundation Discussion Paper No. 1135 and can be obtained on request. Some of the ideas that appear in Section 3 of the paper were first suggested by the author while presenting an overview at a conference on Unit Roots and Cointegration at INSEE/ENSAE in June, 1991. Computations in the paper were performed by the author in GAUSS and the paper was typed by the author in Scientific Word 2.5. The author's thanks go to the co-editor and three referees for comments on the original version of the paper, and to the NSF for research support under Grant No. SBR 94-22922. 1299

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P. C. B. PHILLIPS

What makesthe phenomenondramaticis that it occurseven when the data are otherwiseindependent. In a prototypicalspurious regression,the fitted coefficients are statistically significantwhen there is no "truerelationship"between the dependentvariable and the regressors.The statisticalsignificanceis deemed spuriousand misleading because there is no meaningfulrelationshipbetween the variables.Using Monte Carlo simulations,Grangerand Newbold (1974) showed that this phenomenonoccurswhen independentrandomwalksare regressedon one another. Phillips (1986) gave an analytic theory of regressions of this type for quite generalstochastictrends,showing,inter alia, that the t- and F-ratio significance tests have divergentasymptoticbehavior in such regressions.Therefore, such outcomes are inevitablein large samples. Similarphenomenaoccur in regressions of stochastictrends on deterministicpolynomialregressors,as shown in Durlaufand Phillips(1988).The simple heuristicexplanationfor phenomenaof this type is that conventionalstatisticaltests do nothing more than reveal the presence of a trend in the dependentvariableby makingthe fitted coefficients significantfor all regressorsthat themselveshave trends.Thus,the commonality of trendingmechanismsin data is the source of these spuriousregressions. The purpose of this paper is to develop some new tools for analyzingand understandingsuch regressions.These tools help us explain why significant regressioncoefficientsoccur in what seem to be manifestlyincorrectregression specificationsrelatingvariablesthat are statisticallyindependent.The common theme, of course, is that all the variables share the common feature of a trendingmechanism,even though they may otherwisebe unrelated an\deven though the trendingmechanismsthemselvesmay be very different.We develop an asymptotictheory to explain this phenomena.A fascinatingfeature of the theory is that, just as we may model a continuousfunctionby Fourierseries in terms of different orthonormalsystem coordinates, so too we may validly represent a trending stochastic process in various ways, including the use of trendingregressorsthat are independentof the time series being modelled.The fact that the fitted regressioncoefficientsare significantin such cases is shown to be a statisticalmanifestationof the existence of this underlyingrepresentation. It is importantto recognizethat such representationsas we will discussin this paper do not take the place of temporalpredictivemodels.Nor do they serve as mechanisms for understandingtemporal causal relationships between time series. In an importantrespect,the limit theorywe present is a limit theoryof a "sampleperiod fit,"in which the sampleperiod can be viewed as a snapshotof an infinitetime series. Such asymptoticanalysisis alreadyused, albeit implicitly, in econometrics.One example, for instance, is the derivationof trend break limit theory,whereinthe breaksare consideredto occur at some fractionof the sample that turns out, in the limit, to be the same fraction of the infinite trajectory.In this respect,therefore,the "snapshotof infinity"asymptotictheory of this paper is not a radicaldeparturefrom some establishedlines of asymp-

SPURIOUS REGRESSION

1301

totic analysis in econometrics.It will turn out to be an importantmode of analysisin the general developmentof misspecification-robust asymptoticsfor trendingtime series. The startingpoint in the approachthat we adopt is a general orthonormal representationtheoryof a continuousstochasticprocess,and the theorythat we use here is outlined in Section 2 of the paper. Our theoreticaldevelopmentis primarilyfocussed on stochastictrends and their associated Brownianmotion limits,but manyof our resultshold for other limitingstochasticprocesses(such as diffusions) that are amenable to an orthonormalrepresentation,and to deterministicfunctionsof time other than polynomialsand trigonometricfunctions. Section 3 shows how the orthonormalrepresentationof a stochastic process is accurately reproduced by a fitted regression, and is completely capturedwhen the numberof regressorsgrowswith the sample size. Section 4 shows that the Weierstrassapproximationtheorem can be extended to give a theory of approximationof continuousfunctions by independentWiener processes, gives some illustrations,and appliesthe theoryto the case of the classic spurious regression of independent random walks. Section 5 concludes the paper.Proofs are collected together and notation is listed in an Appendix.

2.

SOME PRELIMINARY REPRESENTATION THEORY

We start by makinguse of the general representationtheory of a stochastic processin termsof an orthonormalsystem.Severalforms are available,the most common of which is the Loeve-Karhunenrepresentation,which is given in Lemma2.1 below. This result ensuresthat any randomfunctionthat is continuous in quadraticmean has a decompositioninto a countablelinear combination of orthogonalfunctions.The representationis analogousto the Fourier series expansionof a continuousfunction.Thus, suppose X(t) is a zero mean stochastic process that is continuousin quadraticmean on the interval[0,1] and has covariancefunction y(r,s). Let {Pk}yk=l be a complete orthonormalsystem in L2[0, 1] with the propertythat these functions serve as the eigenvectorsof the covarianceoperator,i.e., Ak (Pk(r) = fly(r, S)pk(S) ds, where Ak is the eigenvalue of y(r, s) correspondingto the eigenfunction p,k.Mercer'stheorem(e.g., Shorack and Wellner (1986, p. 208)) ensures that the covariance function can be decomposedas (1)

y (r, s)=

E

Ak (Pk(r) Pk (S),

k=1

where the series convergesabsolutelyand uniformlyon [0,1].2 The corresponding decompositionfor the stochastic process X(t) is most often called the Loeve-Karhunenexpansion,althoughthe stationaryGaussiancase is sometimes attributedto Kac and Siegert (1947).The followingstatementof the expansion

1302

P. C. B. PHILLIPS

is given in Loeve (1963, p. 478): 2.1 LEMMA: A random function X(t) that is continuous in quadratic mean on the interval [0,1] has on this interval the orthogonal expansion co

(2)

X(t)

9 'k (P (t)(k

= k=1

with E(

(j)

| f(Pk(S)p1j(s)ds-

a=kj

8kj'

iff the Ak are the eigenvalues and the (Pk are the orthonormalizedeigenfunctionsof the autocovariance function y(r, s). The series (2) converges in quadratic mean uniformlyon [0,1]. The orthogonal random quantities (k that appear in (2) can be

representedin the form

k =

'k-12f'lx(s)Pk(s)ds. The 8kj above is Kronecker's

delta.

Just as Fourierseries of continuousfunctions do not alwaysconvergepointwise (but do convergein mean), the representation(2) of the stochasticprocess X(t) convergesin quadraticmean but not necessarilypointwise.For this reason, the equivalencein (2) is sometimesrepresentedby the symbol"- ", signifying that the series is convergentin the L2 sense and that distributionalequivalence applies. There are many differentrepresentationsof standardBrownianmotion W(r) that originate in the general form (1). The simplest is the Loeve-Karhunen expansionitself, which is obtainedby using the eigenvaluesand eigenfunctions of the covariancekernel -y(r,x) = r A s, viz. 4 (3)

=Pk(r) Ak=(

(2k -i)

2

2,

=

2'

V sin[(k-1/2)'7r]

directlyin (2), givingthe following L2-representation Go

(4)

W(r) =

i> E k-1

sin[(k

-

1/2)'ngr]

(k -1/2)7

Tr

where the components 4k are independentlyand identicallydistributed(iid) as N(O,1). It is easily seen by applyingthe Martingaleconvergencetheorem (for square integrable martingales)that the series representation(4) of W(r) is convergentalmostsurelyand uniformlyin r E [0, 1], so this series does converge pointwise. Another commonlyused representationis developed as follows. Let V(r)= W(r) - rW(1)be the Brownianbridge process correspondingto the Brownian motion W(r). The covariance function of V(r) is -y(r,s) = r A s - rs, which can be decomposedas in (1) above with eigenfunctionsgiven by the orthonormal

1303

SPURIOUSREGRESSION

system {W sin(kg r)}Y=1 and corresponding eigenvalues Ak = (k7)-2 -e.g., Shorack and Wellner (1986, pp. 213-214). This leads to the following L2-representation of V(r): (5)

co sin(kgr) V(r) = '2 E

with

(k

sin(kgs) (k ='2ifV(s)

k=1

o

ds.

The components 4k in this decomposition are also iid N(0, 1), as can be verified by direct calculation. The representation (5) gives rise to a corresponding expansion for the Brownian motion W(r), viz. (6)

co sin(kTr) k

+ W(r)=rro+V2

k=-

with =

W(1),

1 sin(kgs) 2k =

k

Okg

(W(s)

-

sW(1)) ds.

Again, the (k are iid N(0, 1). The series (6) is known to converge almost surely and uniformly for r E [0, 1]-e.g., Hida (1980, p. 73, Remark 2), and Brieman (1992, p. 261), where the series are defined over the intervals [0, 2X], and [0, 4]. They representation (6) has a linear trend component with the random coefficient (0, and shows that W(r) can be written in terms of both polynomial and sinusoidal functions. In fact, (6) is one of many alternative functional representations of Brownian motion. For instance, we may write each of the trigonometric functions in the orthonormal system {v2Isin(kgr)}k= 1 in terms of another orthonormal basis, such as orthonormal polynomials in r, and then substitute these orthonormal series for the sinusoidal functions in (6), giving a new representation of W(r) in terms of the new basis. The coefficients in this new representation are still random and normally distributed, but no longer necessarily independent. Another popular representation of W(r) is in terms of Schauder functions (orthogonal tent functions) and here again the convergence is uniform in r E [0,1] almost surely-see Karatzas and Shreve (1991, Lemma 3.1, p. 57). In all of these different representations to the continuous stochastic process W(r) is written as an infinite linear combination of deterministic functions with random coefficients. What distinguishes the Loeve-Karhunen expansion, is that the random coefficients as well as the deterministic functions form an orthonormal sequence. These expansions may be used directly to create representations for linear diffusion processes like Jj(r) which satisfy the stochastic differential equation dJ,(r) = cJ,(r) dr + dW(r) for some constant c. With initial condition Jj(r) = 0, the solution of this equation has the form:

(7)

J,(r)

= f

e(rs)c dW(s) = W(r) + cf e(r-s)cw(s)

ds.

1304

P. C. B. PHILLIPS

Substituting(4) into (7) we find 1

co

J~() r) = Jcf 42_E 1 (-/) (k - 127 =~'k= X

sin[(k - 1/2)7nr]

co

=

+

cf

~~1 1~\

2

2

k=1 (k - 1/2) 27T2 + C2

+ (k - 1/2),7

e(r-s)c

sin[(k - 1/2)gs]ds]

4k

ce cr ccos[(k -1/2)iTr]

sin [(k - 112)rr]]r

The substitutionis valid because the series (4) is uniformlyconvergentalmost surely and can be integrated term by term. Another representationcan be obtainedby using(6) insteadof (4) in (7), andyet anotheris the Loeve-Karhunen expansion(2) itself, based on the eigenvaluesand eigenfunctionsof the covariance kernel y(r, s) = e(r+s)c/2c[1 -e-2(r A s)c] of Jj(r). 3. REPRODUCTION OF THE ORTHOGONAL REPRESENTATION BY SPURIOUS REGRESSION

The existence of expansionslike (4)-(6) indicates that continuousprocesses such as Brownianmotion can be representedand, indeed, generatedby deterministicfunctionsof time with randomcoefficients.To the extent that standardized discrete time series with a unit root convergeweaklyto Browniannpotion processes,we infer that deterministicfunctionsof the same type maybe used to model such time series. This brings us to the study of prototypicalspurious regressions in which unit root nonstationarytime series are regressed on deterministicfunctions,a topic first studied analyticallyin Durlauf and Phillips (1988) for the case of a linear trend. We are concerned to ask the following question. Consider the time series Yt= Eu, where u, is a stationarytime series with zero mean and finite absolute momentsto orderp > 2. What are the propertiesof a regressionof the form (8)

Yt=

bk(Pk(n) + ut

or, equivalently (with (9)

Y

ak = n 1/2bk),

kl(n)

k

+

when the limitingbehaviorof the dependentvariableis a Brownianmotion,i.e., (10)

Y[n]=B(-)

BM(o-

SPURIOUSREGRESSION

1305

and the regressors Pk form a complete orthonormalsystemin L2[0,1]. In what follows, we assume that the functional central limit theorem (10) holds (see Phillips and Solo (1992), for primitiveconditions);and, to be specific and to relate outcomes directlyto those of the previoussection, we take (Pk and Ak to be the eigenfunctionsand eigenvaluesof the covariancekernel o 22rA s of the limitingBrownianmotion B, which are obtained from (3) above by scaling Ak by o-2.

In view of (2) and (10), we mayverywell expect that the regressorsin (9) take on the role of the deterministicfunctionsin the associatedorthonormalrepresentationof the limitingBrownianmotion B(0).Perhaps,we can go even further tihan this. If K -0oo as n

-*

oo, could (9) succeed in reproducing the entire L2

orthonormalrepresentationof B(-)? We now proceedto examinewhetherthese heuristicnotions can be made more precise. Let aK = (ak) be the coefficients and (PK=

((Pk)

be the K-vector of regressors

be the usual least squares in (9). Let CK E RK be anyvectorwith CKCK = 1, t regression t-ratio for the linear combinationof coefficients caa,' and let R2 and DW bQ the regression coefficient of determinationand Durbin Watson statistics,respectively.The followingtwo theorems give the asymptoticproperties of these statistics when K is fixed and when K -* oo. 3.1 (a)

THEOREM:

For fixed K, as n -0oo we have: KB] -N(0,

CK[

CKaK

=

2cKf

|p

(r)(r A s)pK(s)' dsdrcK)

N(0, cK AKCK),

n

(b)

E 2=fB2

n-2 t=

(c)

O

B

n-1/ t2

2=>f

(d)

R2

1-

^CK

2 fB7fB

B

2pK

B22

2,

DW-_0,

where B9OK() = B() - (fIOB 4)(fl pK 'p)-poK( ) is the L2-projection residual of B on 4pK'AK = diag(A1,..., AK), and Ak is the eigenvalue of the covariancefunction o 2r A s correspondingto (pk

3.2 REMARKS: (a) Theorem 3.1(a) shows that the fitted coefficients in the regression (9) tend to random variables in the limit as n -> oo.Moreover, the random limits are equivalent in distributionto the correspondingrandom elements in the Loeve-Karhunenrepresentationof the limit process B(0). Thus, (9) reproducesaccuratelyin the limit the appropriateelements in the orthogonal representationof the limitingform of the dependentvariableprocess.In this sense, we can interpret (9) as a partial but nonetheless correctly specified empiricalversion of an orthogonalrepresentationof Brownianmotion. We use

1306

P. C. B. PHILLIPS

the word"partial"here because(9) has only K regressors,i.e., (PK = (Cp)JY-. The model is correctlyspecifiedbecause the regressorsthat are omittedfrom (9), viz. ?p=(cpKjY

1, are all orthogonal to the included variables. Hence, (9) is

indeed well suited to least squaresregression.All of the above holds in spite of P the fact that the DurbinWatsonstatistic DW 0, indicatingthat the residuals in the fitted model are seriallydependent.Thus, conventionalwisdomthat the regression model (8) is spurious and that the low DW statistic signals that inference is hazardous is mistaken here. On the other hand, conventional wisdomthat the low DW statisticmay signalpoor predictiveperformanceof the model maywell be appropriate. (b) Part (c) of Theorem 3.1 shows that the usual regression t ratios of the fitted coefficientsdivergeat the rate OP(n (l/2), and, therefore,ultimatelyexceed any finite criticalvalues as n increases. Hence, such tests indicate statistically significantregressioncoefficientswith probabilitythat goes to one as n -* oo. The fitted coefficientsin (9) are not spuriouslysignificantbecause the significant t ratios correctlyindicatethe presence of the orthonormalrepresentation co

(11)

B(r)=

E

where

bk(r)(k,

(kiidN(O,1),

k= 1 CO

=

E k=1

cPk(r)k,

where

1kiidN(0,Ak).

In effect, the fitted regression (9) is an empirical model for (11). Setting K (K)1~, we have (12)

CKaK = N(0, CAKCK)

C

The significant t ratios signal that the regressorsplay an importantrole in representingthe dependent variable-or its limiting version, the stochastic process B(r). (c) The t ratios, tCdK6Ks that are analyzedin Theorem3.1 are computedusing the conventional least squares regression formulae. In their place, robust t ratios which accommodateserial dependence in the residuals could be computed and it is of interest to examinewhether the remarksmade above in (b) continue to apply.Serial correlationrobust t ratios for the coefficient CK a in d, where (9) are based on the formulajC A = CdKf/Ca c KA aK

=C

(K(

(PK

irv ( n'/2Ka,P(P

(n ) )

C

K

and M

lrvar(Xt)=

Ek k()c(j,X), j= -M

XtXt+

c(],X)=n'1 1
Here, lrvar() signifies a kernel estimate of the long run variance of its argument, k(-) is a lag kernel, and M is a bandwidth parameter for which M -* oo,

1307

SPURIOUS REGRESSION

and M/n

-O 0

as n -* oo.Using the same approach as that developed in Phillips

(1991)for analyzingthe asymptoticpropertiesof kernel estimatesbased on M(1) data, it can be shownthat -Irvar n

(PK

/Ut

j)n M

k

EMP (

- c

ds )

( kI-1

t

102

It follows that

n

-SC/a

k(s) ( ds)( -2~~~~~~~~~

mKaK

)(PK

=(lk(s)

BP (,|BPK

(PK @ F

(PKWK)(,6|(P

K

ds)(fBPKPK~)

and thus the serial correlationrobust t ratio tCdKa has the asymptoticbehavior c KaK cKaK

SC'd

Op(l) 1/

KaK O

(n

nl/2

M112J

l/2)

We deduce that the robust t ratios of the coefficients also diverge as n -* oo,but at the rate (n/M)1/2, which is slower than the conventionalt ratio tCKdK by a factor, M112, which depends on the bandwidthM. Hence, the conclusion of

Theorem 3.1 regardingthe inevitablestatisticalsignificanceof the coefficients applieseven when serial correlationcorrectionsare made to the standarderrors of the estimatedcoefficients. (d) An importantfeature of the true model (11) is that the coefficientsrqk are randomvariables,whereas the variables pDk(r)are deterministic.The empirical regression (9) correctly reproduces this feature of the true model as n -* oo, as is

clear from (12). (e) With some changesin notation,Theorem3.1 holds if the limitingbehavior of the dependentvariableis a generalcontinuousstochasticprocess X(r) rather than Brownianmotion.Supposethat for some a > 0, n - y n X( ), a continuous stochastic process on [0,1] with continuous covariance function -y(r,s) whose eigenfunctionsand eigenvaluesare givenby PDkand Ak.Insteadof (9), we run the empiricalregression Yt n

k aa

l(f

k(P

n)fl

1308

P. C. B. PHILLIPS

Then, in place of (a), (b), and (c) of Theorem3.1, we have the followinglimiting behavior: (a')

f

N( [c X CK f K(r) (r, S) PK(S)] ~~~~~=N(,CSK CK K

CKaK = ck [fK k

CK ds dr)

=N(O, cfAKc) n

(bf)

n -(1 +2 a )

A

t=

2 =t|xP2K

1

I

where X9K(*)= X( )-( |OX(PK)

PK )(PK (,

(Cf)

]

n-1/2t

C

[1

and

1X2)

Thus, the empiricalregressionasymptoticscorrectlyreproducethe form of the random coefficients in the general Loeve-Karhunenrepresentationof X(0) given by (2) and correctly signal their significance.These results apply, for example,to the linear diffusionprocess Jc(r) = fre(r-s)c dW(s) in (7) for some constant c, and thereby (i)-(iii) above cover the importantcase of near integratedtime series Yt(i.e., time series with a root, 1 + c/n, that is near to unity) for which we have

n-l/2y[n]

=Jc(.)

(f) As mentionedin the Introduction,the type of limit theorywe are using in Theorem3.1 can be characterizedas a limit theory of the sample period fit or "snapshot asymptotics."The terminologycan be explained as follows. The dependent variable Yt in the empirical regression (8) is transformedinto a standardizedrandomelement n- 1/2Y[n.] in the functionspace C[0, 1].According to (10) the sample behavior of n-1/2y[n.] is approximated by the limit Brownian

motion process B(). Indeed, as discussedin the Section 6 (see (22) below), the probabilityspace can be expandedso that the samplepath can be approximated by the Brownian motion up to an error of Oas (1)' In effect, the sample trajectoryof n-l/2y[n.] on C[0, 1] is a snapshotof the full limitingtrajectoryof B(O) on the same space. From the Loeve-Karhunenrepresentationof B(-), we know that there is a representationof B(O) in terms of the deterministic functions (Pk with randomcoefficients.Likewise,the regression(9) gives us an empirical"snapshot"of this limitingrepresentation. We now proceedto considerwhat happenswhen the numberof regressorsin (9) tends to infinitywith n. 3.3 THEOREM: As K -oo, C'K KCK tends to a positive constant oc2 = c'Ac, where c = (Ck), A = diag(Al, A2,...) and c'c = 1. Moreover, if K -* oo and K/n -O 0 as n -> oo, we have:

(a)

CKaK =N(0,

(b)

n-2

A2

t= 1

9c)

SPURIOUS REGRESSION

(c)

n-

(d)

R2f*1.

122tc,a

1309

diverges,

3.4 REMARKS: (a) Part(a) of Theorem3.3 gives the limitingdistributionof the coefficients of both K and n -* oo.In this case, CK becomes infinite dimensional and c' a becomes an 12 inner product.As in the finite dimensionalcase, c' a^

convergesweaklyto a randomvariable,but in place of (12) we now have CKaK

=N(O, cAc)

c l,

and the limit distributionis the same as that of the variate c'ij = ElcCkl,k from the orthonormalrepresentation(11). (b) Part (c) of Theorem3.3 showsthat the t ratio tc a divergesas both K and n -* oo. As in the fixed K regressor case, all of the fitted coefficients are statistically significant as n -* oo, according to the usual regression t tests.

However, the rate of divergenceof the t ratio is greater in the case where than it is when K is fixed. In other words, the regressioncoefficients appear more significant,not less significant,with the additionof regressorsas n -* oo.This is explained by the fact that the residual variance in the regression

K -0oo

(9) tends in probability to zero when both K and n -* oo,i.e., there is no residual

variance from this regression in the limit, as indicated in part (b) of the Theorem. In effect, as K, n -> oo,the regression (9) succeeds in reproducing the

entire Loeve-Karhunenrepresentationof the limit process B(\) and thereby fully representsthe dependentvariablein the limit. The fact that the empirical regressionfully capturesthe series representationin the limit is confirmedby the limitingregressionR2 of unity. (c) In the same way as for Theorem 3.1-see Remark3.2(d)-Theorem 3.3 can be extended to apply to more general stochasticprocesses than Brownian motion. The proof of Theorem3.3 relies on the use of an extendedprobability space in which a stronginvarianceprincipleapplies-see (22) in the Appendix. Strong invarianceprinciples like (22) have been proved in the literature for standardizedpartialsums that convergeto Brownianmotion. These results can be extended to apply to linear diffusions,like Jj(r) in (7) above, as shown in Lemma 6.3 below. It follows that Theorem 3.3 is valid for both integratedand near integratedtime series with the appropriatechangesin notation. (d) Theorems3.1 and 3.3 give results for empiricalregressionslike (9) where the regressors are the eigenfunctions that appear in the Loeve-Karhunen representation.As discussed in Section 2, there are other representationsof limiting processes like Brownianmotion and diffusions that are of a similar form, but for which the coefficients may not be independent and/or the functions may not be linearly independent. In such cases, it is possible to develop a limit theory for the empiricalregressions,but effects such as the possible collinearityof the regressorsin the limit as K -- oo need to be taken into account. The earlier version of this paper (1996) explored such conse-

1310

P. C. B. PHILLIPS

quences for the case of an empirical version of equation (6) and can be obtained from the author on request.

4. WIENER

PROCESS APPROXIMATION THEORY

The above analysis uses series of deterministic functions with random coefficients to represent stochastic processes like Brownian motion. It is of some interest to ask if the reverse is possible, viz. can we represent an arbitrary deterministic function on a certain interval in terms of stochastic processes? To deal with this question we will take a slightly different approach and try to approximate an arbitrary continuous function on the [0,1] interval in terms of independent Brownian motion processes. The idea is analogous to that of the uniform approximation of a continuous function by polynomials or trigonometric functions. The following shows that there is, in fact, a Wiener process version of the famous Weierstrass approximation theorem.

4.1 THEOREM: Let ff() be any continuous function on the interval [0, 1], and let > 0 be arbitrarilysmall. Then we can find a sequence of independent standard }/1, and a sequence of random variables{di}' 1 such that as Brownian motions {WiE N --oo, ?

N

(a)

f(r) -

sup

a.s.,

=1

re [0, 1]

(b)

diWi(r) <8

E

f'[f(r)-

NdiWi(r)

dr<

a.s.

4.2 REMARKS: (a) The Weierstrass approximation theorem tells us that any continuous function f(r) can be uniformly approximated on the interval [0, 1] by a trigonometric polynomial of the form K

(13)

ao + E (aksin(2Tkr) +P8kcos(27kr)). k=1

In this series approximation, the coefficients {ak, 1k) are nonrandom and the functions are deterministic continuous functions. In an analogous way, Part (a) of Theorem 4.1 shows that we can find a set of N independent Wiener processes on C[O,1] and a sequence of N random variables such that, with probability one as N -* oo, the functionf(r) can be uniformlyapproximatedon the interval[0,1] by the linear combination E= 1diWi(r) of Wiener processes. (b) Part (b) of Theorem 4.1 is sufficient to ensure that the system of Wiener is complete in L2[0, 1] with probability one (e.g., see Tolstov processes {WiJ'JL=1 (1976, p. 58)). It follows that, given any continuous function f(r), we can find a 1 such that with probability one sequence {Wi(r),di}T=

(14)

lim

N?

1

r

f (r) oL

N

i=l

-2

diWi(r) dr = O,

1311

SPURIOUSREGRESSION

and thus (15)

f(r) -

diWi(r) i=l1

in L2. We mayreplacethe WienerprocessesWi(r)by orthogonalfunctionsVi(r) in L2[O,1] using the Gram-Schmidtprocess,i.e.,

f V1 = W1V

(16)

!V2 W2 -(w W2V,()(f1Vy

|V3

W3

)-1V, aa

W3V)( Va

(|

Va,

Va

I [Vl, V2 ]

In place of (14), we then have -r

lim N?

f Lf(r)

-2

N

-

? eiVi(r)

dr = O

i=l

with probabilityone. By virtue of the orthogonalityof the functions {VJ(r)}in L2[0, 1], we get the followingstochasticFourierrepresentationin L2:

(17)

f(r) -

EeiVJ(r),

with

ei=(f1Iy)(f11K2)

and, with probabilityone, we have Parseval'sequality, ff2=

E ei2(fv2),

holding,but now with randomcoefficients. (c) We can apply the approximationtheory of Theorem 4.1 to the sample path of an arbitraryBrownian motion B(O) on the interval [0,1]. Since the sample path of B is continuous, we can find a probabilityspace such that Theorem4.1 appliesand then we have B(r) -'E= 1 diWi(r)in the L240, 1] sense. We formalizethis as follows. 4.3 THEOREM: Let B( ) be a Brownian motion on the interval [0,1], and let e > 0 be arbitrarilysmall. Then we can find a sequence of independent standard Brownian motions JitIN1 that are independent of B, and a sequence of random

1312

P. C. B. PHILLIPS

variables{di}N 1 defined on an augmentedprobabilityspace (PX,i N

F) such that, as

oo, N

(a)

B(r) -

sup r E[O,1]

(b)

f1[B(r)

(c)

B(r) -

dWi(r) < e

a.s.(P),

i= 1

-E

diWJ(r)] dr < e

a.s. (P),

00

diW(r) in L2[0, 1]

a.s. (P).

i=l1

4.4 REMARKS: (a) Part (c) of Theorem 4.3 shows that an arbitraryBrownian motion B( ) has an L2 representation in terms of independent standard Brownianmotions with randomcoefficients.As is clear from the proof of this theorem,the coefficientsdi are statisticallydependenton B(). (b) Part (c) of Theorem 4.3 also gives us a model for the classic spurious regressionof independentrandomwalks. In this model, the role of the regressors and the coefficientsbecomes reversed.The coefficientsdi are randomand they are co-dependentwith the dependent variable B(r). The variables W,(r) are functionsthat take the form of Brownianmotion sample paths, and these paths are independentof the dependentvariable,just like the fixed coefficients in a conventionallinear regressionmodel. Thus, instead of a spuriousrelationship,we have a model that servesas a representationof one Brownianmotionin terms of a collection of other independentBrownianmotions. The coefficients in this model providethe connectivetissue that relates these randomfunctions. (c) Let us now replace {Wi(r)}by the orthogonalsystem {V,(r)} defined in (16). Then, in place of Part (c) we have, as in (17), (18)

B(r) -

eiVJ(r),

with

ei = (f1BVi) (fv

-

(d) When we run an empiricalregressionof one randomwalk on a set of independentrandomwalks,we reproducea finite sample version of the model given in Part (c) of Theorem6.3. Or, equivalently,if we transformthe regressors so that they are orthogonal,then we reproducea finite sample version of the representation(18). 4.5 EXAMPLE: As an illustration,consider the quadraticfunction fi(r) = r2 for - iT < r < iT, combinedwith its periodicextensionoutside this interval.The Fourierseries for this functionis (c.f. Tolstov(1976, pp. 24-25)) iT2

r2 '-

43cosr-

cos2r 22

+

cos3r 32

and this series convergesto f(r) = r2 in the interval [Textensionoutside of this interval.

i,

7T] and

to its periodic

1313

SPURIOUS REGRESSION

The function together with four terms of its Fourier series are shown in Figure 1. Figure2 showsthe same functionwith its approximationin termsof N independentWienerprocesseswith N = 150.The coefficientsin the approximation are calculatedusing least squaresregressionof f1(r) on 1,000 observations generatedfrom 125 independentrandomwalks.With this numberof terms,the Wiener process series capturesthe shape of the periodicquadraticfunction comparablywell. The purpose of this example is simply to illustrate the feasibility of the approximationby Wiener processes and to give some idea of the number of termsthat are needed to achievea level of approximationcomparableto that of a Fourier series for a simple continuousfunction like f1(r). Needless to say, empiricalregressionsof the type (15) are not being recommendedfor practical use, nor do we develop a theory for the selection of the number of Wiener processesin such regressions. 4.6 EXAMPLE: Finally,we considerthe standardGaussianrandomwalk Yt= E>=1u0, where uo -iid N(O,1). Let x= = (xk)= (= lUkAj)/1 be K indepen-

dent Gaussianrandomwalks, all of which are independentof Yt.Considerthe linear regression yu= b xt + it, based on n(> K) observations of these series.

The large n asymptoticbehaviorof bx is given by (Phillips(1986)) WXWXI ]

bx;[1

W[1 WxY]

where Wx and WYare the standard Brownian motion weak limits of the standardizedpartialsum processes n 1/2X[n.] and n-1/2 [n.], respectively. 10

JL I

4

2-

0

2

4

6

8

10

r FIGURE1.-Fourier series f1(r): 3 terms.

12

14

1314

P. C. B. PHILLIPS

10 80 6L4

4 -

1

2 -

0 -2

0

2

4

6

8

10

12

14

r FIGURE2.- f1(r): 125 Wiener terms.

Supposewe orthogonalizethe regressors{Xk = GramSchmidtprocess

(Xkt)i?:

k = 1,..., K} using the

Zlt =Xlt, z2t = X2t -(x

xltX

21-)(x'-xi-)

Z3t =X3t-(X3.Xa)(X.Xa)

F Xa := [X1., X2.]

Xat'

[X'a.],

etc.

By standardweak convergencearguments,we find n-l1/2Zl[n-]

Now let Z = writing bZK =

(Zkt)K,

(bzK),

bZk=t [f

'

V1(),

n-l1/2Z2[n-]

; V2

and consider the regression Yt = bZKzt+ we have the limit v ]j[f

l

yj

V3(-), etc.

f, n-1/2z3[n.] Ut.

In this case,

ek

as in (18). Thus, the empiricalregressionof Yt on Zt reproducesthe first K terms in the orthonormalrepresentationof the limit Brownianmotion WYin terms of an orthogonalizedcoordinate system formed from K independent standardBrownianmotions.The regressiont ratios are tbk = bzk/sbk and these have the limitingbehavior f

f1W2/

j 1/2

'

1315

SPURIOUS REGRESSION

where WVK(*)= WY(.)-

(0

diverge at the rate

(shown in Phillips (1986)), indicating certain significance

1VK(*), and VK(*) = WYVK)(f|OVKVK)

(Vk(*))k

1. As in

the case of deterministicregressors(cf. Theorem 4.1), the regression t ratios n172

of the regressorsin the limit. Moreover, in view of (18), JoW2K-4 0 a.s. as K -, oo,and we can expect the divergence rate of these t ratios to increase when both K, n -* oo. Figure 3 shows the sampling densities of the t ratio, tb, with K = 1,10,20 and n = 100 based on 30,000 simulations. The increase in the

divergencerate of the t ratio as K increasesis apparentin these graphs. Finally, the behavior of the R2 in the regression Yt= b uKzt+ lt is R2 1- f 0 2p R2._'1

W2K/f1JY2 for fixed K, 10

when

K ?- o

as

n ->

.

It follows that the empiricalspuriousregressionfully explains yt in the limit when the numberof independentrandomwalk regressorsgoes to infinity. 5. CONCLUSION

This paper shows that there are mathematicalmodels underlyingthe classic spuriousregressionsof a randomwalk on deterministictrends and the regression of a randomwalk on randomwalks.The empiricalregressionsjust pick off the first few terms in the series representationof the stochasticprocess that is the weak limit of a suitablystandardizedversion of the dependentvariablein the regression.Moreover,it is shown that, if the numberof regressorsin such regressionsis allowedto growwith the sample size (n), these empiricalregres0.040 0.036 0.032

K=1 --- K=10 .......K=20

0.028 0.024 :

0.020 0.016 0.012 0.008

0.004 0.000

-200

............. -100 -50

FIGURE 3.-Densities

0

50 100 150 200

of t ratio tbl:

=

100.

1316

P. C. B. PHILLIPS

sions succeedin accuratelyreproducingthe full series representationin the limit as n -* oo and that the regression R2 tends to unity. The theory also explains why it is natural in these regressionsfor the fitted coefficientsto be random variablesin the limit-they are exactlythis in the underlyingmodel! Thus, not only is there a valid mathematicalmodel underlyingsuch regressions,but the completemodel is consistentlyestimablein the limit as n -* Qo. While these results are definitivein that they fully explainwhat happens in regressions of this sort and why it happens, there is room for considerable debate about the implicationsof these results for empirical research. One viewpointwas clearlystated by a referee of the paper in the followingway: "We use the term spurious regression in contrast to say the concept of cointegrated regressions, i.e., the possibility that certain sets of variables explain the trend of the dependent process in an economically sensible way. The fact that trending time series have valid representations in terms of other independent processes or deterministic functions of time is not of much interest from an economic viewpoint, unless it helps separate the wheat from the chaff."

In commentingon this orthodoxview, I will make only two points here and leave it to future debate to take the discussionfurther.First, it needs to be emphasizedthat cointegratingregressionsdo not explain trends. Instead, they relate trends in multiple time series and thereby pass the trending behavior along to secondaryvariablesthat are usuallyalso endogenous,leavingthe trends themselvesto be explainedby unit roots,time polynomials,and trendbreaks.As this paper shows,the trends themselvescan be validlymodelled in a varietyof ways.Thus,the centralissue addressedin this paper remainspresentin modern cointegration-basedmodels of nonstationarytime series. The second point is that the nature of trendingmechanismsin economics is little understoodand econometricianshave little guidancefrom economictheorymodels about meaningful economic specifications.Were this not so, we would not be as heavily dependent as we presently are on unit root models, time polynomials,trend breaks, kernel regression fits and such like in capturingtrends in empirical research.Against this backgroundand with the currentclass of nonstationary modelsused in econometrics,it is virtuallyinevitablethat the trendingprocesses that appearin econometricmodels have little intrinsiceconomicmeaning,even though the trends themselves may be of considerableeconomic interest. This paper showsthat, even in the impoverishedclass of trendingmechanismthat we currentlyemployin empiricalresearch,a limit theoryof the trendingprocessEs possibleand that it will often be based, in part at least, on a "limittheoryof the sample period fit."This limit theory bringswith it attendantqualificationssuch as those in the Introductionabout the use of these mechanismsin a predictive context. The results presented here have some implicationsfor unit root modelling and testing. In recent years much of that literaturehas emphasizedthe importance of setting up a general maintainedhypothesisthat includes "alternative" specificationsto a unit root model, such as deterministictrends and trend breaks.The resultsof this paper show that such specificationsare not necessar-

1317

SPURIOUS REGRESSION

ily alternativesto a unit root model at all. Since unit root processeshave limiting representationsentirely in terms of these functions,it is apparentthat we can mistakenly"reject"a unit root model in favor of a trend "alternative"when in fact that alternativemodel is nothing other than an alternaterepresentationof the unit processitself. A developmentof the asymptotictheoryin this case and a study of the impact of such considerationson empiricalwork are left for a future paper. Cowles Foundation for Research in Economics, Department of Economics, P.O. Box 208281, New Haven, CT 06520, U.S.A.; [email protected]; http:// korora.econ.yale.edu ManuscriptreceivedJuly, 1996; final revision receivedNovember, 1997. APPENDIX PROOFS A.1 PROOF OF THEOREM3.1: Since cpK([n*]/n) --K

() we have

n

n

I

f

( tln oK( tln) (PK

E

0

t=1

=IK

(PK PK

!

Then, using (10), we obtain n1 EnoK(t/n)yt/l xn

folKB. Let OK be the observation matrixof

the regressorsand let y = (Yt)nin (8). Then, we have (19)

~~~~~~~~~~~~~~ K9 CK=

cK(

N(O,c'K

n)(

(

n

1f

XDK(r)(r

KfK )l/)

As)9K(sYdsdrcK),

givingthe stated result. Now let the orthonormalrepresentationof the Brownianmotion B() be where the 4k are iid N(0, 1) and Ak is the eigenvalue of the given by B(-)= E'jr/7Pk()4k, covariancefunction o 2r A s correspondingto 'Pk*Writethis representationin the form (20)

B(-) = (pK(-YA12(K + (pl (- A 12 l ,

where the functions in pj are all orthonormaland orthogonalto those in the vector (PK,the elements of ( , are all iid N(0, 1) and AK = diag(A1,...,AK), A1 = diag(AK,AK,...). Using this representationof the Brownianmotion B(-), we get CKB =

(PK(pK,)A/K

= CKAK/

N(0

CKAK CK)

as required for part (a). Note that the limiting form of the distribution also follows from a direct

reductionof covariancematrix,viz. J2f1 fIp(r)(r

00

A S)PK(s) dsdr =

f

PK(r)pK(r ' drAK= AK.

1318

P. C. B. PHILLIPS

For parts(b) and (c), define tCCK (21)

52, A = (n-

=

where

c aK/SC K,

1 E (n- 1/2U )

KCKScKiK

1

A simplecalculationrevealsthat n

n

2

E

ut=J B(PK, - (f10B(PK)(f10(KKok)-1'K(-) is the L2-projection residual of B on

where B(() =(B(-)

'PK'

giving

part (b). Further, 2 2 ns,cK aK = (nn 2

E

B

K

I

P

and we deducethat C~K

n

C4f1~KB] l

1/2

cKaK

(1

n)ECuK-bK

(n-1/n>)E(

K

BPt

(l/2

IK

)E,n

n1

2

P/U)

r

<00,

and ERAk <2h

?.

=

1, aelyAk and foy(r, =

r) dr = lfordr.Hence,

I followsthat

C:K AKCK-=EC2Ak
1/

??I/2?? 1/2

/K1/2

A.2 PROOF OF THEOREM 3.3: First, note fthat ll terC

2

Ek

tbfl\

<

E

Ek

<.

ln oKt that is bounded Thus, CKAKCK is an increasingsequence above and is thereforeconvergent.We write limK_ AKMlCK= (TC2 c'Ac, say, where c = (Ck), A = diag(A1,A2,...) and c'c = 1. To prove part (a) we write, as in (19), CAKaK= c(n'I4PKY')(n"I4KYK/n'/2). Using the Hungarianstrong approximation(e.g., Csorgo and Horvaith(1993)) to the partial sum process Yk= _ik 1u1, we can constructan expandedprobabilityspacewith a Brownianmotion B(-) for which

(22)

sup IYk-B(k)KO= a.s.(fl/P), O?k?n sunk<()

1nnna.s.(1)

or

1319

SPURIOUSREGRESSION This gives the representation ( [nr])

Yt-

for (t - 1)/n < r < t/n, t 2 1. It follows that we may write, as n n1E

((PK) = ( 1q,(r)B(r)

(PK()

oo,

dr + Oa.s.(1)

0 as n - oo,we have

Also, since K/n

f = J

(PK(n

(PK (

n

K(r)qoK(r) dr + o(l)

=IK +o(l),

leadingto (23)

CKaK = CK[IK +

[1f

o(1)]

= CKJ pK(r)B(r) K

dr +

(r)B(r)

Oa.s.(1)]

dr + Oa.s (1)

Now use the orthonormalrepresentation(20) of the Brownianmotion B() in (23), and since the series convergesuniformlywe may integrateterm by term,leadingto K

r (PK(r)

(rf

K=

K/

K +

(P,

(rf

A1/2

jI

dr + Oa.s.5 (

0

c

c' fK1oK o (rY drA /f2 ,

APG+

dr+ Oa.s(1)

C'K AKj2GK + Oa.s.(1),

by virtueof the orthogonalityof C'

Ak/2K

=

N(0,

c

(pK

and the elementsof o

AKCK) ='

.

N(0, cAc),

as K - oo.Thus, in the original probability space, when K -o (24)

+ +Oa.,(1)

CKa KC ACA

=

Now

N(O, c

AKCK)

as n oo with K/n

+ Oa.,.(1)

=

0, we have

N(0, c'Ac),

as requiredfor part (a). For parts (b) and (c), we have

n1

/2tc'Ka

s CK).

K=CKK/(n/2

The behavior of the numerator is

given in (24). The squareof the denominatoris ns2KK = nnnC'Ka^K

E

It2c)

n-00)

K(n4)4))c

C

Now

Vn

4OkOK)

CK=

[+

CK =1

+ O

,

1320 as n

P. C. B. PHILLIPS oo for all K such that K/n -O 0. Next

nt=

2 Yt )

n

u(t2'Kn)P n tn n2~~~(

n

rP

PK(n))

dr +

= (f1B(r)2

Oas(1))

(P)(K

'Kn)

B((r) f1K

(ry

( nt

dr + o(1))

X (flJoK(rxpK(ry

=

Yt K(

f~BqK(r)2 dr +

m

dr +

-1

(f

()B(r)

oas5 (1))

dr + oa.s (1))

oa.s.(1),

where (25)

B'K

= B(r)

(r)

( fBP -

=B(r)

) p )

(1B

g)

(j

-1

(PK(r)

(r)

K

E (f'

=B(r)-

1B(s) Pk(S)dsJ(Pk(r).

k=1

But, (d'k)i is a completeorthonormalsystemin L2[0, 1] and, by virtueof Lemma2.1, we have (26)

E(Pk(r) [(Pk(s)B(s)

B(r) =

ds]

in quadraticmeans.It followsfrom(25) and (26) that,as K ??, BPK- 0 in quadraticmean.Hence, as K--o,

--O,

B,K(r)2 dr]

E[f

and it followsthat n

Ea2

n2

0

nS2

t= 1

givingpart (b). In consequence, CKaK

- 1/2 t

CKaK

diverges as n

-oo

when K

oo and K/n

0, thereby establishing part (c). Part (d) follows directly

from(b). A.3 LEMMA:Let yt = Et e(t -j)c/nu. be a near integratedtime seriesfor some constant c, where ut is stationary with zero mean, finite absolute moments to orderp > 2 and has partial sums that satisfy the invarianceprinciple n- 1/2E nr]U. = W() - BM(1). Then, there exists a probabilityspace containing {Yt} and a diffusionprocess JC(r) = fre(r-s)c dW(s) in which Yt satisfies the strong approximation sup re[O,1] 1

Yn

-JC(r)

=Oa.s

(1).

1321

SPURIOUS REGRESSION

A.4 PROOF OF LEMMA6.3: It is known (cf. Phillips (1987, Lemma 11)) that Yt satisfies the invariance principle n-1/2YrnrI = JC(r), and from (7) we have (27)

Jc(r)

=

W(r) + cf e(r-s)cw(s) ds. 0

As in the proofof Theorem3.2, use the Hungarianstrongapproximation (e.g., Csorg6and Horv'ath (1993))to the partialsumprocessELi= 1uj and constructan expandedprobabilityspacethat contains holds: {u,, y,} and the BrownianmotionW(Q)and for whichthe followingstrongapproximation sup

re[O 1] Set XJ()

=

EIt-rlu. 5 i - W(r)

n 1/E2

|=

n

o..(1)

]uj, and write [nzr]

[nr]

(28)

n 1/2Ynr] =n-1/2

n- 1/2 Ee([tnr]-j)c/n

Ee([nr]-i)c/nu

fl/t

dX (s)

[nr]

= E

dX,(s) (j 1)/n

e([nr]-i)c/tfln/il

1 [nr]

=

E

r

f/lt

e('-s)c+?{([tr]n-r)+(ijn-s)}cdX,,(s)

(r-1s)/nc

= jre r

= fre(rs)c 0

dX,,(s)[1

+ oa.s.(1)]

dX,,(s) + Oa.s.(),

1 + o(1) uniformlyin r E [0,1], and in s E [(j - 1)/n, j/n] since e{([nr]/tr)?(/ns)c =-= and uniformlyover j = 1. n. Next, apply integrationby parts to the first term of (28) which is justifiedbecause e(r-s)c is continuousand Xn(s) is of boundedvariationfor finite n. Hence (29)

n -1/2Yrnr]

=XJ(r) + cfe(r-s)cX(s)

ds + oa.s.(1).

0

It followsfrom(27) and (29) that IYrnr1 -h(r) sup | c rE[o,l] Xrn

?

sup IX,,(r)-W(r)I E[O1] +

e(r s)c ds]

sup [cf

sup IX,,(s)-W(s)

I+ oas.(1)

=O.s.(1),

for n -/2Yrnr] in termsof the diffusion Jc(r). givinga strongapproximation A.5 PROOFOF THEOREM 4.1: Let {WT(r)} be any sequence of independentWienerprocessesin the [0,1] interval.Using the series representation(4) for each processWi(r)in the sequencewe may write (30)

JJ7i(r)= Erk

sn(k

-

1/2) r

k

P. C. B. PHILLIPS

1322

where the (ij are independent N(0,1) variates. It is well known (cf. Tolstov (1976)) that the continuousfunctionf(r) can be uniformlyapproximatedon the interval[0,1] by a trigonometric polynomialof the form K

ao+

(aksin (2vkr)+ P8kcos(2vkr)).

E k= 1

Since {V2sin[(k- 1/2)vr]Ik= 1 is a completeorthonormalsystemfor L2[0,1], a slightmodification to the proofof this approximation theorem(usinga piecewiselinearapproximation to f(r) and the fact that the Fourierseriesof a continuous,piecewisesmooth,and arbitrarily close approximation to f(r) is convergentuniformly-see, e.g., Tolstov(1976, Theorem2, p. 81)) shows that the function f(r) can also be uniformlyapproximated by a trigonometricpolynomialof the form (V sin[(k-1/2)vr]

K

say,

=aK fK(r),

K

(k-1/2)b

kYakt 1 k=

for some K; i.e., given ?> 0, there exist coefficients(ak)kK K31

(31)

|

)2

sup f (r) r CE[O,1]

(V FlakI(k<

k=1

sin[(k -1/2)vr] k- /

\

2)

1

and some K for which

)

.

/

2

-

We now seek to combine(30) and (31) to producean arbitrarilyclose approximationto f(r) by Wienerprocesses.Given a fixed K for which(31) holds,we take a probabilityspace on which the sequence{Wi(r)}of Wienerprocessesand the randomvariables{ jj} are definedandwe employthe representations (32)

where CIK=

sin[(k

E

WJ(r)

2

[IC1

qi

E,

CiK], C

(r) =

-

1/2)vr]

iK +

i qiK (r)

(k -12)

= [IiK+?1, CK+2

'

I

and

...]

sin[(K +1/2)vrr (K + 1/2)s

qf1 (rf

sin[(K +3/2)vrr (K + 3/2)v

As discussedearlierin connectionwith (4), the series (32) convergealmostsurelyand uniformlyin r E [0,1]. Takingthe linear least squaresapproximation to the firstterm of (32) based on N observations and (i= 1. N), we obtain ) 15{kN

&K

K

( N)

=*-KN

(J

N )

KN-

P

IN

x

N =

,

say,

I N = [ (.*N l], and WN = (Rh)NX 1. The randomvariables where EKN =[ 1K. *NK] ij in ['IKN, ?IN] are iid N(0, 1). Hence, by the strong law of large numbers,as N -? 0, we have .1 a.s. a.s. a.sKN K's - qYK 0. Moreover,the N- 1kN N0, so that XN-- 0, and YJK IK, and N 'KN A N E strongconvergenceof qJKto YJKis uniformin r [0, 1]. To see this, write IYK qKI

OJ JK ) YK YJ)]

[ J<j(XNXN )qJ1] )

[

LT2lE

sin[(k +

2

? [<

)Vr]

2

)]12 1 II ]1/2[AmaXm(XNXN

\2]

1(

1/2

/

SPURIOUSREGRESSION

1323

where Amax(-) signifiesthe largesteigenvalueof its argumentmatrix.Since XN 0 and Amax(XNXN) is a continuousfunctionof the elementsof XN, we have Amax(XNXN)a4 0. It followsthat sup I~JKfJKI?[SUP I KI 1K < [

c

2

111/2 ]

2

rfe [0, 1]

k=K+

1

k2 k

[Amax(XrXN)]

1/2

f1 < (-

12as /0O,

[Amax(XkXNA)]

\3

/

as N -- oo.Hence, given 8 > 0 there exists(by Egoroffs theorem)a set C6 with P(C) > 1 - 8 and a numberN5> 0 for which SUP kIK-KI < 2EK IaI k=

re [0, 11]

K

for all N > N5. Then,we have If(r)

-a 4&(r)I

<

If(r)

-

a'qK (r)I + Ia'qK (r)

-

aKK(r)I

and (33)

sup If(r)

-a

iK

sup If(r) - a' tK(r)I +

(r)I <

reE[0,1]

sup Ia'K ff(r) - aKf(r)I re [0,1]

re[0,1] 8

K

2

k=1

<- + E

lakI

sup IqK(r)-8(r)

< e,,

re [O,li

for all coE C,. Now note that we can write N

(34)

ac4jf(r)

KN ='KN)

=a'(

KNWN= E d1W(r) i=l

with di = a'(C5iN IKN) _%K It follows from (33) and (34) that N

sup

f(r) -

re [0,1

E

diT(r)

a.s.


i=1

as N oo, givingpart(a) of the requiredresult.Replacing8 by 81/2 in the above,part (b) follows immediately. A.6 PROOFOF THEOREM4.3: Let (Qb = C[O,11,9b, Pb) be the probabilityspace on which the Brownianmotion B() is defined. Let B(-, (b) be a sample path of B. There exists a C with Pb(C) = 1 such that, for all (Ob E C, the samplepath B(r, LOb) iS continuous.Further,there exists a compactset CK of C[O,1] (under the sup norm) such that with arbitrarilylarge probabilitythe samplepaths B(r, LOb) lie in CK.Take any such LObE CK.We can applyTheorem4.1 to B(r, LOb) notingthat the theoremholds uniformlyfor continuousfunctionsin a compactset like CK. We expandthe probabilityspace to the productspace (-Q,, P)

=

(Tb X QW,9b

X.9w,

Pb X PW)

to includea sequenceof independentstandardBrownianmotions{W'}l)1 (definedon Ww, w, Pw) and independentof B) and a sequenceof randomvariables{dj/} 1 (definedon (Q,Y P)) for which N

(35)

sup

B(r,

cob) -

diWi(r)

<8e,

i=1

re[0,1]

f1B

E

b)-

l B(r, (0b)-

asN dj

diWi(r)

2 ]d<8

dr <

e

a.s. (Pw)

1324

P. C. B. PHILLIPS

as N - oo.This is possiblefor all Co)bE CK and, as is clear from the constructionof the coefficients cow) on the sample path di in the proof of Theorem 4.1, we have the dependence di= di(cotb, B(, cob) as well as cowE- 2w, but the functions{Wi(r)}are invariantto cotb-We also have the on the samplepath cobof the Brownianmotion B and cowE Q2w.But, dependenceN = N(cob, Cow) since Theorem4.1(a)holdsuniformlyfor f in a compactset, (35) holdsfor all cotbE CK and N large enough.Since Pb(CK) is arbitrarilyclose to one, we deduce that given the Brownianmotion B(-), there exist independentBrownianmotions{Wi(r)}and randomcoefficients{di} that are definedon the augmented space (Q,P, F) for which, as N

-

NN

f LB(r)-

sup B(r) - , diWi(r) <e, re[O,]

oo,we have

0

i=l

2 EdiWi(r)

dr <

a.s. (P),

=

giving(a). Parts(b) and (c) follow directly.

NOTATION

C[O,1]

space of continuousfunctionson [0,1].

L2[0, 1] space of squareintegrablefunctionson [0,1].

~

[.]

r As a.s.

->

d 0a.s.(1) p

->

weak convergence. integer part of. min(r, s). almost sure convergence.

distributionalequivalence. definitionalequality. tends to zero almostsurely. convergence in probability.

REFERENCES L. (1992):Probability.Philadelphia:SIAM. BRIEMAN, CSORGO,M., AND L. HORVATH(1993): WeightedApproximations in Probability and Statistics. New

York:Wiley. DURLAUF,S. N., AND P. C. B. PHILLIPS (1988): "TrendsVersus Random Walks in Time Series Analysis," Econometrica, 56, 1333-1354. GRANGER,C. W. J., ANDP. NEWBOLD (1974):"SpuriousRegressionsin Econometrics,"Journal of Econometrics, 74, 111-120. GRENANDER, U., AND M. ROSENBLATr (1957): StatisticalAnalysis of StationaryTime Series. New York: Wiley. HENDRY,D. F. (1980): "Econometrics: Alchemy or Science?" Economica, 47, 387-406. HIDA,T. (1980): Brownian Motion. New York: Springer-Verlag. KAc, M., AND A. J. F. SIEGERT(1947): "An Explicit Representation of a Stationary Gaussian Process," Annals of Mathematical Statistics, 18, 438-442. KARATZAS, I., ANDS. E. SHREVE(1991): Brownian Motion and Stochastic Calculus, Second Edition. New York: Springer-Verlag. LOEVE,M. (1963):ProbabilityTheory.Third Edition. New York: Van Nostrand. PHILLIPS, P. C. B. (1986): "UnderstandingSpurious Regressions in Econometrics,"Journal of Econometrics, 33, 311-340. (1987): "Towards a Unified Asymptotic Theory for Autoregression," Biometrika, 74, 535-547. (1988): "Multiple Regression with Integrated Processes," in StatisticalInferencefrom Stochastic Processes, ContemporaryMathematics, ed. by N. U. Prabhu, 80, 79-106.

SPURIOUS REGRESSION

1325

(1991): "Spectral Regression for Cointegrated Time Series," in Nonparametricand Semiparametric Methods in Economics and Statistics, ed. by W. Barnett, J. Powell and G. Tauchen. New York: Cambridge University Press. P. C. B., ANDVICTORSOLO(1992): "Asymptotics for Linear Processes," Annals of Statistics, PHILLIPS, 20, 971-1001. SHORACK,G. R., AND J. A. WELLNER (1986): Empirical Processes with Applications to Statistics. New York: Wiley. G. P. (1976): Fourier Series. New York: Dover. TOLSTOV, YULE,G. U. (1926): "Why Do We Sometimes Get Nonsense Correlations Between Time Series? A Study in Sampling and the Nature of Time Series," Journal of the Royal Statistical Society, 89, 1-69.

Econometrica,

Vol. 66, No. 6 (November, 1998), 1327-1351

BOOTSTRAP METHODS FOR MEDIAN REGRESSION MODELS BY JOEL L. HOROWITZ1 The least-absolute-deviations (LAD) estimator for a median-regression model does not satisfy the standard conditions for obtaining asymptotic refinements through use of the bootstrap because the LAD objective function is not smooth. This paper overcomes this problem by smoothing the objective function. The smoothed estimator is asymptotically equivalent to the standard LAD estimator. With bootstrap critical values, the rejection probabilities of symmetrical t and x2 tests based on the smoothed estimator are correct through O(n - Y) under the null hypothesis, where y < 1 but can be arbitrarily close to 1. In contrast, first-order asymptotic approximations make errors of size O(nW-). These results also hold for symmetrical t and x2 tests for censored median regression models. KEYWORDS:

Asymptotic expansion, smoothing, L1 regression, least absolute deviations.

1. INTRODUCTION

A

LINEAR MEDIAN REGRESSION MODEL has

(1.1)

the form

\ Y= Xf + U,

where Y is an observed scalar, X is an observed 1 X q vector, 18 is a q X 1 vector of constant parameters, and U is an unobserved random variable that satisfies median(U IX = x) = 0 almost surely. The parameters 18 may be estimated by the method of least absolute deviations (LAD). Bassett and Koenker (1978) and Koenker and Bassett (1982) give conditions under which the LAD estimator is n1/2-consistent and asymptotically normal. Koenker and Bassett (1978) treat general quantile regressions. Bloomfield and Steiger (1983), Koenker (1982), and Koenker and Bassett (1978), among others, discuss the robustness properties of the LAD estimator. The asymptotic normality of the LAD estimator makes it possible to form asymptotic t and x2 statistics for testing hypotheses about 18in (1.1). However, first-order asymptotic approximations can be inaccurate with samples of the sizes encountered in applications. As a result, the true and nominal probabilities that a t or x2 test rejects a correct null hypothesis (rejection probabilities or RP's hereinafter) can be greatly different when critical values based on first-order asymptotic approximations are used. The true and nominal coverage probabilities of confidence intervals for components of 18 can also be very different. Buchinsky (1995), De Angelis, et al. (1993), Dielman and Pfaffenberger (1984, 1988a, 1988b), and Monte Carlo results that are presented in this paper provide numerical evidence on the accuracy of first-order approximations. 1Research supported in part by NSF Grants SBR-9307677 and SBR-9617925. I thank Peter Bickel, Moshe Buchinsky, Oliver Linton, Paul Ruud, and Gene Savin for helpful comments and discussions. 1327

1328

JOEL L. HOROWITZ

This paper shows that the bootstrap provides asymptotic refinements to the RP's of t and x2 tests of hypotheses about 18in (1.1). As the sample size, n, increases, the differences between the true and nominal RP's converge to zero more rapidly with critical values obtained from the bootstrap than with critical values obtained from first-order asymptotic theory. It is well known that under suitable conditions the bootstrap provides asymptotic refinements to the RP's of tests and coverage probabilities of confidence intervals (see, e.g., Beran (1988); Hall (1986, 1992); Horowitz (1997)). However, the standard theory of the bootstrap does not apply to t and x2 statistics based on the LAD estimator. This theory uses a Taylor series to approximate the statistic of interest by a smooth function of sample moments that has an Edgeworth expansion. The LAD objective function is not smooth, however, and Taylor series methods cannot be used to approximate the LAD estimator by a smooth function of sample moments. Indeed, De Angelis, et al. (1993) have shown that the distribution of the LAD estimator has a nonstandard and very complicated asymptotic expansion. This paper solves these problems by smoothing the LAD objective function to make it differentiable. The resulting estimator will be called the smoothed LAD (SLAD) estimator. It is first-order asymptotically equivalent to the standard LAD estimator but has much simpler higher-order asymptotics. Use of the SLAD estimator greatly eases the task of obtaining asymptotic refinements and, thereby, makes it possible to obtain results that go well beyond those obtained in previous research. Previous research by De Angelis, et al. (1993) has shown that when U is independent of X and certain other conditions are satisfied, the erroi in the bootstrap approximation to the cumulative distribution function (CDF) of the LAD estimator is OP(n-2/5). Hahn (1995) showed consistency of a bootstrap approximation to the CDF without assuming independence of U and X. De Angelis, et al. and Hahn did not investigate the bootstrap's ability to correct the RP's of t and x2 tests based on the LAD estimator. Falk (1992) discusses bootstrap approximations to the distribution of a population median (no covariates). Janas (1993) considered a t test of a hypothesis about a population median. He showed that when a suitable version of the bootstrap is used to obtain the critical value, the difference between the true and nominal RP's of a symmetrical t test of a hypothesis about a population median is o(n - 7), where y < 1 but can be arbitrarily close to 1 if the populatioP density is sufficiently smooth. By contrast, first-order approximations make an error of size O(n-7). The bootstrap accounts for a term of size O(n-7) in the asymptotic expansion of the distribution of the test statistic, whereas first-order approximations ignore this term. This paper extends the results of previous research in three ways. First, it gives conditions under which the bootstrap provides asymptotic refinements to the RP's of t and x2 tests of hypotheses about 18in (1.1). Second, in contrast to De Angelis, et al. (1993), it is not assumed that U and X are independent. Any form of dependence is permitted as long as median(U IX = x) = 0 almost surely

1329

BOOTSTRAP METHODS

and mild regularity conditions are satisfied. Third, it is shown that the bootstrap also provides asymptotic refinements for t and x2 tests of hypotheses about 18 in the censored median regression model of Powell (1984). Under the conditions that are given here, the differences between the true and nominal RP's of symmetrical t and x2 tests with bootstrap critical values are o(n-7) for a suitable y satisfying 7/9 < y < 1. By contrast, the differences are O(n -7) with critical values based on first-order approximations. As in Janas (1993), the bootstrap accounts for a term of size O(n -7) in the asymptotic expansion of the t or xY2statistic, whereas first-order approximations ignore this term. The value of y depends on the smoothness of the conditional density of U at zero and can b'e arbitrarily close to 1 if the density is sufficiently smooth. Although this paper treats explicitly only the RP's of symmetrical t and x2 tests, the results also apply to coverage probabilities of symmetrical confidence intervals and, with suitable modifications, to equal-tailed and one-sided t tests and confidence intervals. In addition, the methods used here can easily be extended to show that the bootstrap provides asymptotic refinements for tests and confidence intervals based on smoothed versions of the quantile-regression estimator of Koenker and Bassett (1978) and the censored quantile-regression estimator of Powell (1986). Section 2 of the paper describes the SLAD estimator and gives its first-order asymptotic distribution. Section 3 describes the test statistics and procedures for obtaining bootstrap critical values. Section 4 presents theorems giving conditions under which the bootstrap provides asymptotic refinements to the RP's of symmetrical t and x2 tests. Section 4 also describes the extension to censored median regressions. Section 5 presents the results of a Monte Carlo investigation of the numerical performance of the bootstrap, and Section 6 gives concluding comments. The proofs of theorems are in the Appendix. 2. THE SMOOTHED LAD ESTIMATOR

This section describes the smoothed LAD estimator and establishes its asymptotic equivalence to the standard LAD estimator. Let {Yj,Xi: i = 1,... , n} be a random sample of (Y, X) in (1.1). The standard LAD estimator solves n

(2.1)

minimize: Hn(b) = n-1 XIY -Xibl beB

i=l

n

= n-1

(Yi -Xib)[2I(Yi

-Xib

> 0) - 1],

i=l1

where B is the parameter set and I(-) is the indicator function. Hn(b) has cusps and, therefore, is not differentiable at points b such that Y - Xib = 0 for some i. The SLAD estimator smooths these cusps by replacing the indicator function in Hn with a smooth function.

1330

JOEL L. HOROWITZ

To do this, let K be a bounded, differentiable function satisfying K(v) = 0 if v < -1 and K(v) = 1 if v 2 1. Additional requirements that K must satisfy are given in Section 4a. Let {h,} be a sequence of positive real numbers (bandwidths) that converges to zero as n -> oo.The SLAD estimator solves

(2.2)

(Y - XIb) 2K ()1I.

minimize:Hn(b) =n1 n

LhnJ

K is analogous to the integral of a kernel function for nonparametric estimation. K is not a kernel function itself. Let bn be a LAD estimator (a solution to (2.1)) and bn be a SLAD estimator (a solution to (2.2)). Intuition suggests that bn and bn are asymptotically equivalent if hn converges to zero sufficiently rapidly. Theorem 2.1 below shows that this intuition is correct. Regularity conditions for the theorem are given in Section 4a. They are stated in the form that is used to obtain this paper's main objective, which is to show that the bootstrap provides asymptotic refinements for tests based on the SLAD estimator. The regularity conditions are stronger than would be needed if the only objective were to prove that bn and bn are asymptotically equivalent. THEOREM

2.1: UnderAssumptions 1-6 of Section 4a, n1/2(bn - b) = op(l).

To state the asymptotic distribution of n1/2(bn - 13), let ff( Ix) denote the density of U in (1.1) conditional on X = x. Assume that f(O Ix) exists for almost all x. Define D = 2E[X'Xf(O IX)], and assume that D is nonsingular. It follows from Theorem (2.1) and asymptotic normality of the LAD estimator (se'e, e.g., Buchinsky (1995)) that n112(bn-13) --*dN(o,V), where V=D-1E(X'X)D-1. To obtain a consistent estimator of V, let KV1)(v)= dK(v)/dv. Define (2.3)

Dn(b) =2(nhn)

n~~~~~~ E3Xi'XiK(l)(YAhi).

It is not difficult to show that Dn(bn) --P D under the conditions given in Section 4a. E(X'X) can be estimated consistently by the sample average of X'X. However, the asymptotic expansion of the distribution of the SLAD t statistic has fewer terms and is easier to analyze if E(X'X) is estimated by the sample analog of the exact finite-sample variance of dHn(b)/8b at b =,8. This estimator is Tn(bn),where Tn(b) = n

EXIXi

/

2 [K

1

~LhnJ +2(

h

jKM1(

h

)

Under the conditions given in Section 4a, Tn(b,,)--P E(X'X). It follows that V is estimated consistently by Vn Dn(bn)1 Tn(bn)Dn(bn)1.

1331

BOOTSTRAP METHODS

3.

TESTING A HYPOTHESIS ABOUT

1

A. The Symmetricalt and Chi-SquareTests Let bni and f3i, respectively, be the ith components of bn and 18(i = 1,... , q). Let Vni be the (i, i) component of Vn.The t statistic for testing Ho 08i = 130i is t = n12(bni - n30i)/JK2Y2. If Ho is true, then t ,d N(O, 1). The symmetrical t test rejects Ho with asymptoticprobabilitya if ItI> Za /2, where Za /2 is the 1 - a/2 quantile of the N(O, 1) distribution. Now let R be an l x q matrix with l < q, and let c be an l X 1 vector of constants. Consider a test of the hypothesis Ho: R 1 = c. Assume that the matrix RD- E(X'X)D-1R' is nonsingular. Then under Ho, the statistic x2

=

c)

n(Rbn-c)'(RVnR')Y1(Rbn

is asymptotically chi-square distributed with l degrees of freedom. Ho is rejected with asymptotic probability a if x2 exceeds the 1 - a quantile of the chi-square distribution with l degrees of freedom. Section 4 gives conditions under which the bootstrap provides asymptotic refinements to the RP's of the symmetrical t and x2 tests. B. The BootstrapProcedure The bootstrap estimates the distribution of a statistic by treating the data as if they were the population. The bootstrap distribution of a statistic is the one induced by sampling the data randomly with replacement. The bootstrap critical value of a symmetrical t test or x2 test with nominal RP a is the 1 - a quantile of the bootstrap distribution of Itlor X2. The bootstrap distributions of Itl and x2 can be estimated with arbitrary accuracy by Monte Carlo simulation. To specify the Monte Carlo procedure, let the bootstrap sample be denoted by {Y7*,Xi*: i = 1,..., n}. Define the following bootstrap analogs of Hn(b), Dn(b), and Tn(b):

(b)n1 n Hn*

(Yi*-Xi*b) 2K

-1

h

n~~~~~~~~ Dn*(b) = (nhn)

EXiXi

K(l) h(

and T*(b)n

xiXi*

{[2K(y*

y+ *

- Xi*)

X, b

] yi* Xi*b

2

Let b* be a solution to (2.2) with Hn replaced by Hn*.Let Vn*ibe the (i, i) component of the matrix Dn* (b*)- 1Tn* (b*)Dn*(b1) .

1332

JOEL L. HOROWITZ

The Monte Carlo procedure for estimating the bootstrap critical value of the symmetrical t test with nominal RP a is as follows. The procedure for estimating the bootstrap critical value of x2 is similar. 1. Generate a bootstrap sample by sampling the estimation data randomly with replacement. 2. Compute the bootstrap t statistic for testing the hypothesis Ho: f3i= b"j, - b n)/( 112) where bn solves (2.2). The bootstrap t statistic is t*- n W2(b* where b* is the ith component of b*. 3. Estimate the bootstrap distribution of It*Iby the empirical distribution that is obtained by repeating steps 1 and 2 many times. The bootstrap critical value is estimated by the 1 - a quantile of this empirical distribution. 4.

MAIN RESULTS

This section presents theorems giving conditions under which the bootstrap provides asymptotic refinements to the RP's of symmetrical t and x2 tests based on the SLAD estimator. As in other applications (see, e.g., Beran (1988), Hall (1992)), the proof that the bootstrap provides asymptotic refinements is based on showing that the distributions of the test statistics and their bootstrap analogs have asymptotic expansions that are identical to sufficiently high order. The main technical problem that must be solved is establishing the existence of these expansions. This is done in Theorems 4.1 and 4.2. Given the existence of the expansions, it is a relatively easy matter to show that the use of bootstrap critical values provides asymptotic refinements to the RP's of symmetrical t and x2 tests. This is done in Theorem 4.3. A. Assumptions This section presents the assumptions under which it is proved that the bootstrap provides asymptotic refinements. Let r 2 4 be an even integer. Let K(')(v) = d'K(v)/dv'. Let F( Ix) denote the CDF of U conditional on X=x. Let f(. Ix) denote the conditional density of U with respect to Lebesgue measure whenever this function exists. The assumptions are as follows: is a random sample of (Y, X), where ASSUMPTION 1: {Y, Xi: i =1,...,n} Y = X,8 + U, X is a 1 x q vector of observed random variables, U is an unobsernmd random scalar, and 18 is a q X 1 constant vector. ASSUMPTION

2: 18 is an interiorpoint of B, which is a compact subset of Rq.

ASSUMPTION

3: X has bounded support, and E(X'X) is positive definite.

ASSUMPTION 4: (a) F(O Ix) = 0.50 for almost every x. (b) For all u in a neighborhoodof 0 and almost everyx, f(u Ix) exists, is bounded away from zero, and is r - 1 times continuously differentiablewith respect to u.

BOOTSTRAP METHODS

1333

5: (a) K(a) is bounded, K(v) = 0 if v < -1, and K(v) = 1 if v 2 1. (b) K is 4-times differentiableeverywhere,K(1)(v) is symmetricalabout v = 0, and K) (i = 1, ... , 4) is bounded and Lipschitz continuous on ( - oo,oo).(c) Let K(v) be a vector whose components are [2K(v) - 1] and its derivatives through order 3, vK(1)(v) and its derivativesthrough order 3, and [2K(v) - 1 + 2vK(1)(v)]2 and its = 1, there is a partition of [- 1, 1], first derivative. For any 06c10 satisfying 11011 -1 =a1
vVK1)(v)dv =

if i < r, CK (nonzero)

Ifi =r.

ASSUMPTION6: hn a n K, where 2/(2r + 1) < K < 1/3.

Assumptions 1-5(b) define the model and insure that 18is identified, n1/2(bn -,8) is asymptotically normal, and the Taylor series expansions used to obtain higher-order asymptotic approximations to t and x2 exist. The assumption that X has bounded support is not essential and can be dropped at the expense of more complex proofs. Assumption 5(c) is used to establish a modified form of the Cramer condition of Edgeworth analysis (Lemma 9 of the Appendix). Assumption 5(d) requires K(1) to be a "higher-order" kernel. The need for a higher-order K(1) is explained below. Functions K satisfying Assumption 5 can be constructed by integrating kernels given by Muller (1984). The requirement K < 1/3 in Assumption 6 insures convergence of Taylor series remainder terms that depend on fourth derivatives of Hn. The requirement K > 2/(2r + 1) is needed to control the asymptotic bias of bn. This bias adds terms of size O(nh2r) to the asymptotic expansions of Itl and x2 but not their bootstrap analogs. The remaining higher-order terms are O[(nhn)-1]. If K > 2/(2r + 1), the extra bias terms are o[(nhn)- 1] and, therefore, asymptotically negligible. The need for a higher-order K(1) can now be understood by observing that Assumption 6 cannot be satisfied if r = 2. The size of the asymptotic bias of bn is max[O(n01),0(hr )]. If Kr> 1, the asymptotic bias of bn is 0(n-1), which is the same as the size of the asymptotic bias of the unsmoothed LAD estimator. B. Theorems This section gives theorems that establish conditions under which the bootstrap provides asymptotic refinements for symmetrical t and x2 tests based on the SLAD estimator. The following additional notation is used. Let P and 4, respectively, denote the standard normal distribution and density functions. Let Pn*denote the bootstrap probability measure. This measure places mass 1/n at each data point (Y, Xi). The second and fourth cumulants of t can be approximated with accuracy 0[(nhn)-1] by using Taylor-series expansions that are

1334

JOEL L. HOROWITZ

described in the Appendix. Denote the approximate cumulants by the 2 X 1 vector v,* The second and fourth cumulants of t* conditional on the estimation sample can be approximated with accuracy O[(nhn)- 1] almost surely. Let vn*be the 2 x 1 vector containing the approximate bootstrap cumulants. The following theorem establishes the existence of Edgeworth-type expansions of the distributions of ItIand It*1. THEOREM 4.1: Let Assumptions 1-6 hold. Let v be an arbitrary2 X 1 vector. There is a function q(r, v) such that: (a) q(-, v) is a polynomial; (b) q(r, vn) and q(r, vn*)consist of terms whose sizes are O[(nhn)1] (almost surely in the case of q(7, vn*)); (C)

(4.1)

P(ItI< 7) = 2@(T)

-

1 + q(r,

vn)0(r)

+ o[(nhn)y]

unifornly over -, and (d)

Pn*(It*I< 7-) = 2@(7z)- 1 + q(7-, vn*)?>(z) + o (nhn) ]

uniformlyover r almost surely. The coefficients of r in q are functions of vn and vn*.These, in turn, are functions of asymptotic forms of moments of products of derivatives of Hn( 1), Dn(,1), and Tn(13)with respect to the components of 18.Because the number of such moments is very large, obtaining an analytic expression for q is not feasible. It is possible, however, to calculate the rates at which the moments converge to zero, and this is sufficient to prove the theorem. The proof takes place in two main steps. The first step is to show that t and t* can be approximated with asymptotically negligible error by functionals of derivatives of Hn( ,1), Dn( ,1), and Tn(,8) (or their bootstrap analogs in the case of t*). This is done in Propositions 1 and 2 of the Appendix. The second step is to show that the distributions of the approximations have asymptotic expansions through order (nhn)-l. This step is carried out using methods similar to those used to prove Theorems 5.5 and 5.6 of Hall (1992). Now consider the x2 test. Let x2* be the bootstrap version of the x2 statistic. The first two moments of x2 and x2* can be approximated through and denote the vectors of approximate moments. Let O[(nhn)-]. Let Fx, denote the chi-square distribution function with 1 degrees of freedom. The following theorem, which is a modified version of Theorem lb of Chandra and Ghosh (1979), gives conditions under which the distributions of x2 and x2* have Edgeworth expansions through O[(nhn)- 1]. 4.2: Let Assumptions 1-6 hold. Let v be an arbitrary2 X 1 vector. v* ) consist of terns whose sizes are O[(nhn)1] (almost surely in the case of q(&, vn*,)), THEOREM

Thereis a function qx(r, v) such that q(G, vn,) and q(r, (A2)

p(

X2


=8

dr A [11+

~,

7J vn)

F 7(t11[^81

1335

BOOTSTRAP METHODS

uniformlyover z, and

Pn 2* < z) =|

dl[l + q'y( , vn*x]F,(

)+o[n)]

uniformlyover z almost surely. The final theorem shows that the use of bootstrap critical values yields asymptotic refinements to the RP's of symmetrical t and x2 tests. Let t* denote the 1 - a quantile of the bootstrap distribution of It* 1.Let c* denote the 1 - a quantile of the bootstrap distribution of x2*. THEOREM4.3: Let Assumptions 1-6 hold. UnderHO: f3 = I30i' 0

(a)

t*I )=a +[(nh P(NW

)1]

If RD- 1E(X'X)D 1R' is nonsingular, then under Ho: R,8 =c, (b)

P(X2>c*)

= a+o[(nhnYl].

Theorem 4.3 implies that with bootstrap critical values, the RP's of symmetrical t and x2 tests based on the SLAD estimator are correct through O(n-7) under'the null hypothesis, where y 1 - K < 1 but can be arbitrarilyclose to 1 if ff Ix) has sufficiently many derivatives. In contrast, first-order asymptotic approximations make errors of size O(n- 7). This is because first-order approximations drop the terms q4 and qxFx in (4.1) and (4.2). C. CensoredMedian Regressions This section describes the extension of the foregoing results to the censored median regression model of Powell (1984). The model is (4.3)

Y= max(O,X,8 + U),

where X, 13, and U are as defined in (1.1). The censored LAD (CLAD) estimator of 13,bcnI solves n

minimize: n1 beB

i= 1

Yi - max(O,Xib)I

where B is the parameter set. Equivalently, bcn solves n

minimize: Hcn(b) -n1 beB

? {(Y i=l

Xib)[2I(Yj - Xib > 0) - 1] - YI

x I(Xib > 0).

Under regularity conditions, n112(bcn -8) -d N(O,Vc), where Vc= Dc TCDJ1, DC= 2E[X'Xf(O IX)I(X1 > 0)], and TC= E[X'XI(Xf > 0)] (Powell (1984)).

1336

JOEL L. HOROWITZ

Like the objective function of the LAD estimator, HCnhas cusps. The smoothed CLAD estimator (SCLAD) removes them by replacing the indicator functions in Hcn with smooth functions. The SCLAD estimator, bcW solves n

minimize: H n(b) -n 1 Egc(Yi i= 1

beB

Xi, h

where

gC(Y',xh ,b) = ((y -xb)[2K(

- 1] Y)K h

)

and K and hn are as in (2.2). The smoothed version of I(xb > 0) is K(xb/h - 2) instead of K(xb/h) for technical reasons relating to prevention of asymptotic bias. Under conditions stated below, n1/2(bcn - bcn) = op(1) as n >oo. To form t and x2 statistics based on bcn it is necessary to have consistent estimators of Dc and Tc.Define

) Dcn(b) = (nhn)Y

Xi'XiKl) (

)

I(Y >O).

It is not difficult to show that Dcn(bcn) --P Dc. Tc can be estimated consistently by the sample average of X'XI(Xbcn > 0) (Powell (1984)). As in SLAD estimation, however, the asymptotic expansion of the distribution of the t statistic is easier to analyze if Tc is estimated by the sample analog of the exact finite-sample variance of dH n(b)/8b at b = 8. This estimator is T (b where n

Tcn(b)= n-1 [9gc(Yi, Xi, hn, b)ldb][ dgc(Yi Xi, hn, b)ldb]'. i=l1

Vc is estimated consistently by Vcn Dcn(bcn) 1Tcn(bn)Dcn(bcn)

The formulae for t and x2 statistics for testing hypotheses about 18 in (4.3) are the same as in Section 3A but with Vn replaced by Vcn.The procedure for obtaining bootstrap critical values for these statistics is the same as in Section 3B but with Dn, Tn, D*, and TI* replaced with Dc n, and their bootstrap analogs. To establish the ability of the bootstrap to provide asymptotic refinements for t and X2 tests based on the SCLAD estimator, it is necessary to modify Assumptions 1 and 3 as follows: ASSUMPTION1': {Y, Xi: i = 1,...,n} is a random sample of (Y, X), where = Y max(O,X,8 + U), X is a 1 X q vector of observed random variables, U is an unobservedrandom scalar, and 18 is a q X 1 constant vector.

8)]

ASSUMPTION3': X has bounded support, P(X,8 = 0) = 0, and E[(X'X)I(Xb is positive definitefor some e > 0 and all b in a neighborhoodof 18.

>

BOOTSTRAP METHODS

1337

The following theorem shows that the SCLAD and CLAD estimators are asymptotically equivalent and that the bootstrap provides asymptotic refinements to the RP's of symmetrical t and X2 tests based on the SCLAD estimator. THEOREM

(a)

4.4: Let Assumptions 1', 2, 3', and 4-6 hold. Then

n1/2(bcn - bcn)= op(l)

as n

> oo.

Let t* and c*, respectively,denote the 1 - a quantiles of the bootstrapdistributions df the SCLADversionsof It*I and X*2. UnderHO:f80= I30i'

(b)

P(ItI t* )= a+ o (nhn

]

If RDc 1TcDc 1R' is nonsingular, then under Ho: R,B = c, (c)

1].

P(X2>c*)=a+o[(nhn

5.

MONTE CARLO EXPERIMENTS

This section describes the results of a Monte Carlo investigation of the numerical performance of the SLAD t test with bootstrap critical values. The numbers of experiments and replications per experiment are small because of the very long computing times they entail. Each experiment evaluates the RP of a symmetrical t test using asymptotic or bootstrap critical values. The nominal RP is 0.05. The hypothesis being tested is Ho: 81 = 1 in the model Y= 80 + ,811X+ U, where I80 and I81 are scalar parameters whose true values are (,80, ,81) = (1, 1) (so Ho is true), and X U[1, 5]. There are 3 different distributions of U. One is Student t with 3 degrees of freedom scaled to have a variance of 2. Another is Type 1 extreme value, which is skewed, centered and scaled to have median 0 and variance 2. The third is U = 0.25(1 + X)V, where V N(0, 1). This U is heteroskedastic. The smoothing function K is the integral of a fourth-order kernel for nonparametric density estimation (Muller (1984)). This function is (0 K(v

)0.5

if v< -1, + (105/64)[v

if IvI?<1,

-

(5/3)v3 + (7/5)v5 - (3/7)v7]

1 if v>1. K does not satisfy Assumption 5(b) because it has only two derivatives at v = ? 1. This problem can be overcome by smoothing K in neighborhoods of v = + 1, but doing so has no effect on the results of the experiments. The experiments with the SLAD estimator consisted of computing the RP of the symmetrical t test of Ho with bootstrap critical values. To provide a basis for evaluating the performance of the bootstrap, experiments were also carried

1338

JOEL L. HOROWITZ

out with the unsmoothed LAD estimator. These consisted of computing the RP of the symmetrical t test of Ho with the asymptotic critical value. The LAD variance was estimated by Dn(bnY1En[(1,X)'(1,X)]Dn(bn)1, where bn is the LAD estimator of (,Io, 81), Dn is as in (2.3), and EJ(n)is the sample average. Dn was computed using the 2nd-order kernel K2(v) = (15/16)(1 - v2)21(IvI < 1). Computing the SLAD and LAD t statistics requires choosing a bandwidth for each. Existing theory provides little guidance on how to do this for hypothesis tests. It is possible, however, to construct informal, rule-of-thumb bandwidths. The bandwidth ho that minimizes the asymptotic mean-square error of the LAD standard error can be obtained using methods described by Hall and Horowitz (1990). It converges at rate n-1 /5 and can be used with the LAD t statistic. The bandwidth h1 = ch0n-1110 for any c > 0 converges at rate n-3/10. This bandwidth satisfies Assumption 6 and can be used with the SLAD t statistic. For specificity, c = 0.5 is used in the following discussion but, as will be seen, the RP of the SLAD t test varies little over a wide range of c values. The asymptotic biases of the SLAD estimator based on h1 and the LAD estimator are both 0(n-1). There is no reason to believe that ho and h1 are in any sense optimal for hypothesis testing, but they provide rules of thumb that can be implemented by the plug-in method in applications. The experiments use ranges of bandwidths that include ho for the LAD t test and h1 for the SLAD t test. The experiments used a sample size of n = 50 and were carried out in GAUSS with GAUSS pseudo-random number generators. There were 500 replications per experiment with the SLAD estimator and 1000 with the LAD estimator. There were fewer replications in the SLAD experiments because of the long computing times required for Monte Carlo simulations with bootstrapping. Each experiment consisted of repeating the following steps 500 or 1000 times: A. Generate an estimation data set of size n = 50 by randomly sampling (Y, X) from the model under consideration. Obtain the SLAD or LAD estimate of (,8o, 81), and compute the t statistic for testing Ho: 81 = 1. Call its value ts if it is based on the SLAD estimator and tL if it is based on the LAD estimator. B. In experiments with ts, compute the bootstrap critical value by following steps 1-3 in Section 3B. Denote the bootstrap critical value of ts by t0*05. tO*05 was computed from 100 bootstrap samples. .C. Reject Ho based on ts if ItSI > t*05. Reject Ho based on tL if ItLI> 1.96, the asymptotic critical value. The results of the experiments are summarized in Figure 1, which shows the empirical RP's of the SLAD t test with bootstrap critical values (right-hand panels) and the LAD t test with the asymptotic critical value (left-hand panels) as functions of the bandwidth. The dashed lines are Bonferroni uniform 95% confidence bands for the RP's. These were computed by connecting (1 - 0.05/m) pointwise confidence intervals, where m is the number of points at which the RP was estimated. In all of the experiments, the empirical RP of the LAD test is below the nominal RP when the bandwidth is near ho. The empirical RP is sensitive to the bandwidth and can be made equal to the nominal RP by

1339

BOOTSTRAP METHODS

I''1

I

:1-

.3

_

0

LAD

LA

LAD

0-~

with

>X-

with

LAD with

~

T(3)

Extromo

-

-

-

*en~~~~~~~t

Voluo

Hatoskedastle

~

Drstribion of U

Distribution FIGURE

Distribution of U

of U

~~~I

I

I

_I

-

SLA

/--

SLAD

experiments.

with

with

_

Ihi-

DLAO wih

rm

VoluMn

Heteroskedstic

Ctraoxtpon oe

X

U

---

hi

U:

0

Ditribution of U

T(3) _ _ _ _ _ _ _ _ _ _ -

Ditribution of U

6 c.N cm RI-

.r.

I

Carlo

c

---------

I.-Results of of the the Monte

UqI CWQd

7

1340

JOEL L. HOROWITZ

choosing an appropriatebandwidth.It is beyond the scope of this paper, however,to investigatewhethera useful estimatorof the appropriatebandwidth can be found. In contrastto the LAD case, the empiricalRP's of the SLAD test with bootstrapcriticalvalues are close to the nominalRP over a wide range of bandwidths.Thus,use of the SLAD test reducesthe importanceof choosingthe "correct"bandwidth.Moreover,the SLAD test performswell with the rule-ofthumbbandwidthhl. This is an importantpracticaladvantageof the SLAD test. 6.

CONCLUSIONS

This paper has shown how the bootstrapcan be used to obtain asymptotic refinementsfor tests of hypotheses about the parametersof uncensored and censoredlinear median regressionmodels with or withoutheteroskedasticityof unknownform.The method is based on smoothingthe objectivefunctionof the relevantestimator.This approachcontrastswith previousresearchon bootstrap methods for median regressions,which has achievedless general results under more restrictiveassumptionsby smoothingthe data instead of the estimator. Department of Economics, University of Iowa, Iowa City, L4 52242, U.S.A.; [email protected]; http:// www.biz.uiowa.edu/faculty / horowitz receivedSeptember, Manuscript 1996;final revisionreceivedOctober,1997. APPENDIX This Appendix provides proofs of Theorems 4.1-4.4. Assumptions 1-6 hold unless otherwise stated. Define Ui = Yi - Xi ,3 and

Gn(b) n1'

{(Yi -Xib)[2K(

1h

)-1]-iui}.

The SLAD estimator minimizes both Hn(b)and Gn(b)over b E B. G,, is used for the proofs because it is a sum of bounded terms. Let 11-11denote the Euclidean norm. Let X(j) denote the jth component of X. For b eB, define G(b) =E[IY-XbI - IUI]and n Gn(b) =n-l

E (IYi-Xibl -lUil). i=1

The proof is presented in two steps. The first is to approximate t and t* by functionals of derivatives of Hn,Dn,Tn, and their bootstrap analogs. The second is to show that the distributions of the approximations to t and t* have asymptotic expansions that are almost surely identical through O[(nhn) - 1]-

A. Step1: Approximating t andt* LEMMA 1:

sup IGn(b) - G(b) I< o(n-112

heB

log n) + 2h

a.s.

1341

BOOTSTRAP METHODS

PROOF:By Lemma 22 of Nolan and Pollard (1987) and Theorem 2.37 of Pollard (1984), o(n-12 log n) a.s. uniformly over b E B. Also,

IGn(b)- G(b)I =

Gn(b)-Gn(b)

=2n-

K(

,(Yi-Xib) i=1iL

h

)-I(Yj-Xb

n

> O)]

The summand differs from zero only if IYi- XibI < hn. Therefore, n

IGn(b) - Gn(b)l < 2n_1 E iY -XjbII(IY -XjbI < hn) < 2hn. i=l1

Q.E.D.

The lemma now follows from the triangle inequality. LEMMA 2: Given any r > 0, II bn -

/11< r a.s. for all sufficientlylarge n.

PROOF: Let Nr= {b E B: lb - , 11> r}. By Assumptions 3 and 4, ,3 uniquely minimizes G(b) over B. Therefore, G(b) > G( ,8) + 8 for all b EeNr and some 8 > 0. By Lemma 1 and hn -O 0, there is a finite no such that Gn(b)> Gn(8) + 8/2> Gn(/) almost surely for all b E Nr if n > nO. But Q.E.D. G,,(bn) < Gn( /3). Therefore, bn 0 Nr almost surely if n > nO.

For i,j,k,l=1,...,q, define Gnj(b)=dGn(b)/dbj, Gnijk(b) = Gnjj(b)=d2G,j(b)/db8dbj, 93Gn(b)/9bj01,9bk, and Gnijkl(b)= 94G,Z(b)/dbidbjdbkdbl. Also, define Dji(b) =dDn(b)1db, Dnij(b) = d2Dn(b)/8 bj,9bj, and TjI(b) = 9Tn(b)/d bj. LEMMA 3: For all i, j, k, 1= 1. (a)

q, the following relations hold almost surely as n oo:

sup Gni(b) -EGni(b)I

=o[(log

n)/nl/2],

beB

(b)

sup Gnij(b)-EGnij(b)I = o[(log n)/(nhn)1/2], beB

(c)

SUpIGnijkk(b) -EG, ijk(b) = o [(log n)/(nh3)/]

(d)

sup IGnijkl(b) -EGnijkl(b)I =o[(logn)/(nh')1/2]

(e)

beB sup

(f)

beB

(g) (h)

=o[(log n)/(nhn)1/2],

IDn(b) -EDn(b)

sup IDni(b)-EDni(b)

beB

sup

=o[(log n)/(nh3)1/2], =o [(log n) /(nh')

Dn ij(b) -EDn ij (b)

sup ITn(b) -ETJ(b)

2

=o[(log

],

n)/nl/2]

beB

(i)

sup Tni(b) -ETni(b)I =o[(log n)/(nhn)1/2], beB

where (e)-(i) apply to the individual components of the matricesDn, Dni, Dnij, Tn, and Tni. In addition, for all i, j, k, l = 1. q, (j)

EG,i( /3) = 2[(1

-

r)/r!]CKhrE[X(i)f(r-l)(OIX)]

+ o(hr),

n

(k)

n112Gnj(3) =

n - 1/2

X1(j)[2I(U > 0) -1] + OP(n1/2h + h/2), i=l1

(1)

EGnij( /3) = 2E[X(i)X(j)f(OIX)]

(m)

EGnijk(b), EGnijkl(b), EDn(b), EDnf(b), EDnij(b), ET1,(b), and ETni(b)

+ O(hr),

are 0(1) as n oo for all b in a neighborhoodof /3.

1342

JOEL L. HOROWITZ

PROOF:Parts (a)-(i) are proved by using Lemmas 2.14 of Pakes and Pollard (1989) and 22 of Nolan and Pollard (1987) to show that the summands of the G, D, and T functions form Euclidean classes and then applying Theorem 2.37 of Pollard (1984). Parts (j)-(m) are proved using methods like those used to obtain the asymptotic means and variances of kernel density estimators. Q.E.D. Define SfG to be a vector containing the unique components of Gi(il3), G1ij( /3), GflJk( ,3), and q). Order the components of SnG so that the first q are the G"j(/3). Gnijkl(/3) (i, j, k, 1 = 1. LEMMA4: Let SG = plimn SnG. There is a function AP(SnG) taking values in gWq such that = 0 and bn - /3= AP(SnG) + o[1/(n3l2h )] a.s. as n -* oo. -O

A/3(SG)

PROOF:Define An=bb - , and 8ni =b i - 8i3 (i= 1,...,q). Let Gn.(/3) be the vector whose components are the unique components of Gni(,/3) (i =1. q). For fixed j, k, and 1, define Gn.j( ,/), Gn.jk( ,/), and Gn.jkl( /3), respectively, to be the q-dimensional vectors whose components are G,ij((/3),Gnijk(/3), and GnIjkl(/3) (i = 1,.q). Let Qn be the matrix whose (i,j) element is Gn1j(,/). By Lemma 2, bn satisfies the first-order condition Gn.(bn) = 0 almost surely for all sufficiently large n. By Assumptions 3-4 and Lemma 3, Qn( /3) has an inverse almost surely for all sufficiently large n. Therefore, a Taylor series expansion of Gn.(bn) = 0 about bn = /3 yields (Al)

(bn-,B)

+ (1/2)Gn.jk(/3)8nj8nk

-Qn1[Gn.(/3) +(1/6)Gn

.jkl(

,B)Anj8nk6nl

a.s.

+Rn]

for all sufficiently large n, where the summation convention is used, Rn=(1/6)[Gn.jkl(bn)Gnjkl /)]8nj8nk 8ni, and bnis between bn and /3. Arguments similar to those used to prove Lemma 3 show that E[Gn .jkl(b) - Gn.jkl(/)] = O(b - /3) for b in a neighborhood of /3. This result and Lemma 3(d) imply that ? IIRnil
(log n)/(nh')'/1]

+ O(libn-

lI)}lbn-_ 113 a.s.

Given any v > 0 and c > 0, suppose that II,An II< cn -1/2 + v. Then it follows from Lemma 3 that the right-hand side of (Al) is less than cn -1/2 + v almost surely for all sufficiently large n. In addition, Lemma 3(b) and Assumptions 3-4 imply that the consistent solution to Gn.(b) = 0 is almost surely unique for all sufficiently large n. Therefore, application of the Brouwer fixed point theorem to the right-hand side of (Al) shows that for any c > 0, v > 0, (A2)

lib -/311
a.s.

for all sufficiently large n. Application of the implicit function theorem to (Al) shows that there is a differentiable function A3 such that A/ (SG) = 0 and (A3)

a.s.

(bn-/l3)=Ap(SnG+Rn),

where Rn is a vector such that dim(Rn) = dim(SnG), Rn forms the first q components of Rn, and the remaining components of Rn are 0. Application of the mean value theorem to (A3) combined with (A2) shows that

(A4)

(bn-/3) =Ap(SnG)+O[(logn)(n4ht51/2

n3v] a.s.

for any v > 0. The lemma follows from Assumption 6 by making v sufficiently small.

Q.E.D.

PROOF OF THEOREM 2.1: It follows from Lemma 3 that Qn-* D almost surely. Therefore, by (Al), (A2) and a further application of Lemma 3, n (A5) n112(bn)=D- ln- 1/2 X(jM[2I(Uj> O)-1] + o&(l). i=l

(A5) is the Bahadurrepresentationof the LAD estimator.

Q.E.D.

1343

BOOTSTRAP METHODS

Let Sn denote the vectorconsistingof the uniquecomponentsof SnG,D,,(/3), Dnif(3), Dni( /3), Tn(38), and Tni(/3). LEMMA5: For each i = 1. q, there is a real-valuedfunction Avi(Sn) such that nKi/2- Avi(Sn) + 4n,where n=o[(nhn)-1] almost surely.

PROOF:ExpandDn(bn)and Tn(bn)in Taylorseries about bn= /3 throughorders llbn- /8112and IIbn- /3I1,respectively, and use (A4) to obtain (A6) (A6)

ni2- =Avi(SnX,SlG Vnli2

+ cvf)n+ o[(nhf)F

] a.s.

for a suitable differentiable function Avi, where c,, theorem to (A6).

=

o[(nhn)1].

Now apply the mean value Q.E.D.

PROPOSITION1: Define A(Sn) = Ap(SnG)/Avj(Sn). Then

lim sup (nhn){P(t
noo

z

-

P[n1/2A(Sn)
=

0.

PROOF:This is proved by using a version of the delta method. See Hall (1992, p. 76).

Q.E.D.

Let En denote the expectation with respect to Pn*.Define Gn* (b) by replacing (Yi,Xi) with (Yi*, Xi*) in the definition of Gn(b). LEMMA6: For any b EB, define U1* = Y*- X* b and n Wn(b) = n

1

[(Ub*i/hn)dg(Xi*

)f(Ub*i/hn)

- En (Ub*/hn)dg(X*

)f(Ub*/hn)],

i=l1

where g is bounded for bounded values of its argument, d = 0 or 1, and f is a bounded, Lipschitz continuous function of bounded variation with support [-1,1]. (a) Define en = [(hn/n)log n]1/2. There is a finite C0 > 0 such that for all C > C0 and any -y2 0

oo n --*

I> Cn) = 0 (nh )'P,* (sup IWn(b)

a-s- (P).

b eB

(b) Define en = [(logn)/n]1/2.

There is a finite C0 > 0 such that for all C > C0 and any -y2 0

lim (nhn)'Pn*(sup Gn*i(b)-EnGni(b)I > Cn) =0

oo n --*

beB

a.s. (P)

/

and lim (nhn)'Pn*( sup | Tn* (b)0EnTn (b) I >

oo n --*

(c) For any -y? 0 and

Cen

=

a. s. (P).

beB 7 >

0,

lim (nhn)yPn* (sup IGn*(b)-Gn(b)I > 7 =0.

oo n --*

beB

PROOF:Only part (a) is proved. The proofs of parts (b) and (c) are similar. Partition B into subsets {Bj: j = 1. J} such that Ilb1- b211< en2 whenever b1 and b2 are in the same subset. For

1344 each j

JOEL L. HOROWITZ =

1.

(A7)

J, let bj be a point in Bj. Observe that J= Q(&-2q). Then PI*,sup I n(b) I> Cst bE:B

= P*

j=j J

U

1

)( sup IKn(b) I> C&t} bEbBBj

J

<

E Pn`

sup IWn(b ) I > C&n) b E:Bj

j=l

There is an M < co such that sup{[n(b): b E B1}< 2M(log ciently large n,

n)/n

+

Therefore, for all suffiIJWJ'(bj)I.

( Wn(bj)I> C/2). Pn* sup IW4(b)I> Cn) < Pn*

(A8)

By using Lemma 22 of Nolan and Pollard (1987) and Theorem 2.37 of Pollard (1984), it can be shown that En[nWn(bj)2] < clhn almost surely (P) for some cl
P,I

I> C&n/2)< 2exp(-Cd log n) = (IJWn(bj)

2n-Cd

for some d > 0 and all sufficiently large n. Combining (A7)-(A9) yields

(nh )P~(

sup In(b)l > Ce) < 2(nh

)Yn-CdO(

2q) -o(1)

as n -* oofor all sufficientlylargeC. LEMMA7: For any y > 0 and

Q.E.D.

e>0

e) = lim (nh) P,* (IIbn-bnII> n- oo PROOF:

For

a.s. (P).

This is the bootstrap version of Lemma 2 and is proved using similar arguments.

i,j,k,l=

1,..,q,

Q.E.D.

= define Gn*j(b)=aGn*(b)/dbj, Gn*ij(b)=d2Gn*(b)/dbidbj,Gn*ijk(b)

dbk, Gn*ijkl(b)= a4Gn*(b)/dbidbj dbk dbl, Dn*i(b)= dDn*(b)/dbi,

Dn*ij(b)= = aTn*(b)/db,.The bootstrapversionof Lemma3 is as follows: d2Dn*(b)/ab abj, and Tn*i(b) d3Gn*(b)/dbjabj

LEMMA 8: For all i, j, k, 1 = 1, . . ., q, any y > 0, and all sufficiently large C > 0, limn o(nhn)P,n*(An) = 0 almost surely (P), whereAn is any of: - E Gn*i (a) sup Gn*iG(b) (b)I > C[(log n)/n1/2], beB

(b)

sup Gn*ijk (b)-EnGn*ij(b) I> C[(log n)/(nh)12], beB

(c)

sup IGn*ijk( b)-EflG*ij1k (b)> C[(log n)/(

(d)

supIG,lIkl(b)-E11Gn*ikl(b)I| > C[(log n)/(nh51)1/2],X

(e)

nhn )1],

supIDl( b)-EnDn*(b)I|> C[(log n)/(nhn)'

],

beB

(f)

sup ID,n*(b)-EnDn' (b) > C[(log n)/(nh

n)1/2],

beB

(g)

sup Dn*ij(b)-EnD,*ij(b)I > C[(log

n)/(nh)12],

beB

(h)

sup Tn* (b) -EnTI* (b) I > C[(log

n)/n'1/2],

beB

(i)

supITni(b) - ET,i(b) I= o[(log n)/(nh,)1/2], beB

1345

BOOTSTRAP METHODS

In addition, and (e)-(i) apply to the individual components of the matrices D*, D*i, D*ij, T*, and Tn*I. 1. q, foralli,j,k,l= EnG*i(bn) =0

(j)

],

withprobability 1-o[(nhn)

E,, G*il EnD* (b), E D*j(b), E D*ij(b), EnGnij(b), EnG*ikb and are almost 0(1) (P) as n -*0o for all b in a neighborT*(b), surely En EnT*i(b) hood of 3.

(k)

PROOF:Parts(a)-(i) are immediateconsequencesof Lemma6. Part(j) is the first-ordercondition Q.E.D. for the bootstrapestimationproblem.Part(k) followsfromLemma3. Define S*G and S* as SnG and Sn except with (Yi, Xi) replaced by (Yi*, X*) and by bn.

P3replaced

PROPOSITION2: Let A be the function defined in Proposition 1.

a.s.(P).

lim sup (nhn){P* (t*
noo

z

PROOF:This is the bootstrapversionof Proposition1. It is provedusingthe same argumentsthat are used to proveLemmas4-5 and Proposition1 but with SnG,Sn, bn,and /3, respectively,replaced Q.E.D.

by S*G, S*, b*, and bn.

B. Step 2: AsymptoticExpansions

For h > 0, let W(u, x, h) be a vector whose componentsare terms of the form g(x)Kj(u/h), whereg(x) is the productof (not necessarilydistinct)componentsof x that maybe differentin each use of g, and Kj is the jth componentof the vector K definedin Assumption5. LEMMA9: Let r be a vectorwith the same dimension as W. Define fw(r, h) = E{exp[ tr' W(X, U, h)]} where =- ( 1)1/2. For any e > 0, some C > 0, all r satisfying hjrjj> e, and all sufficiently small h, Iiw (,h)<

1 -Ch.

PROOF:Let r index components of K and W. Each component of K satisfies IKr(v)l = 0 or 1 if IvI2 1. Let 8r-= K,(v) if v < -1 and 5,r= Kr(v) if v 2 1. Then using the summation convention

(withthe sums inside the exponents) fw(r, h) =

f exp[vtrg,(x)K,(u/h)]f(u1x)

=Aj(h)

dudP(x)

+A2(h),

where

A1(h) = E{F(-hIX)exp[

tsrg,(X)87-] + [1 -F(hIX)]exp[

trg,(X)8,

]},

and A2(h) =

f (fhexp[Lrrg,(x)K,(u/h)]f(ulx)du}

dP(x).

ConsiderA1(h). 1A1(h)I< EIA1(h,X)I, where A1(h, x) = F( -hlX)exp[ ltrg,(X) 5- ] + [1 - F(hIX)]exp[ trg,(X)8

t].

JOEL L. HOROWITZ

1346 Let

r =,+

if

r+

= - 67-= 1. Note that

8r- otherwise.

r+,=

Therefore,

IAj(h, x)I = IF(-hIX)exp[ ur,g,(X)8r] + [1 - F(hIX)]exp[

< 1 -F(hlx) +F(-hlx) = 1 - 2hf(Olx) - (1/2)h2[f(1)(h1Ix)

-

tr,gr(X)8r]

-f(1)(h2lx)],

whereh, and h2 are between0 and h, and the last line is obtainedby a Taylorseries expansion.Let Ef(OIX)= Cl. By Assumption4(b), C1> 0 and EIf(1)(h1IX)-f(1)(h2IX)I < M for some finite M and all sufficientlysmall h. Therefore, IAj(h) I < El Aj(h, X) I < 1

-

Clh

for all sufficientlysmall h. Now considerA2(h). By a changeof variables A2(h) = hf

(f '

exp[urgj(x)K4(; )]f(h; Ix)d'

dP(x).

}

Given e > 0, choose h sufficiently small that

If(h' Ix)-f(OIx) l d'dP(x)

<

ef1 f(OIx)d4dP(x)

=

2C1.

Then fw(r, h) I< 1-hC1(1-2e)

(Al0)

+ IA3(r, h)I

for all r, e > 0, and sufficientlysmall h > 0, where A3(0, h) = h (f

exp[ trg,(x)Kr( )]f(OIx)d'

}

dP(x).

Since g&(x)= 0 for every r only if x = 0 and P(X= 0) < 1, there are 7j> 0 and yi < 1 such that f(Olx) dP(x) = y7Cj.

2f

IlxII<Xi

Suppose,as will be provedpresently,that for some C2< 1, (All)

sup

f

lexp[trgr(x)Kr( )]d I= C2

uniformly over x such that lix I2I? -. Then for IIrII

1A3(0, h) I
(A12) where

?

72 =

[Yl + (1 -

Y1)C2] < 1-

Combining (AlO) with (A12) yields

sup lqfw(Gr,h)l
1-

72)

Ch

IIrT|> ?

for all sufficientlysmall h > 0 and e > 0, therebyestablishingthe lemma. To prove (All), define t = l1-Il.Fix r/IIrll and x with lixll # 0. Using the summation convention, - 1 = a, < < aL = 1 be a partitionof [- 1,] that satisfies Assumption 5c when Or= gr(x). Then

define ff ) = Trgr(X)Kr( f )/1111-Let

L

d

-1 expI qf(

1J aL 1=2 a1-1

It sufficesto provethat sup (a, -a a,

Itd>

jlj

exp[ttf(;)]d|

al 1

?C3

1347

BOOTSTRAP METHODS

for any 8 > 0 and some C3 < 1 that does not depend on x or r/1 r II.To do this, make the change of

and set v(e) = 1/{df[ 2(e)]/d }. Then

variablese =f()

exp[ttfQ)] d;

{.*(t-a,

=Jf(al) edev(e(()de.

al-

(a,-,1)

The right-handintegralcan be approximatedarbitrarilyaccuratelyby replacingv(-) with a step function.Therefore,it is enoughto provethat sup IJ e'tt de < (a2-1a)C3 Itl>e a1

for all a, < a2 and some C3 < 1 that does not dependon a1 or a2. But

f

sin2[0.5t(a2 - a)]

<

a2

[0.5t(a2-1)] t)/t]. The proofis completedby settingC3= inftI>Y[(sin2

Q.E.D.

Define W* as in Lemma 9 except with 13 replaced by b,. Define t4(6r, hn) = E,{exp[it' W*(U, X, hn)]}.The bootstrapversionof Lemma9 is as follows. LEMMA10: For any 8

>

0 and c > 0, some C* > 0, all T satisfying ? < IirII< n', and all sufficiently

largen, Iq (T, hn)I < 1 - C*hn, almostsurely(P). PROOF:Let BnT= {i: e < II II< nc}. Then

sup I4 (r, hn)I< sup Iiw(rwhn) I+ sup Iqf*(T,hn)-ipw(T h E-BnT ||T11E IIT 11 -BnT

IIT 11eBn,

By argumentslike those used to prove Lemma 6 together with the Borel-CantelliLemma, Iq4(-, hn) - qw(r, hj)I = o(hn) a.s. uniformlyover r E B,,,. Lemma10 followsby choosingC as in Q.E.D. Lemma9 and C* so that that C < C* < 1. Let Wn, be a column-vectorconsistingof the unique componentsof n'/2[Gni(1) - EGj( 13)] (i= 1. q) and n1/2[Tn(13)-ETn(3)]. Let Wn2be a column-vectorconsistingof the unique componentsof (nh )1/2[Gnij( ) - EG1ij(13)],(nh' )1/2[Gljk( 13)-EGnijk(13)],(nh5)1/2[Gfljkl( ,3) - EGfnIjkl(13)],(nh )1/2[Dn(13)- EDn(13)], (nh3)1/2[Dni(,) - EDni(13)], (nh5)1/2[Dnij(13)Define WJ' q). Set W W=[Jn4l,Wn'2]'. 13)](i,j,k,l=1. EDnjj(13)],and (nh and Wn*2 similarlyexceptwith (Y, Xi) replacedby (Yi*,Xi*) and 13replacedby bn.Orderthe Wn*l, componentsof Sn and Sn*conformablywith those of Wnand Wn*.Let V1 be the covariancematrix Let wn1,wn2, relativeto Pn*h. of [W1,Jn'J'2/hn]'and J7* be the covariancematrixof [Wn*4', Wn*2/hn]' These and Wn*2. respectively,be the summandsof the componentsof Wnl,Wn2,Wn*4, wn*l,and wn*2, have the forms gj(X)Kj(U/h) and gj(X)Kj(Un/h), where Un==Y-Xbn. For any 7= (r,T r)' conformablewith (wn1,wn2)',define (A13)

p1(r) =-

(A14)

P2(-) =-(t/6)){E[(Gwn

(A15)

p3()

=

t1(6hn)]E[(r'

t/(2h

)3],

1)3] + (31hn)E[(rTwnj)(Xr'

2]},

2Wn2)]

and (A16)

p4(7r)= [1/(24h )]E[(,r )4'

(/2)h7'[rWn

2)]32]}2

1348

JOEL L. HOROWITZ

Define pi*(v) (i= 1. 4) by replacing w,, with wI* and E with En in (A13)-(A16). Let transformsare (i = 1,...,4) be the signedmeasureswhose Fourier-Stieltjes (A17)

f exp(ir'( ) d7i()

=

i

exp( - 0.5r'Vnj ) Pi(')-

For anyset a Define 7,i (i = 1. 4) by usingVn*and pi*in place of V,,and pi. Let dw = dim(WJ'). in Rdw,let da denote the boundaryof a and (da)& denote the set of all points whose distance from da does not exceed e. Let P denote probabilitymeasureunder the N(O,V') distribution. Define OvP*similarly. LEMMA11: Let A denote a class of Borel sets in sup

J'dw

that satisfy

exp(-0.511 4112)d4 = O(e)

f

aEA (da)'

as e?O+.

Then

sup |P(W, (Ea) - Pv( a) - (nh,,)

1/27T,( a) -

n-

1/21T2( )

caEA

- (h,,/n)1/2 n3(a)

) - 1,

- 17T4(a) =o[(nh

- (nhn)

and

sup |P* (Wa*)-a

v* (a )

- (nh,F)-1/2 r7T(a) -

n- 1/27*

(a)

caEA

a.s. (P).

- (nhn)'i*(a)l =o[(nh)T4] (h,In )1/273* (ae)

PROOF: This is a slightlymodifiedversionof Theorem5.8 of Hall (1992)and is provedusingthe Q.E.D. same argumentsafterreplacingHall'sLemma5.6 withLemmas9 and 10 above. PROOF OFTHEOREM 4.1: Onlyparts(a), (c), and the part of (b) pertainingto q(r, vn)are proved here. The proofsof the remainingpartsare similar.To begin, invert(A17) to obtain (A18)

7Ti(4)

= ki()

V,(0)

where for each n and i, zrnj-i)is a multivariatepolynomial,and 4v, is the multivariatenormal densitywith mean 0 and covariancematrixVn.Let Sn(Wn)be the mappingfrom WJ'to Sn. Define t(Wn)- (nh,,)112A[S,,(WJ)]. By Proposition1 it sufficesto considerP(t < r). Define a1 = (nhn)-1/2 a2 = n-112, a3 = (hn/n)1/2, and a4 = (nhn)-1. By Lemma 11 and (A18)

(A19)

P(

/TI)

d[v,,(~ ?)+ Eaijni(4)4v()]

uniformlyover r. Orderthe componentsof

(

o+[(nhn)]

and Wnso that the firstcomponentscorrespondwith'

(nh, )1/2[G,i( /3) - EG,,i(l3)], where i is the component of ,3 for which t is the t statistic. Let

4

denote the vector consistingof all componentsof 4 except the first, (1. Changevariablesin the integralof (A19) so that the variableof integrationis (t, 4')', therebyobtaining (A20)

P(i
difd(J[(fi,(),(] 4

X (nv,[ hahn[)k[ t, ?

o[(nhnyKl]

), ( ]

?

E

7T

U,

),

l

g)]

1349

BOOTSTRAP METHODS

uniformly over r, where J() is the inverse Jacobian term associated with the change of variables. Taylor series expansions in powers of n1 of the terms involving ( 4) in (A20) yield 5

(A21)

P(
r)

= ?(T)

E

+

cnAq(T)O(r) + o[(nhn)

]

i=l1

_Gn(T) + o[(nhn)]

uniformly over T, where P and 4, respectively, are the univariate standard normal distribution and density functions, the 4q's are polynomial functions of one variable, cn1 = n- /2, Cn2=(nhn) Let qfi and fG, respectively, denote the cn3 = hnn-1/2, cn4 = (nh3/2)-1, and cn5 = (nhn)'. characteristic functions of the distributions of t and Gn. Then If() - G(r)j = o[(nhn)']. A Taylor series expansion shows that t in (A21) can be replaced by a multivariate polynomial in components of Sn - E(Sn). The cumulants through order 4 of this polynomial may be approximated through O[(nhn)- 1] using standard Taylor series methods of kernel estimation. Let kni denote the approximate jth cumulant. Expressing qfi in terms of the approximate cumulants yields fT(k) = frF() + o[(nhn)-] uniformly over r, where q(T)

= [exp( - r2/2)] x {1 + irknl

+ (1/2)(ir)2(kn2

-

1) + (1/6)(ir)3kn3

+ (1/2)[(ir)knl

+(1/24)(ir)4kn4

+ (1/6)(ir)3kn3]2}.

Setting\IfG = qfi,taking the inverse Fourier transform of the result, and setting P(Itl < r) - P(t < - r) yields (4.1) with q(vn,Xr) = -T[k 21 + (kn2 -1) +(1/36)k23

+ (1/12)(4knlkn3 + k 4)(2

(4

10r2

-

P(t < r)

-3)

+ 15)].

A straightforward but lengthy calculation shows that kn1, kn1kn3,and kn3 are o[(nhn)-], kn2 - 1 and kn4 are 0[(nhn)-

=

whereas Q.E.D.

]I

PROOFOFTHEOREM 4.2: Under Ho, c = R,3, so x2 = (nhn)(bn

-

3 )'R'(RVn,R')

1

R(bn-

13).

By arguments similar to those used to prove Propositions 1 and 2 followed by a Taylor series expansion, there is a multivariate polynomial Ax such that p( x2

< z)-P[(nhn)

Ax(Sn)

< z] = o[(nhn)F]

uniformly over z and lim sup(nhn){Pn*,(X2*
a.s.(P).

z

Set 7(Wn)= (nhn)Ax[Sn(Wn)]. By arguments similar to those used to obtain (A19), P

)d

dZ[PV()

+

Eairni

(

))v,(,

)]

+o[(nhn)

1].

Now transform to polar coordinates and proceed as in the proof of Theorem lb of Chandra and Ghosh (1979). A similar argument applies to P( X2* < Z). Q.E.D.

1350

JOEL L. HOROWITZ

PROOF OF THEOREM4.3: Only part (a) is proved here. The proof of part (b) is similar. Let ta and t*, respectively, denote the exact and bootstrap critical values of the symmetrical t test with RP a. Let k* denote the bootstrap version of knt (i = 2 or 4). This is obtained from kni by replacing 13 with bn and expected values with sample averages. By Theorem (4.1),

IP(Itl

>

t*)-

at < sup IP(Itl> r) -Pn* (It* I> r) I

S

sup I[q(G,vn)-q(G, vn*)]0(r)I+ o[(nh- )] Ir

=

n-kn2)

+ O(kn*4 -n4)-

Now use methods similar to those used in proving Lemma 3 to show that k*2 - kn2 and k*4 -n4

are o[(nh) 1 a.s,

QE.D.

PROOFOFTHEOREM4.4: Repeat the proofs of Lemmas 1-11 and Theorems 4.1-4.3 with HCn(b) in place of Hn(b) and Assumptions 1' and 3' in place of 1 and 3. Q.E.D.

REFERENCES BASSETr, G., ANDR. KOENKER (1978): "Asymptotic Theory of Least Absolute Error Regression,"

Joumal of the American StatisticalAssociation, 73, 618-621. R. (1988): "Prepivoting Test Statistics:- A Bootstrap View of Asymptotic Refinements," Joumal of the American StatisticalAssociation, 83, 687-697. BLOOMFIELD,P., AND W. L. STEIGER (1983): Least Absolute Deviations: Theory, Applications, and Algorithms. Boston: Birkhauser. BUCHINSKY, M. (1995): "Estimating the Asymptotic Covariance Matrix for Quantile Regression Models: A Monte Carlo Study," Joumal of Econometrics, 68, 303-338. T. K., AND J. K. GHOSH(1979): "Valid Asymptotic Expansions for the Likelihood Ratio CHANDRA, Statistic and Other Perturbed Chi-square Variables," Sankhya, 41, Series A, 22-47. DE ANGELIS, D., P. HALL, AND G. A. YOUNG (1993):"Analyticaland BootstrapApproximations to Estimator Distributions in L' Regressions," Joumal of the American Statistical Association, 88, 1310-1316. DIELMAN, T., AND R. PFAFFENBERGER(1984):"Tests of LinearHypothesesand L1 Estimation:A Monte Carlo Comparison," American Statistical Association Business and Economic Statistics Section Proceedings, 644-647. - (1988a): "Bootstrapping in Least Absolute Value Regression: An Application to Hypothesis Testing," Communications in Statistics-Simulationand Computation, 17, 843-856. (1988b): "Least Absolute Value Regression: Necessary Sample Sizes to use Normal Theory Inference Procedures," Decision Sciences, 19, 734-743. FALK,M. (1992): "Bootstrapping the Sample Quantile: A Survey," in Bootstrapping and Related Techniques, Lecture Notes in Economics and Mathematical Systems, 376, ed. by K.-H. Jockel, G. Rothe, and W. Sendler. Berlin: Springer-Verlag, pp. 165-172. HAHN, J, (1995): "Bootstrapping Quantile Regression Estimators," Econometric Theory, 11, 105-121. HALL, P. (1986): "On the Bootstrap and Confidence Intervals," Annals of Statistics, 14, 1431-1452. -~ (1992): The Bootstrap and EdgeworthExpansion. New York: Springer-Verlag. HALL, P., AND J. L. HOROWITZ (1990): "Bandwidth Selection in Semiparametric Estimation of Censored Linear Regression Models," Econometric Theory, 6, 123-150. HOROWITZ, J. L. (1997): "Bootstrap Methods in Econometrics: Theory and Numerical Performance," in Advances in Economics and Econometrics: Theoryand Applications, Vol. 3, ed. by D. Kreps and K. W. Wallis. Cambridge: Cambridge University Press, pp. 188-222. JANAS, D. (1993): "A Smoothed Bootstrap Estimator for a Studentized Sample Quantile," Annals of the Institute of Statistical Mathematics, 45, 317-329. BERAN,

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R. (1982): "Robust Methods in Econometrics," Econometric Reviews, 1, 213-255. KOENKER, KOENKER, R., AND G. BASSETT(1978): "Regression Quantiles," Econometrica, 46, 33-50. (1982): "Robust Tests for Heteroscedasticity Based on Regression Quantiles," Econometrica, 50, 43-61. MOLLER, H.-G. (1984): "Smooth Optimum Kernel Estimators of Densities, Regression Curves and Modes," Annals of Statistics, 12, 766-774. NoLAN,D., AND D. POLLARD (1987): "U-Processes: Rates of Convergence," Annals of Statistics, 15, 780-799. PAKES,A., AND D. POLLARD(1989): "Simulation and the Asymptotics of Optimization Estimators," Econometrica, 57, 1027-1057. D. (1984):Convergence of StochasticProcesses.New York:Springer-Verlag. POLLARD, J. L. (1984): "Least Absolute Deviations Estimation for the Censored Regression Model," POWELL, Journal of Econometrics, 25, 303-325. (1986): "Censored Regression Quantiles," Journal of Econometrics, 32, 143-155.

Econometrica, Vol. 66, No. 6 (November, 1998), 1353-1388

EFFICIENCY AND VOLUNTARY IMPLEMENTATION IN MARKETS WITH REPEATED PAIRWISE BARGAINING BY MATTHEW 0.

JACKSON AND THOMAS R. PALFREY 1

We examine a simple bargaining setting, where heterogeneous buyers and sellers are repeatedly matched with each other. We begin by characterizing efficiency in such a dynamic setting, and discuss how it differs from efficiency in a centralized static setting. We then study the allocations which can result in equilibrium when the matched buyers and sellers bargain through some extensive game form. We take an implementation approach, characterizing the possible allocation rules which result as the extensive game form is varied. We are particularly concerned with the impact of making trade voluntary: imposing individual rationality on and off the equilibrium path. No buyer or seller consumates an agreement which leaves them worse off than the discounted expected value of their future rematching in the market. Finally, we compare and contrast the efficient allocations with those that could ever arise as the equilibria of some voluntary negotiation procedure. KEYWORDS:Bargaining, implementation, matching.

1. INTRODUCTION

implementation theory to study decentralized contracting in markets that are limited to bilateral bargaining. To this end, we employ a simple model of matching and search with an infinity of buyers and sellers, who wish to trade one (indivisible) unit of a good. There is a known distribution of seller and buyer valuations. Trade occurs in a finite number of discrete periods. In the first period, buyers and sellers are randomly matched into pairs and then play a bargaining game that either results in a trade at some price, or no trade. If a buyer-seller match does not result in a trade, then each is randomly rematched with a new potential trading partner in the next period. The cost of search comes from each agent having only a finite number of opportunities to trade and discounting between periods. We characterize the efficient allocations and identify the set of allocations that can be achieved by general bargaining procedures. Our main departure from past work in this area is that we approach the problem from the implementation theory perspective. On the one hand, consistent with much of the previous literature on decentralized bilateral trade, the THIS PAPER USES

'This project was initiated while Jackson was visiting the California Institute of Technology and continued while he was at Northwestern University and while Palfrey was visiting CREST-LEI and CERAS; we are grateful for their support. We are also grateful for financial support provided under NSF Grant SBR-9507912. We thank Nabil Al-Najjar, Larry Ausubel, Eddie Dekel, Ray Deneckere, Larry Jones, Dilip Mookherjee, Mike Peters, Larry Samuelson, and Asher Wolinsky for helpful conversations and suggestions. We have benefited from the careful comments and suggestions of an editor and three anonymous referees on an earlier draft. 1353

1354

M. 0. JACKSON AND T. R. PALFREY

matching and search technology described above is taken as given. But contrary to past work on decentralized bilateral trade, we do not treat the rules of trade as exogenously fixed. That is, our objective is not to study properties of equilibria under some specific game form according to which bilateral trade is governed (say, the Rubinstein bargaining game, or the Nash bargaining solution), but rather to pose the implementation question: what allocation rules can be implemented as equilibrium outcomes of some finite extensive form bargaining game of perfect information? We first characterize the set of efficient allocation rules in this environment, and then characterize the set of all allocation rules that can be implemented by some bargaining game form. Using the conditions for implementability, we show that there exist robust distributions of buyer and seller valuations in which efficient trading rules cannot be implemented by any bargaining game. Implementation requires that the desired set of allocations coincide exactly with the set of equilibrium outcomes. In fact, we show the stronger statement that efficient trading rules are not even attainable by any bargaining game, where attainability only requires that an efficient trading rule correspond to some equilibrium outcome. The characterization of efficient allocations subject to the matching constraint identifies systematic distortions relative to unconstrained efficiency. The unconstrained first best is to have a match result in trade if and only if the buyer has a value above the competitive equilibrium price trade and the seller has a value below that price. (The transaction price is irrelevant.) Subject to matching, however, first best will generally be unachievable since chance determines which buyers are matched with which sellers. As a result, constrained efficiency can involve trade between buyers and sellers whose values both fall below (or above) the competitive equilibrium price. We show that the constrained efficient allocations are uniquely determined (up to sets of measure zero) and characterize such allocations. We then investigate the implementability of those constrained efficient rules by a general class of finite-length extensive game forms with perfect information. Also, the game form is augmented by appending to each terminal node a signature move for both the buyer and the seller. Both signatures are required, or the mechanism results in no trade for that match. The role of the signatures is to ensure that trade is voluntary, i.e. respects (endogenous) individual rationality constraints. It is assumed that the the buyer and seller in the match have complete information about each others' valuations, so the solution concept we employ is effectively backward induction. In addition to the general characterization, we demonstrate the importance of the implementation approach by showing an example where efficient trades are not attainable when prices correspond to those from Nash Bargaining, but are attainable when prices correspond to Nash Bargaining with a price cap. Finally, based on some of the necessary conditions from the characterizations, we provide a robust example with heterogeneous seller and buyer valuations where

REPEATED PAIRWISE BARGAINING

1355

the efficient allocations are not implementable or even attainable, even if there is no discounting.

2. RELATION TO THE LITERATURE

Because this paper bridges several different areas, we discuss separately how it fits in with previous work in three broad themes: competitive bargaining, search, and implementation. In short, what we are doing here is layering the implementation question on to a standard model of search and competitive =hargaining.Thus, our work relates to each of these areas.

Relation to the CompetitiveBargainingLiterature The underlying model that we study involves a combination of matching, bargaining, search, and rematching over a sequence of trading periods. As such, it is useful., for studying pure exchange economies from a noncooperative, game-theoretic perspective. Past work in the area2 has typically assumed both the technological features underlying the matching and search technologies and also has assumed the formal rules according to which bargaining between paired agents is required to follow. It is this latter set of assumptions that marks the first key difference between what we are doing and what has been done before. While the bargaining rules usually are modeled as a specific process of offers and counteroffers such as one based on Rubinstein (1982) and Stahl (1972), we explicitly do not assume a particular game form for the bargaining process. Rather, we are trying to identify the set of allocation rules (Walrasian or otherwise) that can be achieved as unique Nash equilibrium outcomes of some bargaining mechanism. The second difference between this paper and earlier work is that we do not focus on the question of the equivalence between Walrasian and competitive bargaining outcomes when market frictions are small. In fact, our main focus is not the case of frictionless markets per se, but rather on the properties of markets in which frictions exist, despite the large numbers of traders. To this end, we characterize efficient allocation rules subject to the matching constraints, and show how these differ in systematic and interesting ways from competitive allocations. Our interest then turns to whether these efficient allocations can be attained via any bargaining rules. 2By now the collection of papers in this area is too large to summarize exhaustively. The most closely related papers include Gale (1986a, b), Rubinstein and Wolinsky (1985), Binmore and Herrero (1988), and McClennan and Sonnenschein (1991) which follow in the footsteps of the early work on search and matching by Butters (1980), Mortensen (1982), Diamond (1982), and others. The bulk of this work is interested in identifying conditions under which game-theoretic equilibria in these decentralized matching and bilateral bargaining institutions will approximate Walrasian allocations when the frictions (search costs, discount factors, etc.) become infinitesimal. We lump all these together under the general heading of "competitive bargaining."

1356

M. 0. JACKSON AND T. R. PALFREY

Relation to the Search Literature Sattinger (1995) studies the question of efficiency in a search model with two-sided heterogeneity.3 He finds that the equilibria of matching procedures in which trades take place at prices determined by the Nash bargaining solution can be inefficient even taking account of the constraints of the search process. That is, one cannot even attain "second best" efficiency.4 The reason for inefficiency in Sattinger's model is that agents who are faced with a choice of trading in a current match do not account for the effect that their choice has on the future distribution of valuations in the market, and thus the future value from matching of other agents.5 There is a problem of congestion and the prices determined by the Nash bargaining solution do not generate adequate incentives for trade to compensate for this externality. In particular, agents do not consummate some trades that society would like them to. The innovation of our work is to investigate arbitrary bargaining procedures and ask whether any such procedure can be constructed to provide agents with the correct incentives for trading. To do this, we characterize the entire set of pricing and allocation rules that can be implemented by some bargaining procedure and compare this set to the set of constrained efficient allocations. The previous work in this area assumes Nash bargaining to determine transaction prices. We demonstrate that in some environments, this kind of pricing is suboptimal since it creates adverse incentive problems which can easily be avoided by resorting to alternative trading mechanisms. Specifically, we provide an example where the efficient trading rule is not attained when prices correspond to Nash bargaining, but can be attained by a simple variation where prices correspond to Nash bargaining with a price cap. Thus, inefficiency under Nash bargaining is not necessarily evidence that efficiency is not attainable. However, we go on to show that there are examples where efficiency is not attainable via any bargaining procedure. Actually, this example identifies a different source of inefficiency that is complementary to the congestion problem identified in Sattinger (1995) (and Shimer and Smith (1994)). There, agents are too patient and pass up efficiency-enhancing trades. In our example, some agents are overly impatient given their anticipated prospects for trade under any mechanism and so they trade too soon. This reduces the future prospects for other agents below the socially efficient level and creates further impatience. 3Also related are papersby Lu and McAffee(1995) and Peters (1991)who studythe allocation rules generated by specific processes of noncooperativecompetitivebargainingconstrainedby matchingand Ponsati and Sakovics(1995)who studybargainingwith outside options.Shimerand Smith(1994, 1996)studythis matchingproblemand obtainadditionalresults and characterizations about efficientsortingsubjectto the constraintsof the matchingprocess and the Nash bargaining solution. 4This contrastswith earlier work of Mortensen(1982) and Hosios (1990) who showed that efficiencycouldbe achieved,but in modelswith homogeneousagents. 5We thankan anonymousreferee for directingus to the Sattinger(1995) and Shimerand Smith (1994)papers.

REPEATED PAIRWISE BARGAINING

1357

Relation to the ImplementationLiterature

Implementationtheory formallymodels tradingmechanismsas game forms and tries to obtain general characterizationsof the allocationrules that can or cannot be achieved as noncooperativeequilibriumoutcomes. Although the necessaryconditionsthat come out of this literaturemust be taken seriously, there is somewhat less consensus about the practicalityof many of the sufficiency results,where very general and abstractmechanismsare constructedin order to demonstratethat a certain class of allocation rules can be implemented.The canonicalmechanismshave been criticizedfor a varietyof reasons relatingto their artificiality,relianceon threats,discontinuities,lack of balance, lack of well definedbehavioron parts of the mechanisms.6 In this paper, we want to avoid the problems of artificialityas well as the problemsinherentin mechanismsfor which behavioris not alwayswell-defined relative to the solution concept. In addition,we wish to begin to remedy two other shortcomingsof the existingwork in implementationtheory. First,we wish to avoid the use of implausiblethreats,used either to enforce certainactions in equilibrium,or to preventcertainstrategyprofilesfrom being "undesirable"equilibria.An extreme exampleof such a threat (which appears often in sufficiencyconstructions)is for the plannerto destroyall or part of the social endowment, if a particularout-of-equilibriummessage profile is announced. The problem with this is that such outcomes may not actually be carriedout, and agents should anticipatethis when decidingon strategies.Such mechanisms seem particularlyfar-fetched in cases where the players have inherent propertyrights (such as an initial endowmentor outside option) that providea lowerbound on the utilitythe agent can expect in the mechanism,for all message profiles.In our model, because the buyerand seller in a matchwill be rematchedin the next period, should they fail to agree to exchange, this places a naturalindividualrationality,or voluntaryparticipation,constrainton the process:no buyer or seller will consummatea trade that leaves him or her worse off than the discountedexpectedvalue of their future rematchingin the market.7We call this voluntaryimplementation. Voluntaryimplementationis relatedto implementationin the face of renegotiation since renegotiationalso providesagentswith an option outside of what is immediatelyprescribedby the mechanism.For example,the approachin Maskin and Moore (1987) is to specify an arbitrary,exogenous, and state dependent 6There ia a growingliteraturerelated to these points and some representativereferencesfor variousaspects of the problemare: Postlewaiteand Wettstein(1989),Jackson(1992), Abreu and Matsushima(1992),Dutta, Sen, and Vohra (1995),Saijo,Tatamitani,and Yamato(1993),Jackson, Palfrey,and Srivastava(1994),and Sj6str6m(1995). 7See Ma, Moore, and Turnbull(1988) for a look at implementationwith an exogenousoutside option for each player. In our paper, individualrationalityis more involved since voluntary implementationtakes the formof an endogenousindividualrationalityconstraintthat is determined by the value of futurerematching,whichin turn dependson the bargainingmechanismitself.

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M. 0. JACKSON AND T. R. PALFREY

renegotiationfunctionthat convertsinefficientoutcomesinto efficientones.8 In contrast,our approachdoes not allow agents to negotiate outside of the explicit rules of the mechanism,so there is no "renegotiation"per se. In particular,we considerfinite horizonmechanismswhere one or both of the agentsmay opt for "no trade,"effectivelywalkingawayfrom the currentmatch,afterwhichthere is no furtherinteractionbetween those two agents. Given this availableoption for no trade (whichexcept at the last date leads to rematching),no mechanismcan impose an outcome of trade between two matchedagents. However,given that agents cannot negotiate outside of a mechanism,it is possible for a mechanism to impose an outcomeof no tradebetween two matchedagents even when those two agents have mutualgains from currenttrade. As for any process of renegotiation,we use the mechanism to represent whateverthe protocol for negotiationbetween the parties is. Our viewpointin this paper is thus differentfrom the usual implementation"plannerimposes a mechanism"viewpoint.Insteadour point of view is more positivein that the full interaction between any agents includingany renegotiationthat they might undertakecan be modeled as a game form. Thus, any interactionbetween the agents is a process that can be described in full by a game form, and any distinctionbetweennegotiationand renegotiationbecomes a questionof semantics. Moreover,after this full process has concluded,the outcome is not final until both agents have signed a piece of paper acknowledgingany agreement that they have reached. Our approachimposes restrictionsthat (i) the whole process of interaction can be modeled as a finite length game, and (ii) the process itself is not state dependent.Point (i) is inessentialto our results (see footnote 30) and u~sedfor simplification.Point (ii) represents an importantdifference between our approach and that of, say, Maskinand Moore (1987). The set of availablemeans for negotiation (i.e., the mechanism) is the language, pieces of paper, and timing, etc., availablefor interactionbetween agents. These same means are availableregardlessof the preferencesof two matched agents. What differs is what agents choose to do as it depends on the state (their preferences,match, time, etc.). We think it is essential that the renegotiationprocess be formally modeled as part of the game form, and be independentof the state (although the actions chosen may be state dependent).This is consistentwith the seminal work on mechanismdesignby Hurwicz(1972). We should add that this approachwill have some importantimplicationsfor examplesof marketswhere the option for agents to walk awayfrom a current match is present. For instance, consideringthe U.S. market for single family homes, there is a standardprocessof negotiationby whicha price is posted, and 8Rubinstein and Wolinsky(1992) adopt a differentapproach,"renegotiation-proof implementation," which requiresPareto efficiencyof the continuationoutcome at all outcome nodes of the implementingmechanism.A related constraint is "credibility,"or the inability to commit to off-equilibrium-path outcomes that the planner (as opposed to the players)would not wish to impose.See, for example,Chakravorti, Corchon,and Wilkie(1992),Baliga,Corchon,and Sjostr6m (1997),and Baligaand Sjostrom(1995).

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then offers and counter offers are made, lawsuits are brought, escrow accounts are impounded, etc.-and these are the same set of available actions that any further negotiation or "renegotiation" also follow, and constitute an overall game form. Any tentative agreement is not binding until the proper signatures are put to paper. The same is true in many security markets (e.g. NASDAQ or the Chicago Mercantile Exchange) and in fact, most of these exchanges prohibit negotiation between member parties outside of the given rules (i.e. mechanisms) for trade. The second issue where we depart from past work in implementation theory is to study dynamic allocation rules. The importance of intertemporal tradeoffs is critical since many problems in which economists are interested, such as bargaining, investment, and growth, are dynamic. Unfortunately, implementation theory thus far has had little to contribute to questions of mechanism design in this large arena. Extensive form games have been examined, but only in the context of using them to implement static allocations.9 Finally, we emphasize that the notion of implementation we examine here is stronger than simple implementation by subgame perfect equilibrium. Our implementation results are for mechanisms that are constructed as games of perfect information, so our concept of equilibrium is actually "backward induction" (as in Herrero and Srivastava (1992)). Summarizing our contributions relative to the implementation literature: using a competitive bargaining model with rematching, we are able to characterize implementability in a dynamic environment, with an endogenous voluntary participation constraint, and without imposing implausible threats or using mechanisms with artificial or suspicious features. Thus, we obtain a characterization of what is implementable in this class of dynamic allocation problems, without resorting to the usually cumbersome methods of proof in implementation theory. Remarks on the Information Structure In our model, agents know the value of the agent with whom they are currently matched and there is a central authority who enforces the rules of the mechanism independent of any knowledge of the values of the agents. This assumption is common to each of the literatures discussed above, as well as the contract theory literature.'0 This approach permits the analysis of mechanism design to focus on incentive problems without introducing the complications of prior beliefs, strategic information transmission, and Bayesian equilibrium. Clearly, most bargaining settings involve some asymmetry of information between negotiating buyers and sellers, and such asymmetric information further compounds the incentive problems and introduces additional potential sources 9Two recent exceptions are Kalai and Ledyard (1995) and Brusco and Jackson (1996).

l?This literature is too large to survey here. For example, see Hart and Moore (1988), Moore (1992), Aghion, Dewatripont, and Ray (1994), and the references they cite.

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M. 0. JACKSON AND T. R. PALFREY

of inefficiency. Our choice is to use a model with symmetric information between bargainers and to focus on a particular source of social inefficiency that arises independently of asymmetric information. From a practical standpoint, in many markets, including some real estate and specialized labor markets, informational asymmetries may play a relatively small role compared to the fundamental problems of value-specific matching and negotiation on which we focus. The remainder of the paper is organized as follows. The model and definitions are presented in Section 3. Constrained efficient allocation rules are characterized in Section 4. Section 5 provides characterizations of voluntary attainability and voluntary implementability. Section 6 combines the results of Sections 4 and 5 to study the implementation of constrained efficient allocation rules. Section 7 contains some concluding remarks. 3.

DEFINITIONS

The Economy There are two goods. One good is indivisible and the other is divisible. Each seller is endowed with one unit of the indivisible good, and each buyer is endowed with one unit of the divisible (numeraire) good. Preferences Agents' preferences are characterized by a reservation value of the indivisible good, v E [0,1]. There are a finite number of dates, t E {1,... , T}, at which trade can take place, and a common discount parameter 8 e [0,1]. A seller with reservation value s who sells her indivisible good for p units of the numeraire good at time t receives (net) utility 8 t(p - s), and a buyer with reservation value b who buys a unit of the indivisible good for p units of the numeraire good at time t receives utility 8t(b -p). An agent who never trades receives utility 0. Distributionsof Values Initially, there is a continuum of buyers and of sellers. The distribution of reservation values of the agents remaining in the economy at the beginning of a time t E {1, ... , T} is summarized by the following functions. Bt(b)-the

mass of buyers at time t with value no more than b;

St(s)-the

mass of sellers at time t with value no more than s.

These are not cumulative distribution functions, since, for instance, it may be that St(1) = 1. The corresponding distribution functions (for St(1) > 0 and Bt(1) > 0) are S,(v)/St(1) and Bt(v)/Bt(1). The initial mass of buyers and sellers is the same, B1(1) = S1(1), so it will always be true that Bt(1) = St(1), for all t. This

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is without loss of generality, since we can model other cases by adding buyers or sellers who should never trade.1" We assume that at least one of the two distributions is atomless. Specifically, we will assume that the initial distribution of buyers, B1, is continuous and increasing at all b > 0. This rules out masses of buyers with identical valuations and assures that there are buyers with values in any open subinterval of [0,1]. This assumption simplifies the analysis in that we do not have to worry about rationing agents with the same valuation, or randomizing. We also assume that S1(0) < S1(1), to rule out the trivial case where all matches should be consummated immediately in the first period. PairwiseMatching At the beginning of each period, the remaining buyers and sellers who have not yet traded are pairwise matched with each other. The matching12 is described by a probability measure gt on [0,1]2 where for any measurable At C [0, 1]2 Hr

s=lb: ( (sEb)eAt

~dBt(b) dSt(s) B,(1)

) St(1)

The distribution over values with which any seller with valuation s will be matched at time t is dBt(b)/Bt(1). Similarly, the distribution over values with which any specific buyer with valuation b will be matched at time t is dSt(s)1St(1)-

Matched buyers and sellers are fully informed of each other's valuation.

AllocationRules Allocation rules describe which buyers and sellers will trade at each time, and what price will be paid (i.e., what transfer is made). We restrict our attention to allocation rules that depend only on the time and on the buyers' and sellers' valuations (but not their names). This restriction reflects our interest in anonymous processes. A tradingrule is a collection, A = (A1,..., AT), of measurable subsets At of [0, 11x [0, 11. A pair (s, b) eAt indicates that any seller with valuation s and buyer with valuation b who are matched at time t should trade. "For instance, B1(1) > S1(1), is handled by adding sellers with s = 1. There is a measurability problem associated with a law of large numbers over a continuum of i.i.d. random variables (see Judd (1985) and Feldman and Gilles (1985)). For any finite economy which approximates ours, we could describe a matching process (which would not be i.i.d.) with the above specified properties, but there would necessarily be some (small) dependence in the random variables. Instead, we work directly at the limit distributions and simply note that we could come arbitrarily close to finding a matching process that formally justifies the assumed one. See Gretsky, Ostroy, and Zame (1992) and Al-Najjar (1996) for more discussion of this. 12

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M. 0. JACKSON AND T. R. PALFREY

A price rule is a collection of measurable functions p = (P, i... PT), where Pt: A, -> [0, 1]. A price rule indicates that if a buyer and seller trade, then the buyer transfers pt(S, b) units of the divisible good to the seller. An allocation rule consists of a trading rule and a price rule. Cutoff Rules One type of trading rule that will play an important role in our results is a cutoff rule. This is a rule such that the set of buyers who trade with any given seller form an upper interval of the set of buyer types, and the set of sellers who trade with a given buyer form a lower interval of the set of seller types. More formally, A is a cutoff rule if for all t and s,13 (i) either {bI(s, b) EAt} = {b E [0, l]Ib ? b'} or {bI(s, b) eAt} = {b E [0, llb > b'} for some b'e [0,1], and (ii) {bI(s, b) eA-} c {bI(s',b) eAA} whenever s > s'. In many cases it will not matter whether the inequalities in (i) are weak or strict (see the definition of equivalence below), and we represent a cutoff rule by functions It(s) (corresponding to b' in (i)). Evolution of Distributionsof Valuations Any trading rule A and initial distributions S, and B1 induce S2, .. I ST and B2,..., BT, according to the matching process. The resulting distributions are defined recursively by (1)

St+ l()

I

= St(v)-

)f

<s?Vb(s,

~~dBt(b)

B,() b) E-AtB())dk

and (2)

Bt+l(v)=Bt(V)

-f

(I b
dSt( ) )dBt(b).

(s, b) EAt St (l)

Equivalence of Tradingand Allocation Rules Given the continuum of agents, we define an equivalence over allocation rules that differ only on sets of measure 0. The trading rules A and A are equivalent if Aut(AtnA') = At(At UAt) for each t, where Autis the measure defined in (0) induced by A according to (1) and (2).14

The allocation rules (A, p) and (A,p) are equivalent if A and A are equivalent and pt({(s, b) eAItpt(s, b) P't(s, b)}) = 0 for each t. The definition can equivalently be stated from the buyer's perspective. . Notice ty that in this case the measure 11 induced by At willperspctive coincide with At

14

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REPEATED PAIRWISE BARGAINING

Expected Utility

The expected utility u'(s; A, p) of a seller with valuation s under an allocation rule (A, p) at the beginningof time t conditionalon not havingtradedyet is given by Us(s; A, p)

[1 -

(l

T E

x

()]

F

K (s (b)EA

(b) s) ~~~dB7 )'

b)

s,b-s

BT(l)

where HtZ ] is taken to be 1. Similarly,the expressionfor the expectedutility u'(b, A, p) of a buyerwith valuation b under an allocationrule (A, p) is given by (r

T 5T

U b(J, A, p) =

1

rd____B [

x

I: sbA (b -pT(s,

ES:(s b) A,ST(l)

S()]

b)) dST (s)

)

ReservationPrices It will often be useful to work with the reservationprices, -s(s; A, p) and Pt3(b;A, p), induced by an allocation rule. The reservationprice at time t is

simplythe price at whichan individualwouldbe indifferentbetweentradingand not tradingat time t. These follow immediatelyfrom above: P(s; A, p) -s = us+ 1(s; A, p) b -Pt (b; A,p)

=

ub+1(b; A,p)

(t=1,

**,T-)

(t =1,",T-

1),

PT(S; A, p)=s, -b(b;A,p)=b.

When (A, p) is fixed,we may simplywrite us(s), s(s), etc. ConstrainedEfficiency

We say that a tradingrule A is constrainedefficientif there exists a price rule p such that (A, p) maximizesthe total expectedsurplus: fus(s; A,p) dS1(s) + fub(b; A,p) dBj(b). S

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M. 0. JACKSON AND T. R. PALFREY

Notice that constrainedefficiency is a propertyof trading rules, and thus is independentof the choice of a price rule p. The "constraint"in constrained efficiency is embedded in the definition of trading rule, which respects the matchingprocess. Constrainedefficiencyis the same as constrainedPareto efficiencyif ex-ante transfersof the divisiblegood can be made among the buyers,and among the sellers. Without such transfers,constrainedefficiencyas we have defined it is utilitarianand thus may rule out some constrainedPareto efficient allocations. To see the difference, consider an examplewhere some sellers are forced to trade with any buyer that they meet in the first period whose valuation falls below a certain level, even if the buyer'svalue is less than the seller's. Such tradescan be part of a constrainedPareto efficientallocationif no transfersare permitted,since these sellers take low valued buyersout of the market,which leads to higher expected utilities for the other sellers because the remaining pool of buyers has a higher average valuation.This sort of tradingfails our definition of constrained efficient allocation since it does not maximize the overallgains from trade. Our definitionof efficiencytakes the set of agents in the system as given. If one allowed control of the set of agents present, then a perfectly informed plannercould induce the extra-marginaltradersto leave, by mandatingthat all trade be consummatedin the firstperiod, at the competitiveprice. We rule out such a scheme by taking the agents present in the initial matchingprocess as exogenous.Moreover,any of a numberof embellishmentsof the model would nullifyschemes of this sort. For example,if there is some aggregateuncertainty about the distributionof buyers or sellers (e.g., a finite number of, traders sampled from a known distribution),then the competitiveprice is not known with certaintyand everytradercould have some probabilityof being on the right side of the market clearing price.15Alternatively(as in Shimer and Smith (1994)),if there is some match-specificcomponentof the valuations,so that the value we model is only the expectationof a value whichmayvarywith the match (or just over time), then even traderswho have a low expectedgain from trade may still have a significantoption value and an incentiveto stay in the market. Finally,admittingconvex preferences and divisibilities(as in a classical Edgeworth box) would offer potential gains from trade to almost all agents even though some could be very small. Rather than complicatethe model in one of these wayswe simplytake the matchingprocess to be exogenous.

4.

CHARACTERIZATION OF CONSTRAINED EFFICIENCY

Our analysisof constrainedefficiencyis restrictedto the case of T = 2. (See Jackson and Palfrey (1997) for some results on arbitraryfinite horizons.)We begin with an illustratingexampleand then turn to the characterizationresult. 15Ouruse of the continuum as a simplifying tool is responsible for the departure from this.

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EXAMPLE 1: Consider a case where 8= 1, buyers'valuationsare uniformly distributedacross [0,1] with a total mass of 1, and a mass 0 < m < 1 of sellers have valuation 0 and the remaining mass, 1 - m, have valuation 1. This is represented by Bl(b) = b for all b and Sl(s) = m for all s < 1. In the absence of matchingconsiderationsor any frictions,Pareto efficient allocations would involve the assets going to the buyers with value at least 1 - m. The competitiveallocationsare an obvious choice, where sellers sell to

the buyers with values above the competitive price, p = 1 - m. In our model,

trade is constrainedthroughthe matchingprocess, and the characterizationof an efficient allocation becomes complicated since some of the higher value buiyersmight never be matchedto a seller with whom they can trade, and it is sometimes better to clear a trade with a low-valuedbuyer than to wait for a buyerwith a higher expectedvalue. It is straightforward, but instructive,to derivethe constrainedefficientallocation rule for this example.In the second (last) period, all positive value trades shouldbe cleared,since there will be no furthermatching.It is also clear that a constrainedefficienttradingrule will be a cutoff rule, so it sufficesto specifythe minimumvalue of a buyerthat shouldtrade in the firstperiodif matchedwith a 0 value seller. (These and other claims in this exampleare provedin Theorem 1.) For any value c set as a cutoff today, the remainingdistributiontomorrow will be B2(b) = b for b < c, and B2(b) = (1 - m)(b - c) + c for b > c. The gain

from ctearinga trade todaywith a buyerof value b, is simplyb. Sellerswho do not trade today are rematchedin the second period.The expectedvalue of the buyer with whom they will trade in the second period is simply the expected value of b under the distributionB2(v)/B2(1), which is 1 - m(l - c)(1 + c) 2(1 - m(l - c))

The constrainedefficient tradingrule is obtained by equatingthe cutoff value equal to the expected value of rematching.That is, on the margin, a trade should be cleared today if (and only if) it offers at least as much total value as could be expected by waiting and clearingthe trade tomorrow.Solvingfor c*, the efficientcutoff rule is *

c*

=

|1-rn

~__

__

__

- (1-rn)

_

__

_

M__ _

m

The cutoff rule is decreasingin m. As the mass of sellers m increases,the current cutoff has less of a reduction effect on tomorrow'sexpected trading value. Also notice that the cutoff value is always lower than the competitive price (1 - m). The efficientsolutionin the above examplehas an easily characterizableform since sellers are effectivelyhomogeneous,but in manyways it is representative of the characterizationwhich is providedbelow for the case of general distributions of buyer and seller valuations.

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M. 0. JACKSON AND T. R. PALFREY

THEOREM 1: Thereexists a unique (up to sets of measure 0) constrainedefficient trading rule. It is described by cutoff rules, with associated functions ,Q(s) and /32(s). These cutoff rules uniquelysatisfy the following equations. I62(S) =s,

Vs E [0,1],

and (3)

813(s)- s = 8 1max[ b' -s, 0 dB2(b') + a

3max[P(s)-s',

0] dS2(S')

dB2(b1 )dSS(s' -

8f'(flmax[ b' - s',O] -B2(1)

B0)

S(f

S2(1)

if this is feasible with p31(s)< 1, and /31(s) = 1 otherwise, where S2 and B2 are determined by (1) and (2), respectively. Furthermore, /31(s) is continuous and is strictlyincreasingat values of s such that 813(s)< 1.

Let us examine the intuition behind (3) as a characterizationof efficiency. Consider a planner deciding whether to clear a currentlymatched pair, with valuationss and f31(s). Since /31(s)is the cutoff value for s, the plannershould be indifferentbetween clearingthis trade or not. If this trade is cleared, then the left-handside representsthe marginalvalue16of consummatingthat trade today.If this trade is not cleared,then s and 131(s)will be put back in the pool in the second period. The right-handside gives the marginalexpected value from throwingboth playersback in the pool to be rematchedtomorrow.This marginal expected value has three components. Throwing the players back means that they are matched with two other players who would have been otherwise matched. Given that those two other players will be randomlyselected, on averageone can treat the opportunitycost for matchingthem with s and /31(s)as being the averagetradevalue in the second period.This is the last expressionin (3). The net value that comes from the randomrematchingof s and p1(s) is then the expected value from each of their rematchings(the first two expressionson the right-handside of (3)), less the opportunitycost of the agentswith whomthey are rematched(the last expressionon the right-handside of (3)). 5. NECESSARY CONDITIONS FOR VOLUNTARY IMPLEMENTATION AND ATTAINABILITY

Next, we turn to the issue of voluntaryimplementationand considerthe case of arbitrary(finite) T. Characterizations of voluntaryimplementationand attainabilityprovideus with the completecollectionof allocationrules that could ever 16Each trade is in fact of measure 0, so a calculus of variations argument is used in the formal derivation.

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be the equilibriumoutcomesof such a dynamicinteraction-under any negotiation process (whichis representableby a finite extensivegame form of perfect information).With such characterizationsin hand, we will return to check whetherconstrainedefficient allocationsare attainable. Negotiation and Game Forms

The formal, or informal,negotiation process that goes on between a buyer and seller who are matchedat time t is representedby an extensivegame form yt. This game form is the same across all pairs matched at time t. The game form y, is a finite stage extensivegame formof perfectinformation.(The results extend to infinitestage game forms,but finite ones are all that are needed.) Since y, can depend on time, in equilibriumit can also depend on the measures of agents remaining.However, y, cannot depend on the history of play. This is essentiallyan anonymityrestrictionso that the mechanismcannot respondto the particularactionsof any agent,whichis motivatedby our interest in modeling markets.If one permits the mechanismto depend fully17on the history,the implementationproblem can become trivial.The future stages of the mechanismcould then be chosen to enforce no trade if any agent deviates from prespecified actions. This defeats the idea of individualrationalityas capturingvoluntarytrade with an endogenous outside option, as the outside option could be controlled as a function of any single agent's actions. If that were the case, the mechanismwould then simplyreduce to a forcingcontract. One can argue that we should use the strongerassumptionthat the mechanism be the same in each period. That is, the form of negotiationavailableat any time shouldbe the same if it is representingsome primitiveset of available actions. While we agree with this in certain contexts (for instance in a richer model where there are balancinginflowsof agents too), allowingfor the larger set of mechanismsstrengthensour impossibilityresult,and is congruentwith the fact that the stock of agents in our model is nonstationary.It is possiblethat the bargainingprocedure could depend on market conditions (for instance, by conventionwho makes the first offer in an alternatingbargainingprocedure might depend on the relativeexcess supplyor demand).Of course, a stationary mechanismis a special case of the ones we considerhere, and our characterization of implementationcan be specializedto that case. The SignatureStage

The heart of our analysis is the assumption that no agreement becomes bindinguntil it is signed by each of the two agents.After negotiationshave led to a suggestedtrade and price, the trade does not take place unless both agents "sign"the agreement.This is captured as follows. Consider, yt, an extensive game formwith perfect recallto be playedbetween an arbitrarybuyerand seller 17Wecould allowy, to dependon the historyof playup to sets of measure0.

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M. 0. JACKSONAND T. R. PALFREY

at some time t, such that each terminalnode suggestseither a trade and price, or no trade. Given yt, let us define a dynamicversion, F(y,), as follows. First, replace any terminalnode of y, which recommendsa trade and price, with a node that has a binarychoice node (yes,no) for the buyer. Let "no" lead to a terminalnode with no trade as the outcome. Let "yes"lead to a binarychoice node (Yes,No) for the seller. Let "No"lead to a terminalnode with no trade as the outcome, and "Yes" lead to a terminalnode with the originallyprescribed trade and price.We have simplyaugmentedy, by additionalmoves that require both the buyerand seller's "signature"before completingthe trade. At any time t, each matchedbuyer and seller play the augmentedversion of y,. If the outcomeof F(y,) is trade,then the trade is consumatedand the buyer and seller are removedfrom the matchingprocess.If the outcome is no trade, then the buyerand seller are returnedto their respectivepools to be rematched in the next period. As an example,consider a simple dictatorialmechanismy, where the seller simply announcesa price p E [0,1] and the outcome is then trade at price p. The augmentedversion F(y,) has the seller announce p E [0,1] as the first stage. Next, the buyer, having observed p, chooses from (yes,no). Finally,the seller, having observed p and the buyer'smove, chooses from (Yes,No). The outcome of F(y,) is trade at price p if the choices in the "signature"stages are yes and Yes; and no trade (returnfor rematchingat time t + 1), otherwise.

Equilibrium A buyer's strategyfor time t is a measurable function, 0,b(s, b), mapping pairs of buyerand seller valuationsinto the set of behavioralstrategiesfor the buyer role in F(y,). A seller's strategy for time t, o.S(s, b), is similarly defined. A collectionof pure18 strategiesv- (U ,..., Ub; U1,..., SO-) inducesan allocation rule (A, ,pO').

An equilibriumof the augmentedsequenceof mechanismsis a specificationof

strategies U- such that for each t and (s, b): (i) Utb(s, b) and Uts(s, b) form a subgameperfect equilibriumof F(y,), where the utility of no-trade is evaluated as 8u+ 1(b, A , p9) for buyers and 8ust+1(s,A,,p,) for sellers (O if t = T),"9 and

(ii) at any node where an agent's actions may lead either to currenttrade"at some price or to rematching,the agent chooses an action leadingto rematching only if it offers an expectedutilityhigherthan any of the other availableactions. '8In this model,mixedstrategieswill not play a role in any equilibrium. 19Thisdefinition is stronger than simply defining an equilibriumto be a subgame perfect equilibriumof the overallgame form with the continuumof playersand T periods.The overall game form has manyinterlacedinformationsets (as agents do not know the play of all the other agents in precedingperiods) and so it does not have proper subgames-so subgameperfection applied overall would simply boil down to Nash equilibrium.The definitionof equilibriumwe employappliessubgameperfectiondirectlyto each time and matchand thus avoidssuch a problem.

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Part (i) of the definitionof equilibriumimposes sequentialrationalityin the form of subgameperfect equilibrium.Part (ii) of the definitionof equilibriumis a tie-breakingrule when an agent is indifferentbetweentradingtodayor waiting and being rematched. The particular form of the tie-breakingrule is not important:we could have definedit to have agents alwaysfavoringdelay in such situations.One can think of this as being equivalentto a lexicographicpreference assumptionthat eliminatesindifference.20This simplifiesthe analysis,as it producesa uniquepredictionof an outcomeof a given extensivegame form as a functionof endogenousreservationprices(althoughthere can still exist multiple equilibriabecause of the endogeneityas in Example2).

VoluntaryAttainabilityand Implementability V An allocation rule (A, p) is voluntarilyattainableif there exist (y.. YT) such that at least one equilibriumof the augmentedsequence of mechanisms results in an allocationrule that is equivalentto (A, p). The difference between attainabilityand implementabilityis uniqueness. Attainabilitydoes not require uniqueness, and hence is a very weak form of implementation.21More generally,one may be interested in knowing all the equilibriaof a mechanism,which motivatesthe definitionbelow. An allocationrule (A, p) is voluntarilyimplementableif there exist ('Yi,l**, YT) such that each equilibriumof the augmentedsequence of mechanismsresultsin an allocationrule that is equivalentto (A, p). Alternatively,we may simplybe concerned that an efficient tradingrule be implemented(or attainable)and not concernedwith the particularprices that are realized.We say that a tradingrule A is voluntarilyimplementableif there exists a sequence yt such that for each equilibriumthere exists a price rule p such that the equilibriumresults in an allocationrule equivalentto (A, p). A tradingrule A is voluntarilyattainableif there exists a sequence yt such that there exists some equilibriumand price rule p such that the equilibriumresults in an allocationrule equivalentto (A, p). Rubinstein and Wolinsky(1992) provide a mechanismfor subgameperfect implementationin a pairwisebargainingmodel, but in their model there is no possibilityof rematching.Thus, in their model the agents'reservationvalues are fixed. Given the possibility of rematchingwe consider here, the reservation values of the agents become endogenous to the equilibrium.This provides serious complicationsto the implementationproblem. We end up having a necessaryconditionof nondecreasingprices which is similarto Rubinsteinand Wolinsky's,except that it is stated relativeto the endogenousvaluations.Also, 20We did not model it that way since it would preclude a utility representation. 21This is roughly equivalent to what has been known in the literature as "truthful" implementation in the case where implementation is possible in direct mechanisms (Dasgupta, Hammond, and Maskin (1979)). We do not consider direct mechanisms given the dynamic and voluntary nature of the problem, so we have defined attainability.

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M. 0. JACKSON AND T. R. PALFREY

we end up with a strongversionof an additionalindividualrationalitycondition that relates the entire set of prescribedprices(thus the prices availablethrough the mechanism)to the endogenousreservationvalues. First, let us examinethe conditionwhich is the appropriategeneralizationof the Rubinsteinand Wolinsky(1992) conditionto say that the prices be nondecreasingin the endogenousreservationvalues. This distinctionbetween valuations and endogenous reservationvalues is very important since reservation values are not always nondecreasing in an agent's primitive valuation, as reservationprices dependon futureprospectsfor tradeunderan allocationrule. NondecreasingPrices: An allocation rule (A, p) has nondecreasing prices as a functionof reservationprices,if for each t, (s, b), and (s', b') in A,:

p, (s, b) 2 p,(s', b') whenever Ns(s; A, p) ?s(s'; 2 A, p) and jb5(b; A, p) > jb(b'; A,p). Notice that an implicationof the above condition is that the price rule can only varywith the reservationprices of the agents. The necessityof nondecreasingpricesis verifiedas follows.Since the bargaining game has a finite extensiveform with perfect informationand agents have strictpreferencesover outcomes,the equilibriumoutcomefor every(s, b) (fixing reservationprices) is unique. Let p = p(s, b) be the outcome for the pair (s, b), and we consider b' with a higher reservationvalue p-b > jb. Since only the buyer'svaluationhas changed,either p is still an equilibriumoutcome,or there is a new equilibriumand the b' buyer must have at least one strictlyimproving deviationsomewherein the game tree. Given the change in preferencesof the buyer,the only way a deviationcan be improving(and not have been improving before) is for the deviation to lead to a price between pb' and pb, while the previousoutcomewas no trade.For this to have an effect furtherup the tree, it must be that an agent chooses this price rather than another one or no trade furtherup the tree, and so the change in the highersubgamemust result in this price. This logic is iterated back to the equilibriumpath, which implies that changes can only result in a higher price. Increasingthe seller's reservation value has similarimplications. In additionto the nondecreasingprice condition,an additionalconditionwill be necessary. Given the individual rationality that is at the heart of our definition of voluntaryimplementation,it is clear that the trades suggested under an implementable(or attainable) allocation rule must be individually rational:the price p of any trade consummatedbetween s and b in period t must lie between the correspondingseller and buyer reservationvalues. Individual Rationality: An allocation rule (A, p) satisfies individual rationality if for any t and (s,b)eAt -nS(s A, p)
REPEATED PAIRWISE BARGAINING

1371

Although it is obvious that individual rationalityis necessary for voluntary attainability,it is more subtle that a strongerconditionis necessaryfor voluntary attainability.This strongerversionof individualrationalitystates that there is no price at which some pair of agents trade at time t that is simultaneously individuallyrationalfor some other pair of agents who should not trade under the allocationrule. StrongIndividual Rationality: An allocation rule (A, p) satisfies strong individual rationalityif it satisfies individual rationality, and for each t, (s', b') At, and (s, b) eAt, either pt(s,

b) > pb(b'; A, p)

Pt(s,

b) < pb(S'; A, p)

or

To understandthe necessity of this condition suppose that there are agents who should not trade under the desired allocationrule, and there is a mutually individuallyrationalprice given their anticipatedvalues from rematching,and this price is availableat some terminalnode in the tree. Tracingthe path from this terminalnode back up the tree, one can find a best responsefor each agent at eac'hnode and this must leave them at least as well off as trade at this price. In this way one can show that there exists an equilibriumthat involves trade between these two agents. However, from the uniqueness of the equilibrium outcome for a given pair of agents in their round of bargaining(fixing their anticipatedreservationprices under equilibriumrematchings),they must trade in every equilibrium,whichwould contradictattainability. We can summarizethe conditionsthat are necessaryfor voluntaryattainability, and thus for voluntaryimplementability. THEOREM 2: Consider a trading rule, A, that is voluntarily attainable (or implementable), and (A, A), an allocation rule correspondingto one of the equilibria of an implementing mechanism, where A is equivalent to A. Then (A, p) satisfies strong individual rationalityand has nondecreasingprices.

We make two remarkson Theorem 2. First, these conditionsare necessary even when one just considersattainability.In other words,these conditionsare needed simplyto ensure that (A, p) can arise as an equilibriumof any mechanism. The conditionsare not arisingfrom multiple equilibriumconsiderations. Second,these conditionsare still necessaryfor voluntaryimplementabilitywhen one admitsinfinitestage mechanisms.Details on this are given in a footnote to the proof (see Appendix). The conditionsof nondecreasingpricesand strongindividualrationalityplaya centralrole in the full characterizationsof voluntaryattainabilityand implementation. The full characterizationtackles difficulties associated with possible

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M. 0. JACKSON AND T. R. PALFREY

discontinuitiesin the implementedprice function, as well as the usual implementationchallengeof rulingout equilibriawhich do not result in an allocation rule equivalentto (A,p). However,in some cases of interest the conditionsof nondecreasingprices and strong individualrationalityare sufficientfor voluntary attainability.Let us describea mechanismthat will show this. Given a set of prices P c [0,1], denote IR(P) = {(q, r) e [0, 1]213p E p, q

Stage 2: The buyer announcesps',

The Outcome Function: pb and (p s, pb) E IR(Rt), then the outcome is sup{p E RtIp < pb'}.

If pb

Otherwise,the outcome is no trade. Considerthe case where B1 and S, are continuousand increasing.Let (A, p) be an allocation rule such that At is a continuous cutoff rule,23and Pt is continuouson At for all t. Giventhese continuityand monotonicityconditionson the allocationrule and the buyer and seller distributions,nondecreasingprices and strong individual rationalityare sufficientfor voluntaryattainability.One can easily verify that there is an equilibriumof the above mechanismwhere the announcementof ps and pb in equilibriumshouldbe the true (endogenous)reservationpricesof the agents. To see how this works, note that the announcementof ps' allows the buyer to challenge the seller's announcementif, for instance, the seller announces ps > js. In that case, the buyercan revise the seller'sannouncementby saying pSt
REPEATED PAIRWISE BARGAINING

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A Characterizationof VoluntaryImplementation

Theorem 3 characterizedvoluntary attainabilityfor situations where the distributionsand allocation rule are well behaved. We now consider general distributionsand allocationrules, and deal explicitlywith the multiple equilibriumproblemthat is inherentin the endogeneityof reservationprices and thus voluntary implementation.

First,we extend the necessaryconditionsfor the case of general distributions and allocationrules. allocation rule (A, p) satisfies voluntary trade if for each t there exists JJ such that: (Vi) [Reservation Price Measurability] For every s, b E At, p,(s, b) =

-. An

Pt c [0,1] and P [0,1]2 Pt(Pt (s,Pt(b)

(V2) [Individual Rationality] (j5'(s), 15(b)) e IR(Pd), for every s, b e At, and -s

< p(pstpb)

< pb,

for every (ps,pb)

E IR(Pt).

(V3) [StrongIndividualRationality]( Ns(s), p-b(b))e IR(Pt), for every s, b e At.

(V4) [NondecreasingPrices]Jt is nondecreasingover the domain IR(Pt). (V5) [SeparatingPrices]for every( ps', pb) E IR(Pt) and jS such that (ps, pb) E IR(Pt), if Pt(Ps, pb) <j3(Ps, pb) then there exists p E Pt such that -' ?'p < pb) then there exists p ePt Sirhilarly, for every pb if p (ps, b)>pt(ps, pS. such that pb ?p >_pb Let us discuss some of the differences in the above condition from the conditionsstated previously.The conditions(V2)-(V4) are direct extensionsof the correspondingprevious conditions. The set Pt correspondsto the set of prices that are reachable by the implementingmechanism. Sometimes it is necessaryfor this to be larger than the set of prices which are supposedto be traded at in equilibrium,as off equilibriumbehavior will be important in determiningequilibriumbehavior(see Example 2, below). Then, for instance, the strongindividualrationalityconditionmust be satisfiedrelativeto all of the prices in Pt. If some price in Pt is individuallyrational, then an equilibrium which results in trade will exist. So (V3) must hold relativeto all of Pt. Condition(Vi) is new relativeto the nondecreasingprices and strongindividual rationality.The functionAt has as its domainreservationprices,as these are what matter in determiningequilibriumactions. It is necessarythen that the implementedprice function be measurablewith respect to reservationprices, which is condition(Vi). The last condition(V5) is also added for the general case. It states that the implemented price function can only be increasing in places where we can distinguishthe reservationprices of the agent in question. If, for instance, pS < PS, but there are no availableprices from Pt in between PS' and s, then these two types would have exactlythe same preferencesover trades in Pt (the only ones possible from the implementingmechanism).In such a case, the equilibriumactions of these two types must be the same.

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M. 0. JACKSON AND T. R. PALFREY

The voluntary trade condition is thus necessary both for voluntary attainability and voluntary implementability. However, voluntary implementability requires an additional necessary condition to avoid multiple equilibria, as illustrated in the following example. EXAMPLE 2: Consider the constrained efficient trading rule defined in Example 1, when m = 1/2. Consider a fixed price of c = C - 1 in the first period, which corresponds to c*. So 0-valued sellers trade with all buyers with values above x1 - 1 in the first period at a price of x1 - 1. In the second period let 0-valued sellers trade with all buyers, and trade at a price equal to the buyer's valuation. The voluntary trade condition is satisfied relative to this A, p by setting P1 = 2- 1), P2 = [0, 1], P1(Ps, pb) = - 1, and P2(P, pb) = pb. It is then simple to verify (i)-(v). There exists a mechanism that has A, p as an equilibrium: In the first period trade is simply at price x1 - 1. In the second period the seller makes take-it-or-leave-it offers to the buyer (and the seller can name any price in [0, 1]). First, we check that there is an equilibrium that results in (A, p). It is the obvious one. Buyers approve trade in the first period if and only if b 2 - 1, and the 0-valued sellers approve trade in the first period. Notice that from the characterization of constrained efficiency (and from Example 1), we know that a 0-valued seller's reservation price is exactly x2I - 1 in the first period. In the second period, 0-valued sellers make the offer of b to the buyer with which they are matched, and it is approved. But there is another equilibrium relative to the above mechanism! It involves all of the sellers rejecting the first period price. The second period is as before. This is an equilibrium, since if all the sellers reject in the first period, then the full mass of buyers is still there in the second period. The average value of the buyers is then 1/2 in the second period. Since this is larger than x2_- 1 (see Example 1), the sellers are indeed acting optimally. Since there are two equilibria, this does not implement the efficient solution.24 Nonetheless, the efficient allocation rule can be fully implemented by an alternative mechanism which is a simple variation on the above mechanism. Consider the following change: In the first period the buyer makes a take-it-orleave-it offer to the seller from the set of prices [k2 - 1, 1]. Any buyer with"A value above V2 - 1 would rather trade in the first period, since they expect to have their full value extracted in the second period. High valued buyers can offer sellers enough to get them to trade in the first period, even if the sellers expect a value above c in the second period. This means that the trades will occur in the first period that should. Given that they occur, the buyers will be able to offer - 1 and get it. 24

In fact, the efficient equilibrium to the above mechanism is fragile: even a small variation in the expectations makes it better for the sellers to wait.

REPEATED PAIRWISE BARGAINING

1375

The mechanism works because it has a range of available prices in the first period that is larger than just x2I - 1. This illustrates the important role of P, in the voluntary trade condition. It also gives us insight to the full characterization of implementation and the relationship to attainability: it must be that (A, p) is attainable, but other (nonequivalent) (A, p)'s are not attainable. If we set P1 = [v2I- 1, 1], then the voluntary trade condition is not satisfied relative to the undesired allocation rule where all of the agents wait until the second period to trade, and so that allocation rule will not be an equilibrium outcome. In particular, (V3), strong individual rationality, is violated in this example relative to this P,. TFor the characterization of implementation, we restrict attention to mechanisms that have the property that there exists a subgame perfect equilibrium of the augmented mechanism for each t relative to every set of reservation prices jS, pb. We call these mechanisms closed. This avoids the use of controversial implementation theory "tricks" which exploit nonexistence of best responses in some portion of the message space. THEOREM 4: If an allocation rule (A, p) is voluntarilyimplementableby a closed mechanism, then: (i) there exists (A, p) which is equivalent to (A, p) and satisfies the voluntary trade condition, and (ii) for each (A', p') not equivalent to (A,p), (A',p') fails to satisfy the voluntarytrade condition relative to the same p and Pt as (A, p3). Conversely, if (i) and (ii) hold and Pt is closed for each t, then (A,p) is voluntarilyimplementableby a closed mechanism.

We remark that (i) is necessary for voluntary attainability as well as implementability. This is proven in the Appendix. Condition (i) states the necessity of voluntary trade, which we have discussed earlier. Condition (ii) is the condition ruling out undesired multiple equilibria, as illustrated in Example 2. The implementing mechanism used to prove sufficiency is a simple variation on the one described in the previous section, prior to Theorem 3. We know that it is not necessary that Pt be closed. It is an open question whether (i) and (ii) are sufficient in the absence of this condition, or whether there are additional necessary conditions.

6.

ATTAINING OR IMPLEMENTING CONSTRAINED EFFICIENT RULES

Given the characterizations of attainability and implementation, we turn to the issue of attaining or implementing efficient trading rules. Let us begin with an example that illustrates that the consideration of all bargaining procedures is important. There are efficient allocation rules that are not voluntarily attainable when one considers a procedure that results in Nash bargaining solutions, but

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M. 0. JACKSON AND T. R. PALFREY

are voluntarily attainable (and implementable) via alternative bargaining procedures. EXAMPLE 3: This is a variation on Example 1, where there is discounting (8 < 1) and where m = 2. Again, buyers' valuations are uniformly distributed across [0,1] with a total mass of 1. A mass 2 of sellers have valuation 0 and the remaining mass have valuation 1. This is represented by Bl(b) = b for all b and Si(s)= 2 for all s < 1.

Using Theorem 1 one can compute the cutoff value c* (i.e., c* = ,1(0)) as the unique solution to the equation c* =

(

2-c* E2[b],

which is the unique root in [0,1] to the cubic equation (C* + C*2)(2

-

8c*) = -(1

2

+ c* 2)(2

-

c*).

Notice that if prices are set by the Nash bargaining solution so that pt(O,b) = b/2, then a seller with s = 0 will choose to trade with any b ? c where c satisfies c/2 = 8E2[b/2], or c = 8E2[b]. This is inefficient, since sellers fail to consummate all efficient trades in the first period. Thus, if one restricts attention to prices corresponding to Nash bargaining, then the efficient trading rule cannot be voluntarily attained. The source of the adverse incentives under the Nash bargaining price rule is that it splits the buyer and seller surplus in half. As a result, a seller matched with a buyer whose valuation is close to the efficient c* would prefer to wait because the trading prospects are more attractive tomorrow. However, the efficient trading rule can be voluntarily attained when one considers a pricing rule that reduces the sellers' prospects tomorrow. A very simple modification of the Nash bargaining price rule accomplishes this: place a price ceiling on the transaction, equal to some value P < 1. This changes the pricing rule from pt(O, b) = b/2 to pt(O, b) = min[P, b/2]. For 8 close to 1 there will exist a ceiling P which creates the right incentives, where sellers matched with a buyer of valuation c will be exactly indifferent between trading and waiting. To see this, note first that the right incentives will be provided as long as P is chosen so that P > (c*/2) and E2[p2(0, b)] = ((2 - c*)/(2 8c*))E2[b/2]. This will guarantee that the ceiling is not binding in the first period and that the cutoff value is chosen optimally. It is easy to see that P can be chosen to accomplish this: E2[p2(0, b)] varies continuously in P, ranging from 0 to E2[b/2] so, for any value of c E [0,1], we can choose P so that E2[p2(0, b)] = ((2 - c)/(2 - 8c))E2[b/2]. When 8 = 1 the solution for the opti-

REPEATED PAIRWISE BARGAINING

1377

mum is c* = V2 - 1 and the choice of P = .5 > F2 - 1 works, and so for 8 close to 1 the appropriate value of P will satisfy P > c*/2, as required. This stationary pricing rule, which is a simple modification of Nash bargaining, offers exactly the right incentives to satisfy strong individual rationality, and together with the efficient trading rule is voluntarily attainable (and in fact implementable). Example 3 shows that the efficient trading rule may not be attainable with a pricing rule determined by Nash bargaining, but could be attainable in conjunction with some other natural pricing rule (here Nash bargaining with a price cap). This illustrates why it is important to consider general bargaining procedures and general pricing rules in these matching/exchange environments. Consideration of only a single pricing rule, such as Nash bargaining, can significantly constrain the set of attainable or implementable allocations. The Proposition below, however, shows that even admitting general bargaining procedures and pricing rules does not allow one to attain efficient allocation rules in some situations. Generally, there is a rich set of constraints imposed by strong individual rationality, and these can be difficult to satisfy when the distribution of sellers is more general than the one in the examples above. PROPOSITION 1: There exists a robust set of continuous and increasing distributions of buyerand seller valuationsfor which the constrainedefficient tradingrule is not voluntarilyattainable (and hence not voluntarilyimplementable).

The robustness mentioned in the Proposition refers to the fact that the result is true for any distributions satisfying the following25 for small enough 0 < E < 1/2: 2-e?B1(e)E<

2

and

B1(l-E)

<

2 +

e

and likewise 2-E

< S1(E)
and

S1(i-e)? <2+ In other words, for small e these distributions have nearly half their mass on values close to 1 and nearly half their mass on values close to 0. A sketch of the proof (details of which are found in the Appendix) is as follows: For such distributions, an efficient solution will clear trades in their first period between low valued sellers and high valued buyers and so the resulting distributions the second period will have approximately 1/3 high value buyers and 2/3 low valued buyers, and similarly 1/3 low value sellers and 2/3 high valued sellers. Thus, low valued sellers and high valued buyers (the only agents really generating gains from trade) have a chance of only 1/3 of meeting a successful match in the second period. Since one side can get no more than half of the surplus of a successful match in the second period, either the low valued seller or the high valued buyer has an expected value of no more than 1/6 from trading tomorrow. Say it is the low valued seller. The combination of individual 25In fact, the onlynontrivialexampleof whichwe knowwherethe constrainedefficientallocation is voluntarilyattainableis in the case of homogenoussellers.

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M. 0. JACKSON AND T. R. PALFREY

rationality and strong individual rationality imply that individual rationality is almost exactly binding for both the lowest valued seller (0) and her cutoff match (,p1(O)),which in turn implies that the low valued seller can get no more than 1/6 from her cutoff trade today. Since her cutoff match 81(O)is approximately 1/3 (as derived from Theorem 1), the buyer with value 1/3 must get at least 1/6 from the trade today. However this buyer 81(O) can expect at most (1/3) x (1/3) = 1/9 from waiting and thus strictly prefers to trade today, which contradicts the fact that the individual rationality constraint should be binding at 81(O).The robustness follows from the fact that the efficient solution varies continuously with the distribution, so we can work with any distribution satisfying the above conditions for small E. The rough intuition is that first period trades create an externality on the distribution of traders who are rematched in the next period. Thus in the optimal solution, it is possible that some "good" trading pairs26 (in the example a low valued seller and a low-middle valued buyer) should not trade and instead be left in the market to offset this externality. This can be true even though the expected surplus from that transaction in the first period exceeds the sum of the expected surpluses of the two transacting parties were they to search one more period. For any game that tries to implement this efficient solution, some of these trading pairs would prefer to trade in the first period, which prevents the efficient solution from being an equilibrium outcome.

7.

CONCLUDING REMARKS

There are three main contributions in this paper. First, we provide a characterization of constrained efficiency in a setting with random matching and search. In situations where markets are truly decentralized, standard notions of efficiency are inappropriate since goods mnaynot be transferable arbitrarily from one agent to another. The matching process imposes constraints on the set of feasible allocations, and introduces search externalities across agents. These constraints and externalities are at the heart of the characterization of constrained efficiency. Second, we provide characterizations of attainability and implementation in situations where mechanisms cannot impose trade on agents. The characterization is intuitive in terms of the (strong) individual rationality conditions that naturally arise from the voluntary choice of agents either to accept the outcome of the mechanism, or to reject it and search for a new trading partner in the next period. The implementation is shown to be achievable by simple mecha26This is the flip side of examples in Sattinger (1995) and Shimer and Smith (1994), where a congestion externality leads to too few trades taking place and removing low value trades to reduce congestion can be an efficiency gain. Here we find that the opposite problem can also occur: social gains can come from having some agents with attractive valuations stay in the market, while they may be too impatient.

REPEATED PAIRWISE BARGAINING

1379

nisms using alternating move games with perfect information, with a structure similar to standard bargaining games. Third, we show that it is often the case that constrained efficient allocations are inconsistent with voluntary decentralized trade under any bargaining game. Even with atomless agents, the externalities cannot be overcome, regardless of the mechanism by which agents negotiate and trade. Thus, in spite of the fact that trading pairs share complete information about each others' valuations, the strong necessary conditions imposed by voluntary trade are incompatible with overcoming the externalities and achieving efficient allocations. The strength of the first two27 results we obtain is, of course, tempered by the fact that we have worked in a specific setting. The specific nature of the preferences of the agents (i.e., the "bargaining" structure), the way in which agents may accept or reject the suggestion of the mechanism, and the particular matching technology are important in terms of the clean and intuitive characterizations we obtain. Relative to the implementation literature, this suggests exploring how the nonimposition restriction behaves in more general environments, especially those where one admits the possibility of some choices in matching, such as those offered by a centralized exchange. Relative to the competitive bargaining problem, it would be interesting to examine how the analysis extends to an infinite horizon, and to situations where there are inflows of agents. In our introduction, we discussed our view that any negotiation and renegotiation should be modeled as part of the given game form. This viewpoint strengthens the conclusions of Proposition 1, since the result is true regardless of the form of negotiation that takes place. However, since we have not taken any stand on the particular process that may govern such interaction, our admissible class of game forms is still quite large. Although we impose restrictions of perfect information, finite length, signature stages, and lack of integer games, etc., we do not impose a priori restrictions on the specific structure of negotiation or renegotiation.So, if one considers an environment where there are natural or exogenously determined restrictions on how this process can take place, so that only some of the mechanisms that we have allowed are feasible, then additional conditions could come out of the characterization. We point out, however, that in spite of the larger class of mechanisms we have admitted, our theorems are proven without resorting to complicated or unnatural mechanisms. The implementing mechanisms used to prove the characterization results are extremely simple and involve only a sequential announcement of a reservation price by each agent, and an opportunity for the other agent to challenge this announcement with another price. Thus, in order for any a priori restrictions on negotiation (or renegotiation) to have an impact, they would have to rule out such mechanisms. Nevertheless, such mechanisms do allow for the imposition of no trade as an outcome even when there are mutual gains from trade to agents. 27The last result (the impossibility of implementing the efficiency rule) still holds in more general settings.

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M. 0. JACKSONAND T. R. PALFREY

Althoughthis maybe reasonablein some markets(e.g., securitymarkets),it may not be in others (e.g., housing markets)where it will be necessaryto rule out such mechanismsbefore one can take the sufficientconditionsfor implementation seriously. Finally,in this model there are no transfersthat are made exceptbetween the pairedagents.Havinga centralizedauthoritythat could execute transfersacross agents and time could help avoid some of the negative externalitiesand help achieve efficiency.This is an importantquestion for future investigation,and suggestsinterestingcomparisonswith centralizedmarkets. Division of Humanities and Social Sciences, 228-77, CaliforniaInstitute of Technology, Pasadena, CA 91125, U.S.A.; [email protected]; html http:// www.hss.caltech.edu/ jacksonm/IJackson. and Division of Humanitiesand Social Sciences, 228-77, CalifomiaInstituteof Pasadena,CA 91125, U.S.A.;[email protected] Technology, receivedNovember,1996;finalrevisionreceivedOctober,1997. Manuscript APPENDIX PROOF OF THEOREM1: First,any efficienttradingrule mustbe (up to sets of measure0) a cutoff rule for secondperiodtradeswith 8B2(S)= s, So A2 = {(s, b): b > s}. Thatis, almosteveryindividually rationalsecondperiodtradewill be consummated. Assumingthis form for 132,and using pit(s, b) = b (since W is independentof p), we can write

W(A1)=

f

(fA

(b-s)dBi(b)) E=-Al(s)

B(b) dS2(s)

dS,(s)+8 f'fl(b-s) s

~~~~~2(l)

where A1(s) = {b: (s, b) EA1}, and B2 and S2 are determinedby (1) and (2). Let us rewritethis as

WMI)= f'(f1,g(s,b)(b-s)dB1(b))

dS(s)+r

I

'(b

-

bs) B(b) dS2(s),

where ir(s, b) = 1 if (s, b) eA1 and 7r(s,b) = 0 if (s, b) =A1. We maximizeW(T) with respect to all measurableir's and show that the unique solution correspondsto the claimed cutoff function in (3). MaximizingW is a vector space optimization problemwith the constraintiT(s,b) E [0,1], for all (s, b). A necessaryconditionfor an optimumis that directional(Gateaux)derivativesare either 0 or point inwardfrom the boundaryfor almostall (s, b). For our problem,this impliesthat, for almostall (s, b), ir = 1 when (d[W(r)]/d[ir(s, b)])> 0 and 7r= 0 when(d[W(Gr)1/d[ir(s, b)O)< 0.28 Sufficiencyof these conditionsfollowsfromthe uniqueness of the solution,the continuityof W and the compactnessof the set of admissible1(s, b) (in the weak* topology). 28To be more explicit,the directionalderivativewith respectto some measurablefunctionh(s, b) in this case works out to be f(s,b)(d[WT)11/d[LT(s, b)])h(s, b)d(s, b). So, requiring that b)])h(s,b)d(s, b) < 0 for any h such that ir(s, b) + h(s, b) E [0,1] for all (s, b), f(s,b)(d[W(&r)I/d[Lr(s, is equivalentto saying that for almost all (s, b), ir = 1 when (d[W(rT)1]/d[L(s,b)1) > 0 and r = 0 when (d[W(r)1/d[ir(s,b)1)< 0.

1381

REPEATED PAIRWISE BARGAINING Recall that S2(V) = Sl(V) - f

(f1

b) dBl(b)) dS,(s)

T(,

and B2(V) =Bl()

-

L?(fV1T (s, b)dS,(s))

dBj(b).

Differentiate W(G) with respect to ,r(s, b) for any (s, b), which leads to A) (Al)

d[W(Ti-)]

dd[WT(s, b)]

(b - s) dBj(b) dS,(s)

-8

d[B2(l)]

d[dS2(s)]

1

+

1

-s)

fIf(b' B2(l) d[Tr(s,b)] oo sJ5

dTr (s, b)]

f

dB2(b')

(

) B2(1)

dB2(b') dSI B2(l) ) d[dB2(b)]

+d dr(s,b)] T

'

b

o

dS2(s') -)

B2(l)

Next, observe that

d[ dS2(s)] d[Tr(s,b)] d[ dB2(b)] d[Tr(s,b)]

dB(b)dS1(s), _dBj(b)dS,(s),

and d[B2(l)]

d[Tr(s,b)]

dB1(b)dS1(s).

Substituting these expressions into (Al) provides d___W__T ),

s[> ) dS,(s) dBj(b)bSk[b d[iT-(s, b)]BI

dB2(b') f (b' -s') B()) (b -s) +b5 S + j B2(l) o S

s) dB,2(W) B (1) -f f1(O'(b

-

^ Ub

dS2(S') 2

JB2 (l)

)]. (b - s' ) dS2(S')1 B

Recall that we must have (almost everywhere) i= 1 when (d[WO-)]/d[L7T(s,b)]) > 0 and ir= 0 when b)]) < 0. To see that the solution should be a cutoff rule, fix s and notice that the (d[W(Or)]/d[Lr(s, b)] is strictly part inside the brackets on the right-hand side of the expression for d[W(&-)]/d[H-(s, b)])= 0 implies (3). The increasing in b since we assume S1(O)< S1(l).29 Setting (d[W(Or)]/d[Trr(s, continuity and increasing properties of I1(s) follow directly from inspection of the right-hand side of Q.E.D. (3). We next present the proofs of Theorems 2-4. We do this in the order: Theorem 4, Theorem 2, Theorem 3. This is different from the order in the body of the paper, but it is the natural order to present the proofs, since the results in Theorem 4 are used to prove Theorems 2 and 3. The claim in Section 6 and Proposition 1 are proved at the end.

8 equals 1, and there are multiple 29 IfS1(0) = S1(), then this derivative is constant in b when solutions for the optimal first period allocation rule.

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M. 0. JACKSON AND T. R. PALFREY

PROOF OF THEOREM4: We begin by demonstrating the necessity of the conditions. Suppose that (A, p) is implemented by (yl,. . ., YT) which satisfies the equilibrium existence condition stated prior to Theorem 4. Let Pt be the set of prices that correspond to some terminal node of yt. LEMMA: For any t, and for any (s, b) pair, there is a unique subgameperfect equilibriumoutcome of

F(y,) satisfying (ii) in the definition of equilibriumas a function of (ps, and only if ( 5s, pb) E IR(Pt).

pb).

It is trade at some price if

Consider any (jpS, pb). By part (ii) of the definition of equilibrium, an agent's choice from a set of outcomes is uniquely determined. The subgame perfect equilibrium outcomes (which always exist under the existence condition on the mechanism) can thus be found by backward induction, which results in a unique outcome. Only If: By the veto power that each agent has under the augmented mechanism, the unique equilibrium outcome must be no trade if (pS, pb) e IR(Pt). If: We now show that if (ps, pb) E IR(Pt), then the unique equilibrium outcome must be trade at some price. Suppose the contrary, so that for some (ps, pb) E IR(Pt), the equilibrium outcome is no trade. Consider a pair of equilibrium strategies for (ps, pb) when they are matched at time t and denote these o-. These lead to no trade at time t. Consider also some strategies which lead to the outcome of p at time t and denote these o-'. Alter o- at each node on the play path of o' to match the action under o-' at that node, and leave the actions at other nodes under o- unchanged. Call this new strategy v". Since '" results in trade at p, it must not be an equilibrium for ps and pb.30 Find the last node along the play path of v" such that there is an improving deviation for the agent choosing at that node. Find a best response for that agent at that node.31 The new play path must lead to trade at some price since it is improving for that agent and both agents weakly prefer p to no trade. The new strategy combination is now a Nash equilibrium in all subgames from this node on (and all subgames off the current play path). Iterate this logic up the nodes of the play path. This results in a subgame perfect equilibrium that has an outcome of trade at some price, which is a contradiction. Q.E.D. With the lemma in hand, we can conclude the proof of necessity in the theorem. Define A to be the equilibrium price of F(yt) as a function of (ps, pb). By the lemma, this is a well defined function on the domain IR(Pt). Let (A, p3) denote an allocation rule corresponding to some equilibrium. By the definition of implementability, it is equivalent to (A, p). Define pt relative to the equilibrium strategies leading to (A,,5). We first verify (i). We show that voluntary trade holds relative to (A, p3),for the Pt and Pt defined above. (VM)and (V3) follow directly from the lemma. (V2) follows from the lemma and the fact that agents will never accept a price that is not individually rational in an equilibrium of the mechanism. (V4): A sketch of the proof of this case was given in the text for increasing buyer's reservation values. The case of an increase in seller's reservation values is analagous, except the type of improving deviations is to have no-trade replace trade at some price.32 (V5): By (V4) we know that ps' < ps. Suppose the contrary of (V5). Then for all p E Pt, either p >Ps and p >ps, or p
REPEATEDPAIRWISEBARGAINING

1383

Next, let us verify(ii). Consideran (A', p') whichis not equivalentto (A, p). Considerthe Atand Pt definedfor each t as above.Notice that (V4) and (V5) are satisfied,as they are independentof the allocationrule. We must show that one of (V1), (V2), and (V3) fail for (A', p') relativeto the 't and Pt definedabove. By the lemma,for each t and (s, b) there is a unique subgameperfect equilibriumoutcome of R(yt) relative to the reservationvalues (Pjs(s;A', p'), pb5(b;A',p')). Select a subgame perfect equilibriumpairof strategiesfor each t and (s, b). By the implementationof (A, p), these strategies cannotresult in (A', p'). Thus,there exists t and (s, b) such that either: Case 1: (s, b) A't and the outcomeis trade at some price p, or Case 2: (s, b) EA't and the outcomeis trade at some price p 0 p'(s, b), or Case 3: (s, b) EA't and the outcomeis no trade. In Case 1, it followsfromthe lemmathat (P]s(s; A', p'), itb(b; A', p')) E IR(PF), whichmeansthat (V3) fails. In Case 2, it followsfromthe definitionof A that p'(s, b) OAt(ps(s;A',p'), P'(b; A', p')), which means that (V1) fails. In Case 3, it followsfromthe lemmathat (Ps(s; A', p'), p'b(b;A', p')) 4 IR(P), whichmeansthat (V2) fails. Sufficiencyis establishedby constructinga mechanismthat will implementany (A, p) satisfying the voluntarytradecondition.The mechanismyt at time t is the one describedin Section5: Stage1: The seller announcespS. Proceedto Stage 2. Stage2: The buyerannouncesps,p b. Proceedto Stage 3. Stage3: The seller announcesp The Outcome Function: If pb pS and (ps',pb') E IR(Pt), then the outcome is Pt(ps', pb'). If pb pS'}. If pb' >pb and (ps, pb') E IR(Pt),then the outcomeis sup{p EPt Ip < pb'}.

Otherwise,the outcomeis no-trade. We now provesufficiency.Assume that (i) and (ii) hold. Considerthe implementingmechanism, (7Y,. .. I YT), describedabove.The remainderof the proof consistsof verifyingthree claims. CLAIM 1: Consider t and a subgame perfect equilibriumof the augmented version of the mechanism describedabove underpart (ii) of the definition of equilibrium,when reservationvalues are (ps, pb). The outcome is unique and: (a) if (pS, pb)e IR(Ft), then the outcome is trade at p Q5s, pb); (b) if (ps, pb) 4 IR(FP), then the outcome is no trade.

PROOFOF CLAIM1: The set of possible outcomes from the above mechanismis Pt. Thus (b) followsby the same logic as the lemma,notingthat in this case a subgameperfectequilibriumexists because no price is ever approvedby both agents. Similarly,if (ps, pb) E IR(Pt), then the unique subgameperfect equilibriumoutcome is trade at some price, providedan equilibriumexists. We need to show that a subgameperfectequilibriumexists and it is trade at pt(pb, pS). Considerthe followingstrategieswhichresult in pt(pb, pS). It is easilycheckedthat given these expectations,these form a subgameperfectequilibrium. On the equilibriumpath behavior:The seller announcespS =5S. The buyerannounces(ps', pb) = (ps, pb).

The seller announces pb' =pb. Both approve this.

Off the equilibriumpath behavior: Each playerapprovesany price that is individuallyrational,and vetos others. If the seller announcespS <]pS, then the buyerannounces(ps',pb) = (ps,pb). If the seller announces

pS < pS,

then the buyer announces

(ps',pb)

= (ps, pb).

If the buyerannouncespb
1384

M. 0. JACKSON AND T. R. PALFREY

conceding some surplus to the buyer. If the seller tells the truth, then the buyer cannot claim the seller has a lower reservation price, or this will lead to no trade. Similarly, the buyer cannot gain from understating his reservation price, since the seller could then correct this announcement to the Q.E.D. true buyer reservation value, and win all the surplus. CLAIM 2: There exists an equilibriumwhich results in At, fit. PROOF OF CLAIM2: If we fix the reservation prices of the buyers and sellers, then there is a unique subgame perfect equilibrium outcome for any matched pair for a specific stage. So fix the reservation prices at those generated by the allocation rule, At, fit. We will verify that the subgame perfect equilibrium outcome in this case results in At,fA. If (s, b) tAt, then no trade is the only subgame perfect outcome of the augmented mechanism. This follows from (V3) and Claim 1. If (s, b) EAt, then from Claim 1 and (VM) and (V2) it follows that the outcome is trade at Q.E.D. Pt(pts(s;A, p), ptb(b;A, p)). CLAIM 3: If (A', p') is not equivalent to (A, p), then (A', p') is not the result of any equilibriumof

the mechanism. PROOF OF CLAIM3: Suppose to the contrary that there is an equilibrium that results in (A', p').

Consider (s, b) t A't. It must be that the outcome of R(-yt) is no trade. From Claim 1, it then follows that (V3) holds relative to A and Pt. Consider (s, b) EAl. It must be that the outcome of R(-yt) is trade at p'(s, b). It then follows from Claim 1 that (Vl) and (V2) hold relative to A and Pt. This contradicts (ii), which implies that (A', p') fails to satisfy (V1), (V2), or (V3) relative to Pt

and Pt.

Q.E.D.

After Theorem 4, we claimed that (i) would be necessary even if one only considers attain'ability, and also if one drops the requirement of a closed mechanism. This is the same as the above proof of the necessity of (i), except that lemma is only stated for (jpS, pb) relative to which equilibrium exists. Then one needs to extend A to satisfy (V2), (V4), and (V5), for (ps, pb) E IR(Pt) relative to which there does not exist an equilibrium. For such a ( ps, pb), define pt(jps, pb) by setting it equal to the max of pS and the sup of Atover (ps'Xpb') E IR(Pt) such that pS ?ps pb
Q.E.D.

PROOF OF THEOREM 3: It follows from Theorem 4 (and the proof above and the continuity so that if any equivalent allocation rule satisfies strong individual rationality and has nondecreasing prices, then (A, p) does as well) that the conditions are necessary. To see that they are sufficient w&j show that the voluntary trade condition of Theorem 4 is satisfied relative to (A, p). Then the result follows from Claims 1 and 2 in the proof of Theorem 4. Under the assumptions of Theorem 3, reservation prices are continuous and nondecreasing functions of s and b. Given the continuity of p and A it follows that individual rationality must hold with exact equality for cutoff pairs.33 Thus, if b = 8t(s) (where 8t is the cutoff defined by the cutoff rule At), then pt(b) =Pt(s) =pt(s, b). To see this, consider a cutoff pair s, b. By individual rationality pt(b) 2 pt(s, b) > Pt(b). For any b' < b we know that either Pt(b') < pt(s, b) or pt(s) > pt(s, b). We know that the second cannot hold, so it must be that Pt(b')
33Given the continuity, all claims that were "almost every," no longer have that qualifier.

REPEATED PAIRWISE BARGAINING

1385

Let st be the min{sI(s, b) EAt for some b}, and st be the max{sI(s, b) EAt, for some b}. Similarly define bt and bt. Next, notice that pt(st) = Pt(bt) =pt(st, bt) and similarly, pt(3t) = pt(bt) =Pt(3t, bt). Given the assumptions on At and Pt, the range of Pt over pairs in At is [ pt(st, t),pt(3t, bt)], since Qt, Pt) EAt and (3t, bt) EAt given that At is a cutoff rule and cutoffs are nondecreasing in value. So, let Pt = [pt(st, bt), pt(, bt)]. For pS, pb E IR(Pt) define Pt(ys, pb) through pt(s, b) by setting p (

pt (s', b'

sp)=

where s' = min{sIps(s) ? max{fjs,j5s(st)}} and b' = max{bIps(b) 2 min{pb,p4b(jt)}}. Using this we verify that the voluntary trade condition is satisfied relative to (A, p). (V1) holds since if s, b eAt, then s' = s and b' = b in our definition above. (V2) holds by the construction of 't and the individual rationality assumed in Theorem 3. (V3) holds by the strong individual rationality assumed in pb), Theorem 3. (V4) holds by the construction of At.To see (V5), notice that for Pt(Ps, pb)
the individual rationality in Theorem 3, we know that (s', b) EAt and (s, b) EAt. By continuity of p, we can find s", b EAt with pt(s', b)
0]) -E2(max[b

-s, 0]).

As E becomes small, E2(b) converges to (B2(1) - B2(.5))/B2(1), E2(max[ ,1(0) - s, 01) converges to B1(O)S2(.5)/S2(1) and E2(max[b - s, 0]) converges to (B2(1)- B2(.5))S2(.5)/(S2(1)B2(1)). So at the limit (as E becomes small), B2(1) -B2(.5) B2(1)

f31(0)S2(.5)

S2(1)

(B2(1) -B2(.5))S2(.5) S2(1)B2(1)

Solving for ,1(0) '810

=-B2 (1) - B2 (.5) B2(1)

This means that ,1(0) is at most 1/2, since none of the b = O'sare cleared in the first period. So, as E goes to zero, it must be that B2(.5) = 1/2 and B2(1) = 3/4. Thus, ,1(0) converges to 1/3 as E goes to zero.35 Similarly, the cutoff for buyers with a valuation of 1, o-r(1),converges to 2/3 as E goes to zero. Let A be equivalent to the efficient trading rule and consider any price rule p. We show that for small enough E the necessary conditions for voluntary attainability cannot be satisfied for (A, p). Suppose to the contrary that they are for all small E. Pick some small y> 0, and apply strong individual rationality to b' = ,1(0) - y and s' = 0. Since we know that individual rationality is satisfied for s close to 0 and some b close to ,1(0), it follows that for some small enough 7y36 ,X1(?- y should not trade in period 1 so that 1(0) -

S2(.5)

7

Pi(0,

p1(o))<S2(1)

S2(1)

( 1(0) Y 7-P2(11 ,31(0) - y)) + 2E

34Since the efficient solution is only defined up to sets of measure 0, one can find s close to 0 and work with that. Uniqueness of the efficient solution and continuity of the welfare function in changes in distribution of buyer and sellers' values imply continuity of the efficient solution in the distribution of buyer and sellers' values. 36We proceed as if this is satisfied for s = 0 and b = ,1(0), while for any trading rule equivalent to the efficient one this can be redefined for some s, b arbitrarily close to these.

1386

M. 0. JACKSONAND T. R. PALFREY

wherethe right-handside boundsthe expectedvalue to rematching:there is a probabilityof at most S2(.5)/S2(1) that the buyeris matchedwith someone with a value between -y and 2, and the best pricethey can get is then P2(', p1(0) - y). For y smallenough,there is at most 2E chancethat they are matchedwith a seller with value smallerthan -y. A similar argument for s' = y and b' = 81(0) leads to y<

pj01310-

B2(1) - B2(.) B2(1)

(P2(7,

y) +

-)-

2E.

Given the symmetryof the distributionsand thus the efficientsolution, S2(.5)

B2(1) -B2(.5)

S2(1)

B2(1)

Summingthe two previousinequalitiesand simplifyingleads to (1()

-

2y) 1-

2

S2(1)1)]

<

S2(1)

S2(1) (P2(7

-Y)

-P2(Y,

p1() -y))

+ 4E.

We can followthe same argumentsin a neighborhoodof the buyerwith value 1 and that buyer's cutoff seller, o-(1), to find that S2(.5)

(1a(l - y)1((

S2(1) / < S2(.5)-y)4E S2( (P2(o1(1)

S2(1)

S2)

+ y, 1 - y)

P2(Y7

- y))4e.

Summingthese two inequalities,for small E and -y this is approximately (,1(0

-1

-(1

1

S2(.5)

S2)

<

S2(5)

S2(1)

(P2011

1Y ys

y)

P2(Y, 010

Y)

or + y,1 - y) -P2(7, fl1(0) - Y)) 9 < 3(P2(0(1) whichis impossibleto satisfysince the right-handside is at most 1/3.

Q.E.D.

REFERENCES AGHION,P., M. DEWATRIPONT,AND P. REY (1994): "Renegotiation Design with Unverifiable Information," Econometrica, 62, 257-282. ABREU, D., AND H. MATSUSHIMA(1992): "Virtual Implementation in Iterative Undominated Strategies: Complete Information," Econometrica, 60, 993-1008. AL-NAJJAR, N. (1996): "Aggregation and the Law of Large Numbers in Economies with a Continuum of Agents," Mimeo, Northwestern University. BALIGA, S., L. CORCHON, AND T. SJ6STROM (1997): "The Theory of Implementation when the Planner is a Player," Journal of Economic Theory, 77, 15-33. BALIGA, S., AND T. SJOSTROM(1995): "Interactive Implementation," Mimeo, Harvard University. BINMORE,K., AND M. HERRERO (1988): "Matching and Bargaining in Dynamic Markets," Review of Economic Studies, 55, 17-32. BRUSCO,S., AND M. JACKSON(1996):"TheOptimalDesign of Security Markets," Mimeo, Northwestern University. BuTTERs, G. (1980): "Equilibrium Price Distributions in a Random Meetings Market," Mimeo, Princeton University. CHAKRAVORTI,B., L. CORCHON,AND S. WILKIE (1992):"CredibleImplementation," forthcomingin Games and Economic Behavior. DASGUPTA, P., P. HAMMOND,AND E. MASKIN (1979): "The Implementation of Social Choice Rules: Some General Results on Incentive Compatibility," Reviewof EconomicStudies,46, 185-216.

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DIAMOND,P. (1982): "Wage Determination and Efficiency in Search Equilibrium," Review of Economic Studies, 49, 217-227. DuTTA,B., A. SEN,ANDR. VOHRA(1995): "Nash Implementation Through Elementary Mechanisms in Economic Environments," Economic Design, 1, 173-204. FELDMAN, M., ANDC. GILLES(1985): "An Expository Note on Individual Risk without Aggregate Uncertainty," Journal of Economic Theory, 35, 26-32. GALE, D. (1986a): "Bargaining and Competition Part I: Characterization," Econometrica, 54, 785-806. (1986b): "Bargaining and Competition Part II: Existence," Econometrica, 54, 807-818. GRETSKY, N., J. OSTROY, AND W. ZAME (1992): "The Nonatomic Assignment Model," Economic Theory, 2, 103-127. LIART,O., AND J. MOORE(1988): "Incomplete Contracts and Renegotiation," Econometrica, 56, 755-785. HERRERO,M., AND S. SRIVASTAVA (1992): "Implementation via Backward Induction," Journal of Economic Theory, 56, 70-88. Hosios, A. (1990): "On the Efficiency of Matching and Related Models of Search and Unemployment," Review of Economic Studies, 57, 279-298. L. (1972): "On Informationally Decentralized Systems," in Decision and Organization,ed. HURWICZ, by C. B. McGuire and R. Radner. Amsterdam: North Holland. JACKSON, M. (1992): "Implementation in Undominated Strategies: A Look at Bounded Mechanisms," Review of Economic Studies, 59, 757-775. JACKSON, M., ANDT. PALFREY (1997): "Efficiency and Voluntary Implementation in Markets with Repeated Pairwise Bargaining," Social Science Working Paper #985, California Institute of Technology. JACKSC,N,M., T. PALFREY, AND S. SRIVASTAVA(1994):

"Undominated

Nash Implementation

in

Bounded Mechanisms," Games and Economic Behavior, 6, 474-501. JUDD, K. (1985): "The Law of Large Numbers with a Continuum of I.I.D. Random Variables," Journal of Economic Theory, 35, 19-25. KALAI,E., AND J. LEDYARD(1995): "Repeated Implementation," Mimeo, California Institute of Technology. Lu, X., AND P. McAFEE (1995): "Matching and Expectations in a Market with Heterogeneous Agents," in Advances in Applied Microeconomics, ed. by M. Baye. Connecticut: JAI Press. MA, C., J. MOORE,ANDS. TURNBULL (1988): "Stopping Agents from Cheating," Journal of Economic Theory, 46, 355-372. MASKIN,E., ANDJ. MOORE(1987): "Implementation and Renegotiation," Mimeo, Harvard University. McLENNAN, A., AND H. SONNENSCHIEN(1991): "Sequential

Bargaining as a Noncooperative

Founda-

tion for Walrasian Equilibria," Econometrica, 59, 1395-1424. MOORE,J. (1992): "Implementation, Contracts, and Renegotiation Environments with Complete Information," in Advances in Economic Theory, ed. by J.-J. Laffont. Cambridge: Cambridge University Press, pp. 182-282. D. (1982): "The Matching Process as a Noncooperative Bargaining Game," in The MORTENSEN, Economics of Information and Uncertainty,ed. by J. J. McCall. Chicago: University of Chicago Press. PETERS,M. (1991): "Ex Ante Price Offers in Matching Games Non-Steady States," Econometrica, 59, 1424-1444. PONSATI,C., AND J. SAKOvICS(1995): "Rubinstein Bargaining with Two-Sided Outside Options," Mimeo, Universitat Autonoma de Barcelona. POSTLEWAITE, A., ANDD. WETTSTEIN (1989): "Continuous and Feasible Implementation," Review of Economic Studies, 56, 14-33. A. (1982): "Perfect Equilibrium in a Bargaining Model," Econometrica, 50, 97-109. RUBINSTEIN, RUBENSTEIN, A., ANDA. WOLINSKY (1985): "Equilibrium in a Market with Sequential Bargaining," Econometrica, 53, 1133-1150.

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(1990): "Renegotiation-Proof Implementation," WP14-90, Tel Aviv University. (1992): "Renegotiation-Proof Implementation and Time Preferences," American Economic Review, 82, 600-614. ANDT. YAMATO(1993): "Toward Natural Implementation," forthcoming SAIJO,T., Y. TATIMATANI, in InternationalEconomic Review. M. (1995): "Search and the Efficient Assignment of Workers to Jobs," International SATTINGER, Economic Review, 36, 283-302. SHIMER,D., AND L. SMITH(1994): "The Normative Implications of Heterogeneity in Search," Mimeo, MIT. (1996): "Assortative Matching and Search," Mimeo, MIT. T. (1995): "Implementation by Demand Mechanisms," Economic Design, 1, 343-354. SJOSTROM, STAHL,I. (1972): Bargaining Theory. Stockholm: Stockholm School of Economics.

Econometrica,

Vol. 66, No. 6 (November, 1998), 1389-1404

INFERENCE ON STRUCTURAL PARAMETERS IN INSTRUMENTAL VARIABLES REGRESSION WITH WEAK INSTRUMENTS BY JiAHUi WANGAND ERIC ZIVOT1 We consider the problem of making asymptoticallyvalid inference on structural parametersin instrumentalvariablesregressionwith weak instruments.Using the localto-zeroasymptoticsof Staigerand Stock(1997),we derivethe asymptoticdistributionsof LR and LM type statisticsfor testing simple hypotheseson structuralparametersbased on maximumlikelihood and generalizedmethod of moments estimationmethods. In contrastto the nonstandardlimitingbehaviorof Waldstatistics,the limitingdistributions of certainLM and LR statisticsare boundedby a chi-squaredistributionwith degreesof freedom given by the number of instruments.Further, we show how to construct asymptotically valid confidencesets for structuralparametersby invertingthese statistics. KEYWORDS: Asymptoticdistribution,generalizedmethod of moments, instrumental variables,Lagrangemultipliertest, likelihoodratio test, limited informationmaximum likelihood,weak instruments.

1. INTRODUCTION

with weak instrumentshas recently INSTRUMENTAL VARIABLES(IV) ESTIMATION capturedthe attentionof appliedand theoreticaleconometricians.Researchhas uncovered two problems associated with inference on structuralparameters when IV methods are used and there are weak instruments.First, Nelson and Startz(1990a, 1990b),Buse (1992), and Bound, Jaeger, and Baker (1995) have shown that in the presence of weak instrumentsIV estimates can be strongly biased in the same directionas OLS estimates and IV-based inference can be highly misleading.Second, Dufour (1997), hereinafter referred to as Dufour, shows that in a limited informationsimultaneousequations model (LISEM) with very weak instrumentsWald statistics for structuralparametersare not asymptotically pivotal and thus the standard asymptotically justified "estimate+ 2 asymptoticstandarderror"type 95% confidence interval for a structuralparameterhas zero coverageprobability,at least when the admissible set of parametervalues is unbounded. To understandthese problems, Staiger and Stock (1997), hereinafter SS, derive alternativeasymptoticdistributionsfor TSLS (two-stage least squares) and LIML (limited-informationmaximumlikelihood) estimatorsof structural parametersin a LISEMbased on a local-to-zeroassumptionfor the coefficients on the instrumentsin the reducedform equation.Under fairlyweak conditions, 1The authorswould like to thank Jean-MarieDufour, CharlesNelson, RichardStartz,James Stock,and a co-editorand three anonymousreferees for numeroushelpfulcommentsand suggestions that greatly improvedthe presentationof the results in this paper. The second author acknowledgessupport from the Tinbergen Institute, Erasmus University Rotterdam,and the NationalScience FoundationunderGrantNo. SBR-9711301. 1389

1390

J. WANG AND E. ZIVOT

SS show that the TSLS and LIML estimatorsof the structuralparametersand Wald statistics do not have standardasymptoticdistributionsbut rather have distributionsthat depend on nuisanceparametersin a complicatedway. Inference on structuralparametersbased on the SS resultsis complicatedby the fact that some of the nuisanceparametersappearingin the asymptoticdistributions are not consistentlyestimable.One solution to this problem,which we take in this note, is to follow Dufour and base inferenceon structuralparametersusing test statisticsthat have asymptoticallypivotalor boundedlypivotaldistributions. The problemis to find test statisticsthat have this property. Since SS and Dufour'sresults show that Wald statisticsbased on TSLS and LIML are not asymptoticallypivotal or boundedlypivotalwe consider making inference on structuralparametersusing certain Lagrangemultiplier(LM) and likelihoodratio (LR) statistics.Our approachis motivatedby Dufour'sobservation that some LR statisticsfor hypotheseson nearly nonidentifiedparameters are boundedlypivotalundercertainconditions.We show,usingSS'slocal-to-zero asymptotics,that the asymptoticdistributionsof a particularTSLS LM statistic and the LIML LR statistic for testing simple hypotheses about structural parametersare asymptoticallyboundedlypivotal.Hence, our analysisshowsthat certain LM statisticsas well as LR statisticsprovideasymptoticallyvalid inference on structuralparametersin a LISEM.Further,we show how to construct asymptoticallyvalid confidencesets for structuralparametersby invertingthese statistics. This note is organizedas follows.Section 2 presentsthe statisticalmodel and introducesthe local-to-zeroframeworkused by SS. In Section 3, we derive the asymptoticdistributionsof LM and LR statisticsfor testing a simple hypothesis on the structuralparametersbased on generalizedmethod of moments(GMM) and LIML estimation.In Section 4 we show how to constructvalid confidence sets for the structuralparametersby invertingthe LM and LR statisticsusing the asymptoticbounding distributionsderived in Section 3. In Section 5 we evaluate the finite sample performanceof the LM and LR statisticsby Monte Carlo simulation.Section 6 concludesthe paper. 2.

THE LOCAL-TO-ZERO LISEM FRAMEWORK

The LISEMmodel we consideris the same as in SS: (1)

y=Yf3+Xy+u,

(2)

Y=ZH7+XP+

V,

where y and Y are a T x 1 and a T X n matrix of endogenous variables, respectively,X is a T x k1 matrixof includedexogenousvariables,Z is a T x k2 matrixof excludedexogenousvariables(or instruments),u is a T x 1 vector of structuralerrors, and V is a T x n matrix of reduced form errors. Let Z = [X Z], which is assumedto be full columnrankand uncorrelatedwith u and V.

INFERENCE ON STRUCTURAL PARAMETERS

Let Xt, Zt, and

Zt

1391

denote the tth observations on X, Z, and Z, respectively,

and assume that

E(Z*Z, [E(ZtXt)E(ZtZI)]

Qz

[QZx

1

The error terms ut and Vt are assumedto have zero mean, and to be serially uncorrelatedand homoskedasticwith ]

var[

["

UV]

E

We call the model just-identifiedwhen k2 = n and the model over-identified when k2 > n. Since we are only interested in making inference on 13,it is convenient to transform(1)-(2) using the Frisch-Waugh-Lovell theorem2as follows: (3) y =Y 1 8 + u Y =Z IH+V,

(4)

where A' =MxA

=

(IT - Px)A, for any conformable matrix A, and

PB=

B(B'B)-1B' denotes the matrixthat projectsonto the space spannedby any full rank matrix B. \P ~~~~~~~~~~~d Let "

->"

denote convergence in probability and

-"

" denote convergence in

distribution.As in SS, we use the followingtwo assumptions: C/V , whereC is a fixedk2 x n matrix.

ASSUMPTION

1: H =

ASSUMPTION

2: Thefollowinglimitsholdjointly:

HT =

(i) (u'u/T, V'u/T, V'V/T) -1-

(ii)Z'ZI/T (iii)U(X

p

-'

(oU

vu, Ivv);

Q; U/,Z, U / V,

X, VI V,

Z, VI

V-)

d(X TZU Xst

as N(O,L 0 Q). whereTI (Iu, Tzu vec(Ixv)', vec(Tzv)')' is distributed Assumption 1 is SS's local-to-zeroassumptionon the reduced form coefficients H. It is a device that allows the instruments Z to remain weakly correlatedwith the included endogenousvariablesY as the sample size grows large. As a result, the Wald statisticfor testing H1=0 in equation (2) is Op(, whereasin the standardfixed-H asymptoticsthe Wald statistictends to infinity. The local-to-zeroassumptionon H also impliesthe structuralparameters18are asymptoticallynearlynonidentified.Phillips(1989)showsthat there is a discontinuity in the asymptotic distribution of the IV estimate of 18 when 18 is nonidentified(H1= 0). The local-to-zeroapproachof SS allows the studyof the asymptoticdistributionin the neighborhoodof the parameterconfigurations 2For example, see Davidson and MacKinnon (1993).

1392

J. WANG AND E. ZIVOT

where 18ceases to be identified.The approachis useful in that it illustratesin a straightforward way the dependenceof the distributionon nuisanceparameters in the neighborhoodof points where the identificationconditionsfail. Indeed, SS are able to characterizesituations in which standardasymptotictheory is applicable and conditions in which it is not. Assumption 2 imposes weak momentconditionson the exogenousvariablesand errorterms. Under Assumptions1 and 2, SS show that the convergenceresults of sample momentsbased on (3)-(4) are different from the fixed-H case. The following lemma gives the main resultswe need in our analysis. LEMMA1: Suppose that Assumptions 1 and 2 hold for the model (1)-(2). Then the following convergenceresults hold jointly as T -> oo: crp (a) u"u'I T (b) 1tz1d 1X/v2 cr-1/2 v.1rl/21 =1L/2' (A + ZV)'Zu

(c) Y "Pz Y ' Xv (A+ Z)'(A+ (d) Tu "Pz u /u "u' Z'Zu 0 = = n weeA [2112C'7112, where D=QzQx QZ3v, VV

) +)/_2='vv

p1/2 = f2 Zu Zu=

QX, xxxz

QZXQ-1A

(z-

2'(1f'U

Z =f ,-1/2'(T V= '12 (A +ZV)'(A _QZXQXlT QZXQ_li' )u-1/2 + Zv), and v2 = (A + Zv)'Zu. The random variable (Z',vec(Zv)')' N(O, I

Ik2) and L is = P VV/VU

the (n + 1) x (n + 1) matrixwith 1 = 1, 2 =In, and whereL is partitionedconformablywith L. A1/2

Y21 = 312

3. TEST STATISTICS AND LIMITING DISTRIBUTIONS

In this sectionwe considertesting hypothesesabout 18using LM and LR-type test statisticsbased on GMM and LIML estimationtechniques.Considerfirst GMM estimation. Given the model (1)-(2) is linear, GMM estimation is equivalent to TSLS estimation. Let u ' (,8) = y

-

Y 1,8. Then the sample

moment condition for GMM estimation is given by mT(1) = ZU''u (18)= 0, which has no solution in general if k2 > n. Under Assumption2, T-1/2m( 8) d

->

N(O, W) where W = ouu[2. Let WT be a consistent estimator of W. Then the

efficient GMM estimator of 18 is found by minimizingthe quadraticform (1)Z WT Z u (1). Using WT= UUZ "Z' /T as mT(P)WT MT( 3)=U the weightingmatrix,where &ruuis an estimatorof o-uu,gives P8GMM= TSLS= (y

IlpIpZy

-y

ltp

lyi.

Newey and West (1987) show how to constructthe trio of statistics-Wald, LM, LR-in the context of GMM for testing general nonlinear hypotheses about 18.For the simple hypothesis Ho: 18= 180vs. H1: 1,80 80, the three test statisticstake the form: LRGMM=[U'(130)'PZlu'(1(0)-u'

LMGMM= u' WaldGMM=

(PGMM)PZ1U

( 8o)'Pz Y (Y Pz?Y') 1lyl" (O-GMM)Y

PZ1Y

(0

(GMM)]/&UU, A lu ( 3)/6

GMM)/UU.

1393

INFERENCE ON STRUCTURAL PARAMETERS

Newey and West (1987)show that these three statisticsare numericallyidentical is used. Hence we use S4> to denote the as long as the same estimatorof orUu commonvalue of the three statisticsfor testing the hypothesis18= 180. Typically, orUUis estimated using the TSLS estimator 6r

= u'(JGMM)U

is the TSLSWald statisticconsideredby SS. SS (f,iGMM)/T. In this case, S'> showthat underAssumptions1 and 2 the TSLSestimatorsof 18and o-uuare not consistent.In fact, they convergeto randomvariablesdue to the asymptoticnear nonidentificationof 18.Moreover,the S4> (WaldTSLS) statisticdoes not have a X2(n) limitingdistributionunderthe null hypothesisbut ratherhas a distribution that depends on the nuisance parametersk2, p, and XA/k2 (the noncentralityparameterof the asymptoticdistributionof the Wald statisticfor testing H = 0 in equation (2)). Part of the nonstandardlimitingbehaviorof SLf is attributableto the nonstandardlimitingbehaviorof u. This effect, however, ( which is a consistent can be eliminated by using ru = u (,80)'u (,80)/T, estimatorof o-uuunder the null hypothesis,instead of the TSLS estimate U. statisticcomputedusing crUu Let V/// denote the S o. Note that V/// has the natu'ralinterpretationof an LM statisticsince all of the componentsof the statisticare based on the value of 18under the null hypothesis.In addition, V.// is equal to T times the uncentered R2 from the regressionof u' ( ,80) on Pz l Y ' . The V/// statisticwas not consideredby SS or Dufour. Next considerLIML estimationof 18.SS focus on the k-class formulationof the LIML estimatorand considerWald statisticsbased on this formulation.As with TSLS estimation, SS show that under Assumptions 1 and 2 the LIML estimator is inconsistent,has a nonstandardasymptoticdistribution,and the asymptotic null distribution of LIML-based Wald statistics are nonpivotal. Dufour showsthat the LR statisticfor testing a hypothesisabout some transformation 8 = g( ,y, n H, ) of 13,y, H, and P based on the likelihoodfunctionof the unrestrictedreduced form of (1)-(2) is boundedlypivotal in finite samples assumingnormalerrors.In additionhe shows how this boundingresult can be used to constructa pivotalboundingdistributionfor the LR statistictesting Ho: ,8= 180 based on the structural model (3)-(4). This bounding distribution, however,is based on the distributionof the Wilksstatisticand is not particularly tight in models with many instruments.Instead of constructinga bound for the distributionof the LR statisticfrom the unrestrictedreducedform,we consider the possibilityof constructingan asymptoticallypivotalboundingdistributionfor the LR statistic computed directly from the concentratedlikelihood function for 18. The log likelihood function of (1)-(2) concentrated with respect to the parameters X y, P, and H (cf. Davidson and MacKinnon(1993, p. 647)) is given by A

A

(S)

Y'(,8)

nT - 2 ln

T 2)- -In 0,8)

T 2-njY

MzY'J,

1394

J. WANG AND E. ZIVOT

where Y= [y Y] and (6)

k(f3)

(y -y'f3)'(y

=

ly'/3)

The LIML estimator of ,3 is found by maximizing (5) which is equivalent to

minimizingk( ,8). It can be shownthat the minimizedvalue, k( fLIML)

= kLIML,

is the smallest root of the determinantal equation IY'MxY - kY'MzYI = 0. For testing the hypothesis Ho: ,3 = 80 vs. /8 80, the LR statistic based on (5) is (7)

T ln k(0)

LRLIML=

-T ln kLIML.

The following theorem gives the local-to-zero asymptotic distributions of S101 and LRLIML and shows that these distributions are boundedly pivotal. THEOREM 1: Suppose that Assumptions 1 and 2 hold for the model (1)-(2). Under the null hypothesisHo: /3= 80, as T - oo: (a) V2. S>X0 v 2 1

(b) LRLIMLd ZZU

k

where kVIMLis the smallest root of the determi-

IML,

nantalequationI 0*-k - 1I= O and 0* = [Zu(A+ Zv) [Zu(A+ Zv)].

(c) When k2 =n, the limiting distributions in parts (a) and (b) reduce to a X2(k2) distribution; when k2 > n, the asymptotic distributionsare bounded from above by a X2(k2) distribution. PROOF:

(a) Consider the LMGMMexpression for g o

= (u "u

/T)

1u

'PZI y

I

(Y

A#.///oThen

PZ I y

I

) -lyIPZ

IU

, + ZU(A ZV)[(A + ZY)'(A + ZV) 1(A + ZV)'ZU, where the convergence follows from parts (a)-(c) of Lemma 1. (b) After some algebraic rearrangement, the likelihood ratio statistic (7) may be rewritten as LRLIML= -Tln(1-u"PZ =

Tu Pz U /u"u'

u /ulu"u)-Tln(1 -T (kLIML -1)

+(kLIML-1)) + op(1).

The result follows from Lemma 1 part (d) and SS's Theorem 2. (c) When k2 = n, A + Zv is a nonsingular square matrix and kLIML = 1 so the limiting expressions for 4'A/ and LRLIMLreduce to Z'Zu - X2(k2). When k2 >n, ( A+Zv)[(A+Zv)(A +Zv)] 1(A+Zv)' =PA+ZV,which is the projection matrix onto the space spanned by A + Zv, so ZUPA ZZU
INFERENCE ON STRUCTURAL PARAMETERS

1395

Theorem1 showsthat underweak instrumentsthe asymptoticdistributionsof V.// and LRLIML depend on nuisance parametersbut can be boundedby a X2(k2) distribution.Some intuitionfor this result can be given by linkingthese statisticsto the Anderson-Rubin(AR) statistic,originallyproposedby Anderson and Rubin (1949), for testing the hypothesis Ho: 8 = 180 vs. 18o 1803 The AR statisticis given by u` (,10Y)'PzLu (10)/k2

AR(0)_ = u

0( 3)'Mz,

u(f0)/(T-k1-k2)

which has an exact F distributionin finite samples with normal errors and, under Assumptions 1 and 2, SS show that k2AR(,13) -> x2(k2). Let CT= Tk2/(T - k, - k2) so that k( 1,) = 1 + T-1CTAR( 1). After some algebrait can be shown that S'AV'O= CTk( ,10)1AR(0) - u ' ( 1GMM) 'Z' U ( IGMM)/UuO and LRLIML = CTAR( )- CTAR( 18LIML) + op(l). Now, under Assumptions 1 d and 2 CT->k2, k( 130) 1, cTAR( 30) X2(k2), and the second terms in the

and LRLIML are both positivewith probabilityone in the expressionsfor V'A'AO overidentifiedcase and zero in the exactly identified case. Hence 4'4O, LRLIML' and AR( 10) are asymptoticallyequivalentin the exactlyidentifiedcase and in the overidentifiedcase V/// and LRLIML are less than AR(,13) asymptoticallyand are thereforeboundedby a X2(k2)distribution. The statistics V/// and LRLIML are similar to AR(,10) but there is an importantdifference in the overidentifiedcase. By constructionAR(,13) tests the joint hypothesis 8 = 180and the validityof the overidentifyingrestrictions whereas V./9/O and LRLIML test 18=,80 and impose instrumentvalidity.As a result,joint confidencesets for 18based on invertingAR( ,80)can be misleadingly small if the data reject the overidentifyingrestrictions. 4. INFERENCE ON STRUCTURAL PARAMETERS

We have shown that V'/4'/ and LRLIML are asymptoticallybounded by a valid, but X2(k2) distributionwhen instrumentsare weak and so asymptotically often conservative,inference can be made on the n-dimensionalvector 18.In addition, asymptoticallyvalid inference using the X2(k2) distributioncan be made from k2AR. In principle, we can invert 9>A0,

LRLIML, and k2AR to

obtain valid joint confidencesets for 18.To illustrate,let T( ,80) denote either the K110, LRLIML,or k2AR statisticfor testing 18= 180.Then the asymptotically valid (1 - a). 100%joint confidence set for 18based on invertingT( ,80) takes the form C,,(a)

= {0

: T(1o)

< X?2(k2)}

3We thank a referee for pointing out this result.

={13:fa(10)?O}

1396

J. WANG AND E. ZIVOT

where ,2(k2) denotes the (1 - a) 100% quantile of a X2(k2) distribution, fa(

)=

(y1 Y4aY'

18 - 2y l4aY

+y 'l)aY' 13J

and the matrix

Aa

depends

"Pz? on the particular test statistic used. For , Aa=Pz ?Y'(Y Y YY for LRLIML,Aa =IT-kLIMLexa 2)/TMz ; and PZ -ITXa2(k2)/T; - k, - k2). The function fa(,8) defines for k2AR, A. = Pz, -Mz Xa2(k2)1(T either an ellipsoid or a hyperboloidin an n-dimensionalspace, as long as Y lAY ' is nonsingular.Thus the joint confidenceset C, (a) will be either the "inside"or the "outside"of an ellipsoidof hyperboloidwhere the shape of the confidenceset is determinedby the eigenvaluesof Y 'AaY ', and the sign of the constant y LA Y y -y 1A y L'(YI A Y ')Y1Y IAaY' determines whether the confidenceset lies inside or outside of the shape. To illustrate, suppose we have two right-hand-sideendogenous variables (n = 2). Representativecontourplots of fa( ,8) for four possiblecases are shown in Figure 1 and the joint confidence sets for /3= (011,32)' are given by the shaded areas where fM(3) < 0. In Case I, both eigenvaluesof Y 'iay' are positiveand the confidenceset is given by the inside area of an ellipse. In Case II, both eigenvalues are negative and the confidence set is given by the unboundedoutside area of an ellipse. In Cases III and IV, one eigenvalueis positive and the other negativeand so the confidenceset is given by either the outside or the inside areas of a hyperboladependingupon whether the saddle path of the hyperbolais above or below zero. In contrastto Wald-basedjoint confidencesets, which are alwaysbounded (Case I), the confidencesets determined by invertingthe , LRLIML, and AR statisticscan be unbounded regions (Cases II-IV). Simulationexperiments(not reported)show that these unboundedregions often occur in models with weak instruments.While unfamiliar,the results of Dufour indicate that such confidencesets are appropriate in IV models with nearlynonidentifiedparameters.Indeed, Dufour shows that any valid (1 - a) 100%confidenceset for 18must be unboundedwith probability close to 1 - a for models in which 18 is nearly nonidentifiedwhen the admissibleset of parametervalues is unbounded. For the case of a single right-hand-sideendogenous variable,the different types of confidencesets for 18are discussedby Dufour and Jasiak (1996) and Zivot, Startz, and Nelson (1997). Zivot, Startz, and Nelson (1997) show that confidence regions formed by inverting K110, LRLIML, and AR may be boundedintervals,empty (AR only), cover unboundedregions on the real line or cover the entire real line. In particular,these regions are unboundedwhen the first stage regressionis not significant.4Unboundedconfidencesets in this

4Zivot,Startz,and Nelson (1997)show that in the n = 1 case Cp(a) based on invertingAR and is unboundedwheneverthe F statisticfor testing 7 = 0 is insignificantat level a and that Cp(a) based on invertingS'XZ is unboundedwheneverthe LM-statisticfor testingwhen 7 = 0 is insignificantat level a. In addition,they show that when 7 = 0 the (asymptotic)probabilitythat C, (a) is unboundedis equalto 1 - a. ThisconfirmsDufour'sresultsthatvalidconfidencesets must be unboundedwith probabilityclose to (1 - a) for nearlynonidentifiedmodels.

LRLIML

INFERENCE ON STRUCTURAL PARAMETERS

CaseI: Insideof an ellipse.

CaseII: Outsideof an ellipse. FIGURE l.-Examples

1397

CaseIII:Insideof a hyperbola.

Case IV: Outsideof a hyperbola. of bivariate confidence sets.

case are appropriate since the data suggest that 8l is nearly nonidentified so that almost any value of ,B is consistent with the data. In the case of multiple right-hand-side endogenous variables and weak instruments, the bivariate illustrations in Figure 1 show that unbounded confidence sets can have many different forms and that these sets can still provide important information. That is, the data may be more informative about some parameters or parameter configurations than others and this will be reflected in the shape of the confidence set. For example, Stock and Wright (1997) invert bivariate AR-type statistics for testing hypotheses about structural parameters in several Euler equation asset pricing models with weak instruments and often find unbounded but nevertheless informative confidence intervals. An apparent drawback of using 9.//,0, LRLIML, and AR for inference is that these tests are defined for a joint hypothesis on the full vector ,B. However, if we are interested in making inference on individual elements of ,(3,or some function of ,B, a Scheffe-type projection method as described in Dufour and Dufour and Jasiak (1996) can be employed to make valid inference on the parameters of interest. To illustrate, let C,3(a) be an asymptotically valid se confidence ellipse.ed LR or AR on invertingOutsid0,IMLa (1 - Ca) 100%

1398

J. WANG AND E. ZIVOT

and consider some nonlinear transformation -q=g( 83).Then the confidence set for -q defined by C,(4a) = {7Ro:-qo=g( 3) for some ,BE C o(a)} has asymptotic coverage probability at least 1 - a. If -q= g( /8) = /3i,an individual element of /3, the set CJ( a) is simply the projection of C, (a) on the /3i-axis. For a scalar function -q= g(,3), a valid confidence interval [-qL, % ] can be constructed by solving )L = inf{j0o:0E/- C0(a )) and )H = sup{j0o:0E/- C0(a )}, which are obtained by minimizing and maximizing -ro= g(,/3) subject to the restriction fa( ,80) < 0. The construction of asymptotically valid confidence sets for /8 based on and AR is conceptually and computationally more inverting KW',A LRLIML, than the Bonferroni method suggested by SS because the limiting simple (bounding) distributions of the test statistics do not depend on nuisance parameters.5 However, the choice of which procedure to use in practice ultimately depends on the finite sample size and power of the tests, an issue we take up in the remaining sections. 5. MONTE CARLO RESULTS WHEN

n

=

1

5.1. Comparisonof Asymptoticand Finite Sample Distributions Monte Carlo experiments are conducted to evaluate the asymptotic approximations to the distributions of K',AXAand LRLIML and to compare the coverage probabilities of 95% confidence sets constructed by inverting th6 AR, SX>0, LRLIML, and WaldTSLS test statistics when n = 1. The experiments are set up according to SS's design I. That is, data are generated from (1)-(2) with ,8 = 0, y = 0, X a vector of ones, Z iid N(O,Ik2) and (u, Vi)' iid N(O, ]). Results are reported for k2 = 1, 4; p = 0.5,0.99; A'A/k2= 0, 0.25, 1,10 and are summarized in Table I and Figure 2. The finite sample and asymptotic pdfs of K,'AX0and LRLIML are plotted in Figure 2 for the case p = 0.99, A'A/k2= O. Plots for the other cases are similar. The asymptotic approximations appear very good at the level of detail of the plots and for T/k2> 10 there is very little difference between the finite sample distributions and the asymptotic approximations. Table I gives the finite sample coverage rates of nominal 95% confidence sets formed by inverting the AR L ML, and WaldTSLS test statistics. When k2 = 1, the asymptotic ILRI, are X2(1) whereas the distribution of distributions of AR, A',Ao, LRLIML WaldTSLS depends on the nuisance parameters p, A'A/k2, and k2. The finite sample coverage rate of K',AXAis 95% for both values of p whereas the are close to 85%. The coverage rate of coverage rates of AR and LRLIML is sensitive to the value of p and A'A/k2 and, as noted by Hall, WaldTSLS Rudebusch, and Wilcox (1996), Zivot, Startz, and Nelson (1997) and SS, is worst 50f course,SS's Bonferronimethodcould also be appliedto W,A// and LRLIML.

1399

INFERENCE ON STRUCTURAL PARAMETERS TABLE I FINITE SAMPLECOVERAGERATES OF 95% CONFIDENCEINTERVALS T/k2=

Parameters

p

'A/k2

0.50 0.50 0.50 0.50 >0.50 0.50 0.50 0.50 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99

0.00 0.25 1.00 10.00 0.00 0.25 1.00 10.00 0.00 0.25 1.00 10.00 0.00 0.25 1.00 10.00

x2(l)

k2

WaldTsLs

GMMo

1 1 1 1 4 4 4 4 1 1 1 1 4 4 4 4

0.88 0.88 0.87 0.82 0.81 0.82 0.85 0.91 0.31 0.53 0.67 0.79 0.01 0.13 0.42 0.85

0.95 0.95 0.95 0.95 0.85 0.87 0.91 0.94 0.95 0.95 0.95 0.95 0.51 0.62 0.78 0.93

Parameters p 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99

'A/k2 0.00 0.25 1.00 10.00 0.00 0.25 1.00 10.00 0.00 0.25 1.00 10.00 0.00 0.25 1.00 10.00

5 Switching

x_2(k2)

LR

k2 AR

0.84 0.84 0.84 0.84 0.72 0.75 0.82 0.92 0.84 0.84 0.84 0.84 0.72 0.92 0.93 0.93 T/k2 = 20

0.85 0.86 0.85 0.85 0.90 0.90 0.90 0.90 0.86 0.85 0.85 0.86 0.90 0.90 0.90 0.90

x2(l)

GMMo

LR

GMMo

LR

1.00 1.00 1.00 1.00

0.95 0.96 0.98 0.99

0.98 0.98 0.97 0.94

0.93 0.93 0.93 0.92

0.97 0.98 0.99 1.00

0.95 0.99 1.00 1.00

0.96 0.93 0.86 0.93

0.91 0.96 0.96 0.94

GMMo

LR

GMMo

LR

0.99 0.99 1.00 1.00

0.97 0.98 0.99 1.00

0.97 0.97 0.95 0.94

0.96 0.95 0.95 0.94

0.95 0.96 0.99 1.00

0.98 1.00 1.00 1.00

0.95 0.90 0.83 0.93

0.94 0.97 0.97 0.95

x2(k2)

k2

WaldTSLS

GMMo

LR

k2 AR

1 1 1 1 4 4 4 4 1 1 1 1 4 4 4 4

0.97 0.97 0.96 0.94 0.86 0.86 0.88 0.93 0.36 0.65 0.79 0.90 0.01 0.14 0.44 0.87

0.95 0.95 0.95 0.95 0.86 0.88 0.91 0.95 0.94 0.95 0.94 0.95 0.57 0.66 0.80 0.93

0.93 0.93 0.93 0.94 0.77 0.80 0.87 0.94 0.93 0.93 0.93 0.93 0.77 0.94 0.95 0.95

0.94 0.93 0.93 0.94 0.94 0.94 0.94 0.94 0.93 0.93 0.93 0.93 0.94 0.94 0.94 0.94

Switching

Notes: Number of simulations= 10,000. The column labeled "switching" gives the coverage rates based on choosing the degrees of freedom from a pre-test of the significance of the first stage regression.

when p is close to 1 and A'A/k2 is close to zero. When k2 = 4, the asymptotic distributions of VWX0, LRLIML, and WaldTSLS are nonpivotal but the standard X2(1) distribution works well for V1A>0 and LRLIML if A'A/k2= 10. In all cases, the asymptotic Y2(k2) bounding distribution for W'ASA0and LRLIML holds in finite samples.

1400

J. WANG AND E. ZIVOT

k=1, T=20, p=0.99,X'Vk=O 1.2

4_

1.0

0.8

_

|

_ . .. . .. ----

8

9

!ill ,A

t

_

0.6

_

_

_

_

_

LR (fs) LR (asy) GMM0(fs) GMM. (asy)

0.4 0.2 0.0j 2

1

0

4

3

5

7 6 LR GMMO,

10

11

12

k=1, T=80, p=0.99,X'Vk=O 0.4

-

0.3

-

...... . .. .----GTMM

0.2

/

0.1 -0

%.

__

LR (fs) LR (asy) GMMV(fs) (asy)

_____

___

0.0 0

2

4

6

8

10 12 GMMO,IR

14

16

18

20

Notes: Finite sample and asymptoticdistributionsare based on 10,000 simulations. FIGURE 2.-Asymptotic

and finite sample pdfs of GMMo and LR.

The K',A0 and LRLIML confidence sets based on 2(k2) critical values may be large if k2 is large, so the power of the KWW and LRLIML tests may be poor when instruments are reasonably good (A'A/k2 is large). To avoid such a possibility, Zivot, Startz, and Nelson (1997) suggest inverting K,'AX0and LRLIML subject to a pre-test of the significance of H in the first stage regression (2).

INFERENCE ON STRUCTURAL PARAMETERS

1401

Specifically, let P denote the value of such a pre-test statistic and let Pa denote the critical value such that, under the null H = 0, Pr(P > P) = a. Then, if P < Pa invert VWX0 or LRLIML using Xa(k2) and if P > Pa use instead Xa(1). Zivot, Startz, and Nelson (1997) suggest using F=o and LMHI=o as the pre-test statistics when inverting LRLIML and VW>0, respectively, and they denote these statistics that switch degrees of freedom LRLIJL and . The finite sample coverage rates for these pre-test based confidence intervals are given in Table I under the columns labeled "switching." The coverage rates for LRLIML are almost exact and the coverage rates for f(fosW are off a little only when AA/k2 = 1. These results are in contrast to the results reported in Hall, Rudebusch, and Wilcox (1996) who, based on an extensive Monte Carlo analysis, found no clear benefits from pre-testing and using the Wald statistic to make inference on structural parameters. The reason that pre-testing works in the present context is that the distributions of K'AZA and LRLIML conditional on the realized value of P are much better behaved than the Wald statistic. For a more detailed analysis see Zivot, Startz, and Nelson (1997). 5.2. Power Comparisons Empirical powers of various 5% test statistics were computed using the Monte Carlo design of the previous section under a range of alternatives Ha :3 P=a where Pa = 80 + 6i and 6i ranges from -2 to 2 in increments of 0.2. Powers are computedfor AR, KW',A LRLIML, ? 4/SWJ LRsWL, as well as SS'sasymptotically valid Bonferroni tjBJML and tl%Ls statistics.6 The Bonferroni statistics are computed as follows. Given a 100s (1 - a1)% confidence interval for A'A/k2, and (1- a2)* 100% confidence sets for , given AA/k2,CtIAA/k2(a2) CA'A/kial), the confidence set defined by p

Aa

UAk'/k2 E CA,A/k2(a1) Ca lAA/ k(2

2

has coverage rate at least (1 - a) 100% by Bonferroni's inequality where a = a1 + 27 Power curvesfor the statistics AR, V./f//fsw, LRsWML,and tLIML are illustrated in Figure 3 for the case k2 = 4 and T = 100 with weak (AA/k2 = 1) and moderately good (AA/k2 = 10) instruments. When k2 = 1, the powers of AR, KW>o, and LRLIML are roughly equivalent

and are higherthan the powersof dominates all others but

((4SW

tLBIMLand tl%Ls. When k2 - 4 no test strictly and LRswML appear to have the best overall

6The powers of the 5% test statistics, except Vzz and LRLIML in the overidentified case, are size-corrected using the finite sample critical values under the null. The size-correction for -I/,SW and LRLIwJL is obtained by adjusting the size of the pre-test statistic, using a line search procedure, W and LRsIwJL are 5%. so that the empirical sizes of 7 Specific details on the construction of tLIML and tTSLs are described in SS. For the power results, we set a1 = a2 = 0.025. As noted by a referee, higher power for these procedures may be achieved by using a smaller value of a1. We thank Douglas Staiger for generously providing the GAUSS code to compute these statistics.

1402

J. WANG AND E. ZIVOT

T=100, k--4, p=0.25, VVk-O10

T=100, k'4, p=0.25, MXkI1

ft~~~~~~~~~~~LA

0J.5. 1.0 -1. -0. 0. . ~15 ~10 ~-05.0~~~~~Altenative: ~ ~ v0.5~ ~ ~1.5 ~2.0 ~i o~~~~~~~~~~~~~~06 ~ ~ -2.~ -1,~ ~~~~~~~~A f Alteirnatve< ftD

-20 06 u

0.(~~~~~~~~~~~~~~~~~~~~. T=lO0,k4, p=055X'X/k 0.0~~~~~~~~~~~~~~~~~~~~. 10-.0aD -.

-1.5

-1.0

-0.5

00

. ? <

o5

.

.

1.0

1.5

Alternative:ft

Altemaitive:ft

-2.0

tt

T=lO0,k=4,P=05,X'/lO~1

-

O.

.

. [

0.5

1.0

2.0

2.0

1.5

-1.5

-1.0

-0.5

0.5

0.0

2.0

Notres Number of simulations= 10,,000 The culrvelabeled tLIML gives SS's Bonferroni LIML t statistic. The curves ft~ ~~~~~~~~~~~~~~~~~~~~~~~~tO labeled LRSW and GMMSW give the LRLIML and GMM, statistics based on choosing the degrees of freedom from a pre-test of the first stage regression.

~ ~ ~

0.2

0.4~~~~~~~~~~~~~~~~~~~~. T=100, k=4, p=075, X'Vk-1

1=100, k-4,

1.0

1

Alternative:ft

Alterna0tive:ft

~0.4

p=075,;XVk=10

1;_

......,R

0.8

08...A

GMW

I statistic. of The curve 2.0 labeled tLIML -2.0 gives -5 SS's Bonferroni LIML The curves 2.0 Notes.-2.0Number -05 -10 00 0.5 1.0 1.5 0 - 10,000. 1 1.5 -1. simulations -0.5 .5 -1.5 LRSW and GMMsw give the LRLIML, and GMMo, statistics based on choosing the degrees of freedom from a labeled 0.606 pre-test of the first stage regression. 0.404

T=100,k=4,p=-075,X

3.-Empir

TiI

~~~~~~~~~~~~~~~~~~~~0.2

0.2

-2.0

-1.5

-1.0

0.0 0.5 -05 Alterative ft

1.0

Notes:Number9 siulton ofI--IT= ,100.Te lable LRWadGMWgv1h.RJLad pre-test of the first stage regression

~

0.8

0

00pk:w4e p-o0f testd

1.5

2.0

-2.0

uvelbee

~

...FGUE..Emiicl

M0

~

-1.5

-1.0

1I iesS'sBnfroni9, ttsisbsdo hoin

~

oer

~

~

f es taitis

-0.5 0.5 0.0 Alternative:ft

L

~

h

1.0

1.5

sttsi.Tecre M c= I ffedmfo ere tLD

~

~

2.0

INFERENCE ON STRUCTURAL PARAMETERS

1403

power,with the power of Jjos/W slightlygreater than LRsjW LI 9 Z0 and have the worst power,and the power of tLIML dominatestTSLS. All of the powercurvesare similarin shape and are asymmetricabout /8= 0 for p 0 0 and the degree of asymmetryincreases as p increases. AR exhibits the most asymmetryand tLBIML the least. For the weak instrumentcase, AR and LRswML

LRLIML

have higher power than

tLBIML and J/g/sW

has higher power than

tLBIML except

for a small regionwhen p > 0.25. When p > 0.25, AR has slightlyhigherpower and LRswML than ?7g4sW and LRswML in some cases for 8 > 0 and 2/g/JW have higher power than AR for 8 < 0. Interestingly,for p> 0.5 the power curves are not monotonicbut have peaks near the median value of /8oLS. For and LRswML dominate AR and tLIML and the good instrumentcase J,/(4sw AR has slightly better power than

6.

tLBML for

8 < 0.

CONCLUSION

For nearly nonidentifiedmodels, Dufour argues that Wald-basedconfidence sets based on the familiar "estimate+ 2 asymptoticstandarderror"may be highlymisleading.SS develop a new asymptotictheory to analyzethis problem in instrumentalvariables regression with weak instruments.The asymptotic resultsin SS for t statisticsbased on TSLSand LIML,while highlyinformative, are not straightforward to applyin practicesince they involveunknownvalues of nuisance parameters.In this note, we have shown how asymptoticallyvalid inferenceon structuralparametersin LISEMscan be easily made using certain LR or LM statistics.In contrastto SS, our methods do not require estimating nuisance parameterssince our test statistics are asymptoticallyboundedlypivotal. In addition,we have shown that for a model with a single endogenous variablea degrees-of-freedomadjustedversion of our test statisticsbased on a pre-test of the significanceof a given set of instrumentshas proper size and generallyhigherpowerthan SS's Bonferronit statistics. MathSoftInc., 1700 WestlakeAve., N., Suite 500, Seattle,WA 98109, U.S.A.; com jwang@statsci. and Dept. of Economics, Universityof Washington,302 SaveryHall, Seattle, WA 98195, U.S.A.;[email protected] ManuscriptreceivedJuly, 1996; final revision receivedDecember, 1997.

REFERENCES

T. W., AND H. RUBIN (1949):"Estimationof the Parametersof a Single Equationin a Complete System of Stochastic Equations," The Annals of Mathematical Statistics, 20, 46-63. BOUND, J., D. A. JAEGER, AND R. M. BAKER (1995): "Problemswith InstrumentalVariables Estimation When the Correlation Between the Instruments and the Endogenous Explanatory Variables is Weak," Journal of the American StatisticalAssociation, 90, 443-459.

ANDERSON,

1404

J. WANG AND E. ZIVOT

BUSE,A. (1992): "The Bias of Instrumental Variables Estimators," Econometrica, 60, 173-180. DAVIDSON, R., AND J. G. MACKINNON (1993): Estimation and Inference in Econometrics. Oxford: Oxford University Press. DUFOUR, J.-M. (1997): "Some Impossibility Theorems in Econometrics with Applications to Structural and Dynamic Models," Econometrica, 65, 1365-1388. DUFOUR, J.-M., AND J. JASIAK (1996): "Finite Sample Inference Methods for Simultaneous Equations and Models with Unobserved and Generated Regressors," Mimeo, C.R.D.E., Universite de Montreal. HALL, A. R., G. D. RUDEBUSCH, AND D. W. WILCOX(1996): "Judging Instrument Relevance in Instrumental Variables Estimation," InternationalEconomic Review, 37, 283-298. NELSON, C. R., AND R. STARTZ (1990a): "The Distribution of the Instrumental Variables Estimator and Its t-ratio When the Instrument is a Poor One," Journal of Business, 63, S125-S140. (1990b): "Some Further Results on the Exact Small Sample Properties of the Instrumental Variables Estimator," Econometrica, 58, 967-976. NEWEY, W. K., AND K. D. WEST (1987): "Hypothesis Testing with Efficient Method of Moments Estimators," InternationalEconomic Review, 28, 777-787. PHILLIPS,P. C. B. (1989): "Partially Identified Econometric Models," Economic Theory, 5, 181-240. STAIGER,D., AND J. H. STOCK(1997): "Instrumental Variables Regression with Weak Instruments," Econometrica, 65, 557-586. STOCK, J. H., AND J. WRIGHT (1997): "GMM with Weak Identification," Unpublished manuscript, Kennedy School of Government, Harvard University. ZIVOT, E., R. STARTZ, AND C. R. NELSON (1997):"ValidConferenceIntervalsand Inferencein the

Presence of Weak Instruments," forthcoming in InternationalEconomic Review.

Econometrica,

Vol. 66, No. 6 (November, 1998), 1405-1415

NOTES AND COMMENTS

IMPARTIALITY:DEFINITIONAND REPRESENTATION BY

EDI KARNI1

1. INTRODUCTION IMPARTIALITY IS THE MORAL IMPERATIVE

requiring that conflicting claims be evaluated

without prejudice.In this paper I propose an axiomaticdefinitionof impartialityand examineits implicationsfor the theoryof socialwelfarefunctions.Followingthe seminal work of Harsanyi(1953, 1977) I take as given the set of individualsthat constitutethe society and the set of social alternatives,representingthe constitutions,income distributions, institutions,or policies amongwhichthe societymust choose. Moreover,the moral value judgmentthat shouldgovernthis choice is modeled as a preferencerelationof an ethical observer. However, unlike Harsanyi,who defines the observer's preference relation on the set of all extendedlotteries (i.e., joint probabilitydistributionson social alternativesand individuals)I define the observer'spreference relation on a set of allocationswhose elementsare assignmentsof social-alternativelotteries(i.e., probability distributionson the set of social alternatives)to individuals.As in Harsanyi'stheory,the observer'spreferencerelationis supposedto governthe choice amongsocial alternatives of individualsplaced behind a "veil of ignorance"regardingtheir social position and preferences. Harsanyi assumes that the observer'spreference relation on the set of extended lotteries and the individualpreferencerelationson the set of social-alternativelotteries satisfythe axiomsof expectedutilitytheoryandjointlysatisfythe principleof acceptance.2 He showsthat the observer'spreferencerelationmay be representedas a weightedsum of individualvon Neumann-Morgenstern utilities and defines impartialityas the restriction that the individualutilities are assignedequal weights.This representationmay be interpretedas assigningequal probabilitiesto the events of being each individualin society. Note, however, that given any preference relation that is representableas a weightedsum of individualutilitieswith strictlypositiveweights,a new set of individual utilities may be defined (by multiplyingeach utility function by its weight and dividing throughby the inverseof the numberof individuals)to obtain a new representationof the preference relation with uniform weights. In other words, the same observer's as a weightedsum of individualutilitieswithequalweights. preferencerelationis represented Since this manipulationdoes not changethe underlyingobserver'spreferencerelation,it does not make it impartialexcept in a tautologicalsense. 11 am deeplyindebtedto John Weymarkfor very stimulatingdiscussionsof Harsanyi'simpartial observertheoremduringwhich the need for axiomaticdefinitionof impartialitycrystallizedand to Philippe Mongin, Zvi Safra, Uzi Segal, an editor, and two anonymousreferees for very useful commentsand suggestions. 2The principleof acceptancerequiresthat, when facing a choice amongextendedlotteriesthat assignthe unit mass to individuali, the preferencerelationof the observeragreeswith the actual lotteries. preferencerelationof individuali on the set of social-alternative

1405

1406

EDI KARNI

The analysis that follows departs from that of Harsanyiin an importantrespect, namely, impartialityis defined axiomaticallyin terms of the observer'spreferences. Methodologicallywe adopt the revealed preference approach(i.e., the evaluation of conflictingclaimsis based solelyupon individualordinalpreferenceson social-alternative lotteries).To graspthe main ideas considera society consistingof two individuals,say A and B, facing the problemof allocatinga fixed amount,W, of wealth. Supposethat the two individualsare identical in every respect except for their evaluationsof the social alternativelotteries.Let WAand WB denote the wealth allocatedto A and B; then the set of social alternativesis the set of all pairs of nonnegativenumbers,(WA,WB),whose sum is W. Suppose that each individualhas unique most and least preferred social alternativesand certaintyequivalentfor each social alternativelottery.Then our definition involvesthe followingjudgments:First, the outcome "being individualA when the social alternativeis A's most preferredalternative,"is assumedethicallyequivalentto the outcome "being individualB when the social alternativeis B's most preferred alternative."In other words,facing a choice between A's and B's most preferredwealth distribution,the society shouldbe indifferentbetween these two alternatives.This value judgment,which is central to our approach,is obtainedby default.The methodological frameworkof revealed preference providesno ground for preferringone individual's most preferred alternativeover that of the other. Consequently,strict preference in either directionis either biased or involvesconsiderationsother than the rank order of the alternatives.The same logic appliesto A's and B's worst alternatives.Second, A's certaintyequivalentof a social-alternativelottery that assigns the probabilityA to A's most preferredsocial alternativeand the probability(1 - A)to A's least preferredsocial alternativeis equivalentto B's certaintyequivalentof the social-alternativelottery that assigns the probabilitiesA and (1 - A), respectively,to B's most preferred and least preferred social alternatives.Third, these value judgments are extended to general social-alternativelotteries as follows: Given any social-alternativelottery, 1, friced by individual A, find for every element, x, in its support the (unique) corresponding social-alternativelottery that assigns the probabilityA(x) to A's most preferredsocial alternativeand the probability(1 - A(x)) to A's least preferredsocial alternative,for which x is the certaintyequivalent.Let y(x) be individualB's certaintyequivalentof the social-alternativelotterythat assignsthe probabilitiesA(x) and (1 - (x)) to B's most and least preferredsocial alternatives,respectively,and define 1' to be the social-alternative lotterythat assignsto y(x) the probabilitythat 1 assignsto x. Our definitionrequiresthat if A's and B's rankingsof 1 and 1' do not agree, the societyis indifferentbetween 1 and 1'. Implicitin this is the ethical judgementthat the loss to A from the choice of the lotterypreferred-by B is equivalent,in an ordinalsense,to the loss to B from the choice of the lottery preferredby A. Under these restrictionsimpartialityis shown to be that aspect of the observer'spreferences that requires the weights assigned to individu-6l utilitiesin the representationto be inverselyproportionalto the diameterof the imageof these utilities. Thus, the imposition of impartialityrenders meaningfulinterpersonal comparisonsof variationsin ordinalwell-being. Two recent contributionsto the social choice literature,by Dhillon and Mertens(1996) and Segal (1996),are closely related to the presentwork.3While adoptingphilosophical positionsand using analyticalframeworksthat are differentfrom those used here, they obtain the same social welfarefunction. 3Surveys of the literature dealing with utilitarian social welfare functions are provided by Sen (1986) and Mongin and d'Aspremont (1996).

IMPARTIALITY

1407

Dhillon and Mertens(1996) follow Arrow's(1963) quest for ethical rules that should governthe aggregationof individualvalues, and adopt his multi-profileapproach.They modifyArrow'sframework,weakeningthe axiomof independenceof irrelevantalternatives and introducingthe preference structureof expected utility theory. Within the modifiedframework,they axiomatizea unique social welfare function,which they call "relativeutilitarianism,"representingsocial preferences as the sum of individualvon Neumann-Morgenstern utilities normalizedto have infimumzero and supremumone. Segal (1996) rejects Harsanyi'snotion that individualsare capableof the detachment requiredby the "veil of ignorance"argument.His objectiveis to discernthe implications of a moral principle that can be unanimouslyendorsed by a society with any initial endowment(withoutnecessarilyimplyingan agreementon a social preferenceordering). Segal defines this principleto be that all dictatorshipsare equallybad. In other words, each individual is indifferent between an allocation that assigns the entire wealth possessed by the society to himself and any allocationthat assigns the society'sentire wealth to any other individual.

2.

IMPARTIALITY: A DEFINITION

2.1. Preliminaries Let N= {1,...,n}, o > n 2 2 be a set of individualsand X a topological space. Elementsof X are socialalternatives (e.g., elementsof X maybe alternativedistributions of wealth, in which case X is identifiedwith R+.) Let Z be the Borel o-field on X. A social-alternative lotteryis a probabilitymeasureon the measurablespace (X,X). Let P denote the set of all social-alternativelotteries(i.e., the set of all probabilitymeasureson (X,X)) and assumethat P is o-convexand is closed underthe formationof conditional expectations.4For every x e X let Axdenote the degenerateprobabilitymeasure that assigns {x} the unit mass. Assume that AxE P for all x. For each i E N, let >i be a binary relation on P. We refer to >i as the preference relation of individual i. The strict preference relation, i be the quotient of X with respect to -i (i.e., elements of X/ -i are subsets of X defined by {y eXX1y -i Ax}for every x eX). Denote by pi the natural mapping from X onto X/ i that carries each point of X to the element of XI >i containing it. Suppose that X/ >i is endowed by the quotient topology induced by pi and let X be the corresponding Borel a-field on X/ >i . Clearly, pi is measurable. Given any p E P and i E N, let Pi(p) be a probability measure on X/ >i defined by Pi(p)(Z) =p( pi1(Z)) for each Z E-. Let Pi be the set of all such probability measures (i.e., Pi = { i(p)lp e P}). We assume throughout that any two lotteries over X are indifferent if they induce the same lotteries on a set of indifference classes. Formally, we have the following assumption. 4We define the convex operation on P as follows: For all p,q E P, a E [0,1], and Z e-', = ap(Z) + (1 - a)q(Z). Similarly,u-convexityis defined using the operation = Ek=1 akPk(Z) Ek=1 akPk(Z), where a 2 0, for all k, and Ek= l ak = 1. P is said to be closed underformationof conditionalexpectationsif, for all p E P, the conditionalprobability,p(.Z), is in P for all Z E=' such that p(Z) > 0. (ap + (1 - a)q)(Z)

1408

EDI KARNI

ASSUMPTION (A.0): For all p, q E P, Pi(p) =j3i(q) impliesp -i q, i E N. Define individual i's induced preference relation >_ on Pi as follows, for all Pi(p), qi(q) E Pi, Pi(p) >i qi(q) if and only if p >i q. Note that Pi is a convex subset of a linear space and that a is antisymmetric. Assume the following for each i E N. ASSUMPTION(A.1): >i is a weak order satisfying the Archimedean and independence axioms of expected utility theory and the corresponding strict preference relation, >-i is nonempty.5

The above axioms are necessary and sufficient for the existence of expected utility representation on the set of social-alternative lotteries with finite supports. However, our main result requires the existence of expected utility representation of preference relations over more general sets of social-alternative lotteries. To obtain such representation we follow Fishburn (1970) in assuming that the sets {x E XI >i 15x>-i y) and {x eX/I Iys >-iA x}belong to r for ally eX/ i and i eN. Moreover >i satisfies the following assumption. ASSUMPTION(A.2) (DOMINANCE): For each i E N and Z Et, p(Z) = 1 implies that for all y E X/I i, if by i Ax for all x E Z, then by >i p, and if Ax>i by for all x E Z, then

P >-i 16Y The expectation of a bounded, real-valued X-measurable function f on X with respect to a probability measure p is defined in Fishburn (1970, Section 10.3) and is denoted Jxf(x) dp(x). Then the following expected utility theorem holds (see Fishburqi (1970, Theorem 10.3). The continuity of the utility function is an implication of the richness of the set X/I THEOREM (T.1): Let >i be a binaryrelation on Pi. Then >i satisfies (A.1), (A.2) if and only if there exists a real-valued, continuous, bounded utilityfunction ui on X/ >i such that, for allp, q E Pi, p >i q if and only if Jx/ >, ui(x) dp(x) 2 Jx/,, ui(x) dq(x). Moreover, ui is unique up to positive affine transformation. Having eliminated the symmetric parts of the individual preference relations on the set of social alternatives by moving to the quotient spaces X/ >i, we now reconcile the different preference relations using an analytical construct introduced in Karni (1993). For every x E X 1 and i e N we associate a unique element qii(x) e X/ i such that: (a) it is order-preserving on the set of social alternatives (i.e., 8x >-1 8y implies 8qi(X) >i 8q.(Y) and (b) any probability measure, p, on X/ 1 is associated with a probability measure ri(p) on X/ >i that assigns to qii(Z) the same probability that p assigns to Z EsZ and is order-preserving on the set of social alternative lotteries (i.e., p 1 q if and only if ri(p) >i Vr(q)). The mappings {0Ai} E N capture the differences in individual valuations of social alternatives. The Archimedeanaxiom says that for all p, q, r E Pi such that p >-j q >-i r, for some a, , E (0,1), ap + (1 - a)r >-i q and q >-i &p + (1 - ,3)r. The independenceaxiom requiresthat for all p,q,rePi and a E[0,1], p ai q implies ap + (1 - a)rai aq + (1 - a)r.

1409

IMPARTIALITY LEMMA 1: There exist bijections X/ >X1 functions {Pi: P1 PA}ie N defined {qij: by (1)

= ) -p(Z)

ri(P) (qfi

such that, for all i eN, (2)

}

N

and probabilitytransformation

g Z E=1 Xl

ri is a bijection and, for all p, q E Pl,

Vrj(P) >--Vr,(q) P:p>-1 q.

Moreover, if {Pi i N and {i}i e N are another set of bijections and the corresponding -probabilitytransformationfunctions satisfying (1) and (2), then fi = 'pi for all i E N. The proof is given in the Appendix. 2.2. Observer'sPreferences To formalize the idea of an ethical observer who is sympathetic to the interests of each individual in society, we use the notion of allocation.6 An allocation is an assignment of a unique element of Pi to every individual i eN, e.g., the allocation (85x,..., x) represents the prospect, as seen from behind the "veil of ignorance," of being person 1 in the social-situation represented by the social alternative x1, or being person 2 in the social situation corresponding to the social alternative x2, etc. Let H be the set of all allocations (i.e., H = P1 x P2 x *- x P). Given two allocations p = (p1, ... , pn) and q = (ql,..., qn) in H and a Ec [0,1], define the mixture a p + (1-a)q E H by (a p + (1 - a)q)i(Zi) = api(Zi) + (1 - a)qi(Zi) for all i E N and Zi EX. With this operation, H is a convex subset of a linear space. Let a denote the observer's preference relation on H and assume that it satisfies the Axiom (A.1) properly modified in view of the different domain.7 For any given p E H denote by [pulp] the allocation obtained from p by replacing the ith entry with p (i.e., [p Ip] = (pi, ...*, Pi- 1, pPi+ 1. .., P)). Given the observer's preference relation, a, on the set of allocations, define the induced conditional preference relations, {> _9ieN' on the set of social-alternative lotteries as follows: For all p, q E Pi, p a q if and only if is well defined. The [pilp] > [pIlqI. Since a satisfies the axioms of expected utility, a conditional strict preference relation, >-w, and the conditional indifferent relation, , are defined as usual. Assume that a4 satisfies (A.2), i = 1, .. ., n. Then Fishburn (1970, Theorem 13.1) and (T.1) imply the existence of nonconstant, continuous, bounded functions {wi: X/ >i - Ri N such that, for all p and q in H, n

(3)

p a-q

E| i=l

n

wi(x) dpi(x)2 X/ai

, i=l

A wi(x) dqi(x). X/ai~~~~~~~>_

6Allocations are analogous to horse lotteries, individuals are analogous to states of nature, and social-alternative lotteries are analogous to roulette lotteries in the Anscombe-Aumann (1963) model of decision-making under uncertainty. The potential for exploiting this analogy has recently been suggested, but was not pursued, in Mongin and d'Aspremont (1996). Allocations were used in Karni and Weymark (1998). 7The required modification is the substitution of p, q, r, and a for p, q, r, and >i, respectively, in the statement of the axioms of weak order, Archimedean, independence.

1410

EDI KARNI

REMARK: Modeling moral value judgment as a preference relation on allocations is equivalent to restricting the observer's preference relation in Harsanyi's original framework to a subset of extended lotteries that have the same implicit individual-identity lottery. To grasp this claim, let L be the set of extended lotteries and denote by a ' the observer's preference relation on L in Harsanyi's model. Fix a = (a1,..., a,,) in R' such that, for all i, ai > 0 and E' 1ai = 1 (a is an individual-identity lottery), and let e LIflx (i, x) dx = ai, i e N}. For any 1e La let li(l) be the social-alternative L={ lottery defined by li(l)(x) = l(i, x)/ai, for all i e N, and denote by 1i(l) the induced is an allocation. Next, Then 1(1)=(l(l),...,l(i)) probability distribution on X/ai. as follows: for every given 1 e La, define lotteries a * I(l) on N>X Ui eN XIi = * 1(1)11E La}. But * [a * (l)](i, x) acl(l)(x) is a one-to-one let La {a1(-) and)0 1( correspondence between H and La. Let a on H and -a on La be defined by (1) - (1') if and only if 1 '1' if and only if a * 1(l) -a a * 1(1'). Then, (H, a) and (La, a) are isomorphic. This discussion highlights the fact that modeling moral value judgment as a preference relation on allocations constitutes a departure from Harsanyi's concept of extended lotteries in form only. In both instances the ultimate outcome is an "individual-social-alternative" pair, (i, x) e N X X. The process by which the outcome is determined is different in the two models. Harsanyi's extended lotteries may be regarded as compound lotteries in which the individual is selected in the first stage by an individual-identity lottery, and, conditional on the outcome of the first stage, a social-alternative is selected in the second stage by a social-alternative lottery. In any given extended lottery the implicit individual-identity lottery is objectively known and, for any two extended lotteries, the individual identity lotteries are not necessarily the same. In the present model the social alternative is selected in the second stage in exactly the same way as in the case of extended lotteries. The selection of the individual in the first stage is modeled as a random event whose probability is not specified as a primitive of the model, but i'nstead, as in subjective expected utility theory, is implicit in the structure of the observer's preference relation. i

2.3. Sympathy An observer is said to be sympathetic if, when facing a choice between two allocations that assign the same social-alternative lottery to every member of the society except individual i, the observer's preferences agree with the preferences of individual i. This condition represents a strengthening of Harsanyi's principle of acceptance.8 Formally, we have the following assumption. ASSUMPTION(A.3) (SYMPATHY):For all i E N,

=

i.

Next we introduce the notion of invariance. Let WA E-N be defined as in Lemma 1. Then an observer's preference relation a on H is said to display invariance if, for all p, q E P1 and i E N, Ti(p) >-ciC(q) if and only if p >-c q. Invariance requires that, for any two social-alternative lotteries, the rank order of the transformed lotteries be preserved across conditional preference relations. An immediate 8Assumption (A.3) was introduced in Karni and Weymark (1998), where it is referred to as the strong principle of acceptance.

1411

IMPARTIALITY

implication of Lemma 1 is that sympathy implies invariance.9 Formally, we have the following Lemma. 2: Given the observer'spreferencerelation LEMMA

a

on H, sympathyimplies invariance.

The following representation theorem is an implication of Lemma 2 and Karni (1993, Theorem 3): (T.2): Let a be an observer's preference relation on H. Then a satisfies THEOREM Axioms (A.1)-(A.3) if and only if thereexist bounded, nonconstantfunction {u: X/ >1 - R}, aqprobabilitymeasure iT on N, and bijections {fii: X/ , -1 X/ >i}iE N such that, for all p, q E H,

p

aq

TE (Ti) f ieN

u?(i(x))[dpi(x)

-

dqi(x)]

O0.

XI>i

and probability Moreover, if (Q, r') is another pair of real-valued function on X/I = bu(qi-10) = i-f 1()) then 7T 7T' and in the same sense, that a v( represent measure on N

+ai, b>0. 2.4. Impartiality For any given p E H denote by [pi, p, q] the allocation obtained from p by replacing the ith entry with p and the jth entry with q. (I.e., [piiJp,q] = (P1,.,pi_1,p, Pn).) To formalize the notion of impartiality fix p E H and take Pi+ 1, Pj- 1, q, Pj+ 1, any p, q E P1 such that p >-1 q. Consider the allocation, p', obtained from p by replacing its ith entry, pi, by Ii(p) and the jth entry, pj, by Tj(q) and the allocation, p", obtained from p by replacing pi by Pi(q) and pj by Pj(p) (i.e., p' = [piiJTi(p), j(q)] and Tj(p)]). By Lemma 1, individual i prefers the allocation p', while individp" = [p',ilJi(q),

ual j prefers the allocation p", thus presenting the observer with conflicting claims. Assessing the merits of these claims, the observer notes that the ranking of Ti(p) and Ti(q) according to the preference relation of individual i is the same as that of 1Ij(p) and Pj(q) in that of individual j. Impartiality is the requirement that, in view of sympathy, the observer is indifferent between the allocation p' that favors individual i and the allocation p" that favors individual j. Formally, we have the following assumption. ASSUMPTION(A.4) (IMPARTIALITY): For all p E H and p, q ePj, Pi',ilWi(p), Vrj(q)]

Pi',ilWi(q), Vrj(P)].

Implicit in this definition is a moral value judgment that assigns individual utilities to outcomes in a way that permits interpersonal comparisons of utility differences. To grasp this note that, by sympathy, ui(x) = ai + biu(qf1(x)), bi > 0, for all x E XI i and i E N. Take any x, y E X/ >1 and let p, q E P1 in (A.4) be p = Ax and q = 8y. Impartiality, Theorem (T.2), and the definition of Pi imply that 7T(i)bi[u(x) - u(y)] = IT(j)bj[u(x) u(y)] for all i, j EN. Thus, 7T(i)bi = rT(j)bj, and, consequently, uj(xj(x)) - uj(1%(y)) for any x, y e X/ 1. -

9I am gratefulto Zvi Safrafor suggestingthis result.

ui(qi(x))

-

ui((i(y))

=

1412

EDI KARNI 3. REPRESENTATION

3.1. TheMainResult The followingtheoremassertsthat the preferencerelationof a sympatheticimpartial observer on the set of allocations is representableas a weighted sum of individual utilities,wherethe weightassignedto each utilityfunctionis inverselyproportionalto the diameter of the image of the set of social alternativescorrespondingto that utility function. THEOREM1: For each i E N, let >i be the preferencerelation on Pi of individuali and let

be an observer'spreference relation on H. Then the following conditions are equivalent: (i) For all i E N, -i satisfies (A.1), (A.2), a satisfies (A.1), (A.2), (A.4), and jointly a and {i }E i N satisfy (A.3). (ii) Thereexist bounded nonconstant continuousfunctions {ui: X/ >i R-ie N such that, for all i E N and p, q E Pi, a

(4)

P ai q

f-

L1(p)

dp(x) 2 J ui(x)

ui(x) dq(x)

f

Ui(q)

and, for all p, q E H, (5)(5)

a q E- .Ui(q)] ? 0. [U11(p)

P

p >-

eN SPEX/AUiX-fX SUPXGEX/>ui(x) iE-N

GEXI >-i -infXe/>ui(x)

(X

Moreover, if {V}i E N is another set of functions representing >i and a in the sense of (4) and (5), respectively,then there is a real number b > 0 and a E Rn such that vi(x) = bui(x) +ai, for all i eNandxeX/I.i

The proof is given in the Appendix. 3.2. Discussion The uniqueness of ui in Theorem (T.1) implies that (

-E / a u (X) infxI /i SUPXGE infX E. XI >-i U i(X)

sup

X

(X) XI

>-i uj(X)

is a utility function representing >i . Thus, by Theorem 1, for all p, q E H, (6)

P>q

dp(x) U*(OW

E-[J iE.N

n

Xi

I

>*

W dq(x)]

2O.

I >-i

The significance of this observation is that assigning equal weights to distinct individual utilities is neither a necessary nor sufficient propertyof impartiallybut an implication of the choice of normalization. In general, impartiality requires that the social welfare functions assign proportionally smaller weight to an individual whose utility image has larger diameter. Notice also that the utility functions {U*}in 1 are cardinally measurable unit comparable.10 10See Weymark (1991) or Roemer (1996) for definition and discussion.

1413

IMPARTIALITY

A social welfarefunctionW: X -- R representsthe observer'spreferencesrestrictedto social alternatives.The discussionabove impliesthe existenceof social welfarefunctions that have the functionalformof an arithmeticmean of individualutilities.To see this, let a * on X be the restrictionof the preferencerelation a on H to the set of degenerate constant lotteries (i.e., to the set {p E Hlp = (8x, A6x..., **X), x eX}). Define W(x) = 1nl.=lu*(8) x eX. Then, by Theorem(T.1) and equation(6), for all x,y eX, x a *y if and only if W(x) ? W(y). Dept.of Economics,JohnsHopkinsUniversity, Baltimore,MD 21218, U.S.A. receivedDecember,1996;final revisionreceivedAugust,1997. Manuscript APPENDIX PROOF OF LEMMA 1: (OUTLINE): For each

i e N let (i: [0,1] -> X

i be a function defined by

A)) Asup{ui(x)1x E X/I i} + (1 - A) inf{u(x)jx Ex X/I i}, ui ( GM= where ui is the utility function in Theorem (T.1). Define the mapping i/i: X/I--, X/.>a by l)(x), where (y is the inverse function of (1. Then if1 is the identity function and, qpi(x) = ( ( by Karni (1993, Lemma 1), {Pi}i N are one-to-one and onto. Moreover, Ti in Lemma 1 is a one-to-oneand onto. By Theorem(T.1) PF(p)>i Ti(q) if and only if

(7)

J

u (x Ui

ui(x) dTifp)(x) 2

>_i

>_i

(q)(x) dTi

But, by definition, the last inequality is equivalent to

(8)

f

u(qi (x)) dp(x) 2 >_i

f

uipi(x))

dq(x).

>_i

The boundedness and uniqueness of ui in Theorem (T.1) imply the existence of individual utility functions {Wi} E N representing the preference relations { >i }iE N such that ai(X/ >i) = Qj(X/ >--) for all j, i E N. Since >-i is nonempty, ui is nonconstant and inequality (8) is equivalent to

(9)

J

t W()dp(x) 2 A uj t Wi ) dq(x). uji

But, by definition of equivalent to

(10)

f

i/i

and /ij, ij(6i(x)) = fi(j61(x)) for all x eX/ >1 . Hence, inequality (9) is

dp(x) 2 _j10fji(x))

f

1j j

(x)) dq(x),

which, by the same arguments as above, is equivalent to

(11)

A

uj(x) dTj(p)(x)

2

uj(x) dTj(q)(x).

By Theorem (T.1) inequality (11) is equivalent to Tj(p) >j Tj(q). To prove the uniqueness of i/i suppose that, for some i 7/ 1 and z E X/ >1, i(z) / qpj(z). Using 'ps,define ti analogously to Ti. Without loss of generality, assume that 84iwz) 8i %.(Z). Let {xk} and + 1, k 1,. For each k, {yk} be sequences in X/ 1 such that: x1 >_1y', xk+l s1 Xk, k > let lk=[xk, Ak;yk(1-Ak)] 88z. Then, 'I(8)% ip(lk) and P(8z) cp(lk) But 'i(8z) = %4

EDI KARNI

1414 (Z). Hence,

and '(8z)= k= 1,2,....

f

(12)

ui(x)dTi

k=

ii(1k)

1P(lk)

(lk)(x)>

By Theorem (T.1) this implies that for

1,2,.

ui(x)dr(lk)(x).

fi

But inequalities(12) are equivalentto + (1 -

AkUi(qi(xk))

> AkUi(Di(xk))

Ak)ui(i(yk))

+ (1 - Ak)ui(q~i(yk)),

for all k = 1,2.... Hence, takinglimitson both sides and usingthe continuityof ui, we get Auk**+ (1 -Ak)0 > AuW*+ (1-Ak)u* where

*

=

sup{ui(x)lx E X/ >i } and u* = inf{ui(x)lx E X/ >i }, and

_ . A=lMmk

kk A

=lkmk

u1(z)

- u1(y k) u1(z) -U uk* uj* (k)(k) -u

A contradiction.

Q.E.D.

PROOFOF THEOREM1: (i) - (ii). That (i) implies the representation(4) is an implicationof Theorem(T.1). By Theorem (T.2)a satisfies Assumptions(A.1), (A.2), and (A.3) if and only if there exist bounded, nonconstant,function u: X/ 1-3 R, unique up to positive linear transformation,a unique probability measure ,T on N, and bijections, {i: p, q e H, (13)

pa q

u(4'i-(x))[dpi(x) - dqi(x)]

ir(i) f

E

X/

ieN

-3 X/I >}iE N such that, for all

X/I

O.

i

By sympathyand the uniquenessof ui, u maybe chosen so that, for each i E N, (14)

( )) aGp{/S-

ui(-) - inf,xE xI i Ui(X)

supX E X/ >i Ui(X) - infX E X/

>.i

u(x)

By impartiality,for all p E H and p, q E P1, (15)

[p,iXI'Pi( p), Pj(q) I

[p,iXI'Pi(q), 'Pj(p)] I

Thus,by (13) and the definitionof Ti, (15) implies -iJ

u' i- 1 W(q) [dTi i X/ =TQ(j,

(x(p)(x) dTi(q)(x)

u(0j-l(x))[dTj(

x]

p)(x) - dTj(q)(x)].

Hence, for all x, y E Xl, and letting p = 8x, q = byin the last equation,we get ir(i)[u(x) - u(y)]; u(y)].Thus,for all i E N, ,r(i) = 1/n. Substitutingthis and (14) in (13) we obtainthe equivalentrepresentation(5) in (ii). The fact that the representation(5) implies impartialityand sympathyis immediate.Thus, (ii) implies(i). Q.E.D.

= T(j)[u(x)-

REFERENCES F. J., AND R. AUMANN (1963): "A Definition of Subjective Probabilities," The Annals of Mathematical Statistics, 34, 199-205. ARROW, K. J. (1963): Social Choice and Individual Values, Vol. 12, Cowles Foundation Monographs, Second ed. New Haven: Yale University Press.

ANSCOMBE,

IMPARTIALITY

1415

DHILLON,A., AND J. F. MERTENS (1996): "Relative Utilitarianism: An Improved Axiomatisation," Core Discussion Paper 9655. FISHBURN, P. C. (1970): Utility Theoryfor Decision Making. New York: John Wiley & Sons. HARSANYI, J. C. (1953): "Cardinal Utility in Welfare Economics and in the Theory of Risk-Taking," Journal of Political Economy, 61, 343-435. (1977): Rational Behavior and Bargaining Equilibrium in Games and Social Situations. Cambridge: Cambridge University Press. KARNI, E. (1993): "A Definition of Subjective Probabilities with State-Dependent Preferences," Econometrica, 61, 187-198. KARNI, E., AND J. A. WEYMARK (1998): "An Informationally Parsimonious Impartial Observer Theorem," Social Choice and Welfare,15, 321-332. MONGIN, P., AND C. D'AsPREMoNT (1996): "Utility Theory and Ethics," in Hand-book of Utility Theory, ed. by S. Barbera, P. Hammond, and C. Seidl. Dordrecht: Kluwer (forthcoming). ROEMER, J. E. (1996): Theories of DistributiveJustice. Cambridge: Harvard University Press. SEGAL, U. (1996): "Let's Agree that all Dictatorships are Equally Bad," Mimeo. SEN, A. K. (1986): "Social Choice Theory," in Handbook of MathematicalEconomics: Volume III, ed. by K. J. Arrow and M. D. Intriligator. Amsterdam: North-Holland. WEYMARK, J. A. (1991): "A Reconsideration of the Harsanyi-Sen Debate on Utilitarianism," in Interpersonal Comparisons of Well-Being, ed. by J. Elaster and J. E. Roemer. Cambridge, U.K.: Cambridge University Press.

1998),1417-1425 Econometrica, Vol.66,No.6 (November, STRICT SINGLE CROSSINGAND THE STRICT SPENCE-MIRRLEES CONDITION:A COMMENTON MONOTONE COMPARATIVE STATICS AARON

S.

EDLIN AND CHRIS SHANNON'

1. INTRODUCTION

MILGROM AND SHANNON (1994) clarifythe relationshipbetween order-theoreticmethods for comparativestatics and more traditionaldifferentialtechniquesby developingrelationships between the differential Spence-Mirrleessingle crossing property and the order-theoreticsingle crossing property. Both conditions are central for monotone comparativestaticsanalysisin a numberof settings.In particular,Milgromand Shannon show that the order-theoreticsingle crossingpropertyis necessaryand sufficientfor the set of optimalchoices to be nondecreasingin certainchoice problems,and that a strict form of the single crossing property guarantees the stronger conclusion that every selection from the set of maximizersis nondecreasingin such problems.Milgromand Shannon assert that under appropriateconditions the Spence-Mirrleescondition is equivalentto their single crossingproperty,and that the strictversionsare also equivalent. In this note, however,we give counterexampleswhich show that their strict single crossingpropertymayhold even thoughthe strictSpence-Mirrleesconditionfails. In fact, we show that the strict single crossing property may hold even though the strict Spence-Mirrleesconditionholds only on a set of arbitrarilysmall measure.We also give a correct statementof the relationshipbetween the Spence-Mirrleesconditionand the single crossingproperty. These counterexamplesexplainthe discrepancybetween the monotonicityconclusions that Milgromand Shannon(1994) derivefrom the strictsingle crossingpropertyand the strict monotonicityconclusions that Edlin and Shannon (1998) derive from the strict Spence-Mirrleescondition.In Section 3 we also use these counterexamplesto illustrate the fact that the strict single crossingpropertycan allow both pooling and separating equilibriawhile the strict Spence-Mirrleesconditioneliminatesthe possibilityof pooling equilibria.The eliminationof pooling equilibriain signallingand screening models is more subtle than Edlin and Shannon's(1998) strict monotonicityconclusionsbecause agents need not face a differentiableconstraint. 2. RESULTS

To state our result and examples,we require two definitionsof single crossing:the order-theoreticsingle crossingpropertyof Milgromand Shannon(1994) and the differential Spence-Mirrleescondition. DEFINITION

1: Let X and T be partiallyorderedsets. A functionf: XX T -- R is said

to satisfy the single crossingproperty in (x; t) if for all x' > x*: 1. wheneverf(x', t*) f(x*, t*), then f(x', t') > f(x*, t') for all t' > t*; and 'Thanksto SusanAthey,Paul Milgrom,an editor,and three anonymousrefereesfor their useful comments.Shannon'sworkis supportedin partby NSF GrantSBR-9321022. 1417

1418

A. S. EDLIN AND C. SHANNON

2. wheneverf(x', t*) > f(x*, t*), then f(x', t') > f(x*, t') for all t' > t*. The functionis said to satisfythe strictsinglecrossingpropertyin (x; t) if for all x' > x, wheneverf(x', t*) f(x*, t*), then f(x', t') > f(x*, t') for all t' > t*. 2: Let f: X x T -- R be continuouslydifferentiable,where X c R2. Then f DEFINITION is said to satisfythe (strict)Spence-Mirrlees conditionif fx/lfyl is (increasing)nondecreasing in t, and fy = 0 and has the same sign for every(x, y, t). Milgromand Shannon(1994,Theorem3) assertthat these conditionsare equivalentas long as T = R and the function f is continuouslydifferentiableand what they call completelyregular,whichmeans that the level sets are path-connected.A correctversion of their theoremcan be stated as follows. THEOREM 2.1: Let R2 be given the lexicographicorder, with (x, y) 1 (x', y') if either ? y'. Suppose that U(x, y; t): R3 -* R is completely regular and

x > x' or x = x' and y

0 Then U(x, y; t) satisfies the single crossingpropertyin continuouslydifferentiablewith Uy O. (x, y; t) if and only if it satisfies the Spence-Mirrleescondition. Moreover, U(x, y; t) satisfies the strict single crossingpropertyin (x, y; t) if it satisfies the strict Spence-Mirrleescondition. Althoughthe lexicographicorder may appearto have come out of the blue here, for sufficientlywell-behavedpreferencesthe single crossingpropertyunderthe lexicographic orderis equivalentto the more familiarassumptionthat indifferencecurvescrossat most once, and alwaysfrom the same direction.See Athey, Milgrom,and Roberts(1996)for a discussionof this point. Milgromand Shannon'sproof establishesthat under these regularityconditions,the strictsingle crossingpropertyholds wheneverthe strictSpence-Mirrleesconditionholds, and that the nonstrict versions of these properties are equivalent. That the strict propertiesare not equivalentis demonstratedby the followingexample.Let T = {t*, t'} with t' > t*, and let f(x, y, t*) =y _X2 and f(x, y, t') =y - x2 +X3/10, as illustratedin Figure2.1, whichgraphsff, 0, t') and ff(, 0, t*). Then f satisfiesthe strictsinglecrossing property in (x, y; t), since f(x, y, t') - f(x, y, t*) is increasing in x, but the strict Spence-

Mirrleesconditionfails wheneverx = 0, since fx (Oy t') = 0

fx (0, y,t*)

Since the strict Spence-Mirrleescondition holds almost everywherein the above example,one mightconjecturethat Milgromand Shannonwere almostcorrect.That is, perhapsfor the class of differentiablefunctionsthe strictsingle crossingpropertyimplias~ that the strictSpence-Mirrleesconditionholds almosteverywhere.Surprisingly, however, continuouslydifferentiablefunctions can violate the strict Spence-Mirrleescondition over most of their domainsand still satisfythe strictsingle crossingpropertyeverywhere. To establish this fact, we first show that an analogousconjecturefor one-dimensional problems is also false by constructinga function g(x, t) that has strictly increasing differences,but that has increasingmarginalreturnsonly on a set of arbitrarilysmall measure.For this example,we requireseveraladditionaldefinitions. DEFINITION3: A function g: X x T -- R is said to have strictlyincreasingdifferences if g(x', t') - g(x*, t') > g(x', t*) - g(x*, t*) whenever x' > x* and t' > t*.

1419

MONOTONECOMPARATIVE STATICS

-1

0.5

-0.5

1

-0.2

4 ~~~~~~-0.

/

6 ~~~~~~~-0.

/

~~~~~~~-0.8

/

X

f

~~~~~~~~~~n(X\.~~~ 0

FIGURE2.1.-The strict single crossing propertydoes not imply the strict Spence-Mirrlees

condition.

4: A function g: X x T -- R is said to have increasingmarginalretums at x DEFINITION if gx(x,'t) is increasing in t. The key to the following counterexamples is the fact that for any given ? E (0, 1), there exists a closed, nowhere dense subset of [0,1] having measure e, called the e-Cantor set and denoted C. Like the Cantor set, it is constructed by sequentially removing open intervals from [0,1]. First, the interval [0,1] is split by removing an open interval from its center, leaving two closed intervals of equal length. These closed intervals are likewise split by removing open intervals from their centers, and this process is continued ad infinitum. Then 2 n-1 intervals removed in the nth iteration are each of length (1 - e)/(22n-1) so that the total length removed is (1 - O)En=1(1/2n) = 1 - ?. What remains is the e-Cantor set, which has measure e.2 Consider the function

g(x,t)=tf

h(s)ds,

where

h(s)-

0

inf

Iz-sls

ZE Ce

Since h() is continuous, g(O)is well-defined and continuously differentiable. Furthermore, g(*) has strictly increasing differences. To see this, note first that

g(x',

t) - g(x*,

t) = tf

h(s)

ds.

x*

2See, for example,Aliprantisand Burkinshaw(1981, p. 113) for a further discussionof the constructionof this set and some of its properties.

1420

A. S. EDLIN AND C. SHANNON

Since CEis closed, h(s)> 0 Vs - C,. Hence this integral is positivewheneverx' > x, since CEis closed and nowhere dense.3 However, g() only has increasingmarginal returnsfor x e C_, since gx(x, t) = th(x), which implies that gx(x, t) = 0 if x E C,, and gx(x, t) > 0 if x V CF. Since C, has measuree, whichcan be set arbitrarilyclose to 1, this impliesthat g(-) mayvery rarelyhave increasingmarginalreturns. Next, notice that if r(x, t) is any functionwith strictlyincreasingdifferencesand we define w(x, y, t) = r(x, t) +y, then w satisfiesthe strictsinglecrossingpropertyin (x, y; t) with respectto the lexicographicorder on R2. To see this, supposethat (x', y') >1 (x, y) and w(x', y',It*)2 w(x, y, t*). Either x' > x, or x' = x and y' > y. If x' = x, then w(x', y', t') - w(x, y, t') =y' -y, whichis positivesince y' > y. If x' > x, then since w(x', y', t*) 2 w(x, y, t*), we knowthat y -y' < r(x', t*) -r(x, t*) < r(x', t')-r(x,

Vt' > t*,

t'

since r(x, t) has strictlyincreasingdifferences.Thus w(x', y', t') > w(x, y, t') for all t' > t* in either case, which showsthat w satisfiesthe strictsingle crossingproperty. From this discussion,it follows that f(x, y, t) g(x, t) +y satisfies the strict single crossingproperty.However,if x E C8, then f fails to satisfythe strict Spence-Mirrlees conditionat (x, y) for any y, because fx

l y* (XIYI

IfyI

tt'gt).gf (x,t*) =)-(x,t')

dx

=

0

dx

=

l(x,y,P).

iy

Notably,this failure occurs on a set of measure e, which again can be arbitrarilyclose to 1. Milgrom and Shannon (1994) err by presumingthat if the strict Spence-Mirrlees conditionfails, then it must fail on a set of positive measurewith nonemptyinterior. They then integratealong an indifferencecurvein this interiorto show that if the strict Spence-Mirrleesconditionfails, then so too must the strict single crossingproperty.As these examplesillustrate,however,their presumptioncan be wrong:even though the strictsingle crossingpropertyholds, the Spence-Mirrleesconditioncan fail, and can fail on a set of positivemeasure,as long as that set has an emptyinterior.In our examples, their integrationargumentcannot work because there is no path along an indifference curvewhere the strict Spence-Mirrleesconditionfails. 3. AN ILLUSTRATIVEEXAMPLE

As the results of the previoussection indicate, the strict single crossingpropertyis weaker than the strict Spence-Mirrleescondition,and thus the monotone comparative 3Since C, is nowhere dense, if x' # x*, there exists x 1 C, between x* and x'. Since C, is closed, there is an open interval around i contained in the complement of C8, and h() must be positive on this interval since h(s) > 0 Vs e C,.

MONOTONE COMPARATIVE STATICS

1421

staticsresultsobtainedby Milgromand Shannonare actuallystrongerthan they claimed. The fact that these propertiesdifferalso explainswhyEdlin and Shannon(1998)are able to derive strict comparativestatics results from the strict Spence-Mirrleescondition, while such conclusionscannotbe drawnfrom the strictsingle crossingproperty.Milgrom and Shannon(1994)showthat the strictsinglecrossingpropertyis sufficientto guarantee that every selection from the set of maximizersis nondecreasing,yet this conclusion allows the possibilitythat some selections may remain constant over some range of parameters.This differenceis illustratedin Figure3.1. Signalingand screeningmodels provideanotherexampleof the importanceof distinguishing between the strict single crossing property and the strict Spence-Mirrlees cQndition.In such models, the strict single crossingpropertyallows both pooling and separatingbehaviorin equilibrium,while the strict Spence-Mirrleesconditionrules out pooling equilibria.These models are more complex than the optimizationproblems considered in Edlin and Shannon (1998): here, a screener need not offer agents a differentiablechoice set, so separatingbehaviorcannotbe inferredsimplyby comparing solutions to agents' optimizationproblems.Instead separationresults from equilibrium considerations. As an illustration,considerthe menu of price-qualitycontractsa monopolywill choose to offer to consumers. Let T = {l, h} with h >1. Consumersof type 1 and h have preferencesgiven by U _(q,p) =

- q 33-p,

if q E[0,2], if q > 2,

and -(q -1)-p i~p,

U2q hT

if q E [0,2],

+ 3(q-1),

if q >2,

where q denotes qualityand p denotesprice.The monopolycannotobservea consumer's type. Let R2 be given the lexicographicorder on (q, -p), that is, the order in which (q', p') 2 (q, p) if either q' > q or q' = q and -p' 2 -p. By the same argumentgiven in the previous example, these preferences satisfy the strict single crossing propertyon [0,2] x R+ x T since Uh(q, p)

-

U(q, p)

=

1 -(q 3

-

is increasingin q. The strict Spence-Mirrleesconditionfails wheneverq= 1, however, since Uh __

dUh

/l

dq

dp

/ dul(l -/ dp dq

dul

__U

dUh

llP)

=

2

P).

When the productioncost is 2 per unit, the unique equilibriumis a pooling equilibmonopolywill offer only one contract,qh = q, = 1, rium in which the profit-maximizing

1422

A. S. EDLIN AND C. SHANNON

f(x,t*)

(a)

---

f(XXt) f(x,t*)

(b) FIGURE3.1.-(a) The strict single crossingpropertyholds, so every selection from the set of

maximizersis nondecreasing.Nonetheless,some selectionwill be constantat pointssuchas p where the strict Spence-Mirrleesconditionfails. (b) The strict Spence-Mirrleesconditionholds, so every selection from the set of maximizers is increasing.

MONOTONE

COMPARATIVE

1423

STATICS

p Uh (q,p)

= Uh(qe,p;)

UE(q,p) = U(O,O)

/~~~~~~~~j,S/ (pl)

=q

(
low type high type

q

1

(a) Uh(q P) = Uh(qZ,Pt)

~~~~~~~~~~~~~~(qh* aph),-

.

Ut(q,p)

- Ut(O,O)

iso-profitset

-

-

1

low type high type

q

(b) Pooling equilibrium. It may be optimal to offer only one bundle when both types' indifference curves are tangent to the iso-profit set at the same point. This simultaneous tangency is possible, and may in fact be common, even though the strict single crossing property holds. (b) Separating equilibrium. The optimal contract involves selling higher quality to the high type than to the low type when the strict Spence-Mirrlees condition holds, because the high type's indifference curve is steeper than the low type's at each point. FIGURE 3.2.-(a)

1424

A. S. EDLIN AND C. SHANNON

is possiblebecausethe = 3, as depictedin Figure3.2(a).4The poolingequilibrium strict Spence-Mirrleesconditionfails at q = 1. In contrast,supposeinsteadthat the preferencesof the high type are given by

Ph =PI

3q -(q -1) 5 -p,

Uh(q,p)=

p,

if qE[0,2], ~~ifq >2.

In this case, preferencessatisfynot only the strict single-crossingpropertybut also the strongerstrict Spence-Mirrleescondition on [0,2] x R+ x T. Here it is optimal for the monopolyto offer a separatingcontractwhich involvesselling a higher qualitylevel to the high type than to the low type. Offering two distinct contracts is optimal here because, by standard arguments, (dUI/dq)(q*,Pl) 2 2,5 so that by the strict Spence-Mirr4The monopoly's profit maximization problem is max Pl

+PSI - 2(q1 + qh)

(IR1)

U1(ql, pl) 2 U1(O,O),

(,Rh)

Uh (qh XPh)

(Idz

U, (ql, pl ) 2

subject to

2 Uh(0, O), UI (qh, Ph )-

To find the solution (q*, pI); (q*, p*) to the monopoly's problem, observe first that (IR1) must bind, so that Ul(q, p*) = U1(O,0)= -1; equivalently, p* = 2q* - (q* - 1)2 + 1. Thus we can restrict attention to contracts (ql, pl) such that p, = 2q1 - (q, - 1)2 + 1. Given any such contract, the optimal contract to offer to the high type solves max - 2qh

subject to

+Ph

(qh, P,) Uh (qh, Ph) 2 Uh (?, ), Uh(qh, Ph)

2

U, (ql, pl) 2

Uh(ql,

pl),

UI (qh, Ph)-

The most profitable contract satisfying (ICh) is qhl = 1, Ph = 3 - (1/3)(q1 - 1)3. When q1 < 1, this contract also satisfies (ICQ) and (IRh) because the strict single crossing property and (MRl) hold. Hence this contract is optimal when q1< 1. In contrast, when q12 1, the optimal contract is (qh, Ph) = (ql, pl). Thus given any level q1 offered to the low type, the monopoly's maximum profits will be

(2- (q

- 1)2 _

3(q, - 1)3

22-2(q1-1),

if q1 < 1,

if q1?1.

The solution to this profit maximization problem occurs at q* = 1, and hence the unique solutioti to the monopoly's problem is to offer the pooling contract (q*, pl) = (q*, p*) = (1, 3). < 2, or equivalently if q* > 1, then the monopoly can increase profits by 5If (0U1/dq)(q,p?) selling slightly less to the low type. More precisely, consider changing the low offer to (41, P3l) where E for some E > 0. Clearly (IR1) and (ICQ)continue to hold. This U (1,P U,= (ql,p ) and 41= qnew contract also satisfies (ICh) because: (1) the low type is indifferent between (q*, p*) and (l, j3), so by the strict single crossing property the high type prefers (q*, p* ) to (q1,j3l); and (2) (ICh) holds for the original contract. Under the new contract, monopoly profits change by 2E-2E-

(q*

_ E-

1)2 + (q*

_ 1)2 = 2E(q*

-1)-,2,

which is positive for E sufficiently small since q* > 1, indicating that the monopoly is not optimizing. Hence (dUedq)(qr,pt) 2 2.

MONOTONE COMPARATIVE STATICS

1425

lees condition,(dUh/dq)(q, p,) > 2. Since the high type'smarginalwillingnessto pay at (q*, p*) exceeds marginalcost, unlike the previousexample,the monopolywill offer a second bundle with a higher qualitylevel intended for the high type, as illustratedin Figure3.2(b). Dept. of Economics, University of Califomia, Berkeley, 549 Evans Hall, Berkeley, CA 94720, U.S.A.; [email protected], and Dept. of Economics, Universityof Califomia, Berkeley, 549 Evans Hall, Berkeley, CA 94720, U.S.A.; [email protected];http://emlab.berkeley.edu/users/edlin/index.html receivedAugust,1996;final revisionreceivedJanuary,1998. Manuscript REFERENCES ALIPRANTIs,

C. D.,

AND O. BURKINSHAW

(1981): Principles of Real Analysis. New York: North-Hol-

land. ATHEY, S., P. MILGROM,AND J. ROBERTS (1996): "Robust Comparative Statics," draft. EDLIN, A. S., AND C. SHANNON (1998): "Strict Monotonicity in Comparative Statics," Journal of

Economic Theory, 81, 201-219. MILGROM,P., AND C. SHANNON(1994): "Monotone

Comparative Statics," Econometrica, 62, 157-180.

Econometrica, Vol. 66, No. 6 (November,1998)

ANNOUNCEMENTS NOMINATIONOF FELLOWS,1999 ANY MEMBEROF THE ECONOMICSOCIETYmay nominatecandidatesfor Fellow.Nomina-

tion forms may be obtained from the Society's business office. Informationrequired includesthe name, position, and addressof the nominee; a six-itembibliographyand a nominationstatement(maximum125words)that summarizesthe main scientificachievement of the nominee.Longernominationstatementsof severalpageswill be acceptedfor those nomineeswhose principalscientificwritinghas been publishedin languagesother than English. Requests for nomination forms as well as completed forms should be sent to: Professor Julie P. Gordon, Secretary,The EconometricSociety, Department of Economics,NorthwesternUniversity,Evanston,Illinois 60208-2600. Members or Fellows supplying completed nomination forms should also include writtenevidencesupportingany endorsementsof each nominationby additionalFellows. Either copies of letters or signatureson the nominationform are acceptable.However, the business office will not accept signaturesby fax or by e-mail. Hard copies of fax or e-mailmaybe sent to the originalnominator,who may then collect and forwardthem to the businessoffice. The NominatingCommitteefor Fellows will conduct a rankingof all nominees in order to determine the candidatesto appear on the ballot. If three or more Fellows endorsea candidate,then the candidate'snamewill automaticallyappearon the ballot of that year. Candidateswho fail to win election in a givenyear are no longer carriedover automaticallyto the next year's ballot and must be renominatedby the standard procedurein orderto appearon the ballot in any subsequentyear. Please note that the deadlineof receipt of nominationforms (hardcopy only) and of signaturesin the businessoffice by April 30, 1999,will be strictlyobserved. THE 1998 INDIA AND SOUTH EAST ASIA MEETING OF THE ECONOMETRICSOCIETY ANNOUNCEMENT 1998 INDLAAND SOUTH EAST ASIAMeetingof the EconometricSocietywill be held at the UniversitiMalaysiaSarawak,Sarawak,duringDecember 15-17, 1998.

THE

The ProgramCommitteeconsistsof: Anjan Mukherji,JawaharlalNehru University,Chairperson. MohamadArif, UniversitiMalaysiaSarawak. RomarCorrea,Universityof Mumbai. SanghamitraDas, IndianStatisticalInstitute,New Delhi. SudiptoDasgupta,IndianStatisticalInstitute,Calcutta. Dato ZawawiIsmael,-UniversitiMalaysiaSarawak. M. G. Kanbur,UniversitiMalaysiaSarawak. M. Ali Khan,Johns HopkinsUniversity. T. KrishnaKumar,IndianStatisticalInstitute,Bangalore. H. M. Leung,NationalUniversityof Singapore. LatifahWan Mohamad,UniversitiMalaysiaSarawak. B. Ramaswami,IndianStatisticalInstitute,New Delhi. 1427

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S. Sarkar,IndiraGandhiInstituteof DevelopmentResearch,Bombay. K. Sengupta,JawaharlalNehru University. SudhirShah,Delhi School of Economics. A. Yamazaki,HitotsubashiUniversity. The Local ArrangementsCommitteeis being chairedby Dr. LatifahWan Mohamad, Facultyof Economicsand Business,UniversitiMalaysiaSarawak,94300 Kota Samarahan, Sarawak,Malaysia.

1999 NORTH AMERICAN WINTER MEETING OF THE ECONOMETRICSOCIETY THE 1999 JOINT MEETINGwith the American Economic Association will be held in New

York, January3-5, 1999, as part of the Allied Social Science AssociationMeeting.The programwill consistprimarilyof contributedpapers. The ProgramCommitteeconsistsof the followingindividuals: RandallWright,Universityof Pennsylvania(Chair) John Kennan,Universityof Wisconsin(LaborEconomics) MarianneBaxter,Universityof Virginia(InternationalEconomics) Alan Auerbach,Universityof California,Berkeley(PublicEconomics) Glenn MacDonald,Universityof Rochester(IndustrialOrganization) StanleyZin, CarnegieMellon University(FinancialEconomics) Dale T. Mortensen,NorthwesternUniversity(Informationand SearchTheory) BoyanJovanovic,New York University(Developmentand Growth) Neil Wallace,PennsylvaniaState University(MonetaryEconomics) NarayanaKocherlakota,FederalReserveBank of Minneapolis(MacroeconomicTheory) ChristopherSims,Yale University(EmpiricalMacroeconomics) FrancisX. Diebold,Universityof Pennsylvania(AppliedEconometrics) Rosa Matzkin,NorthwesternUniversity(EconometricTheory) WolfgangPesendorfer,PrincetonUniversity(Game Theory) David Levine, Universityof California,Los Angeles (EconomicTheory,Mathematical Economics) Furtherinformationon registrationand housingwill be sent in due course.Enquiries may be sent to: EconometricSocietyWinterMeetings,c/o Prof. RandallWright,Dept. of Economics,Universityof Pennsylvania,3718 Locust Walk, Philadelphia,PA 19104, U.S.A., or from the society'sweb site: http://www.econometricsociety.org/es99/.

NORTH AMERICAN SUMMER MEETING OF THE ECONOMETRICSOCIETY ANNOUNCEMENTAND CALL FOR PAPERS THE 1999 NORTH AMERICANSUMMERMEETING of the Econometric

Society will be held

in Madison, Wisconsin, on the campus of the University of Wisconsin, from June 23-27, 1999. The Program Chair is Rodolfo Manuelli. The program committee consists of the

ANNOUNCEMENTS

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followingindividuals: Daron Acemoglu,MassachusettsInstituteof Technology FernandoAlvarez,Universityof Chicago AlessandraCasella,ColumbiaUniversity V. V. Chari,Universityof Minnesota ThomasGresik,PennsylvaniaState University John Heaton, NorthwesternUniversity Hugo Hopenhayn,Universityof Rochester BoyanJovanovic,New York University LarryJones, NorthwesternUniversity Ken Judd,HooverInstitution Kala Krishna,PennsylvaniaState University John Kennan,Universityof Wisconsin Per Krusell,Universityof Rochester Rosa Matzkin,NorthwesternUniversity ChristopherPhelan,NorthwesternUniversity Peter Reiss, StanfordUniversity Ennio Stacchetti,Universityof Michigan CharlesWhiteman,Universityof Iowa The programwill include invited lectures and both invited and contributedpapers. Submissionsshouldincludea cover letter, and an abstractto be e-mailedto the Program Chair ([email protected]).Preference will be given to those submissionsthat also include the paper itself. If the paper is submitted electronically(pdf or Postscript formats),the file should be sent to the same e-mail addressas above, attached to an e-mailmessage.If a papercopy is submitted,please mail three (3) copies to the Program Chairat the followingaddress:ProfessorRodolfoManuelli,7470SocialScienceBuilding, 1180 ObservatoryDrive, Universityof Wisconsin,Madison,WI 53706. The cover letter should identify which author will be presenting the paper, and should include the name(s), institutional affiliation(s),address(es),telephone number(s),fax and e-mail address(es)of all coauthors.Includeup to three JEL primaryfield names. For additional informationand updated informationabout the meetings, see announcementsin subsequentissues of Econometricaor on the Internet at the following address:http://www.ssc.wisc.edu/esm99/,or at the EconometricSociety Home Page: http://gemini.econ.yale.edu/es/. The deadlinefor submissionsis Januaty29, 1999.All abstractselectronicallysubmitted by that date that conform to the instructionswill be considered by the Program Committee.

THE 1999 FAR EASTERN MEETING OF THE ECONOMETRICSOCIETY ANNOUNCEMENT THE 1999 FAR EASTERN MEETING of the Econometric Society will be held in Singapore,

July 1-3, 1999. The Department of Economics and Statistics, National University of Singapore is the organizer of the Meeting.

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ANNOUNCEMENTS

The Meeting is open to all economistsand econometricians,includingthose who are not membersof the EconometricSociety. The Programwill consist of invited lectures and contributedpapers in economics and econometrics,both theoretical and applied. The Keynote Speakersfor the Meeting are ProfessorRobert Wilson (PresidentDesignate, 1999), ProfessorClive Granger,ProfessorPeter Phillips,and ProfessorKiminori Matsuyama. The ProgramCommitteeconsistsof: BasantKapur,NationalUniversityof Singapore(Co-Chair) KimioMorimune,KyotoUniversity(Co-Chair) Yiu-KuenTse, NationalUniversityof Singapore(DeputyChair) TilakAbeysinghe,NationalUniversityof Singapore Anil Bera, Universityof Illinois,Urbana-Champaign Soo-HongChew,Hong Kong Universityof Science and Technologyand Universityof California,Irvine BhaskarDutta, IndianStatisticalInstitute Hian-TeckHoon, NationalUniversityof Singapore MaxwellKing,MonashUniversity Chung-MingKuan,NationalTaiwanUniversity Lung-FeiLee, Hong Kong Universityof Science and Technology Kian-GuanLim, NationalUniversityof Singapore MichaelMcAleer,Universityof WesternAustralia KazuoNishimura,KyotoUniversity Sam Ouliaris,NationalUniversityof Singapore Joon Y. Park,Seoul NationalUniversity GarryPhillips,Universityof Exeter David Yeung, Universityof Hong Kong JunxiZhang,NationalUniversityof Singapore Any queries, should be sent to Professor Basant Kapur, Dept. of Economics and Statistics, National University of Singapore, Kent Ridge, Singapore 119260 (fax no: 65-7752646;e-mail:[email protected]). Consult the Internet at http://www.nus.sg/NUSinfo/FASS/webarts/ecs/news.htm for furtherinformation.

1999 AUSTRALASIANMEETINGSOF THE ECONOMETRICSOCIETY ANNOUNCEMENTAND CALL FOR PAPERS THE 1999 AUSTRALASIAN MEETINGS of the Econometric Society will be held in Sydney,

Australiafrom Wednesday7 July to Friday9 July,inclusive.The programco-chairsare Tony Hall and Colm Kearney. The meetings are open to all economists and econometricians, including those not currently members of the Econometric Society. It is hoped that the papers presented at the meeting will represent a broad spectrum of applied and theoretical economics and econometrics. The keynote speakers include Darryl Duffie, Roger Farmer, Christian Gourieroux, Jean-Franqois Richard, and Chris Sims.

ANNOUNCEMENTS

1431

Submissionsshould represent original manuscriptsnot previouslypresented to the EconometricSociety.Jointlyauthoredpapersshouldbe submittedby the personwho will present the paper (should it be accepted) and no person may present more than one paper. Abstractsshouldnot be over 300 wordsin length,typeddouble-spaced,shouldinclude the author'sname(s), affiliation(s),paper title, key words, JEL classification,and the followingdetailsfor the presenter:completeaddress,telephonenumber,fax number,and e-mail address.The abstractand the paper may be mailed or submittedelectronically (Word,WordPerfect,LaTex,Tex, PDF, or Postscriptformats:the files should be sent attachedto an e-mail message). The closingdate for submissionsis March15, 1999. Furtherand updatedinformationabout the meetingsis availablefrom the conference home page, or by writingto ESAM99,School of Finance and Economics,Universityof Technology Sydney, PO Box 123, Broadway, NSW 2007, Australia; Phone: + 6129281 2020; Fax: + 6129281 0364; E-mail: [email protected];Home page: http://www.bus.uts.edu.au/fin &econ/conferences/esam99/esam99.html 1999 EUROPEAN MEETING OF THE ECONOMETRICSOCIETY ANNOUNCEMENTAND CALL FOR PAPERS THE NEXTEUROPEAN MEETINGof the Econometric Society will be held on Santiago de

Compostela,August 29 to September1, 1999. Registrationwill begin on the eveningof Sunday,-August 29. It will be immediately followed by the Congress of the European Economic Association, which will run from September 2 to September 4. Economists, including those who are not currently members of the Econometric Society, are invited to submit papers for possible presentation at the meeting. It is hoped that papers presented at the meeting will represent a broad spectrum of applied and theoretical economics and econometrics. The Program Co-Chairs are: Economic Theory: Professor Klaus Schmidt, Seminar fuer Wirtschaftstheorie, Volkswirtschaftliche Fakultaet, Ludwig-Maximilians-Universitaet Muenchen, Ludwigstrasse 28 (Rgb), D-80539 Muenchen, Germany; Econometrics: Professor Andrew Chesher, Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, United Kingdom. Submission of papers: Two copies of a paper, together with three copies of an abstract, prepared as described below, should be sent to the appropriate Program Co-Chair at the corresponding address given above. Submissions will not be considered if received after March 15, 1999, so authors should allow for potential delays in international mail. Submissions by fax or e-mail will not be accepted. It is expected that decisions on submissions will be made by May 15, 1999. No financial aid will be available. Jointly-authoredpapers: Jointly-authored papers should be submitted by the person who will present the paper, if it is accepted. Please note that each participant may present at most one paper. Preparationof abstract: There will be no printed abstract form. The following information should be supplied, on a single sheet of paper: (i) Conference name: ESEM99; (ii) name(s) of the author(s), with the surname of the paper presenter in capitals; (iii) title of paper; (iv) institutional affiliation of paper presenter and complete mailing address; (v) e-mail address of paper presenter; (vi) abstract of paper of no more than 100 words containing no symbols, references, or equations; (vii) up to six key words.

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ANNOUNCEMENTS

At the same time as a submissionis made to the ProgramCo-Chair,a duplicate abstractshouldbe suppliedto the Local ArrangementsCommittee,either by e-mail,or on a floppydisk in a DOS format(MicrosoftWord,Word Perfect, or as an ASCII text file) accompaniedby a hard copy. (Floppiescannot be returned.)The e-mail addressis: [email protected] mailing addressis: ProfessorAlberto Meixide, Departamentode Fundamentosdel Analisis Economico,Facultadde Ciencias Economicasy Empresariales, Avda.Juan XXIII, s/n, 15704Santiagode Compostela(A Corunia),Spain. Detailed informationon registrationand housingwill be sent in Local arrangements: due course. Enquiriesmay be addressedto: ESEM99Local ArrangementsCommittee, Attention:ProfessorAlberto Meixide,Palacio de Congresosy Exposicionesde Galicia, San Lazaro,15703 Santiagode Compostela(A Corufia),Spain;Tel: +34-(9)81-552420; fax: +34-(9)81-577550;email:[email protected]. Furtherinformationis availableon the Internetat http://web.usc.es/- esem99.

2000 NORTH AMERICAN WINTER MEETING OF THE ECONOMETRICSOCIETY THE 2000 JOINTMEETINGwith the American Economic Association will be held in Boston, January 7-9, 2000, as part of the Allied Social Science Association Meeting. The program will consist primarily of contributed papers. The Program Committee consists of: Wolfgang Pesendorfer, Princeton University (Chair) Joshua Angrist, Massachusetts Institute of Technology (Labor and Applied Econometrics) Steve Coate, Cornell University (Public Economics) Jordi Galli, New York University (Macroeconomics, Monetary Economics) Penelopi Goldberg, PrincetonUniversity (Empirical Industrial Organization) Gene Grossman, Princeton University (International Economics, Political Economy) John Heaton, Northwestern University (Applied Time Series, Financial Economics) Guido Imbens, University of California, Los Angeles (Theoretical and Applied Econometrics) Per Krusell, University of Rochester (Macroeconomics, Inequality) Stephen Morris, Yale University (Economic Theory, Game Theory) Werner Ploberger, University of Rochester (Theoretical Econometrics) Michael Riordan, Boston University (Theoretical Industrial Organization, Information and Uncertainty) Richard Rogerson, University of Minnesota (Macroeconomics, Labor) Christina Shannon, University of California, Berkeley (Economic Theory, Mathematical Economics) Chris Udry, Yale University (Development, Empirical Microeconomics) Jeffrey Zwiebel, Stanford University (Financial Economics, Applied Theory). Prospective contributors are invited to submit abstracts of their papers before April 1, 1999. All submissions should be mailed (fax or electronic submissions will not be accepted) to: Econometric Society Winter Meetings, c/o Prof. Wolfgang Pesendorfer, Dept. of Economics, Princeton University, Princeton, NJ 08544, U.S.A.

1433

ANNOUNCEMENTS

Submitted abstracts should not be over 200 words in length and should be submitted in triplicate. Each person may submit only one paper, or be a co-author on multiple submissions provided that if all such papers were accepted, no person would present more than one paper. Abstracts should represent original manuscripts not previously presented to the Econometric Society or submitted to other professional organizations for presentation at these same meetings. Please type abstracts double-spaced and label at the top with the author's name, affiliation, complete address, telephone number, e-mail address, general field of designation, and paper title. ACCEPTED MANUSCRIPTS

in THE FOLLOWINGMANUSCRIPTS,

additionto those listed in previousissues, have been

accepted for publication in forthcoming issues of Econometrica. NABILI: "Decomposition and Characterization of Risk with a Continuum of Random Variables: Corrigendum." (KGSM/MEDS, Northwestern University, 2001 Sheridan Road, Evanston, IL 60208-2001.

AL-NAJJAR,

CABALLERO,RICARDO J., AND EDUARDO M. R. A. ENGEL: "Explaining

Investment

Dynamics in U.S. Manufacturing: A Generalized (S, s) Approach." (Dept. of Economics, Massachusetts Institute of Technology, 50 Memorial Drive, Cambridge, MA 02142-1347.) CICCONE,ANTONIO, AND KIMINORIMATSUYAMA:"Efficiency

and Equilibrium with Dy-

namic Increasing Aggregate Returns Due to Demand Complementarities." (Dept. of Ecopomics, 549 Evans Hall #3880, University of California at Berkeley, Berkeley, CA 94720-3880.) DEMANGE, GABRIELLE,AND GuY LAROQUE:"Social Security and Demographic

Shocks."

(INSEE, Departement de la Recherche, 15 Bd. Gabriel Peri-BP100-G302, 92245 Malakoff Cedex, France.) FOSTER,JAMESE., AND EFE A. OK: "Lorenz Dominance

and the Variance of Logarithms,"

(Dept. of Economics, Vanderbilt University, Nashville, TN 37235.) GUL,FARUK:"Efficiency and Immediate Agreement: A Reply to Hart and Levy." (Dept. of Economics, Princeton University, Princeton, NJ 08544-1021.) HART,SERGIU,ANDZoHAR LEVY:"Efficiency Does Not Imply Immediate Agreement." (Dept. of Economics and Mathematics, The Hebrew University of Jerusalem, Feldman Bldg., Givat-ram, Jerusalem 91904, Israel.) "An Endogenous JACKSON,MATTHEWO., EHUD KALAI,AND RANN SMORODINSKY:

Repre-

sentation of Priors in Bayesian Learning." (Division of Humanities and Social Sciences, 228-77, California Institute of Technology, Pasadena, CA 91125.) MAGNUS, JAN R., AND J. DURBIN: "Estimation

of Regression

Coefficients

of Interest

When Other Regression Coefficients Are of No Interest." (Center for Economic Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands.) MITRA, TAPAN, AND GERHARD SORGER: "Rationalizing

Policy Functions

by Dynamic

Optimization." (Dept. of Economics, University of Vienna, Briinnerstr. 72, A-1210 Vienna, Austria.) PERRY, MoTry,

AND PHILIP J. RENY: "On The Failure of the Linkage Principle

in

Multi-Unit Auctions." (Dept. of Economics, University of Pittsburgh, 4S01 Forbes Quad, Pittsburgh, PA 15250.)

Econometrica, Vol. 66, No. 6 (November,1998)

NEWS NOTES THENBER-NSF SEMINAR on Bayesian Inference in Econometricsand Statistics,the InternationalSocietyfor BayesianAnalysis,and the ASA Sectionon BayesianStatistical Science are co-sponsoringan annual LeonardJ. SavageAward of seven hundredfifty dollars($750)for an outstandingdoctoraldissertationin the area of BayesianEconometrics and Statistics. To be consideredfor the 1998 SavageAward,two copies of a doctoraldissertation must be submittedby the dissertationsupervisorbefore December31, 1998, and accompanied by a short letter from the supervisorsummarizingthe main results of the dissertation.An EvaluationCommitteewill be appointedby the Boardof the LeonardJ. S'avageMemorialFund(S.E. Fienberg,S. Geisser,J.B. Kadane,E.E. Leamer,J.W.Pratt, and A. Zellner,President)to evaluatethe dissertationsthat are submittedfor the Savage Award. The winnerof the 1997 SavageAwardand those receivinghonorablementionwill be announced at the Savage Award Session, arranged by E. Soofi, of the American StatisticalAssociation'sannualmeeting,August 9-13, 1998,in Dallas, Texas. Dissertations and supportingletters should be sent to Professor Arnold Zellner, GraduateSchool of Business,Universityof Chicago,1101 East 58th Street, Chicago,IL 60637, U.S.A., before December 31, 1998. *

*

*

THEEYROPEAN ECONOMIC ASSOCIATION will hold its annualCongressin Berlin,September 2-4, 1999.Note that EEA99will be preceededon the same site by the Econometric SocietyEuropeanMeeting,whichwill run fromAugust 29 to September1. Contributedpapers (in English) in all areas of economics are actively solicited. Submissionsshould include three copies of the paper and a short abstract(preparedas describedbelow), and must be receivedby March1, 1999, at the office of the Program Chairman,ProfessorPatrickRey, Universitedes SciencesSociales,Place AnatoleFrance, 31042ToulouseCEDEX, France. While no printed form will be provided,an abstractshould provide the following informationon a single sheet of paper:(i) Conferencename:EEA99;(ii) name(s)of the author(s),with the surnameof the paper presenterin capitals;(iii) title of paper;(iv) institutionalaffiliationof the paperpresenterand completemailingaddress;(v) abstract of paperof no more than 100 wordscontainingno symbols,references,or equations;(vi) up to six key words. At the same time as a submissionis made to the ProgramChairman,a duplicate abstractshould be suppliedto the Local ArrangementsCommittee,either by e-mail, or on a floppydisk in a DOS format(MicrosoftWord,Word Perfect,or as an ASCII text file) accompaniedby a hard copy. (Floppiescannot be returned.)The e-mail addressis: [email protected] mailing address is: Professor Alberto Meixide, Departamentode Fundamentosdel Analisis Economico,Facultadde Ciencias Economicasy Empresariales, Avda.Juan XXIII, s/n., 15704Santiagode Compostela(A Corunia),Spain. Furtherinformationis availableon the Internetat http://web.usc.es/-eea99. *

*

*

will be held in Stockholm December17-19, 1998.(EC)2 is a series of annualinternationalconferenceson research in quantitativeeconomics and econometrics.The first conference in the series was THE (EC)2 CONFERENCE ON FORECASTING IN ECONOMETRICS

1435

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NEWS NOTES

organized 1990. The acronym (EC)2 stands for European Conferences of the Econom[etr]icsCommunity.The main aim of these conferencesis to providea convenient forumfor both senior and junior Europeanresearchersin quantitativeeconomics and econometricsto discuss their research results. The 1998 edition of (EC)2 will be arrangedat StockholmSchoolof Economics,Stockholm,Sweden,17-19 December1998. The general theme of the conferenceis Forecastingin Econometrics.Invitedspeakers include FrankDiebold, Henk Don, Clive Granger,David Hendryand SvendHylleberg. The programwill also contain a limited numberof contributedpaperpresentationsand two poster sessions. As in previous(EC)2 conferences,the number of participantsis restrictedto 100. The conference is organizedby the Departmentof Economic Statistics,Stockholm School of Economics.Inquiriesconcerningthe meeting can be sent by e-mail to the followingaddress:mailto:[email protected] The web site of the (EC)2conferenceis: http://www.hhs.se/stat/workshops/ec2/.The informationwill be updatedcontinuously. *

*

*

THE FEDERALRESERVEBANK OF CHICAGOwishes to announce that a new micro

databaseis availablewithoutchargeon the worldwide web. The databaseincludesthe balancesheets and income statementsfor all U.S. banksthat are federallyinsured.The informationstartsin 1976 and continuesuntil the present and is stored as SAS XPORT files. The web site also containsdocumentationon all the variables,a short description on how to form consistenttime-seriesfor many of the majorseries, and separate files that identifythe dates and outcomes of all the U.S. bank mergersthat have occurred duringthis time period. The addressof the site is www.frbchi.org/rcri/rcri-database.html The bankasksyou to identifythe web site in any papersyou publishthat use the data. *

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NEP (NEWECONOMICS PAPERS)is a family of mailing lists that disseminates new papers

in economics.It is organizedalong manyfields, each havinga separatemailinglist. The featuredpaperswill be new additionsto RePEc, a databasethat currentlyholds about 56000. This includespapersfrom the NBER, the CEPR, all US Feds, as well as many other departmentsand institutions.The currentholdingscan be inspectedat WoPEc: http://netec.mcc.ac.uk/WoPEc/WoPEc.html (online papers), BibEc; http://netec.mcc.ac.uk/BibEc/BibEc.html (printed papers) and IDEAS; http://ideas. uqam.ca/ (all). The NET home page is at http://netec.wustl.edu/NEP/

The current reports are: nep-cdm (Collective Decision-Making);nep-dcm (Discrete Choice Models); nep-dge (Dynamic General Equilibrium);nep-ecm (Econometrics); nep-ets (Econometrics-TimeSeries); nep-eff (Efficiency and Productivity);nep-eec (EuropeanEconomics);nep-evo(EvolutionaryEconomics);nep-exp(ExperimentalEconomics);nep-fmk(FinancialMarkets);nep-gth (Game Theory);nep-hea (Health Economics);nep-ifn(InternationalFinance);nep-ltv(Labor:Turnover,Vacancies);nep-mic (Microeconomics);nep-net (NetworkEconomics);nep-pol (PositivePoliticalEconomy); nep-pke (Post Keynesian Economics);nep-pbe (Public Economics);nep-pub (Public

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Finance);nep-reg(RegionalEconomics);nep-spo(Sportsand Economics);nep-tid(Technology and IndustryDynamics). Each report is edited by an individualor an editorial board. In addition, nep-all provides all new papers and nep-ann announces new reports. To subscribe,send a messageto [email protected] with no subjectand the followingbody: oin nep-xxxfirstnamelastname puttingyour name where appropriate,where xxx is the name of the reportto whichyou wish to subscribe.Removeyour signatureto preventerrormessagesfrom mailbase.All NEP reportsare free. NEP is sponsoredby the ElectronicLibraryProgramme. *

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THE1999 MEETINGS of the Society for Economic Dynamicswill be held June 27-30 (Sunday-Wednesday),1999 at the PortoconteResearchCenter,Alghero,Sardinia,Italy. Registrationand welcome receptionwill be held on Saturday,June 26, 1999. Program organizersare TimothyKehoe and Antonio Merlo. The Society for Economic Dynamicssolicits applicationsfor papers in all areas of dynamiceconomicsto be presentedat this conference.Membersand nonmembersare invitedto participate.The deadlinefor submissionsis February1, 1999. Please send an abstractand a paperif available,togetherwith names, affiliations,addresses,and e-mail addressesof all authors,to SED Conference,c/o Wendy Williamson,Departmentof Economnics, Universityof Minnesota,271 19th Avenue South, Minneapolis,MN 55455. Fax or electronicsubmissionswill not be considered.For furtherinformationabout the conference,and submissionand acceptanceof papers,check the conferencewebpageat http://www.econ.umn.edu/sed99.

Econometrica, Vol. 66, No. 6 (November,1998) PROGRAM OF THE FIFTEENTH LATIN AMERICAN MEETING OF THE ECONOMETRIC SOCIETY SANTIAGO, CHILE

AUGUST12-15, 1997 Sponsors Central Bank of Chile IDB IDBR Banco del Estado Ministry of Finance Bank Paribas Estrategia Newspaper Cochrane Corporation Tajamar - travel agency

Alumni Association of School of Economics, University of Chile Referees Manuel Agosin, University of Chile Miguel'Bash, University of Chile David Bravo, University of Chile Carlos Budnevich, Central Bank, Chile Romulo Chumacero, University of Chile Dante Contreras, University of Chile Gustavo Crespi, University of Chile Eduardo Engel, University of Chile Eugenio Figueroa, University of Chile Ronald Fisher, University of Chile Jorge Friedmann, University of Chile Rodrigo Fuentes, University of Chile Alexander Galetovic, University of Chile Jose de Gregorio, University of Chile Osvaldo Larrainaga,University of Chile Leonardo Letelier, University of Chile Carlos Maqueira, University of Chile Patricio Meller, University of Chile Alejandra Mizala, University of Chile Felipe Morande, Central Bank, Chile Patricio Mujica, University of Chile Ricardo Paredes, University of Chile Franco Parisi, University of Chile Luis Riveros, University of Chile Pilar Romaguera, University of Chile Jose Miguel Sanchez, University of Chile Ricardo Sanhueza, University of Chile Klaus Schmidt-Hebbel, Central Bank, Chile 1439

1440

PROGRAM

SebastianValdes, Universityof Chile Jose Yafiez,Universityof Chile SalvadorZurita,Universityof Chile Local Organizing Committee OsvaldoLarrafiaga,Directorof the EconomicsDepartment,Universityof Chile RicardoParedes,Departmentof Economics,Universityof Chile Filipe Morande,CentralBank of Chile KlaussSchmidt-Hebbel,CentralBank of Chile ProgramCommittee Luis Riveros(Chair),Departmentof Economics,Universityof Chile Argentina:Victor Elias, MiguelKiguel,Rolf Mantel,J. P. Nicolini,CarolaPesino Basil:Aloisio Araujo,RicardoPaez, MarildaSotomayor,Pedro Valls, SergioWerlang Chile:VittorioCorbo,EduardoEngel, OsvaldoLarranaga,Rolf Luders,PatricioMeller, Filipe Morande,RicardoParedes,KlausSchmidt-Hebbel Arlito Barletta,Juan R. Vargas CentralAmericaand the Caribbean: Colombia:AlvaroReyes, EduardoSarmiento,RobertoSteiner Europe:SalvadorBarbera,P. A. Chiappori,BernardCornet,AlbertoHolly, J. J. Laffont, AndreuMas-Colell,MarioPascoa,Jesus Seade Mexico:AlejandroHernandez,Pedro Uribe, CarlosUrzua Peru:JavierEscobal,MaximoVega-Zenteno Uruguay:GastonnLabadie,Ruben Tanzini U.S.A.:GuillermoCalvo, Rudiger Dornbusch,Sebasian Edwards,Ricardo Hausmann, Hugo Hoppenhayn, Rodolfo Manuelli, Marc Nerlove, Sergio Rebelo, Jose A. Scheinkman ProgramSummary August12 Registrationand Information August13 08:30-9:30OpenCeremony Mr. RicardoHausmann,Presidentof the LatinAmericanBranchof the Econometric Society; Prof. Luis Riveros,Presidentof the ProgramCommittee; Mr. EduardoAninat,Ministerof Financeof Chile. 09:30-10:15 Prof.JagdishBhagwati,ColumbiaUniversity,"WhereAre We Goingin TradeTheory." CHAIR: Luis Riveros,Universidadde Chile. Sponsoredby CorpBanca. 10:15CoffeeBreak.

PROGRAM

1441

10:30-12:30 Session I 12:30-13:30 Simultaneous Sessions

(1) SpecialSession on EconomicGrowth Prof. Robert Barro, Harvard University, "The Interplay between Political and EconomicDevelopment." CHAIR: Juan R. Vargas,Universidadde Costa Rica. Sponsoredby School of Economics& Business,Universidadde Chile. (2) Prof. DavidHendry,NuffieldCollege-Oxford,"RecentMethodologicalInnovations in Econometrics." RodrigoVergara,Centrode EstudiosPublicos. CHAIR: Sponsoredby The WorldBank. 13:30-15:00 Lunch

Guest Speaker:Mr. Alvaro Saieh, President of the Corp Group, "Policy Making, EconomicsSignalsand PrivateSectorDevelopment." Sponsoredby CorpBanca. 15:00-17:00 Session II 17:00 Coffee Break 1 7:15-18:15 Carlos Diaz Alejandro Lecture

Prof. RicardoCaballero,MIT, "UnprotectedTransactions:Social Costs and MacroeconomicConsequences." CHAIR: AlejandroHernandez,ITAM. Sponsoredby The WorldBank. 18:15-19:15 Miguel SidrauskyLecture

Prof. Rolf Mantel, Universidadde San Andres, "Recent Theoretical Progress in WelfareTheory." PatricioMeller,Universidadde Chile. CHAIR: Sponsoredby Departmentof Economics,Universidadde Chile. August 14, 1997 08:30-9:30 Session III 09:30-10:45 Special Session in Honor of Michael Bruno

WilliamEasterly,The WorldBank. LeonardoLeiderman,Tel-AvivUniversity. Miguel Kiguel,Ministryof Economy-Argentina. Rafi Melnick,CentralBank of Israel. MarioBlejer,IMF. CHAIR: Sponsoredby IMF, Bank of Israel. 10:45 Coffee Break

1442

PROGRAM

11:00-13:00 Session IV 13:00-14:00 Panels (Simultaneous) (1) Public Sector Reform in Perspective

MartinRama,The WorldBank. MarianoTommasi,Universidadde los Andes. AugustoLopez Claro,IMF. MichaelGavin,IADB. CHAIR: CarlosVegh, UCLA. Sponsoredby Ministryof Finance,Chile. (2) Financial Econometrics and Policy Analysis

MichaelBrennan,UCLA. Robert Shiller,Yale University. George Tauchen,Duke University. CHAIR:SalvadorZurita,Universidadde Chile. Sponsoredby Banco del Estado de Chile. 14:00-15:30 Lunch

Guest Speaker:Mr. CarlosMassad,Presidentof the CentralBank of Chile, "Experiencingwith an IndependentMonetaryPolicy." Sponsoredby CentralBank of Chile. 15:30-17:30 Session V 17:30 Coffee Break 177:45-18:45Simultaneous Sessions

(1) Daniel Hamermesh,Universityof Texas at Austin, "The Art of Laborometrics." CHAIR: Gaston Labadie,ORT. Sponsoredby IADB. (2) James Heckman,Universityof Chicago,"New Developmentsin ProgramEvaluation and Estimationof TreatmentEffects." Victor Elias, Universidadde Tucuman. CHAIR: Sponsoredby IADB. 18:45-20:15 Invited Papers (Simultaneous) (1) Early Signals of Currencyand Banking Crises CHAIR: OmarChisari,Universidadde Buenos Aires.

(1.1) "LeadingIndicatorsof CurrencyCrises,"GracielaKaminsky,Boardof Governors of the Federal Reserve System;Saul Lizondo,World Bank; CarmenRienhart,Universityof Maryland. (1.2) "EarlingWarningIndicatorsof BankingCrises:Are They the Same in Emerging Marketsand IndustrialCountries?"LilianaRojas Suarez,IADB. (1.3) "The Case for an InternationalBankingStandard,"MorrisGoldstein,Institute of InternationalEconomics.

PROGRAM

1443

(1.4) "LeadingIndicatorsof Currencyand BankingCrises,"John Welch, Banque Paribas. (2) CentralAmerica CHAIR: MaximoVega-Centeno,PUC-Peru. (2.1) "Prices, Money and Exchange Rate in Costa Rica: An Error Correction Model,"KatiaVindas,CentralBank of Costa Rica. (2.2) "Importsand ExchangeRate in Costa Rica,"Ana Cecilia Kikut,Universidad Nacionalde Costa Rica. (2.3) "EstimatingMonetaryAggregateBalancesusingWeeklySeasonalCoefficients. A MethodologicalProposal,"EvelynMuiioz,CentralBank of Costa Rica. (2.4) "The CostaRican Peace Dividend,"JuanRafael Vargas,Universidadde Costa Rica. 21:00 OfficialDinner Guest Speaker:Mr. Andres Sanfuentes,Presidentof Banco del Estado de Chile. Sponsoredby Banco del Estado de Chile. August15, 1997 08:30-9:45Panels (Simultaneous) (1) Povertyand IncomeDistribution. FranciscoFerreira,The WorldBank. OsvaldoLarrafiaga,Universidadde Chile. Juan Luis Londofio,IADB. Jose de Gregorio,Universidadde Chile. CHAIR: Sara Calvo,The WorldBank. Sponsoredby the EDI, The WorldBank. Economics,In Honor of MarioHenriqueSimonsen. (2) RecentResultsin Mathematical MarildaSotomayor,Universityof Pittsburg. SergioRibeiroda Costa, Getulio VargasFoundation. CHAIR:Aloisio Araujo, PUCRJ.

and Development. (3) CapitalAccountLiberalization John Williamson,The WorldBank. RicardoFfrench-Davis,CEPAL. Toru Yanahigara,Hosei University. HarveyRosenblum,FederalReserve Bank of Dallas. CHAIR: ManuelAgosin, Universidadde Chile. Sponsoredby GraduateSchool of Economics,Universityof Chile. 9:45 Coffee Break 10:00-12:00 Session VT

1444

PROGRAM

12:00-13:30 Simultaneous Sessions (1) Economic Analysis Lecture

Prof. PrestonMcAfee, Universityof Texas at Austin, "RecentDevelopmentsin the Theoryand Practiceof Auctions." CHAIR: Felipe Morande,CentralBank of Chile. Sponsoredby Revistade AnalisisEconomico. (2) Trade TheoryLecture

Prof. EdwardLeamer,UCLA, "EconomicIntegrationWages and Welfare." CHAIR: RodrigoFuentes,Universidadde Chile. Sponsoredby RevistaEstudiosde Economia. 13:30-15:00 Lunch 15:00-17:00 Session VII 17:00 Coffee Break 177:15-18:30Panel: Stabilization,Adjustmentand EconomicPolicy in Latin Americain

the 1990's. RicardoHausmann,IADB. GuillermoPerry,The WorldBank. VittorioCorbo,PUC-Chile. CHAIR: KlausSchmidt-Hebbel,CentralBank of Chile. Sponsoredby CentralBank of Chile. 18:30-19:30 PresidentialLecture

Prof.ArnoldHarberger,UCLA, "The Processof EconomicGrowth." Rolf Luders,PUC-Chile. CHAIR: Sponsoredby Estrategia. 20:00 Closing Cocktail Sessions August 13, 10:30 to 12:30 am Poverty & Inequality, Session I-Room

1

"PolicyOptionsfor PovertyAlleviation,"MiguelSzekely,IADB. "Welfare, Poverty and Income Inequality:Appraisal of Historical Evolution and Regional Disparities,"Ricardo Paes de Barros, Rosane Mendonca, and Renata PachecoNogueira,IPEA, Brasil. "Microfinanzas y Pobreza:El Caso de las CajasMunicipalesdel Peru,"Albert Chong, U. of Maryland,USA, and EnriqueSchroth,GRADE, Peru. "EarningsDispersionand Returnsto Skills after StructuralAdjustment:the Case of Peru,"Jaime Saavedra,GRADE, Peru. "IncomeDistributionin Sao Paulo and the 1994StabilizationPlan,"MariaCarolinade SilvaLeme and Ciro Biderman,FundacionGetulio Vargas,Brasil. CHAIR:Eduardo Engel.

PROGRAM

1445

August13, 10:30to 12:30am Institutional Economics,SessionI-Room 2. "The InformalSector,FirmDynamicsand InstitutionalParticipation," Alec R. Levenson, Milken Institute for Job & Capital Formation,USA, and William Mahoney, Universityof Illinois,USA. "AmnestyProgramsas CorruptionDeterrenceDevices,"James Kahhat,Boston University,USA. "Corruption:Some Elements for the Analysis,"Federico Weinschelbaum,UCLA, USA. "On the DistributionalEffects of Social Security Reform," Mark Hugget, ITAM, Mexico,and GustavoVentura,Universityof Illinois,USA. CHAIR: Victor Gastafiaga. August13, 10:30to 12:30am GrowthTheory,SessionI-Room 3. "An ExportSectorLed EndogenousGrowthModelwith Tradeableand No Tradeable Goods,"EnriqueCasares-Gil,U. AutonomaMetropolitana,Mexico. "A Dynamic Analysis of an Endogenous Growth Model with Leisure," Salvador Ortigueira,ITAM,Mexico. "Patternsof EconomicDevelopmentand the Formationof Clubs,"Alain Desdoigts, UniversiteD'EvryVal, D'Essonne,Francia. "Growth,ExternalDebt and ExchangeReal," Delfim Gomes Neto, DELTA,Francia. "Multiple Equilibrium,Variability and the Development Process," Jose Galdon Sanchezand Luis Carranza,IMF, USA. CHAIR: ElizabethBucacos. August 13, 10:30 to 12:30 am Econometric Theory,Session I-Room

4.

"Stationarityand Structural Breaks: Evidence from Classical and Bayesian Approaches,"Antonio E. NoriegaMuro and Enriquede Alba, Universidadde Guanajuato, Mexico. "Finite Sample Propertiesof the Efficient Method of Moments,"Romulo A. Chumacero,Universidadde Chile, Chile. "NonparametricEstimation of a SurvivorFunction with Across-Interval-Censored Data,"RobertoA. Ayala and MarkYuyingAn, Banco Central,Ecuador. "ThresholdUnit Root Models,"MartinGonzalo-U.CarlosIII de Madrid,Espafia. CHAIR: GustavoCrespi. Applied Finance, Session I-Room

5.

"Financial Derivatives Introductionand Stock Return Volatility in an Emerging MarketwithoutClearinghouse:The MexicanExperience,"FaustoHernandezTrillo, CIDE, Mexico. "Pricingthe Option Adjust Spread of BrazilianEurobonds,"Franklinde Oliveira Goncalvesand Marciode 0. Barros,Banco de Bahia InvestimentosS.A., Brasil.

1446

PROGRAM

"A Theory of CorporateCapital,"Miguel Cantillo Simon, Universityof California, Berkeley,USA. "The EmpiricalTests to the CAPM:Borrowingand LendingRestrictionsand Related Issues,"SebastianA. De Ramon,UniversityCollege London,UK. "ADRs in EmergingEquity Markets:MarketsIntegrationor Fragmentation,"Kent Hargis,U. South Carolina,USA. EduardoFernandez. CHAIR: August13, 10:30to 12:30am CapitalMarkets,SessionI-Room 6. "A Theory of FinancialMarketStructure,"BharatAnand, Yale School of Management, USA, and AlexanderGaletovic,U. de Chile, Chile. "BusinessGroups,StocksMarketsAtrophyand Growth,"GonzaloCastafieda,Universidad de las Americasde Mexico,Mexico. "InvestmentUnder Uncertaintyand Financial Market Development:A q Theory Approach,"SergioLehmann,Banco Central,Chile. "On the Limits to Speculationin Centralizedvs. DecentralizedMarket Regimes," Felipe Zurita,Universidadde Chile, Chile. "On FinancialMarkets,Entrepreneurshipand the Distributionof Wealth,"Alexander Monge N., Universityof Chicago,USA. CHAIR: RodrigoAranda. August13, 15:00to 17:00 FiscalPolicy,SessionII-Room 1. "Flat Tax Reform: A QuantitativeExploration,"Gustavo Ventura, University of Illinois,USA. "Does Fiscal Policy Matter?:The Case of Chile," Yianos T. Kontopoulos,Merrill Lynch,USA, and DimitriosD. Thomakos,ColumbiaUniversity,USA. "PublicDebt Sustainabilityand EndogenousSeignoragein Brazil:Time-SeriesEvidence from 1947-92,"Luis Renato R. de OliveiraLima,FoundationGetulioVargas, Brasil,and Joao Victor Issler,Banco Marka,Brasil. "UnderstandingTax Evasion Dynamics,"Eduardo Engel and James R. Hines, Jr., Universidadde Chile, Chile. CHAIR:Silvia Montoya.

August13, 10:30to 12:30am IndustrialOrganization, SessionII-Room 2. "Detecciondel Poder de Mercadoen el Sector ManufactureroMexicano,"Alejandro CastafiedaSabido,El Colegio de Mexico,Mexico. "AuditConservatism,ManagementIncentiveand ManagerialOptimism,"PradyotK. Sen, SUNY Buffalo,USA. and MarketStructurein the PatentsLicensingRelationship,"Manel "Experimentation Antelo, Universidadde Santiagode Compostela,Espafia. "EntryMistakesHappen,"Luis Cabral,LondonBusinessSchool,UK. CHAIR:Carlos Zarazaga.

PROGRAM

1447

EconomicTheory,SessionII-Room 3. "EquilibriumAllocationsof Fair Quasi-Gamefor Economieswith IndivisibleGoods," CarmenBevia, UniversitatAutonomade Barcelona,Espafia. "A Notion of Subgame Perfect Nash EquilibriumUnder KnightianUncertainty," SergioRibeiro de Costa Werlang,FundacionGetulio Vargas,Brasil. "Distance Metric and the Debreu-FarrellMeasure of TechnicalEfficiency,"Walter Briec and FranciscoF. Ramos,UniversidadFederalde PernambucoPIMES,Brasil. "AsymptoticResult on the Core of a ContinuumEconomy,"Carlos Herves-Beloso, Emma Moreno-Garcia,and CarmeloNuiiez-Sanz,Universidadde Vigo, Espafia. "An Applicationof the CatastropheTheory in General EquilibriumTheory,"Elvio Accinelli,IMERL,Uruguay,and MartinPuchet,UNAM, Mexico. CHAIR:Romulo Chumacero.

August13, 10:30to 12:30am Macroeconometrics,Session II-Room

4.

"ErrorCorrectionand Cointegrationin a DynamicPanel:PurchasingPowerParityin a Flexible ExchangeRate Regime,"Victor Gastafiaga,U. of SouthernCalifornia, USA. "Determinatesdel Tipo de CambioReal: Un Modelo de VectoresAutoregresivopara Chile," PatriciaMujicaand GustavoCrespi,Universidadde Chile, Chile. "An Estimationof a MacroeconomicModel for the Chilean Economy,"Leonardo L6telierand MiguelBasch,Universidadde Chile, Chile. "PurchasingPower Parityfor Brazil.A Test for FractionalCointegration,"Vera L. Fava and DenisradC.O. Alves, FEA/USP, Brasil. CHAIR: Felipe Zurita. August 13, 10:30 to 12:30 am Labor I, Session II-Room

5.

"The Impact of the Labor Market Policies on the Wage Structure:Evidence from Chile and Uruguay,"Gaston Labadie and Steven Allen, North Carolina State University,USA. "EfficientWage Dispersion,"D. Acemogluand R. Shimer,PrincetonUniversity,USA. "6ExisteAlgunaRelacionentre SalarioMinimoy Empleo?Teoria,Dogmay Evidencia Empirica,"David BravoU. and Dante Contreras,Universidadde Chile, Chile. "LaborMarketStructurein DevelopingCountriesTime Series Evidenceon Competing Views,"WilliamF. Maloney,U. Illinois at Urbana-Champaign, USA. CHAIR: MarioTello. August 13, 10:30 to 12:30 am Household Economics, Session II-Room

6.

"Socio-economicFactorsAffecting the Joint Consumptionof Tobacco and Alcoholic Beverages:SpanishEvidence,"Justo Manrique,Saint Louis University,Espania,and Helen H. Jensen, Iowa State University,USA. "A BivariateIntegral Control MechanismModel of Household Consumption,"Jan Spitalsky,Academyof Sciencesof the Czech Republic,Czech Republic.

1448

PROGRAM

"ChildHealth and the Distributionof Household Resources at Marriage,"Duncan Thomas,UCLA,USA, Dante Contreras,Universidadde Chile, Chile, and Elizabeth Frankenberg,RAND. "Does Birth OrderMatter?The Importanceof Birth Orderwhen ParentsSpecialize by Gender in Child'sNutrition:Evidencefrom Chile,"Luis R. Rubalcava,UCLA, USA, and Dante Contreras,Universidad de Chile, Chile. "Who SufferDuringan EconomicCrisis?"GracielaTeruel, UCLA, USA. CHAIR: RonaldFischer. August 14, 8:30 to 9:30 am Information & Uncertainty,Session III-Room

1.

Partnesswith BilateralMoralHazardand BalancedBudgets,"Sonia Di "Risk-Sharing Giannatale,StephenE. Spear,and ChengWang,CarnegieMellon University,USA. "Policy Uncertaintyand InformationalMonopolies:The Case of MonetaryPolicy," LarryE. Jones, NorthwesternUniversity,USA, and Rodolfo Manuelli,Universityof USA. Wisconsin-Madison, "Risk Dominace Selects the Leader. An ExperimentalAnalysis,"Antonio Cabrales and MassimoMotta,Universitat,PompeuFabra,Espafia. CHAIR:Dante Contreras. August 14, 8:30 to 9:30 am Commodities, Session III-Room

2.

"CoffeeExportBooms and MonetaryDisequilibrium-Some Evidencefor Colombia: 1970-1992,"Jesus G. Otero, The Universityof Warwick,UK. "StabilizationMechanismsfor PrimaryCommodityExporters:An Oil Establization Fund for the Venezuelan Case," Alejandro Grisanti,Universityof Pennsylvania, USA. "ComparingTwo Modelling Approaches:An Example Using Fed Beef Supply," CamiloSarmineto,Colombia. CHAIR: StevenAllen. August 14, 8:30 to 9:30 am Capital Flows, Session III-Room

3.

"ControllingCapitalFlows:TargetingStocksvs. Flows,AmartyaLahiri,UCLA,USA. "CapitalFlowsto Brazilin the Nineties:MacroeconomicAspectsand the Effectiveness of CapitalControl,"MarcioG. P. Garciaand AlexanderBarcinski,PUC RJ, Brasil. "What Determines Capital In Flows?: An EmpiricalAnalysis for Chile," Romulo Chumacero,Universidadde Chile, Chile, Raul Laban,GERENS,Chile and Felipe Larrain,PUC, Chile. CHAIR:JavierEscobal.

PROGRAM

1449

August14, 8:30 to 9:30 am Econometrics,SessionIII-Room 4. "Approximating and Simulatingthe Real BusinessCycle:Linear QuadraticMethods, ParameterizedExpectations,and Genetics Algorithms,"John Duffy, Universityof Pittsburgh,USA, and Paul D. McNelis,Universityof Georgetown,USA. "Unit Roots, Exogeneity,and Persistence:A Critical Overview,"Guglielmo Maria Caporaleand NikitasPitis, LondonBusinessSchool,UK. "ConditionalLabor SupplyQuantile Estimates in Brasil,"EduardoPontual Riveros, UniversidadeFederalde Roraima,Brasil. CHAIR:LuisQuintas.

August14, 8:30 to 9:30 am EmpiricalMonetaryEconomics,SessionIII-Room 5. "CurrencySubstitutionand the Moneynessof MonetaryAssets,"EduardoA. Moron, UCLA, USA. "Is the Business Cycle of Argentina'Different'?"Fin Kydlandand Carlos E. J. M. Zarazaga,FederalReserveBank of Dallas, USA. "The Demandand Supplyof MoneyunderHigh Inflation:Brazil1974-1994,"Octavio A. F. Tourinho,IPEA, Brasil. CHAIR: EstebanJadresic. August14, 8:30 to 9:30 am MonetaryTheory,SessionIII-Room 6. "Do PrescribedStochasticMonetaryEquilibriaExist? And Other MonetaryIssues," Haim Abraham,Universityof Cape Town,South Africa. "Global Optimum and ConvergenceRates in Monetary Policy," Hsih-chia Hsich, ProvidenceUniversity,Taiwan. "A DynamicModel of the AsymmetricEffects of MonetaryShocks,"PedroAuger and Paul Beaudry,IADB, USA. CHAIR:Maurice Kugler. August 14, 11:00 to 13:00 pm Exchange Rate I, Session IV-Room

1.

"Riesgo Cambiario y Riesgo Tasa de Interes en la Banda de Flotacion," Umberto Della Mea, Banco Central de Uruguay. "Cyclical Fluctuations in Brazil's Real Exchange Rate: The Role of Domestic and External Factors," P. Richard Agenor, A. Hoffmaister, and C. Madeiros, IMF. "Misalignment and Fundamental Fernando A. Broner Norman Loayza," Humberto Lopez, Massachusetts Institute of Technology, USA. "A Model of Smuggling and Black Markets in Foreign Exchange," Richard C. Barnett, University of Arkansas, USA. CHAIR:Miguel Basch.

1450

PROGRAM

August14, 11:00to 13:00pm MonetaryPolicy,SessionIV-Room 2. "Growth,Money and Debt: The MexicanExperienceof 1989-1995,"FranciscoVenegas-Martinez,CIDE, Mexico. "CentralBank Interventionwith Options,"FernandoZapateroand Luis F. Reverter, ITAM,Mexico. "Sistema Bancario y EstabilizacionMacroeconomica:Peru," Mario ZambranoB., Banco Centralde la Rerservadel Peru, Peru. "Encajey AutonomiaMonetariaen Chile,"Luis Oscar Herreraand RodrigoValdes P., Banco Centralde Chile, Chile. "Transmisionde la Politica Monetariaen Chile,"RodrigoValdes, Banco Centralde Chile, Chile. CHAIR:Alejandra Mizala. August 14, 11:00 to 13:00 pm Applied Econometrics, Session IV-Room

3.

"Money Demand in Uruguay:An ArtificialNeural Network Approach,"Elizabeth Bucacos,Banco Centralde Uruguay,Uruguay. "AlternativeApproachesto TestingHysteresis,"HildegartAhumada,InstitutoTorcuato Di Tella, Argentina. "Is ImpliedCorrelationWorthCalculating?Evidencefrom ForeignExchangeOptions and HistoricalData,"ChristianWalterand Jose A. Lopez,FederalReserve Bankof New York,USA. "Non Linear Discrete and ContinuousModels for the BrazilianShort-TermInterest Rate,"EduardoC. de la Rocque, Bnaco de Bahia InvestimentosS. A., Brasil. CHAIR: RicardoLopez. August 14, 11:00 to 13:00 pm TradePolicy I, Session IV-Room

4.

"EnvironmentalProtection under Bilateral Trade and Imperfect Competition," Roberto Burguet,Institutefor EconomicAnalysis,CSIC,Espafia,and Jaime Sempere, Centrode EstudiosEconomicos,El Colegio,Mexico. "ExportSubsidies,Price Competitionand Vertical Integration,"LeonardoMedrano, Centrode Investigaciony Docencia Economica,Mexico. "TradeLiberalizationand Productivity:A Panel Studyof the MexicanManufacturing Industry,"Talan Iscan,DalhousieUniversity,Canada. "North-SouthCustoms Unions and InternationalCapital Mobility,"Eduardo FernandoArias and MarkSpiegel,IADB, USA. "The Choice of an InternationalCurrencyfor Trade,"Francois Hada, Banque de France,and BenjaminSabel, ENSAE, Francia. CHAIR: Julio de Brun

PROGRAM

1451

August14, 11:00to 13:00pm Game TheoryI, SessionIV-Room 5. "The Theory of Implementationwhen the Planner is a Player," Luis Corchon, Universidadde Alicanta,Espafia. "Negotiationwhenthe Size of the Pie Dependson Howit is Cut,"DavidMayer-Foulkes and Raul Garcia-Barrios,CIDE, Mexico. "The Sources of Randomnessin Model of Social Interaction:A Note on Arthur (1994),"EduardoZambrano,CornellUniversity,USA. "EquilibriumSelection by Finite Automata and StrategicDomination in Repeated Games,"Luis Quintas,IMASL,Universidadde San Luis, Argentina. CHAIR:David Bravo. August 14, 11:00 to 13:00 pm Savings and Investments, Session IV-Room

6.

"Quantifyingthe Effects of Medicaidand Medicareon AggregateSavings,"RonaldW. Gecan, Universityof Minnesota,USA. "CapitalInflows and InvestmentPerformance:Chile in thel990s," Manuel Agosin, Universidadde Chile, Chile. "Los Determinantesdel AhorroPrivadoen el Peru y el Papel de la PoliticaEconomica,"Jorge Barredaand Elmer Cuba,Macroconsult,Peru. "DomesticSavings,Public Savingsand Expenditureson ConsumerDurableGoods in Argentina,"RicardoLopez M. and FernandoNavaja,Fundacionde Investigaciones Argentina. EconomicasLationoamericanas, "Investmentin Colombia:Microadjustmentand Aggregation,"Alberto Isgut, Wesleyan University,USA. ArmandoCastelar. CHAIR: August 14, 15:30 to 1 7:30 pm Empirical Growth, Session V-Room 1.

"El Capital Fisico y Humano en Uruguay,"Victor Elias, UniversidadNacional de Tucumany FundacionBanco Empresarario,Argentina. "Crecimientoy FluctuacionesMacroeconomicsaen America Latina,"Pablo Cotler, ITAM,Mexico. "An ExplorationInto the Sourcesof EconomicGrowthin FourteenLatin American Countries,"EdgardRobles, UCLA, USA. "PotentialOutput Growthin EmergingMarkets:The Case of Chile,"Jorge Roldos, IMF, USA. "Inestabilidad y Continuidad del Crecimiento: Algunas Manifestaciones del Desempefiode la EconomiaPeruana,"MaximoVega-Centeno,PUC, Peru. CHAIR:Patricio Mujica.

1452

PROGRAM

August 14, 15:30 to 17:30 pm StructuralReforms, Session V-Room 2.

"Social SecurityRegimen, Growth,and Income Distribution,"Patricia M. Langoni, Banco Centralde Chile, Chile. "InfractucturePrivatizationin a NeoclassicalEconomy:MacroeconomicImpactand Welfare Computation,"Pedro CavalcantiG. Ferreira,FundacionGetulio Vargas, Brasil. "PensionReform,InformalMarkets,and Long Term Income and Welfare,"William C. Grubenand Robert McComb,FederalReserve Bank of Dallas, USA. CHAIR: Paul D. McNelis August 14, 15:30 to 17:30 am Applied Microeconomics, Session V-Room 3.

"MultipleEquilibriumin the Welfare State with Costly Policies," Alvaro Forteza, Universidadde la Republicade Uruguay,Uruguay. "FiringCosts and Stigma:A ThoreticalAnalysis and Evidence from Micro Data," PatriziaCanziani,LondonSchool of Economics,UK. "Fundacionde ProduccionEducacionaly Eficiencia de la Produccionen Chile," AlejandraMizala,PilarRomaguera,and Dario Farren,Universidadde Chile, Chile. "InteraccionEstrategicaen EleccionesParlamentarias," EduardoEngel and Alejandro Neut, Universidadde Chile, Chile. "Disposiciona Pagar por bienes Ambientales:Una Aplicacion a la Contaminacion Atmosfericaen Santiagode Chile,"RobertoAlvarez,Eugenio Figueroa,and Sebastian Valdes, Universidadde Chile, Chile. CHAIR:Leonardo Letelier. August 14, 15:30 to 17:30 am Regional/Agricultural Economics, Session V-Room 4.

"GeographicalAssociationas an Alternativefor Diffusionin the PeruvianTraditional Agriculture,"MaximoTorero,UCLA, USA. Jun"RegionalHeterogeneityin the Developmentof BrazilianAgriculture,"Juliano queiraAssuncao,CEDEPLAR,U. Federalde MinasGerais,Brasil. "IntegracionEspecialde MercadosAgricolas:Un Analisisde CointegracionMultivariada,"JavierEscobaland JorgeAgiuero,GRADE, Peru. "El Centroy la Periferia,Una ApproximacionExpiricaa la Relacion entre Limay el Resto del Pais," Giovana Aguilar Andia and Gonzalo CamargoCardenas,PUC, Peru. CHAIR: EduardoMoron. August 14, 15:30 to 17:30 pm Productivity& Innovation, Session V-Room 5.

"Dinamicade Empleoy Productividaden Manufactura:EvidenciaMicroy Concecuencias Macro,"Alexis Camhi,EduardoEngel, and AlejandroMicco, Universidadde Chile, Chile.

PROGRAM

1453

"StochasticEstimation of Firm Inefficiency Using Distance Functions,"Scott E. Atkinson,Rolf Fire, and Dan Primont,The Universityof Georgia,USA. "Intra-RegionalInequality,TechnologicalDiffusion,and the Patternof International Trade," Maurice Kugler, U. de los Andes, Colombia, and Josef Zwelmueller, Universityof Zurich. "LosEfectos de la LiberalizacionComericalen Gastose InversionesI + D: El Caso de la IndustriaManufactureraUruguaya,"Walter GarciaFontes, UniversitatPompeu Fabra,Espaiia,and Ruben Tansini,Universityof Uruguay,Uruguay. "MachineReplacement,Network Externalitiesand InvestmentCycles,"Juan Ruiz, Boston University,USA. CHAIR:Juan Luis Londofio.

August14, 15:30to 17:30pm BankingI, SessionV-Room 6. "X-Inefficiency in the Private Banking Sector of Argentina: Its Importance with Respect to Economies of Scale and Economies of Joint-Production," Atrid Dick, Banco Central de la Republica Argentina, Argentina. "Market Power in the Chilean Banking System: Evidence from a Panel Data Study," Antonio Ahumada and Juan Caceres, Universidad de Santiago,Chile. "Estimation of a Cost Function for the Private Banking Sector of Argentina with Panel Data," Tamara Burdisso, Banco Central de la Republica, Argentina. "El M\?argende la Intermediacion Bancaria en Colombia," Roberto Steiner, Adolfo Barajas, and Natalia Salazar, FEDESARROLLO, Colombia. "Determinantes de los Margenes Cambiarios: Evidencia Empirica para Chile," Rodrigo Fuentes and Miguel Basch, Universidada de Chile, Chile. CHAIR:Hildegart Ahumada. August 15, 10:00 to 12:00 am Banking II, Session VT-Room 1. "An Endogenous Growth Model of Money, Banking and Financial Repression," Marco A. Espionsa, Federal Reserve Bank of Atlanta, USA, and Chong K. Yip, The Chinese University of Hong Kong. "Contagion, Banks Fundamental or Macroeconimic Shock? An Empirical Analysis of Argentina 1995 Banking Problems," L. D'Amato, E. Grubisic, and A. Powell, Banco Central de la Republica de Argentina, Argentina. "Public Disclosure and Bank Failures," Tito Cordella, U. Pompeu Fabra, Espania, and Eduardo Levy, IMF, USA. "Systemic Mess, the Interbank Market and the Domino Effect," Guillermo Alger, GRAMAQ, Francia. CHAIR:Pablo Cotler. August 15, 10:00 to 12:00 am Wages,Session V7-Room 2. "Relative Wages and Trade Liberalization in Mexico," Gustavo Cafionero and Alejandro Wegrner, IMF, USA.

1454

PROGRAM

"Inflation, Regulation and Wage Adjustment Patterns: Non-Parametric Evidence from Longitudinal Data," Marcelo Neri, IPEA, Brasil. "Wage Indexation and Macroeconomic Stability," Esteban Jadresic, IMF, USA. "Wage Dispersion and Inflation: Some Empirical Evidence," Alicia Menedez, Boston University, USA. "Investing Rationality in Wage-Setting," Sonia R. Bhalotra, University of Bristol, UK. CHAIR: Tourinho.

August 15, 10:00 to 12:00 am Trade Theory,Session V7-Room 3. "North-South Trade, Labor Supply, and the Wages of Unskilled Workers," John P. Formby and Guangmao Nie, The University of Alabama, USA. "Lobbying, Innvocation, and Protectionist Cycle," Gabriel Sanchez, Columbia University, USA. "Optimal Dynamic Tariffs with Varying Degrees of Commitment," Cristina T. Terra, PUC RJ, Brasil. "La Teoria de la Proteccion Contingente," Ronald D. Fischer, Universidad de Chile, Chile. "Can Nonhomothetic Preferences Explain the Post World War II Growth in Trade," Raphael Bergoeing, ILADES, Georgetown University, Chile. CHAIR:Sergio Ribeiro. August 15, 10:00 to 12:00 am Exchange Rate II, Session V7-Room 4. "Exchange Rate Dynamic and Learning," Pierre-Olivier Gourinchas and Aaron Tornell, Stanford University, USA. "Currency Bands and Stabilization Dynamics," Julio de Brun, Universidad ORT, Uruguay. "Devaluacion, Precios Relativos y Flujos Comerciales: El Caso Peruano 1969-1995," Marco Arena, Jesus Ferreyra, and Pedro Tuesta, Banco de la Reserva del Peru, Peru. "Exchange Rate Determinacy on General Equilibrium Theory," Gerardo Jacobs, Banco de Mexico, Mexico. "A Quarter of Century of Currency Crises in Argentina: Has the Nature of the Crisis Changed," Gracilla Kaminsky, Board of Governors of the Federal Reserve System, USA. CHAIR:Sebastian Valdes. August 15, 10:00 to 12:00 am Stabilization,Session V7-Room 5. "Measuring Credibility Effects During Two Stabilization Attempts in Uruguay: 1978-82 and 1990-95," Andres Massoller, Banco Central de Uruguay, Uruguay. "Politica de Esterilizacion, Choques Externos y Variabilidad de las Reservas en Mexico 1988-1994," Gerardo Esquivel, Harvard University, USA.

PROGRAM

1455

"Can Optimal Fiscal Policy Be Procyclical?" Ernesto Talvi, IADB, USA, and Carlos A. Vegh, UCLA, USA. "On the Effects of US Interest Rate on the Brazilian Foreign Debt," Gyorgy Varga, EPGE/FGV, Brasil. "Crisis, Foreign Aid, and Macroeconomic Reform," Eduardo Fernandez-Arias, IADB, USA. CHAIR:Eugenio Figueroa. August 15, 10:00 to 12:00 am Political Economy, Session VT-Room 6. "Fiscal Federalism, Political Equilibrium and Distributive Politics," Guillaume Cheikbossian, DELTA, Francia. "Do Rich Countries Choose Better Governments?" Costas Azariadis and Amartya Lahiri, UCLA, USA. "Inflation Distributive Struggle and Political Coalitions: Why and When Monetary Disorder Ends?" R. Ahrend, Delta, Sticerd; Th. Verdier, CERAS, DELTA, and C. Winograd, University of Evry Val d'Esonne, DELTA. "The Fiscal Politics of Big Governments," Costas Azariadis and Luisa Lambertini, UCLA, USA. "Subgame Perfect Equilibria in a Ramsey Taxes Model," C. Pheland and E. Stacchetti, USA. CHAig:Ricardo Paredes. August 15, 15:00 to 17:00 pm TradePolicy, Session VII-Room 1. "Structural and Trade Reforms, and the Measure of Price, Quantity and Product Composition Trade Indexes in Developing Countries: The Nicaraguan Case, 1990-1995," Mario Tello, PUC, Peru. "An Evaluation of the Dynamic Gains from MERCOSUR Using Applied General Equilibrium," Jorge Cavalcante, UFRRJ, McGill University, and Jean Mercenier, CRDE, Universite de Montreal, Canada. "North American Economic Integration: Has NAFTA Facilitated Trade and Investment?" David M. Gould, Federal Bank of Dallas, USA. "MERCOSUR: Un Camino a la Apertura o la Consolidacion de un Bloque Cerrado?" Alejandro Nin and Maria Ines Terra, Universidad de la Republica, Uruguay. CHAIR: Ricardo Sanhueza. August 15, 15:00 to 17:00 pm Business Cycle, Session VII-Room 2. "Banks and Macroeconomic Disturbances under Predetermined Exchange Rates," Sebastian Edwards and Carlos A. Vegh, UCLA, USA. "Markup Pricing Strategies and the Business Cycle," Ernesto Felli, University of Roma 3, and Giovanni Tria, University of Roma La Sapienzia, Italy. "Business Cycle in Argentina and Brazil," Jorge Carrera, Mariano Feliz, and Demian Panigo, Universidad de la Plata, Argentina.

1456

PROGRAM

"Learning about Trends and Aggregate Fluctuations in Open Economies," Daniel Heymann, CEPAL, Pablo Sanguinetti, Instituto Di Tella y Universidad de Buenos Aires, and G. Bozzoli, U. Torcuato Di. Tella, Argentina. August 15, 15:00 to 17:00 pm Game TheoryII, Session VII-Room 3. "Optimal Screening Schemes in Auctions," Sara Castellanos, UCLA, USA. "Dynamic Growth Games with Externalities," Ronald D. Fisher and Leonard J. Mirman, Universidad de Chile, Chile. "Auctions of Licenses and Market Structure," Gustavo E. Rodriguez, U. Torcuato di Tella, Argentina. "Renegotiation in Incomplete Contracting Games: An Application to Concessions of Public Works," Eduardo Saavedra, Cornell University, USA. CHAIR:Alexander Galeovic. August 15, 15:00 to 1 7:00 pm Labor II, Session VII-Room 4. "The Performance of Migrants in the Lima Labor Market," Philip Bond, University of Chicago, USA. "Gender Wage Differences in Spain. A Quantile Regression Approach," Jaume Garcia, Pedro J. Hernandez, and Angel Lopez, Universitat Pompeu Fabra, Espana. "Unemployment and Wages In Chile: A Synthetic Cohort Analysis," Osvaldo Larranaga and Ricardo Paredes, Universidad de Chile, Chile. "Analisis del Desempleo Regional en Argentina," Silvia Montoya and Manuel\Willington, IIERAL, Fundacion Mediterranea, Argentina. CHAIR:Jorge Carrera. August 15, 15:00 to 17:00 pm InequalityIssues, Session VII-Room 5. "Equality of Opportunity and Optimal Cash and in-Kind Policies," Leonardo C. Gasparini, Princeton University, USA. "Inequlity, Welfare and Monotonicity," Yoram Amiel, Ruppin Institute, Israel, and Frank A. Cowell, London School of Economics, UK. "Distributional Surprises After a Decade of Reforms: Latin America in the Nineties" Juan Luis Londofio and Miguel Szekely, IADB, USA. "Inequality of Poverty: Job Strategies and Differentials by Gender," Ricardo Paes de Barros, Rio de Janeiro Federal University, Brasil, Ana Flavia Machado, Minasgerais Federal University, Brasil, and Rosane Silva Pinto de M., Rio de Janeiro Federal University, Brasil. "Inequality in the Distribution of Personal Income in the World, How is Changing and Why," Paul Schultz, Yale University, USA. CHAIR: Juliano Junqueira.

Econometrica, Vol. 66, No. 6 (November, 1998)

PROGRAM OF THE 1998 NORTH AMERICAN SUMMER MEETING OF THE ECONOMETRIC SOCIETY MONTREAL,QUEBEC JUNE 25-28, 1998

Program Committee Torben G. Andersen, Northwestern University Susan Athey, Massachusetts Institute of Technology Paul Beaudry, University of British Columbia Marcel Boyer, Chair, Universite de Montreal Jean-Marie Dufour, Universite de Montreal Larry Epstein, The Hong Kong University of Science & Technology Rene Garcia, Universite de Montreal Eric Ghysels, The Pennsylvania State University Bo Honore, Princeton University Joel L. Horowitz, University of Iowa Peter Howitt, The Ohio State University Guido W. Imbens, University of California at Los Angeles Kenneth Judd, Stanford University Thomas Lemieux, Universite de Montreal Tracy Lewis, University of Florida Dilip Mookherjee, Boston University Herve Moulin, Duke University Robert Pindyck, Massachusetts Institute of Technology Jean-Franqois Richard, University of Pittsburgh Paul Romer, Stanford University Fallaw Sowell, Carnegie Mellon University Daniel F. Spulber, Northwestern University

Acknowledgments We thank the members of the program committee for their excellent cooperation. We thank Jacques Blais, Sylvie Barrette-Methot and the staff at CIRANO for their dedicated support and hard work. We gratefully acknowledge the financial support of Aeroports de Montreal, Canadian National, the City of Montreal, Industry Canada, the Institute of Canadian Bankers, National Bank of Canada, Provigo Distribution Inc., St. Laurent Paperboard Inc.; and the following university research centers: CRDE (Centre de recherche et developpement en economique), CREFE (Centre de recherche sur l'emploi et les fluctuations economiques), NCM2 (Network for Computing and Mathematical Modeling), Chaire Jarislowsky in technology and international competition, and CIRANO (Center for Interuniversity Research and Analysis on Organizations). Marcel Boyer, Program Chair Pierre Lasserre, Local Arrangements Chair 1457

1458

PROGRAM

Session Listings Thursday,June 25, 8:30 am-10:00 am Dynamic Agency Relationships

"Shakingthe Tree: An Agency-TheoreticModel of Asset Pricing,"JamsheedShorish, Universityof Aarhus;StephenE. Spear,CarnegieMellon University. DISCUSSANT: Michel Poitevin, Universite de Montreal.

"DynamicRisk-SharingContractwith Non-Commitmentand Savings,"KarineGobert, Universitede Montreal;Michel Poitevin,Universitede Montreal. DISCUSSANT: Stephen D. Williamson, University of Iowa.

"The Dynamicsof Chapter11 Bankruptcy,"Dan Bernhardt,Universityof Illinois at Urbanaand Queen'sUniversity;Dennis Lu, Queen'sUniversity. DISCUSSANT: JacquesLawarree,Universityof Washington. ORGANIZER:Paul Beaudry.

Nabil Al-Najjar,NorthwesternUniversity. CHAIR: StrategicBargaining

"'Money or Reputation'-A RationalTheoryof Blackmail,"MikkoLeppamaki,Universityof Helsinki. DISCUSSANT: Joseph E. Harrington Jr., Johns Hopkins University. "Outsiders' Threat or Consecutive Offers: on Shaked and Sutton's Model," Quan Wen, University of Windsor. DISCUSSANT:Joseph E. Harrington Jr., Johns Hopkins University. "Bilateral Bargaining under Non-Verifiable Information," Anke S. Kessler, University of Bonn; Christoph Liulfesmann, University of Bonn. DISCUSSANT:Zvika Neeman, Boston University. Daniel F. Spulber. ORGANIZER: CHAIR:Gamal Atallah, Industry Canada and Universite de Montreal. Theory "Competitive Equilibrium Growth," David K. Levine, UCLA; Michele Boldrin, University Carlos III de Madrid. DISCUSSANT:David de La Croix, Universite Catholique de Louvain. "Discounting an Uncertain Future," Christian Gollier, Universite de Toulouse I. DISCUSSANT: James W. Friedman, University of North Carolina-Chapel Hill. ORGANIZER:Paul M. Romer.

CHAIR:Emanuela Cardia, Universite de Montreal. Policy in an Open Economy "Growth, Reserve Requirements, and Inflationary Finance in an Open Economy," Richard C. Barnett, University of Arkansas; Mun S. Ho, Harvard University. DISCUSSANT:Pamela Labadie, George Washington University. "On the Gains to Monetary Union," Russell Cooper, Boston University; Hubert Kempf, Universitede Paris I. DISCUSSANT:Young-sik Kim, University of Iowa and Victoria University of Wellington.

1459

PROGRAM

"On Government Credit Programs," Marco Espinosa-Vega, FRB of Atlanta; Bruce D. Smith, FRB of Atlanta; Chong K. Yip, The Chinese University of Hong Kong. DISCUSSANT:Diego Restuccia, University of Minnesota. ORGANIZERS:Peter Howitt and Rene Garcia.

CHAIR:Dean Corbae, University of Iowa. Issues in Dynamic EquilibriumModels "Equity, Bonds, Growth and Inflation in a Quadratic Infinite Horizon Economy," Michael Magill, University of Southern California; Martine Quinzii, University of California at Davis. DISCUSSANT:Costis Skiadas, Northwestern University. Dynamic Equilibrium with Liquidity Constraints," Jerome Detemple, McGill University. DISCUSSANT:Frank Riedel, Humboldt University. "Imperfect Information Leads to Complete Markets if Dividends are Diffusions," Frank Riedel, Humboldt University. DISCUSSANT:Marcel Rindisbacher,

Universite

de Montreal.

ORGANIZERS: Torben G. Andersen, Rene Garcia and Eric Ghysels. CHAIR:Jerome Detemple, McGill University. Model Selection and Specification Errors "Integral U-Statistics for Nonlinear Test of Independence," David Mayer-Foulkes, Centro de Investigacion y Docencia Economicas. DISCUSSANT:Ramazan Genqay, University of Windsor.

"The Relative Measurement Error Bias in Nonparametric and OLS Regression: Implications for Specification Tests," Christopher R. Bollinger, Georgia State University. DISCUSSANT:Jeff Racine, University of Southern Florida.

"Model Section Criteria in Neural Networks for Financial Time Series Forecasting," Min Qi, Kent State University; Guoqiang Zhang, Kent State University. DISCUSSANT:Ramazan Genqay, University of Windsor. ORGANIZER:Jean-Marie Dufour.

CHAIR:Joanna Jasiak, York University. Empirical Game Models "Identification and Estimation of a Class of Game Theoretic Models," Jean-Pierre Florens, Universite de Toulouse I; Jean-Francois Richard, University of Pittsburgh. DISCUSSANT:Tong Li, Washington

State University.

"Learning in Sender-Receiver Games," George R. Neumann, University of Iowa, Andreas Blume, University of Iowa; N. Eugene Savin, University of Iowa; Douglas V. DeJong, University of Iowa and Tilburg University. DISCUSSANT:Holger Sieg, Duke University.

"Estimating a Bargaining Model with Asymmetric Information," Holger Sieg, Duke University. DISCUSSANT:Rafael Tenorio, University of Notre Dame. ORGANIZER:Jean-Franqois Richard.

CHAIR:Robert C. Marshall, The Pennsylvania State University.

1460

PROGRAM

Thursday, June25, 10:30am-12:00 noon Advantagein Timing "Term-LimitLegislationas Response to IncumbencyAdvantage,"Kong-Pin Chen, AcademiaSinica. Dan Bernhardt, University of Illinois and Queen's University. DISCUSSANT: "Second-Mover Advantages in Technological Competition," Heidrun C. Hoppe, Universitat Hamburg; Ultich Lehmann-Grube, Universitat Hamburg. DISCUSSANT:Carolyn Pitchik, University of Toronto.

"Rent Shifting and Inefficiencies in Sequential Bilateral Contracting," Leslie M. Marx, University of Rochester; Greg Shaffer, University of Rochester. DISCUSSANT:Sergei Severinov, Stanford University. ORGANIZERS:Dilip Mookherjee and Tracy Lewis.

CHAIR:Lambros Pechlivanos, Universite de Toulouse I. Rationality in Games "Iterated Dominance and the Implications of Commonly Assumed Rationality," Christian Ewerhart, University of Bonn. DISCUSSANT:V. Bhaskar, University of St. Andrews. "Conditional Dominance, Rationalizability, and Game Forms," Makoto Shimoji, University of California at San Diego; Joel Watson, University of California at San Diego. DISCUSSANT:Geir Asheim, University of Oslo. ORGANIZER:Dilip Mookherjee. CHAIR:Maxwell Stinchcombe, University of Texas at Austin. General EquilibriumI "Tiebout Economies with a Continuum of Traders," John P. Conley, University of Illinois at Urbana-Champaign; Myrna Wooders, University of Toronto. DISCUSSANT:Marcus Berliant, Washington

University.

"Empirical Consistency of General Equilibrium Models," Susan K. Snyder, Virginia Polytechnic Institute and State University. DISCUSSANT:Lin Zhou, Duke University.

"Capital Income Taxation in an Open Economy: Implications on International Business Cycles and World Welfare," Sunghyun H. Kim, Brandeis University. DISCUSSANT:Alessandro Citanna, Carnegie Mellon University. ORGANIZER:Larry Epstein.

CHAIR:Alessandro Citanna, Carnegie Mellon University. Internal Labour Markets "Choice of Technology and Labor Market Consequences: Explaining US-Japanese Differences in Management Styles," Hodaka Morita, Cornell University. DISCUSSANT: Lars Vilhuber, Universite de Montreal and York University. "Strategic Recruiting and the Chain of Command: On the Abuse of Authority in Internal Labor Markets," Guido Friebel, Stockholm School of Economics; Michael Raith, University of Chicago. DISCUSSANT:Pascale Viala, Universite ORGANIZER:Thomas Lemieux.

de Montreal and EIB.

CHAIR:Lars Vilhuber, Universite de Montreal and York University.

PROGRAM

1461

Money and Search

"RationalCreditFluctuationsin a SearchModel of Money,"Dean Corbae,University of Iowa;Joseph Ritter,FRB of St. Louis. DISCUSSANT:Jeff Lacker, FRB of Richmond. "Money and the Law of One Price," Ruilin Zhou, University of Pennsylvania and FRB of Minneapolis; Edward J. Green, FRB of Minneapolis. DISCUSSANT: Stephen Williamson, University of Iowa. "Money and Prices in a Multiple-Matching Search Model," Victor E. Li, FRB of St. Louis and Pennsylvania State University; Derek Laing, Pennsylvania State University; Ping Wang, Pennsylvania State University. DISCUSSANT:Shouyong Shi, Queen's University. ORGANIZER:Peter Howitt. CHAIR:Pamela Labadie, George Washington University.

Cross Section Econometrics "Estimates Qf the Returns to Scale for U.S. Manufacturing," Alpay Filiztekin, Koc University; Sumru Altug, Koc University. DISCUSSANT:Robert Gagne, HEC-Montreal. "Instrument Variable Estimation Based on Mean Absolute Deviation," Shinichi Sakata, University of Michigan. DISCUSSANT:Paul Rilstone, York University. ORGANIZERS:Torben G. Andersen and Fallaw Sowell. CHAIR:Paul Rilstone, York University. Testing "Testing Conditional Distributions of Dynamic Models," Jushan Bai, Massachusetts Institute of Technology. Marine Carasco, Ohio State University. DISCUSSANT: "Bayesian Score Statistics in Linear Models," Frank Kleibergen, Erasmus University Rotterdam; Richard Paap, Erasmus University Rotterdam. DISCUSSANT:Keisuke Hirano, Harvard University. "Monte Carlo Tests for Contemporaneous Correlation of Disturbances in Multi-Equation Regression Models," Lynda Khalaf, Universite Laval; Jean-Marie Dufour, Universite de Montreal. DISCUSSANT:Allan Wiurtz,University of Aarhus. ORGANIZERS: Bo Honore and Eric Ghysels. CHAIR:Allan Wiirtz, University of Aarhus. Advantage in Timing "A Conditional Goodness-of-Fit Test for Time Series," Atsushi Inoue, University of Pennsylvania. DISCUSSANT: Nour Meddahi, Universite de Montreal. "Simple Robust Testing of Regression Hypotheses," Tim Vogelsang, Cornell University; Nick Keifer, Cornell University; Helle Bunzel, Cornell University. DISCUSSANT: Jean-Marie Dufour, Universite de Montreal.

1462

PROGRAM

"Structural Change Tests for Simulated Method of Moments," Alain Guay, Universite

du Quebec 'aMontreal;Eric Ghysels,PennsylvaniaState University. DISCUSSANT:HaroldH. Zhang,CarnegieMellon University. Torben G. Andersen, Rene Garcia and Eric Ghysels. ORGANIZERS: CHAIR:James MacKinnon, Queen's University. Thursday,June 25, 2:00 pm-4:00 pm ComputationalEconomics "Convergence of Trading Networks: A Simulation Approach," Ann Maria Bell, Vanderbilt University. DISCUSSANT:Max R. Blouin, Brown University. "An Empirical Model of Human Strategic Behavior in Economies with Multiple Equilibria," Ernan Haruvy, University of Texas at Austin; Dale 0. Stahl, University of Texas at Austin. DISCUSSANT:Takako Fujiwara-Greve, Keio University. "Magic on the Internet: Evidence from Field Experiments on Reserve Prices in Auctions," David Lucking-Reiley, Vanderbilt University. DISCUSSANT:Matthew 0. Jackson, California Institute of Technology. "Computational Playability of Backward Induction Solutions II," Hidetoshi Tashiro, Hitotsubashi University. Thomas Brenner, Max-Planck-Institute for Research into Economic DISCUSSANT: Systems. ORGANIZER:Kenneth Judd.

CHAIR:Martine Quinzii, University of California at Davis. ContractsI "Contracting with Externalities," Ilya Segal, University of California at Berkeley. de Montreal.

DISCUSSANT:Michel Poitevin, Universite

"The Dynamics of Inequality Under Moral Hazard," Ezra Friedman, Massachusetts Institute of Technology. DISCUSSANT:Karine Gobert, Universite

Montreal.

"Optimal Structure of Agency with Product Complementarity and Substitutability," Sergei Severinov, Stanford University. DISCUSSANT:Uday Rajan, Carnegie Mellon University. ORGANIZER:Dilip Mookherjee.

CHAIR:Tracy Lewis, University of Florida. Auctions I "Independent Private Value Multi-Unit Demand Auctions: An Experiment Comparing Uniform Price and Dynamic Vickrey Auctions," Dan Levin, Ohio State University; John M. Kagel, University of Pittsburgh. DISCUSSANT:Lawrence Ausubel, University of Maryland. "Multiple-unit Demand Auctions: Theory and Experimental Evidences," Jacques Robert, Universite de Montreal; Marie Corriveau, CIRANO; Claude Montmarquette, Universite de Montreal. DISCUSSANT:Dan Levin, Ohio State University.

1463

PROGRAM

"The Optimality of Being Efficient," Lawrence Ausubel, University of Maryland; Peter Cramton, University of Maryland. DISCUSSANT: Susan Athey, Massachusetts Institute of Technology. "The Effect of the Reservation Price on Revenue and Efficiency in a Multiple-Unit Auction," Michal Bresky, The Economics Institute of the Academy of Sciences of the Czech Republic. DISCUSSANT:Claude Montmarquette, ORGANIZER:Jean-Franqois Richard.

Universite

de Montreal.

CHAIR:Jean-Franqois Richard, University of Pittsburgh. Social Choice "Income Convergence and Assimilation: The Economics of Mating Taboos," Linda Wong, University of Iowa. DISCUSSANT:Ruqu Wang, Queen's University. "On the Structure of Choice under Different External References," Wulf Gaertner, Universitat Osnabriick; Yongsheng Xu, University of Nottingham. DISCUSSANT:Walter Bossert, University of Nottingham. "Protecting Minorities through Voting Rules," Alain Trannoy, Universite de CergyPontoise; Regis Renault, Universite des Sciences Sociales de Toulouse. DISCUSSANT:Marcus Berliant, Washington

University.

Herve Moulin. CHAIR:Michel Truchon, Universite Laval. ORGANIZER:

Money and Credit "Coalition-Proof Risk Sharing with Private Information and Side Trading," Jeffrey M. Lacker, FRB of Richmond; Peter N. Ireland, Rutgers University. DISCUSSANT:Dean Corbae, University of Iowa.

"Liquidity, Interbank Market, and the Role of Central Bank," Young-sik Kim, The University of Iowa and Victoria University of Wellington. DISCUSSANT:Richard Barnett, University of Arkansas. "Money, Credit, and Allocation Under Complete Dynamic Contracts and Incomplete Markets," Stephen D. Williamson, The University of Iowa; S. Rao Aiyagari, University of Rochester. DISCUSSANT:Roger D. Lagunoff, Georgetown

University.

"On Aggregate Precautionary Saving: When is the Third Derivative Irrelevant?," Mark Huggett, Instituto Tecnologico Autonomo de Mexico; Sandra Ospina, Instituto Tecnologico Autonomo de Mexico. DISCUSSANT: Christian Gollier, Universite des Sciences Sociales de Toulouse. ORGANIZER:Peter Howitt.

CHAIR:Derek Laing, Pennsylvania State University. Nonparametricand SemiparametricIdentification and Inference "Semi-Nonparametric Estimation of Dynamic Single Duration Model," Xiaohong Chen, University of Chicago; James Heckman, University of Chicago; Ed Vytlacil, University of Chicago. DISCUSSANT:Yuichi Kitamura, University of Minnesota.

1464

PROGRAM

"EdgeworthApproximationsfor SemiparametricInstrumentalVariable Estimators and Test Statistics,"OliverB. Linton,Yale University. Yuichi Kitamura,Universityof Minnesota. DISCUSSANT: "NonparametricEstimationof an AdditiveModel with an UnknownLink Function," Joel L. Horowitz,Universityof Iowa. DISCUSSANT:OliverB. Linton,Yale University. ORGANIZER: Joel Horowitz. CHAIR: N. Eugene Savin, University of Iowa.

Alternative Techniquesfor Diffusion Estimation "A Simple Bayesian Approach to the Analysis of Markov Diffusion Processes," Christopher S. Jones, University of Pennsylvania. DISCUSSANT:Bjorn Eraker, Norwegian School of Economics.

"MCMC Analysis of Diffusion Models with Application to Finance," Bjorn Eraker, Norwegian School of Economics. DISCUSSANT: Christopher S. Jones, University of Pennsylvania. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach," Yacine Ait-Sahalia, Graduate School of Business, University of Chicago. DISCUSSANT:Lars Peter Hansen, University of Chicago. Torben G. Andersen, Rene Garcia and Eric Ghysels. ORGANIZERS: CHAIR:Torben G. Andersen, Northwestern University. Thursday,June 25, 4:30 pm-6:00 pm Invited Plenary Session by Peter Howitt "Economic Growth and Creative Destruction: Recent Developments," Peter Howitt, Ohio State University. CHAIR: Pierre Lasserre, Universite

du Quebec 'a Montreal.

Friday,June 26, 830 am-1i0:00 am AsymmetricInformation and Public Choice "Pivotal Players and the Characterization of Influence," Nabil I. Al-Najjar, Northwestern University; Rann Smorodinski, Northwestern University. DISCUSSANT: Zvika Neeman, Boston University. "Efficiency and Information Aggregation in Auctions with Costly Informationi," Matthew 0. Jackson, California Institute of Technology; Valentina A. Bali, California Institute of Technology. ORGANIZER:Dilip Mookherjee. CHAIR:Paul Beaudry, University of British Columbia. Leadership and Cooperation "An Economic Theory of Leadership Turnover," Carolyn Pitchik, University of Toronto; Maria Gallego, Wilfrid Laurier University. DISCUSSANT: Mikko Leppamaki, University of Helsinski.

PROGRAM

1465

"Cooperativeor NoncooperativeBehavior?A Note to the Testable Implicationsof CollectiveChoiceTheories,"Yves Sprumont,Universitede Montreal. DISCUSSANT:Wulf Gaertner,Universitat Osnabruick. "Endogenous Coalition Formation in Rivalry,"Ruqu Wang, Queen's University; Guofu Tan, Universityof British Columbiaand Hong Kong Universityof Science and Technology. DISCUSSANT:SanchulSuh, Universityof Windsor. ORGANIZERS:Robert Pindyckand Herve Moulin. CHAIR:Roger D. Lagunoff, Georgetown University. Games "Finitistic Games," Maxwell Stinchcombe, University of Texas at Austin; Chris Harris, University of California at Los Angeles; Bill Zane, University of California at Los Angeles. DISCUSSANT: John Conley, University of Illinois at Urbana-Champaign. "Managing a Reputation," Larry Samuelson, University of Wisconsin; George Mailath, University of Pennsylvania. DISCUSSANT:Leslie M. Marx, University of Rochester. "A Theory of Rigid Extremists and Flexible Moderates with an Application to the U.S. Congress," Joseph E. Harrington Jr., Johns Hopkins University; S. Brock Blomberg, Wellesley College. D,SCUSSANT:Sergei Severinov, Stanford University. ORGANIZER:Larry Epstein. CHAIR:William Bentley MacLeod, University of Southern California. Industrial Organization "Entry Deterrence and Entry Inducement in an Industry with Complementary Products," Jeong-Yoo Kim, Dongguk University. DISCUSSANT:Paul Johnson, Universite de Montreal. "Underinvestment and Market Structure," Volker Nocke, London School of Economics. DISCUSSANT:Paul Johnson, Universite de Montreal.

"The Informational Value of Sequential Fundraising," Lise Vesterlund, Iowa State University. DISCUSSANT:Pascale Viala, Universite ORGANIZER:Daniel F. Spulber.

de Montreal and EIB.

CHAIR:Al Slivinski, University of Western Ontario. Open Economy Macro "Slow North-South Convergence When Labor and Capital are Perfectly Mobile," John Coleman, Duke University; Francesco Caselli, University of Chicago. DISCUSSANT:Jeff Campbell, University of Rochester.

"Dynamic Collusion, Pricing to Market, and Real Exchange Rates," Michael Devereux, University of British Columbia; Roberto Chang, FRB of Atlanta. DISCUSSANT:William J. Polley, University of Iowa.

1466

PROGRAM

"Investmentand the CurrentAccountin the ShortRun and the Long Run,"JamesM. Nason, Universityof BritishColumbia;John H. Rogers,Boardof Governorsof the FederalReserve System. DISCUSSANT:EmanuelaCardia,Universitede Montreal. ORGANIZER:Peter Howitt.

CHAIR:Richard Barnett, University of Arkansas. Asset Pricing "Asset Prices with Contingent Risk Preferences," Stephen Gordon, Universite Laval; Pascal St-Amour, Universite Laval. DISCUSSANT:Simon Van Norden, HEC-Montreal.

"Pricing and Hedging Derivative Securities with Neural Networks and a Homogeneity Hint," Ramazan Genqay, University of Windsor; Rene Garcia, Universite de Montreal. DISCUSSANT:Min Qi, Kent State University.

"Habit Formation: A Resolution of the Equity Premium Puzzle?," Charles H. Whiteman, University of Iowa; Christopher Otrok, University of Iowa, B. Ravikumar, University of Iowa. de Montreal.

DISCUSSANT:Rene Garcia, Universite

Torben G. Andersen, Rene Garcia and Eric Ghysels. ORGANIZERS: CHAIR:Rene Garcia, Universite de Montreal. Regressionand Time Series "Bias and Mean Squared Error of Least Squares Estimators in Dynamic Regression Models with a Unit Root," Jan F. Kiviet, University of Amsterdam; Garry D. A. Phillips, University of Exeter. DISCUSSANT:N. Eugene Savin, University of Iowa.

"Estimation in Multivariate Time Series Regression Models with Elliptically Symmetric Errors," Douglas J. Hodgson, University of Rochester; Oliver B. Linton, Yale University; Eugene Choo, Yale University. DISCUSSANT: Guido M. Kuersteiner, Massachusetts Institute of Technology. "Robust Wald Tests in Sur Systems with Adding Up Restrictions: An Algebraic Approach to Proofs of Invariance," Surajit Ray, University of Iowa; B. Ravikumar, University of Iowa; N. Eugene Savin, University of Iowa. DISCUSSANT:Lynda Khalaf, Universite Laval. ORGANIZER:Jean-Marie Dufour.

CHAIR:Lynda Khalaf, Universite Laval. Search and Employment "Job Search and Firm Size Distribution Effects on Wages: Theory and Evidence," Takako Fujiwara-Greve, Keio University; Henrich R. Greve, University of Tsukuba. DISCUSSANT:Marc Van Audenrode,

Universite

Laval.

"Product Market and the Size Wage Differential," Shouyong Shi, Queen's University. DISCUSSANT:Jean Fares, Banque du Canada. ORGANIZERS:Paul Beaudry and Thomas Lemieux.

CHAIR:Pierre Cahuc, Universite de Paris I.

PROGRAM

1467

Friday, June 26, 10:30 am-12:00 noon ContractsII

"Double Moral Hazardand Multi-DimensionalSignalingin Principal-AgentContract Negotiations:A Study of The National Football League," Mike Conlin, Cornell University;PatrickEmerson,CornellUniversity. DISCUSSANT: Patrick Gonz'alez, Universite Laval. "Over-Compensation as a Partial Solution to Commitment and Renegotiation Problems: the Case of Ex-Post Moral Hazard," Martin Boyer, HEC-Montreal. DIsCUSSANT: Thomas Palfrey, California Institute of Technology. "Strategy, Proof Solutions in a Permit Sharing Problem," Sangchul Suh, University of Windsor. DISCUSSANT: Guido Friebel, Stockholm School of Economics. ORGANIZER: Daniel F. Spulber. CHAIR:Nabil Al-Najjar, Northwestern University. Financial Difficulties "A Model of Financial Fragility," Roger D. Lagunoff, Georgetown University; Stacey L. Schreft, FRB of Kansas City. DIscussANT:Jeffrey M. Lacker, FRB of Richmond. "Financial Constraints, Cost of Capital and Net Worth," Pamela Labadie, George Washington University. Dean Corbae, University of Iowa. DISCUSSANT: "Wage Concessions and Debt Forgiveness as Strategic Responses to Financial Distress," Pascale Viala, Universite de Montreal and EIB; David N. Margolis, Universite de Paris I. DISCUSSANT: Dan Bernhardt, University of Illinois and Queen's University. ORGANIZER:Paul Beaudry.

CHAIR:James M. Nason, University of British Columbia. Investment "Technology Choice under Uncertainty on the Demand," Corinne Chaton, Universite de Toulouse I. DIScuSSANT: Alain Trannoy, University de Cergy-Pontoise. "Capital Accumulation through "Controlled Competition": Two Stage Tournaments as a Strategy for Firm Growth," Yutaka Suzuki, Hosei University. Bernard Sinclair-Desgagne, Ecole Polytechnique de Montreal. DIScuSSANT: "Designing Optimal Patents," Corinne Langinier, University of California at Berkeley. Leslie Marx, University of Rochester. DISCUSSANT: ORGANIZER:Herve Moulin.

CHAIR:Matthew0. Jackson, California Institute of Technology. Coordinationand Collusion "Semicollusion in the Norwegian Cement Market," Frode Steen, Norwegian School of Economics and Business Administration; Lars Sorgard, Norwegian School of Economics and Business Administration. DIscussANT:Al Slivinski, University of Western Ontario.

1468

PROGRAM

"ConsumptionExternalities,Coordinationand Advertising,"Ivan Pastine, Bilkent University;TuvanaPastine,BilkentUniversity. Al Slivinski,Universityof WesternOntario. DISCUSSANT: "Collusionin a Model of RepeatedAuctions,"Paul Johnson,Universitede Montreal; JacquesRobert,Universitede Montreal. Al Slivinski, University of Western Ontario. DISCUSSANT: ORGANIZERS:Daniel F. Spulber and Dilip Mookherjee.

CHAIR:James W. Friedman, University of North Carolina-Chapel Hill. Local Public Goods "Local Public Goods and Clubs: A Unified Theory of First Best," Marcus Berliant, Washington University; John H. Y. Edwards, Tulane University. DISCUSSANT: Oscar Volij, Brown University. "Fiscal Externalities," Andrew D. Austin, University of Houston. DISCUSSANT: Robert J. Gary-Bobo, Universite de Cergy-Pontoise. "Tiebout and Redistribution in a Model of Residential and Political Choice," Christoph Liulfesmann, University of Bonn; Anke S. Kessler, University of Bonn. DISCUSSANT: John Conley, University of Illinois at Urbana-Champaign. ORGANIZER:Herve Moulin. CHAIR:John Conley, University of Illinois at Urbana-Champaign. General EquilibriumII "Measurement Error in General Equilibrium: The Aggregate Effects of Noisy Economic Indicators," Antulio N. Bomfim, FRB. DISCUSSANT: Alok Johri, McMaster University. "Walrasian Allocations without Price-Taking Behavior,"Roberto Serrano, Brown University; Oscar Volij, Brown University. DISCUSSANT:Maxwell Stinchcombe, University of Texas at Austin. ORGANIZER:Larry Epstein. CHAIR:Shouyong Shi, Queen's University. Applied Econometrics I "Nonparametric Estimation of Competing Risks Models with Applications," Shiferaw Gurmu, University of Virginia; Jose Canals, University of Virginia. DISCUSSANT:Hans van Ophem, University of Amsterdam. "Instrumental Variables Estimation of Quantile Treatment Effects," Josh Angrist, Massachusetts Institute of Technology; Alberto Abadie, Guido W. Imbens, University of California at Los Angeles. DISCUSSANT:Phil Merrigan, Universite du Quebec 'a Montreal. "2-Step Estimation of Semiparametric Censored Regression Models," Shakeeb Khan, University of Virginia; James L. Powell, University of California at Berkeley. DISCUSSANT:Paul Rilstone, York University. ORGANIZER:Guido W. Imbens. CHAIR:Paul Rilstone, York University.

PROGRAM

1469

Forecasting

"Unit Roots and Forecasting:To Difference or Not to Difference?,"Lutz Kilian, Universityof Michigan;FrancisX. Diebold, Universityof Pennsylvania. DISCUSSANT:Tim Vogelsang,CornellUniversity. "ContentHorizonsfor Time Series EconomicForecasts,"John W. Galbraith,McGill University. Lutz Kilian, University of Michigan. DISCUSSANT:; "Using Information Criteria to Draw Inference on Existence of Unit Roots and Cointegration," Chor-yiu Sin, City University of Hong Kong. DISCUSSANT: John Galbraith, McGill University. ORGANIZERS: Torben G. Andersen, Rene Garcia and Eric Ghysels. CHAIR:Bryan Campbell, Concordia University. Friday,June 26, 2:00 pm-4:00 pm Intermediationand AsymmetricInformation "Adverse Selection with Competitive Inspection," James W. Friedman, University of North Carolina; Gary Biglaiser, FCC and University of North Carolina. DISCUSSANT: Ruqu Wang, Queen's University. "Strategic Behavior and Price Discovery," Luis Angel Medrano, Universitat Pompeu Fabra; Xavier Vives, Institut d'Analisi Economica CSIC. Daniel F. Spulber, Northwestern University. DJISCUSSANT: "Positive Information Rents with Correlated Information," Zvika Neeman, Boston University. DISCUSSANT: Daniel F. Spulber, Northwestern University. "The Effect of Information on the Well-Being of the Uninformed: What's the Chance of Getting a Decent Meal in an Unfamiliar City?,"James Albrecht, Georgetown University; Harold Lang, Royal Institute of Technology; Susan Vroman, Georgetown University. DISCUSSANT:Anke S. Kessler, University of Bonn. ORGANIZERS: Daniel F. Spulber, Dilip Mookherjee and Tracy Lewis. CHAIR:Daniel F. Spulber, Northwestern University. PropertyRights and Incomplete Contracts "Property Rights and Incomplete Contracts: Dealing with Nuisance," Christopher M. Snyder, George Washington University; Rohan Pitchford, Australian National University. DISCuSSANT: Ilya Segal, University of California at Berkeley. "Joint Ventures and Stock Markets as Commitment Devices," Tridib Sharma, Instituto Technologico Autonomo de Mexico. DIScuSSANT: Shira B. Lewin, Iowa State University. "Bargaining Power, Outside Options and the Boundaries of the Firm," Yanni Tournas, Rutgers University; Rudolf Kerschbamer, University of Vienna. DISCUSSANT: Christopher M. Snyder, George Washington University. "The Allocation of Ownership Rights in the Presence of Non-Cooperative Bargaining," Ayse Mumcu, Bilkent University. DISCUSSANT:Patrick Gonzalez, Universite Laval.

1470

PROGRAM

ORGANIZER: Susan Athey. CHAIR:Joanne Roberts, Queen's University. Agency Problems "Ex-Post Choice of Monitoring," Jacques Lawarree, University of Washington; Fahad Khalil, University of Washington. DIscussANT:Martin Boyer, HEC-Montreal. "Restoring Higher-Powered Incentives through Audits," Bernard Sinclair-Desgagne Ecole Polytechnique de Montreal. DISCUSSANT: Lambros Pechlivanos, Universite de Toulouse I. "Why Promotions," Regis Renault, Universite de Toulouse I; Emmanuelle Auriol, Universite d'Aix-Marseille II. Hakan Orbay, Koc University. DISCUSSANT: ORGANIZER:Susan Athey.

CHAIR:Pascale Viala, Universite de Montreal and EIB. Economies with AsymmetricInformation "Moral Hazard and Non-Exclusive Contracts," Danilo Guaitoli, Universitat Pompeu Fabra; Alberto Bisin, New York University. DISCUSSANT: Jamshed Shorish, University of Aarhus. "A Decentralized Market with Common Values Uncertainty: Non-Steady States," Max R. Blouin, Brown University; Roberto Serrano, Brown University. DIScUSSANT: Martine Quinzii, University of California at Davis. "Communication, Credible Improvements and the Core of an Economy with Asymmetric Information," Oscar Volij, Brown University. DISCUSSANT: Michael Magill, University of Southern California. "Competitive Equilibrium with Moral Hazard in Economies with Multiple Commodities," Alessandro Citanna, Carnegie Mellon University; Antonio Villanacci, Universita' di Firenze. DISCUSSANT: John P. Conley, University of Illinois at Urbana-Champaign. ORGANIZER: Dilip Mookherjee. CHAIR:Larry Samuelson, University of Wisconsin. Business Cycles and Money "Aggregate Employment, Real Business Cycles, and Superior Information," Michel Normandin, Universite du Quebec a Montreal; Martin Boileau, University of Toronto. DISCUSSANT: Timothy Cogley, FRB of San Francisco. "Learning by Doing and Aggregate Fluctuations," Alok Johri, McMaster University; Russell Cooper, Boston University. DISCUSSANT: Michel Normandin, Universite du Quebec a Montreal. "Variability in the Effects of Monetary Policy on Economic Activity," Ka-fu Wong, Chinese University of Hong Kong. DISCUSSANT: Bryan Campbell, Concordia University. "Uncovering Financial Market Beliefs about Inflation Targets," Francisco J. RugeMurcia, Universite de Montreal. DISCUSSANT: Angelo Melino, University of Toronto.

PROGRAM

1471

ORGANIZERS:; TorbenG. Andersen,Rene Garciaand Eric Ghysels. CHAIR:Angelo Melino, University of Toronto. Labour MarketFlows and Unemployment "Job Protection Laws and Jobs: Evidence from a Natural Experiment," Marc Van

Audenrode,UniversiteLaval;JimmyRoyer,UniversiteLaval. DISCUSSANT: LarsVilhuber,Universitede Montrealand York University. "SearchUnemploymentwith AdvanceNotice,"Pietro Garibaldi,InternationalMonetaryFund. DISCUSSANT: Shouyoung Shi, Queen's University.

"TechnologyChoice and EmploymentDynamicsat Young and Old Plants,"JeffreyR. Campbell, University of Rochester; Jonas D. M. Fisher, FTB of Chicago and Universityof WesternOntario. DISCUSSANT: Curtis Eberwein, McGill University. "Hours Equations, IV Estimation and Nonlinear Measurement Error," Normal K. Thurston, Brigham Young University; Karl N. Snow, Brigham Young University. DISCUSSANT: George Neumann, University of Iowa. Thomas Lemieux and Guido W. Imbens. ORGANIZERS: CHAIR:Nicole Fortin, Universite de Montreal. NonstationaryMultivariateand Panel Models "Nonstationary Index Models," Yoosoon Chang, Rice University; Joon Y. Park, Seoul National University. DISCUSSANT:Benoit Perron, Universite

de Montreal.

"Permanent-Transitory Decomposition in VAR Models with Cointegration and Common Cycles," Alain Hecq, University Maastricht, Franz C. Palm, University Maastricht; Jean-Pierre Urbain, University Maastricht. DISCUSSANT: Jacques Raynauld, HEC-Montreal. "Frequency Domain Inference for Impulse Responses," Jonathan Wright, University of Virginia. DISCUSSANT: Lynda Khalaf, Universite Laval. "On the Role of Human Capital in Growth Models: Evidence from a Nonstationary Panel of Developing Countries," Peter Pedroni, Indiana University. DISCUSSANT:Jean Mercenier, Universite

de Montreal.

ORGANIZERS: Torben G. Andersen, Rene Garcia and Eric Ghysels. CHAIR:Jean Mercenier, Universite de Montreal. Estimation and Testingof Econometric Models I "An Extended Yule-Walker Method for Estimating a Vector Autoregressive Model with Mixed-Frequency Data," Peter A. Zadrozny, Congressional Budget Office; Baoline Chen, Rutgers University-Camden. DISCUSSANT:Harold H. Zhang, Carnegie Mellon University.

"Testing for Serial Correlation of Unknown Form Using Wavelet Methods," Jin Lee, Cornell University; Yongmiao Hong, Cornell University. DISCUSSANT:Aris Spanos, Cornell University.

1472

PROGRAM

"Multiple Vector Autoregressions and Impulse Response Analysis," Andre J. Hoogstrate,TilburgUniversity. DIScUSSANT: Alain Paquet,Universitedu Quebec a Montreal. "Shape-Preserving Estimationof Diffusions,"XiaohongChen, Universityof Chicago; LarsPeter Hansen,Universityof Chicago;Jose Scheinkman,Universityof Chicago. DIScUSSANT: Eric Renault,CRESTand INSEA. ORGANIZER: Fallaw Sowell. CHAIR:Douglas Hodgson, University of Rochester. Friday,June 26, 4:30 pm-6:00 pm Invited Plenary Session by Lars Hansen

"Risk and Robustnessin General Equilibrium,"Lars Hansen, Universityof Chicago. CHAIR:Torben G. Andersen, Northwestern University. Saturday,June 27, 8:30 am-i10:00 am Contractingand Information

"LimitedAttention and IncompleteContracts,"SharonGifford,RutgersUniversity. DISCUSSANT: Huseyin Yildirim, University of Florida.

"SpecificInvestment,Commitmentand Observability,"PatrickGonzalez, Universite Laval. DISCUSSANT: Ruqu Wang, Queen's University.

"StrategicBiddingby PotentialCompetitors:Will MonopolyPersist?,"YongminChen, Universityof Coloradoat Boulder. DISCUSSANT: Tracy Lewis, University of Florida. ORGANIZER:Tracy Lewis.

CHAIR:Tracy Lewis, University of Florida. Cooperationin Repeated Games with Moral Hazard

"MoralHazardand PrivateMonitoring,"V. Bhaskar,Universityof St. Andrews;Eric van Damme,TilburgUniversity. Nabil Al-Najjar, Northwestern University. DISCUSSANT:

"Discounted Repeated Games with Imperfect Private Monitoring,"Massimiliano Amarante,MassachusettsInstituteof Technology. DISCUSSANT: Stefan Ambec, Universite de Montreal.

"TeamworkManagement:OptimalIncentiveContractsand OptimalTeam Size in an Era of DiminishingCommitment,"LambrosPechlivanos,Universitede ToulouseIe Guido Friebel,StockholmSchool of Economics;EmmanuelleAuriol,Universitede Toulouse I. DISCUSSANT: Fahad Khalil, University of Washington. ORGANIZER: Dilip Mookherjee. CHAIR:Jacques Robert, Universite de Montreal. Science and TechnologyPolicy

"CanGovernment-Industry R&D ProgramsIncreasePrivateR&D? The Case of the SmallBusinessInnovationResearchProgram,"Scott J. Wallsten,StanfordUniversity. DISCUSSANT: Lutz Hendricks, Arizona State University.

PROGRAM

1473

"The Impactof FederallyFundedBasic Researchon IndustrialInnovation:Evidence from the PharmaceuticalIndustry,"AndrewA. Toole, StanfordUniversity. DISCUSSANT: Paul Beaudry,Universityof BritishColumbia. "ThreeWays of ImprovingGrowthand (Maybe)Welfarein a Model of Endogenous TechnologicalChange,"AndreasHofert, SwissFederalInstituteof Technology. DISCUSSANT: TimothyCogley,FRB of San Francisco. ORGANIZER: Paul M. Romer. CHAIR:Timothy Cogley, FRB of San Francisco. Topics in Development

"SpatialPatterns in Local Markets:MalaysianDevelopment,"Timothy G. Cogley, NorthwesternUniversity;Frederick Flyer, New York University;Grace Tsiang, Universityof Chicago. Anne Beeson Royalty,StanfordUniversity. DISCUSSANT: "TechnologyAdoption and Schooling:AmplifierIncome Effects of Policies Across Countries,"Diego Restuccia,Universityof Minnesota. DISCUSsANT: Hugo Benitez-Silva,Yale University. "EndogenousBusiness Networks,"Raja Kali, Instituto Tecnologico Autonomo de Mexico. DISCUSSANT: Joanne Roberts, Queen's University. ORGANIZER: Paul M. Romer.

Joanne Roberts,Queen's University. CHAIR: M?,nufacturingFirms and AggregateOutcomes

"A Deeper Look at Job Destruction:MatchingTheories and Evidence,"Scott Schuh, FRB of Boston;RobertTriest,FRB of Boston. DISCUSSANT: Sumru Altug, Koc University.

"LaborProductivityin U.S. Manufacturing: Does SectoralComovementReflect TechnologyShocks?,"MichaelT. Kiley,FRB. Russell Cooper, Boston University. DISCUSSANT:

"Does Total FactorProductivityFall After the Adoptionof New Technology?,"Sandra Ospina,InstitutoTecnologicoAutonomode Mexico;MarkHuggett,InstitutoTecnologico Autonomode Mexico. DISCUSSANT: Rajeev Dhawan, University of California at Los Angeles. ORGANIZER: Paul Beaudry.

CHAIR: DavidDe La Croix,UniversiteCatholoquede Louvain. StructuralChange and Nonlinear Models

"Linear CovarianceMatrix Lower Bounds for Time Series Estimators,"GuidoM. Kuersteiner,MassachusettsInstituteof Technology. DISCUSSANT: Douglas J. Hodgson, University of Rochester.

"StochasticPermanent Breaks,"Aaron D. Smith, Universityof California at San Diego; RobertF. Engle, Universityof Californiaat San Diego. Valentina Corradi, University of Pennsylvania. DISCUSSANT:

"A Simple Almost Sure Rule for Detecting Multiple StructuralBreaks," Filippo Altissimo,Bank of Italy;ValentinaCorradi,Universityof Pennsylvania. Tim Vogelsang, Cornell University. DISCUSSANT:

1474

PROGRAM

ORGANIZER: Jean-MarieDufour. Jan Kiviet,Universityof Amsterdam. CHAIR: New Developmentsin Theoretical andAppliedEconometrics "NonregularMaximumLikelihoodEstimationin Auction,Job Searchand Production FrontierModels,"Han Hong, StanfordUniversity. DISCUSSANT: XiaohongChen, Universityof Chicago. "The Effect of NuisanceParameterson Size and Power:LM Tests in Logit Models," Allan Wiirtz,Universityof Aarhus;N. Eugene Savin,Universityof Iowa. DISCUSSANT: Xiaohong Chen, University of Chicago.

"Causal Inference in EncouragementDesigns with Covariates,"Keisuke Hirano, HarvardUniversity;Guido Imbens,Universityof Californiaat Los Angeles; XiaoHua Zhou, IndianaUniversity;Donald B. Rubin,HarvardUniversity. DISCUSSANT: Paul Rilstone, York University. ORGANIZER: Joel Horowitz.

Paul Rilstone,York University. CHAIR: Saturday,June 27, 10:30 am-12:00 noon Invited Plenary Session by Herve Moulin

"RandomPriority:A ProbabilisticResolutionof the Tragedyof the Commons,"Herve Moulin,Duke University. Jean-MarieDufour,Universitede Montreal. CHAIR: Saturday,June 27, 2:00 pm-4:00 pm Limited Dependent Models and SemiparametricEstimation I

"EstimatingDerivativesin NonseparableModelswith LimitedDependentVariables," HidehikoIchimura,Universityof Pittsburgh;Joseph G. Altonji,NorthwesternUniversity. DISCUSSANT: Shakeeb Khan, University of Virginia.

"An EmpiricalLikelihood Approach to ConditionalMoment Restriction Models," Yuichi Kitamura,University of Minnesota; HyungtaikAhn, Korea Information SocietyDevelopmentInstitute. DISCUSSANT: Keisuke Hirano, Harvard University.

"NonparametricMaximumLikelihoodEstimationand Inference,"Jeff Racine, Universityof South Florida;ChunrongAi, Universityof Florida. William Horrace, University of Arizona. DISCUSSANT:

"Estimationof TreatmentEffectswith a Quasi-Experimental Regression-Discontinuity Design,"WilbertVan Der Klaauw,New York University;JinyongHahn, University of Pennsylvania;Petra Todd, Universityof Pennsylvania. DISCUSSANT: Joshua Angrist, Massachusetts Institute of Technology. ORGANIZER: Guido W. Imbens. CHAIR:Wilbert Van Der Klaauw, New York University.

PROGRAM

1475

Estimationand Testingof EconometricModelsII "A FrequencyDecompositionof ApproximationErrorsin StochasticDiscountFactor Models,"TimothyCogley,FRB of San Francisco. DIScUSSANT: Eric Renault,CRESTand INSEA. "StatisticalAdequacyand the Testing of Trend Versus Difference Stationarity,"Aris Spanos,Universityof Cyprus;Elena C. Andreou,Universityof Manchester. DISCUSSANT: Peter Zadrozny,CongressionalBudget Office. "SpecificationTests in the EfficientMethodof MomentsFrameworkwith Application to the StochasticVolatilityModels,"HaroldH. Zhang,CarnegieMellon University; Ming Liu, The ChineseUniversityof Hong Kong. DISCUSSANT: ValentinaCorradi,Universityof Pennsylvania. "Calibrationof StructuralModels by IndirectInference,"Eric Renault, CRESTand INSEA; Ramdan Dridi, Universite de Toulouse I; Laurence Broze, CORE and Universitede Lille 3. DISCUSSANT: Xiaohong Chen, University of Chicago.

ORGANIZERS: TorbenG. Andersen,Eric Ghyselsand FallawSowell. CHAIR: XiaohongChen, Universityof Chicago. Panel Data

"UnequallySpacedPanel Data Regressionswith AR(1) Disturbances,"Badi H. Baltagi, TexasA&M University;Ping X. Wu, Universityof Melbourne. DISCUSSANT: Shakeeb Khan, University of Virginia.

"SpuriousRegression and Residual-BasedTests for Cointegrationin Panel Data," ChihwaKao, SyracuseUniversity. DISCUSSANT: Guido Kuersteiner,MassachusettsInstituteof Technology. "Does StockholdingProvide Perfect Risk Sharing?,"Muhammet Fatih Guvenen, CarnegieMellon University. DISCUSSANT: Shakeeb Khan, University of Virginia. Bo Honore. ORGANIZER:

ShakeebKhan,Universityof Virginia. CHAIR: Capital Markets

"CapitalMarketsand the Evolutionof FamilyBusinesses,"B. Ravikumar,University of Iowa, Utpal Bhattacharya,IndianaUniversity. DISCUSSANT: Christian Gollier, Universite de Toulouse I. "American Option Pricing under GARCH by a Markov Chain Approximation," Jean-Guy Simonato, HEC-Montreal;Jin-ChuanDuan, Hong Kong Universityof Science and Technology. DISCUSSANT: Peter Christoffersen,InternationalMonetaryFund Research Department. "IdentifyingBull and Bear Marketsin Stock Returns,"ThomasH. McCurdy,University of Toronto;John M. Maheu,Queen'sUniversity. DISCUSSANT: Stephen Gordon, Universite Laval. ORGANIZERS: Torben G. Andersen, Rene Garcia and Bo Honore.

CHAIR: StephenGordon,UniversiteLaval.

1476

PROGRAM

BusinessCyclesand MonetaryRules "The Robustnessof Simple MonetaryPolicy Rules to Model Uncertainty,"John C. Williams,Boardof Governorsof the FederalReserveSystem;AndrewLevin,Board of Governorsof the FederalReserve System;Volker Wieland,Boardof Governors of the FederalReserveSystem. DIScUSSANT: FranciscoJ. Ruge-Murcia,Universitede Montreal. "Non-SteadyState Equilibriumin a RandomMatchingMonetaryModel:A Dynamical SystemsApproach,"Ted Temzelides,Universityof Iowa;Hector Lomeli,University of Colorado. DISCUSSANT: Steve Ambler,Universitedu Quebec a Montreal. "SeasonalCycles,BusinessCycles;MonetaryPolicy,"Zheng Liu, ClarkUniversity. DISCUSSANT: James M. Nason, Universityof BritishColumbia. ORGANIZERS: Rene Garcia and Peter Howitt.

Paul R. Bergin,Universityof Californiaat Davis. CHAIR: Empirical Labour Economics

"A Discrete Choice Estimator of Workers'Valuation of Fringe Benefits," Anne Beeson Royalty,StanfordUniversity. DISCUSSANT: Jorgen Hansen, Concordia University.

"Examiningthe Link Between Wages and Quality in the Teacher Workforce:The Role of AlternativeLabor Market Opportunitiesand Non-PecuniaryVariation," MarianneE. Page, Universityof Californiaat Davis; SusannaLoeb, Universityof Michigan. DISCUSSANT: Kelly Bedard, McMaster University.

"An EmpiricalAnalysis of the Social Security DisabilityApplication,Appeal, and AwardProcess,"Hugo Benitez-Silva,Yale University;John Rust, Yale University; Hiu-ManChan,Yale University;SofiaSheidvasser,Yale University;Moshe Buchinsky,BrownUniversity. DISCUSSANT: Phil Merrigan, Universite du Quebec a Montreal.

"SubjectiveDiscount Rates, UnobservedAbility,FamilyBackgroundand The Return to Schooling,"ChristianBelzil, ConcordiaUniversity;J6rgenHansen,Universitede Montrealand G6teborgUniversity. DISCUSSANT: Phil Merrigan, Universite du Quebec a Montreal. ORGANIZER: Thomas Lemieux.

ThomasLemieux,Universitede Montreal. CHAIR: Communication, Contractingand InstitutionalDesign

"Autonomy,Contractibility,and the FranchiseRelationship,"Shira B. Lewin, Iowa State University. DISCUSSANT: Regis Renault,Universitedes SciencesSocialesde Toulouse. "Cost ReducingStrategies,"Pau Olivella,UniversitatAutonomade Barcelona;Maite Pastor,Centrode EstudiosUniversitariosSan Pablo. DISCUSSANT: YongminChen, Universityof Coloradoat Boulder. "Plea Bargainingand 'Truth-Telling'," Joanne Roberts,Queen's University. DISCUSSANT: Guofu Tan, Hong Kong Universityof Science and Technology.

PROGRAM

1477

"A Model of Expertise,"John Morgan,PrincetonUniversity;VijayKrishna,Pennsylvania State University. Bernard Sinclair-Desgagne, Ecole Polytechnique de Montreal. DISCUSSANT: ORGANIZERS: Susan Athey and Herve Moulin. CHAIR:Lin Zhou, Duke University. Empirical Investigationsin 1.O.

"InformationSharing and Competitionin the Motor Vehicle Industry,"Maura P. Doyle, FRB of Governors;ChristopherM. Snyder,George WashingtonUniversity. Wallace P. Mullin, Michigan State University. DISCUSSANT:

"StructuralEstimationof OralAscendingBid Auctions,"PhilipA. Haile, Universityof Wisconsin-Madison. DISCUSSANT:MatthewShum,StanfordUniversity. "Price Dispersion, Search, and Switching Costs: Evidence from the Credit Card Market,"Victor Stango,Universityof Tennessee. DISCUSSANT: Shiferaw Gurmu, University of Virginia.

"Exchange Rates, Domestic Market Structure and Foreign Trade: The Case of Turkey,"Benan Zeki Orbay,IstanbulTechnicalUniversity,Oner Giincavdi,Istanbul TechnicalUniversity. DISCUSSANT:WilliamJ. Polley, Universityof Iowa. ORGANIZER:Tracy Lewis.

WallaceP. Mullin,MichiganState University. CHAIRa. Taxation and Growth

"EconomicGrowth,(Re-) DistributivePolicies, CapitalMobilityand Tax Competition in Open Economies," Gunther Rehme, Technische Universitat Darmstadt and EuropeanUniversityInstitute. Cornelis A. Los, Nanyang Technological University. DISCUSSANT:

"Time-ConsistentLinearTaxationand Redistributionin an Overlapping-Generations Framework,"ThierryPaul, Universitede Grenoble II; PhilippeMichel, Universite de la Mediterranee. DISCUSSANT: Susan K. Snyder, Virginia Polytechnic Institute and State University.

"Growth,Death, and Taxes,"Lutz Hendricks,ArizonaState University. Yanni Tournas, Rutgers University. DISCUSSANT:

"CapitalIncomeTaxationin an Economywith Housingand BusinessCapital,"Martin Gervais,Universityof WesternOntario. DISCUSSANT: Uday Rajan, Carnegie Mellon University. Robert Pindyck and Herve Moulin. ORGANIZERS:

CHAIR: Uday Rajan,CarnegieMellon University. Saturday,June 27, 4:30 pm-6:00 pm PresidentialAddress by Jean Tirole

"CorporateGovernance,"Jean Tirole, Universitede Toulouse I. MarcelBoyer,Universitede Montreal. CHAIR:

1478

PROGRAM

Sunday,June28, 8:30 am-10:00 am Politicsand ElectoralCompetition "CampaignAdvertisingand Voter Welfare,"AndreaPrat,TilburgUniversity. DISCUSSANT:GuillaumeCheikbossian,ColumbiaUniversityand WarsawUniversity. "A Positive Model of the Permission to Tax in Regional Public Finance," Paul Rothstein,WashingtonUniversityin St. Louis. DISCUSSANT:Eric Hughson,Universityof Utah. "The Controlof Public Policy:Electoral Competition,Polarization,and Endogenous Platforms,"JakobSvensson,The WorldBank. DISCUSSANT: ChristopherM. Snyder,George WashingtonUniversity. ORGANIZER: Tracy Lewis.

CHAIR: ChristopherM. Snyder,George WashingtonUniversity. and Collusion Cooperation "NarrativeEvidence on the Dynamics of Collusion: The Sugar Institute Case," Wallace Mullin, MichiganState University;David Genesove, MassachusettsInstitute of Technology. DIscussANT: PhilipA. Haile, Universityof Wisconsin-Madison. "Is it Harmfulto Allow Partial Cooperation?,"Pierre Cahuc,Universitede Paris I; Paul Beaudry,Universityof BritishColumbia;HubertKempf,Universitede ParisI. SusanK. Snyder,VirginiaPolytechnicInstituteand State University. DISCUSSANT: "Corruption,Collusion and Implementation:A HierarchicalDesign," Mehmet Bac, BilkentUniversity;ParimalKantiBag, Universityof Liverpool. DISCUSSANT: Guofu Tan, Hong Kong Universityof Science and Technology. ORGANIZERS: TracyLewis and Dilip Mookherjee. CHAIR: Guofu Tan, Hong Kong Universityof Science and Technology. Demographics

"RisingLongevity,UnintendedBequests,and EndogenousGrowth,"Jie Zhang,Victoria Universityof Wellington;Junsen Zhang, Chinese Universityof Hong Kong; RonaldLee, Universityof Californiaat Berkeley. DISCUSSANT: AndreasHofert, SwissFederalInstituteof Technology. "LifeExpectancyand EndogenousGrowth,"DavidDe La Croix,UniversiteCatholique de Louvain;OmarLicandro,FEDEA and UniversityCarlosIII. DISCUSSANT: Matti Suominen, INSEAD. ORGANIZER: Paul M. Romer.

CHAIR: FranciscoJ. Ruge-Murcia,Universitede Montreal. Rationality

"Thoughtor Reflex,"WilliamBentley MacLeod,Universityof SouthernCalifornia. DISCUSSANT: Costis Skiadas, Northwestern University.

"CommonKnowledgeof Rationalityin ExtensiveGames,"Geir Asheim,Universityof Oslo; MartinDufwenberg,UppsalaUniversity. DISCUSSANT: Larry Samuelson, University of Wisconsin.

PROGRAM

1479

"RationalizableTrade,"Costis Skiadas,NorthwesternUniversity;StephenMorris. DIScUSSANT: OscarVolij, BrownUniversity. ORGANIZER: Larry Epstein.

CHAIR: Dan Bernhardt,Universityof Illinois and Queen's University. Mechanism Design

"Strategy-Proof Allocationof FixedCosts,"JamesA. Dearden,LehighUniversity;Karl W. Einolf, LehighUniversity. DISCUSSANT: MichelTruchon,UniversiteLaval. "Implementationin Principal-AgentModels of Adverse Selection," Uday Rajan, CarnegieMellonUniversity;Anil Arya,The Ohio State University;JonathanGlover, CarnegieMellon University. DIScUSSANT: Lin Zhou, Duke University. "FairDivisionin ExchangeEconomies,"WilliamThomson,Universityof Rochester. DISCUSSANT: TayfunSonmez,Universityof Michigan. ORGANIZER: Herve Moulin. CHAIR:Yves Sprumont, Universite de Montreal. Price-Settingand Macroeconomics

"StaggeredPrice Setting and EndogenousPersistence,"Paul R. Bergin,Universityof Californiaat Davis;Robert C. Feenstra,Universityof Californiaat Davis. DISCUSSANT: Alex Wolman, FRB of Richmond.

"Does State DependentPricingImplyCoordinationFailure?,"AlexanderL. Wolman, FRB of Richmond;AndrewJohn, Universityof Virginiaand INSEAD. DISCUSSANT: Ruilin Zhou, University of Pennsylvania and FRB of Minneapolis.

"LiquidityEffects and Market Frictions,"Guang-JiaZhang, Bank of Canada;Scott Hendry,Bank of Canada. DISCUSSANT: John Coleman, Duke University. ORGANIZER: Peter Howitt.

CHAIR: James M. Nason, Universityof BritishColumbia. Applied Econometrics II

"TechnicalEfficiencyand FirmSize:EvidencefromU.S. Panel Data,"RajeevDhawan, Universityof Californiaat Los Angeles. DISCUSSANT: Denis Bolduc, Universite Laval.

"Sector-SpecificOn-The-JobTraining:Evidence from U.S. Data," Lars Vilhuber, Universitede Montreal. DISCUSSANT: Christian Belzil, Concordia University.

"The Role of Unobservablesin the Measurementof Wage Effects of Job Mobility: EvidencefromGermanyand the United States,"VeroniqueSimonnet,Universitede Paris I. DISCUSSANT: Daniel Parent, Universite de Sherbrooke. ORGANIZER: Thomas Lemieux.

CHAIR: Daniel Parent,Universitede Sherbrooke.

1480

PROGRAM

Modeling High-FrequencyMarketDynamics

"ModelingHigh-FrequencyData in ContinuousTime,"Nour Meddahi,Universitede Montreal;Eric Renault,CRESTand INSEA;Bas J.M. Werker,CREST. DISCUSSANT:Eric Ghysels,PennsylvaniaState University. "TradingVolume and the Short and Long-RunComponentsof Volatility,"Roman Liesenfeld,UniversitatTiubingen. DISCUSSANT: Jean-Guy Simonato, HEC-Montreal.

"StochasticVolatilityDurationModels,"JoannaJasiak,YorkUniversity;Eric Ghysels, PennsylvaniaState University;ChristianGourieroux,CRESTand CEPREMAP. DISCUSSANT: Peter Christoffersen,InternationalMonetaryFund ResearchDepartment. ORGANIZERS: TorbenG. Andersen,Rene Garciaand Eric Ghysels. CHAIR:Peter Christoffersen,InternationalMonetaryFund ResearchDepartment. New Empirical Models in Macroeconomics

"Multifractalityof Deutschemark/US Dollar Exchange Rates," Adlai Fisher, Yale University;LaurentCalvet,Yale University;Benoit Mandelbrot,Yale University. Tim Conley, Northwestern University. DISCUSSANT:

"A New Approachto the Analysisof Shocksand the Cycle in a Model of Outputand Employment,"Hans-MartinKrolzig,Universityof Oxford;Juan Torro, European UniversityInstitute. DISCUSSANT: John Coleman, Duke University.

"InternationalInterest Rate Differentials:The Interactionwith Fiscal and Monetary Variablesand the BusinessCycle,"Alain Paquet,Universitedu Quebec a Montreal; Paul Fenton, Bank of Canada. DISCUSSANT: Christine Parlour, Carnegie Mellon University. ORGANIZER: Fallaw Sowell. CHAIR:Lutz Kilian,Universityof Michigan. Sunday, June 28, 10:30-12:00 noon Walras-BowleyLecture by Adrian Pagan

"Bullsand Bears,"AdrianPagan,AustralianNationalUniversity. CHAIR: Eric Ghysels,PennsylvaniaState University. Sunday, June 28, 2:00 pm-4:OOpm Information and Stock Prices

"GradualIncorporationof Informationinto Stock Prices:EmpiricalStrategies,"Sara Fisher Ellison, MichiganState Universityand MassachusettsInstitute of Technology; Wallace P. Mullin, MichiganState Universityand MassachusettsInstitute of Technology. DISCUSSANT: Thomas H. McCurdy, University of Toronto.

"ExternalHabit and Cyclicalityof StockReturns,"HaroldH. Zhang,CarnegieMellon University;ThomasTallariniJr., CarnegieMellon University. DISCUSSANT: Rene Garcia, Universite de Montreal.

PROGRAM

1481

"Timingof Orders,OrdersAggressivenessand the OrderBook at the Paris Bourse," Thierry Kamionka,Universite de Toulouse I; ChristopheBisiere, Universite de Toulouse I. DIScUSSANT: JoannaJasiak,York University. "TradingVolume and InformationRevelation in Stock Markets,"Matti Suominen, INSEAD. DIScUSSANT: Nour Meddahi,Universitede Montreal. ORGANIZERS: TorbenG. Andersen,Rene Garciaand Eric Ghysels. Nour Meddahi,Univeristede Montreal. CHAIR: Limited Dependent Models and SemiparametricEstimation II

"A SemiparametricError ComponentDensity EstimationTechnique for Stochastic FrontierModels,"WilliamHorrace,Universityof Arizona. Han Hong, Stanford University. DISCUSSANT:

"New Evidenceon Returnsto Scale and ProductMix amongUS CommercialBank," Paul W. Wilson,Universityof Texas;David C. Wheelock,FRB of St. Louis. DISCUSSANT: James McIntosh, Concordia University.

"The Duration of Higher Education:SemiparametricEstimation of a Dependent CompetingRisks Model," Hans Van Ophem, University of Amsterdam;Nicole Jonker,Universityof Amsterdam. DISCUSSANT: Shiferaw Gurmu, University of Virginia.

"EfficientEstimationof CensoredDurationModelswith UnobservedHeterogeneity," Paul Rilstone,York University;Peter Bearse,Universityof Tennessee;Jose Canals, Universityof Virginia. DISCUSSANT: Hidehiko Ichimura, University of Pittsburgh. ORGANIZER: Guido W. Imbens.

CHAIR: HidehikoIchimura,Universityof Pittsburgh. Auctions II

"The Impactof Synergieson Biddingin the GeorgiaSchool Milk Market,"Robert C. Marshall, PennsylvaniaState University;Matthew E. Raiff, PennsylvaniaState University. DISCUSSANT: Leslie Marx, University of Rochester.

"ConditionallyIndependent Private Values in OCS Wildcat Auctions," Tong Li, WashingtonState University;Isabelle Perrigne,Universityof SouthernCalifornia; QuangVuong,Universityof SouthernCalifornia. DISCUSSANT: Olivier Armantier, University of Pittsburgh.

"ReinforcementLearningin 2 x 2 Games and the Concept of ReinforcablyStable Strategies,"Thomas Brenner, Max-Planck-Institutefor Research into Economic Systems. DISCUSSANT: Takako Fujiwara-Greve, Keio University.

"A StructuralEconometricAscendingAuction Model,with an Applicationto the PCS SpectrumAuctions," Matthew Shum, Stanford University;Han Hong, Stanford University. DISCUSSANT: Jacques Robert, Universite de Montreal. ORGANIZER: Jean-Franqois Richard.

CHAIR: Jean-PierreFlorens,Universitede ToulouseI.

1482

PROGRAM

DistributivePolitics

"Distributive Politics and Representative Democracy," Guillaume Cheikbossian, ColumbiaUniversityand WarsawUniversity. DISCUSSANT: Jakob Svensson, The World Bank.

"SocialLimits to Redistribution,"Giacomo Corneo,Universityof Bonn; Hans Peter Griiner,Universityof Bonn. DISCUSSANT: Andrea Prat, Tilburg University.

"Term Limits and Pork Barrel Politics,"Dan Bernhardt,Universityof Illinois and Queen's University;Eric Hughson, Universityof Utah; Sangita Dubey, Queen's University. DIScUSSANT: Paul Rothstein,WashingtonUniversityin St. Louis. ORGANIZER:Tracy Lewis.

CHAIR: Paul Rothstein,WashingtonUniversityin St. Louis. Foreign Investment

"OptimalTradePolicyin the Presenceof DFI and InternalTax Competition,"Laurel A. Adams, NorthwesternUniversity;Pierre Regibeau, UniversidadAutonoma de Barcelona. DIScuSSANT: Raja Kali, InstitutoTecnologicoAutonomode Mexico. "InternationalProjectFinanceunderLenderHeterogeneity,"Sule Ozler,KoqUniversity;SumruAltug, KoqUniversity;HakanOrbay,KoqUniversity. DISCUSSANT: Yutaka Suzuki, Hosei University.

"ForeignDirect Investmentand the Dynamic Effects of Tariff Policy,"William J. Polley,The Universityof Iowa. Corinne Langinier, University of California at Berkeley. DISCUSSANT:

"Relative FDI Incentives-The Principle of ComparativeAdvantageOnce Again," LarryD. Qiu, Hong Kong Universityof Science & Technology. Michael Devereux, University of British Columbia. DISCUSSANT: ORGANIZER: Robert Pindyck. CHAIR:Richard Barnett, University of Arkansas. Information and Uncertainty

"IncreasedRisk Takingwith Multiple Risks,"Ed Schlee, Arizona State University; ChristianGollier,Universitede Toulouse I. Susan Athey, Massachusetts Institute of Technology. DISCUSSANT:

"The Value of Informationin Monotone Decision Problems,"Susan Athey, Massachusetts Institute of Technology;Jonathan Levin, MassachusettsInstitute'6of Technology. Dirk Bergemann, Yale University. DISCUSSANT:

"InformationAcquisitionand ResearchDifferentiationPriorto an Open-BidAuction," Serge Moresi,GeorgetownUniversity. Ezra Friedman, Massachusetts Institute of Technology. DISCUSSANT:

"The Demand for Sweepstakes,"Guofu Tan, Hong Kong Universityof Science and Technology;Soo Hong Chew,Hong Kong Universityand Universityof California. DISCUSSANT: Christian Gollier, Universite de Toulouse I. ORGANIZER:Susan Athey.

CHAIR: ChenghuMa, McGillUniversity.

PROGRAM

1483

Axiomatics and Mechanism Design

"Strategyproofand ImplementableMechanismsfor the Provisionof a Menu of Public Goods,"Eiichi Miyagawa,Universityof Rochester. DISCUSSANT: Roberto Serrano,BrownUniversity. "RationalizableSolutionsto Pure PopulationProblems,"WalterBossert,Universityof Nottingham;Charles Blackorby,Universityof British Columbiaand GREQAM; David Donaldson,Universityof BritishColumbia. Herve Moulin,Duke University. DIsCUSSANT: "RecapturingEfficiencyin House Allocationwith ExistingTenants,"TayfunSonmez, Universityof Michigan. Yves Sprumont, Universite de Montreal. DISCUSSANT:

"ChoquetBargainingSolutions,"Lin Zhou, Duke University. Herve Moulin, Duke University. DISCUSSANT: ORGANIZER:Herve Moulin.

CHAIR:Herve Moulin,Duke University.

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Econometrica, Vol. 66, No. 6 (November,1998)

SUBMISSION OF MANUSCRIPTS TO ECONOMETRICA 1. Four copies of the original manuscript should be sent to: Professor Drew Fudenberg Editor, Econometrica, Dept. of Economics, Harvard University Cambridge, MA 02138, U.S.A. They must be accompanied by a letter of submission and be written in either French or English. Submission of a paper is held to imply its contents represent original and unpublished work, and that it has not been submitted for publication elsewhere. If the author has submitted related work elsewhere, or does so during the term in which Econometrica is considering the manuscript, then it is the author's responsibility to provide Econometrica with details. There is neither a submission charge nor page fee; nor is any payment made to the authors. 2. Papers may be rejected, returned for specified revision, or accepted. Currently, a paper will appear approximately nine months from date of acceptance. 3. All submitted manuscripts should be typed on bond paper of standard size, preferably 8.5 by 11 inches, and should have margins of at least one inch on all sides. Double space all material in the manuscript, including captions, headnote, footnotes, references, and so forth. Please submit only high-quality reproductions (not ditto, thermofax, or verifax). 4. Specific instructions on the form in which to prepare the manuscript can be found in the "Manual for Econometrica Authors, Revised" written by Drew Fudenberg and Dorothy Hodges, published in the July, 1997 issue of Econometrica and posted on the The Manual deweb at http://www.econometricsociety.org/es/manual/index.html. scribes how footnotes, diagrams, tables, etc. should be prepared. In addition, it explains editorial policy regarding style and standards of craftsmanship. It will facilitate the submission and review process for both the author and referees if all submitted manuscripts conform as closely as possible to these guidelines. 5. Papers should be accompanied by an abstract that is full enough to convey the main results of the paper. On the same sheet as the abstract should appear the title of the paper, the name(s) and full address(es) of the author(s), and a list of keywords. 6. If you plan to submit a comment on an article that has appeared in Econometrica, please follow these guidelines: First send a copy to the author and explain that you are considering submitting the comment to Econometrica. Second, when you submit your comment, please include any response that you have received from the author. Comments on published articles will be considered only if they state you have been through this process. If an author does not respond to your letter after a reasonable amount of time, then this should be explained in your submission. Authors will be invited to submit for consideration a reply to any accepted comment of this kind. 7. Manuscripts on experimental economics should adhere to the "Guidelines for Submission of Manuscripts on Experimental Economics" written by Thomas Palfrey and Robert Porter, and published in the July, 1991 issue of Econometrica. 8. Econometric Society business to be published in Econometrica as an Announcement, Report, Program, etc. should be submitted to the Econometric Society Secretary, Julie Gordon (address on inside back cover). Announcements of academic conferences, workshops, etc., conducted by other societies or organizations, the availability of fellowships, postdoctorates, etc., and honorary awards for publication in the News Notes section of the journal should be sent to the Editor, Drew Fudenberg (address above). Deadlines for all such endmatter are the fifteenth day of the fourth month prior to publication, i.e., September 15 for the January issue, November 15 for the March issue, etc. Please refer to the "Manual for Econometrica Authors, Revised," by Drew Fudenberg and Dorothy Hodges, published in the July, 1997 issue for further information on format and content.

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THE

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SOCIETY

An InternationalSocietyfor the Advancementof Economic Theoryin its Relation to Statistics and Mathematics FoundedDecember 29, 1930

1998 OFFICERS JEANTIROLE,IDEI,Universitedes Sciences Sociales de Toulouse,PRESIDENT ROBERT WILSON,Stanford University, FIRSTVICE-PRESIDENT ELHANAN HELPMAN,Harvard University and Tel Aviv University, SECONDVICE-PRESIDENT ROBERT E. LUCAS,JR., University of Chicago, PASTPRESIDENT JULIEP. GORDON,Northwestern University, SECRETARY ROBERT J. GORDON,Northwestern University, TREASURER

1998 COUNCIL IMPA, Brazil ALOIsIoARAUJO, ANGUSS. DEATON, Princeton University, U.S.A. (*)AvINASH K. DIXIT, Princeton University, U.S.A. ' BHASKARDUTTA,Indian Statistical Institute, India ROBERT F. ENGLE,University of California, San Diego, U.S.A. (*)BIRGITGRODAL,University of Copenhagen, Denmark OLIVER HART,Harvard University, U.S.A. FUMIOHAYASHI, University of Tokyo, Japan DAVIDHENDRY, Oxford University, U.K. (*)BENGT R. HOLMSTROM,Massachusetts Institute of Technology, U.S.A. TAKATOSHI ITO, International Monetary Fund, U.S.A. MURRAYC. KEMP, University of New South Wales, Australia A. KiNG,Bank of England, U.K. MERVYN DAVIDM. KREPs,Stanford University, U.S.A. GuY LAROQUE,INSEE-CREST, France ROLFR. MANTEL,Universidad de San Andres, Argentina

Pompeu

ANDREU MAS-COLELL, Universitat

Fabra of Cambridge, U.K. ROGER B. MYERSON, Northwestern University, U.S.A. J. PETER NEARY, University College Dublin, Ireland JAMES MIRRLEES, University

KAzuo NISHIMURA,Kyoto University,

R. PAGAN, Australian University, Australia

ADRIAN

Japan National

University,

ROBERT PORTER, Northwestern

U.S.A. ARIEL RUBINSTEIN,Tel Aviv University, Israel THOMAS J. SARGENT, Hoover

Institution

and

University of Chicago, U.S.A. RIENHARD

SELTEN, University

of

Bonn,

Germany Bank for NICHOLAS H. STERN, European Reconstruction and Development, U.K. of Chicago, NANCY L. STOKEY, University U.S.A. KOTARO SUZUMURA, Hitotsubashi University, Japan

The ExecutiveCommitteeof the Society consistsof the Officers,Editor,andthe starred(*) membersof the Council.