Tilburg University

The continuous multivariate location-scale model revisited

Fernández, C.; Osiewalski, J.; Steel, M.F.J.

Publication date:1993

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Citation for published version (APA):Fernández, C., Osiewalski, J., & Steel, M. F. J. (1993). The continuous multivariate location-scale modelrevisited: A tale of robustness. (CentER Discussion Paper; Vol. 1993-80). Unknown Publisher.

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~~ ~Z Discussionfor a er3414 ~mic Research

199380

II Ih I III N N I I I N I N IU II I N II III IN II ~ N I~ I

~ K.U.B.D giBLIOTHEEK

TILBURG-~~~~

Centerfor

Economic Research

rf ~'-

No. 9380

THE CONTINUOUS MULTIVARIATELOCATION-SCALE MODEL REVISITED:

A TALE OF ROBUSTNESS

by Carmen Fernandez,Jacek Osiewalski

and Mark F.J. Steel

December 1993

ISSN 0924-7815

'he Continuous Multivariate Location-Scale Model Revisited:A Tale of Robustness

By CARMEN FERNANDEZDepartment of Matheraatics, Uníversidad Autónoma, 28049 Madrid, Spain

JACEK OSIEWALSKIDepartmr.nt of Econametrica, Academy of Economics, 31-510 Kraków, Poland

and MARK F. .l. STEELCcntEi and Dc.pt. of Econometrics, Tilburg University, 5000 LE Tilburg, The Netherlands

SummaryWe considcr the location-scale faniily of all nrultivariate continuous sampling distribu-

tions. Any tnodel from t.he subclass induced by fixing a particular distribution on the unit.sphc~re,, combine~d with a standard improper prior on the scale paranteter, will lead to thesatne inícrcure ou location ancí observ~ables. This finding sheds light on many robustness

rosults in lcss general rnodels and its implications are illustrated in a number of examples.

Some key words: Bayesian inference; Cardioid distribution; Model robustness; Multivariatedistribution theory; Nuisance pazameter.

1. Introduction

In the location-scale model we are often interested in the location parameter, whereas

the scale may just be a nuisance pararneter. Inference on missing values of the observablescould also be a reason for studying these models. In a Bayesian context, these considera-tions translatc to focussing our attention on the joint density of the observables, a, and thelocation parametcr, ~1. Bayesian modcls that lcad to the same p(x, ~a) are called mazginallyequivalent, following Osiewalski and Steel (1993c).

The main result of this paper is that all sampling models from the general continuousmultivariate location-scale faznily, combined with a standard itnproper reference prior onthe scale parameter, are marginally equivalent, provided we fix the distribution over theunit sphcre. In other words, the conditional distribution of the radius given the point on thesphere does not affect p(x, p) and thus the ensueing posterior density of te or the predictive

density. Some examples based on the cardioid circulaz distribution clearly illustrate that

many very different sampling densities can still lead to mazginally equivalent models, i.e.the same inference on p and unobserved elements of r.

We consider general continuous multivariate distributions which constitute the final

atage of a series of subsequent generalizations. In decreasing order of generality, we men-

tion v-spherical distributions, as introduced in an, as yet, unpublished paper by the presentauthors (Cent.ER Discussion Paper 9374, Tilburg University), henceforth denoted by FOS,

2

ly-spherical ciistributions, cíefined in Osiewalski and Steel (1993b), and spherical distribu-tiou,, ati discussc~d in Ke~lker (1970), Cambanis et aL (1981), Dickey and Chen (1985) andFaufi ct a.l. (1990, Chapters 2-4). Similar robustness results in thcsr special cases arcgeneralized arrd put in a broader perspective here.

2. Marginal Equivalence

In this paper we shall analyze the general location-scale family of continuous mul-tivariate distributions. In particular, we consider the sampling density p(xl~s,r), wherex- p}- r-~y, ~ E Ji", r E~~ and P(ylW,r) - f(y) is a proper density in Ji" (n 1 2).Thus -

P(xlp,r) - r"f{r(x - p)}, (2.1)

wit.h x-(xr,... ,x") E ~li", p a location pararneter and r-r a scale parameter. Through-out. the papc~r, all densitics are taken with respect to Lebesgue mcasure in the correspondingspacc.

A full Bayesian model will consist of a sampling density from (2.1) combined with aprior ciensity on the parameters (p, r), either proper or improper. In practice, our interestwill focus on the location parameter, {r, which may be reparameterized in terms of a lowerdimensional vcctor, whercas r will typically be a nuisance pazameter. In addition, we maybe interested in prediction of unobserved elements of x.

Tlrese cuusiderations naturally lcad to the concept of marginal equivalence, as intro-duced in Osiewalski and Stcel (1993c). In particular, two Bayesian models with differentsampling densities from thc family (2.1) and potentially different priors are marginallyequivalent for x and f~ if both models lead to the same density p(x, ~e). Note that p(x, p)simply corresponds to the joint density of r-ry and p, evaluated at (x - p,~). Therefore,our attention will be focussed on p(r-~y,~).

We rrow express y E~" as y- u h(w), where u- IIyII E~t is the Euclidean norm of yand h(w) - y~IIJII E S"-~, the unit sphere. In order to facilitate variable transformations,we represent the sphere in terms of w E S2 C 32"-r through a one-to-one function h(.) withcontinuous first order paztial derivatives. For example, we shall often characterize pointsin S"-r through angulaz polar coordinates.

We factorize the joint density function of (u, w), into the marginal density of w, denotedby f2(w), and the conditional density of u given w, fr(u;w). Observe that then propernessof f(y) is equivalent to both f2(w) and fr(u;w) being proper. Clcarly, for a given h(.),f1(w) uniquely determines a probability measure over the unit sphere.

Wc rernind thc rcader that the Bayesian ruodcls considered here can differ in the choiccuf f(.), :wd tlicrcforc fr(.) and fZ(.), and in thc choicc of p('c,r).

3

3. Inference Robustness under an Improper Prior

In a location-scale model, we often do not possess real prior information regarding theuuisance parautet.er r. A conmion solution to this problem is to specify a prior densitywhich is wiifornt on lug r and has thc following product struc-turc:

P~l~, r) - P(I')1'(r) - p(l~)c'r-~, (c ) 0). (3.1)

Aclopting this refercnre prior in our framework has an interesting conseqnence, as is for-mailizecí iu the following thcorem.

Theorem. If p(x~p,r) - r"f{r(~ - p)} and p(p,r) - p(p)cr-' (c ~ 0), then the jointdensity p(a,N) does not depend on f~(u;w).

Proof: Considerz - r-'u, sothat p(u,w, p,z) - f~(u;w)f2(w)p(p)cz-'. Thus, p(w,tt,z) -fz(w)p(p)cz-~ and does not depend on f~(u;w). Therefore, p(zh(w),p) and p(a,tr) do notdepend on f~(u; w) either. ~

The key to this result is the fact that the density p(r) - cr-' is invariant under thetransfortnxtion z- r-' u for arty value of u, 0, so that u disappears once we make thevariable transformation from r to z. No other prior than (3.1) will induce this particularproperty.

Thc, Thcoretn itnplies that if we fix the distribution on the unit sphere by choosing apata-ticular h(w) and fi(w), the conditirntal distribution of the radius, given by f~(u; w), doesnot. inHuence t.hc resulting cíensity p(a,Ec). This rneans that all Bayesian models consistingof a satnpling model from a subclass in (2.1) induced by a particular f2(w), given i~(w),artd a prior corresponding to (3.1) are marginally equivalent. Choosing a distributionon the unit sphere, S"-', in combination with (3.1) suffices to obtain perfect robustnessof posterior inference on Fi with respect to any choice of a proper density f~(u;w). Ofcourse, in orcíer to deduce a proper posterior density p(p~x) from p(x,ta), we need that theprcdictive density p(x) is finite. In addition, we achieve robustness of predictive inferenceon tmobserved elements of x, say af, given the observations, say xo, where properness ofp(x f~xo) rcquires that p(.zo) G oo.

Clearly, by varying fl(.), we cazt obtain dramatically different sampling models thatwill, however, still lead to the same inferences on x and p. Section 5 will provide someillustrative examples.

4. The Case of General Priors

In order to examine marginal equivalence under other prior structures than (3.1), anexplicit expression for p(:c, p) proves useful.

Denoting the Jacobian of the transformation from (u,w) to y by u"-'s(w), where s(.)depends on the form of h(.) adopted, we can express the density f(y) as the product offl(u;w)fZ(w) and the inverse of this Jacobian. This immediately leads to the following

4

expression for p(r., fe):

1-nIlx - ~II x -P(x,~) -S h- ( ll f~ ~L ~ II ~II

{ ~ l r-uIIJx-

J ~ Ïi S r~~x - 1~~~~ h~`~~x - lr~~ ~~rP(l~, r)dr.

o l

Thus, gíven h(w), atiy set of combinations of fz(w), fi(u;w) and p(Fi,r), the latter eithcrproper or icnproper, that leads to the same p(x, fi) in (4.1) defines a class of marginallyequivalent Bayesian modcls. In particular, if we fix the distribution ovcr the unit sphere,margina] equivalence is characterized by pairs of f~(u; w) and p(p, r) that lead to the samevalue of the integral in (4.1). Examples of such marginal equivalence can be constructed

cíirectly from the univariate integral identities in Appendix B of FOS.From thc~ Theorem we know that under the invariant prior (3.1) any choice of a proper

f~(u;w) will give us a mazginally equivalent model. The latter class will be characterizedby

P(x, Fc) - c ~~x - Fc~~-n f2 ! h-'` x- 1~~~ ~~

P(Fc)-g{h- ~lr~r)} l II

5. Examples

The thcory will now be illustrated by some examples in the bivariat.e case (n - 2).

A well-known circular distribution on the unit sphere, S', is the cardioid distribution,which corresponds to the following density function in polar coordinates

1 f 2pcos wf~(w) - 2~ I[o,z,c)(w),

where ~p~ c 1~2. Nachtsheim and Johnson (1988) combine (5.1) with an independentGamma(n~2,1~2) distribution on u~, leading to

fi (u;w) - uexp(-u2~2)I(0.~)(u). (5.2)

They use many other nonuniform distributions on the sphere, keeping f~(u;w) fixed at(5.2), and call the resulting distributions anisotropic. In this Section, however, we fixf2(w) to be the cazdioid density in (5.1) with p - 0.3 and vary the choice of fl(u;w).

Thus, following our Theorem, we generate sampling distributions that all lead to

marginally equivalent Bayesian models, provicícd we use the impropcr prior in (3.1) for

P(li, r).Pauel a of Figure 1 shows thc density fiinction of y- r(x - p) corresponding to

(5.2) in Cartesian coordinates, whereas panel b gives its isodensity contours. The latter

5

convcutiou will bc followcd throughout. this Scctiou. Fígttres 2 throttgh 4 corresponcl toGaututa, Im-c,rtcd Bcta and Bcta spccifications for ft(u;w). In part.icular, wc Lavc chuscn

filrc:w) - fr.(u~20, 1) - {r(20)}-~urscxp(-u)Ita,~t(rt)r

fi (rt:w) - fie(u]1,?,~~ f 1) -(1 -~ w'~)~ ~1 ~ 1 f w~~ 1(c,~)(u)

fi(tt; w) - Ía(u~(w -~r)~ f 1~2,1~2, 10)

I'{(w - n)2 f 1} u(W-~1'-tl2 u-tlz

- l0ar~t I' {(w - n)~ -F 2 } ~10~ ~1 - 10~ 1(o'to}(u)~

Insert Figures 1 to 4

Note that fb(u;w) retains the iudependence between u and w found in (5.2), yet leadsto vcry diffcrcnt. isodensity contours. This forccfully illustrates that fr(u;w) is not thelabclliug fitnction, even uncíer iucíepencíenee. Eqtilvaleut.ly, choosing fz(w) generally doesnot define the isodensity sets. In the special case of v-spherical distributions as introducedin FOS, where f(.) - g{v(.)} and v(-) and g(.) are subject to certain restrictions, choosingfl(w) is equivalent to choosing v(.), which determines the isodensity sets, whereas ft (u;w)uniquely identifies thc labelling function g(.) given v(.). In particular, we then obtainfz(w) oc s(w)~c~{h(w)}]-" and fr(u;w) a u"-tg[uv{h(w)}]. If we specialize further to !y-spherical distributions, introduced in Osiewalski and Steel (1993b), we choose v(.) to bethe Iq-nonn, where q- 2 corresponds to the usual definition of sphericity.

The last two examples introduce dependence between u and w. In addition, fi(u;w)does not take the same values for w- 0 as for w - 2a and thus leads to discontinuities inthe density function in Cartesian coordinates. The last example concentrates all the masswithin the circle with radius 10 and does not lead to a bounde.d density function. Thedensity plot in Figure 4a is thus truncated.

In contrast to v-spherical distributions, isodensity sets are not necessarily locatedaround a common point, as is illustrated by Figures 2 and 4, corresponding to f~(u;w)and Íi (u~w).

We reiterate that posterior inference on the location parameter of the correspondingsampling densities of z - p~- T- t y, as well as predictive inference on 2 itself, is exactlythe same in all the examples discussed above. Any choice of a proper density ft(u;w) incombination with (5.1) will constitute a model that is marginally equivalent to the cardioidmodel of Nachtsheim aud Johnson (1988), provided we use prior (3.1), howevcr differcntthc shape of thc density fimction and the isodensity sets may be. Of course, the Thcoremapplics to any other proper density fz(w) as well.

6. Concluding Remarks

In the general class of location-scale models with continuous multivariate distributions(2.1), we find that using a common reference prior on the scale will result in the samc

s

posterior inference on the location parameter and the same predictive inference on ~,within a subclass of (2.1) induced by a commou distribution over the tmit sphere. Thisvery general result implies that we can gene.rate a continuum of potentially very differentsampling distributions, by varying the proper conditional density of the radius, keeping thedistribution on the spherc fixed, that all lead to marginally equivalent Bayesian models.

So fa.r, wc havc bcen interested iu the fonn of p(x,p). Let us uow consider p(r~x,(i),t.ho rcmniniul; factor of our Baycsiau mocicl P(:r,{e,r). If x is fitlly obscrvcd, P(r~r,lc) iathc~ conelitional posterior dcnsity of r given (i. Under the conditions of the Thcorem, wcobtain

P(r~x, tl) - ~~2 - f~~~ft~rll x - l~ll; h-t `~~y - f~~~ ~~

(6.1)l x -

The latter dettsity sunmiarites the entire irtfluence of the choice of f t(.) and does notdcpcnd ou thc forrn of fz(.). Thus, given (i, posterior infcrence on t.he scale pararneteris uua.ffi~ctcd hy t.hc dist.ribut.ion ou thc sphcrc. Howcvcr, this couditional robustncss islost whcn wc calculate the nrarginal postcrior p(r~a). In addition, r is updated by thcsample information and its conditional posterior iu (6.1), which is a proper density, doesnot replicate the functional form of the prior in (3.1). Therefore, our Theorem does nothold for more than one independent vector observation from (2.1) and achieving inferencerobustness u~erns itnpossible in the case with repeated sampling.

The ThMirem can straightforwazdly be generalized to the case where we sample fromt- Az with x distributed according to (2.1) and A a nonsingular matrix function of aparazneter vector r). In that case, the conditions of the Theorem combined with any priorp(N., rO will lead to a density p(t, (a, p) which does not depend on fr (.). Thus, the robustnessresult.s extend to thc posterior distribution of r( as well, if the latter exists.

This papcr gc~neralizes ancl puts in perspective a host of earlicr papers in multivariatcdistribution thc~ory. Spccial ca,~s of our thcorcrn wcrc discussecí in Zcllncr (19ï6) for thctnultivariatc Stucíent-t regression model, whercas Jammalamadaka et al. (1987), Chib etaL (1988, 1991) and Osiewalski (1991) treated the case of scale mixtures of Normals,generalized to ellipticity in Iz in Osiewalski and Steel (1993a,c). Finally, (y-spherical andv-spherical distributions were introduced and examined by Osiewalski and Steel (1993b)ancí FOS, respectively. The result in this paper clarifies the nature of the robustnessresults found in the papers mentioned above: we achieve robustness with respect to theconditional distribution of the radius, and not, in general, with respect to the labellingfirnction. In tlie previous papers, both types of robustness were confounded but the generalframework considered in this paper allows us to clearly establish the underlying nature ofall these tnarginal equivalence results. In addition, the fundamental cause is seen to residein invariance of the improper prior on the scale parameter.

AcknowledgementsSpecial thanks to Julián de la Horra for many insightful comments. We also acknowl-

edge useful suggestions from the Editor and stimulating discussions with Michel Mouchartand Raouf Jaibi. The first and third authors received support from D.G.LC.Y.T. undergrant numbers PB91-0014 and PB92-0246, respectively, and have benefited from a visit tothe Institute for Empirical Macroeconomics, Federal Reserve Bank of Minneapolis. The

first antbc,r .c;r:ctc,fnll,V acknowlydt;os thc hospit:clit.y of t.}ic~ Dcpart,uc~nt of Erououcctric.s,Tillnn~; l~ni~~c~rnit.y.

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Cambanis, S., Huang, S. dr Simons, G. (1981). Ou thc thcory of clliptically contourcddist.ributions. .1. Mult. A~anl. 11, 368-85.

Chib, S., Osicwalski, J. êc Steel, M.F.J. (1991). Posterior inference on the degrees offreedom parameter in rnultivariate-t regression models. Er.on. Letters 37, 391-7.

Chib, S., 1'iwari, R.C. X~ Jammalamadaka, S.R.. (1988). Bayes prediction in regressionswith clliptical errors. J. Econometrics 38, 349-60.

Dickey, J.M. 8t Chen, C.H. (1985). Direct subjective-probability modelling using ellipsoidaldistributions. In Ilayesian Stati~tice 2, Ed. J.M. Bernardo, M.H. DeGroot, D.V.Lindlcy and A.F.M. Smith, pp. 157-82, Amstcrdam: North-Holland.

Fang, lí.-T., Kot.z, S. C Nt;, Ií.W. (1990). S;ryiu.ncct.rir. Mnltivnrintc and Rr.Intccl Dietrióu-tàons. Luudon: Chapman aud Hall.

Jammalamadaka, S.R.., Tiwari, R.C. óc Chib, S. (1987). Bayes prediction in the lineazrnodel with spherically symmetric errors. Econ. Letter~ 24, 39-44.

Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale gen-eralization. Snnkh,yá A 32, 419-30.

Nachtsheim, C.J. 8t Johnson, M.E. (1988). A new family of multivariate distributions withapplications to Monte Carlo studies. J. Am. Stntiet. Assoc. 83, 984-9.

Osiewalski, .I. (1991). A note on Bayesian inference in a regression model with ellipticalcrrors. J. Ecotzometrics 48, 183-93.

Osicwalski, J. 8t Stcel, M.F.J. (1993a). Robust Bayesian inference in elliptical regressionmocíels. J. Econometricy 57, 345-63.

Osiewalski, J. ~t Steel, M.F.J. (1993b). Robust Bayesian inference in ly-spherical models.I3inraetrika 80, 456-60.

Osiewalski, J. i~t Steel, M.F.J. (1993c). Bayesian marginal equivalence of elliptica] regres-sion models. J. Econometric9 59, 391-403.

Zellner, A. (1976). Bayesian and non-Bayesian analysis of the regression model with mul-tivariate Student-t error terms. J. Am. Stntiet. A~eoc. 71, 400-5.

Figure 1: Densitc and contour plots for f~ (u:-!

-z~~-~

Figure 2: Densit}- and contour plots for fb(u:.:)

(b1

z

Figure 3: Drnsit~ and coutour pluts for f~ (u;.;)

z

c.s

-0.5 J 0.5

Figure -l: Densit}- and contour plots for fi (u:.;)

(b

No. Author(s)

9301 N. Kahana and

S. Nitzan

9302 W. Guth andS. Nit7an

9303 D. Karotkinand S. Nitran

9304 A. Lusardi

9305 W. Giith

9306 B. Peleg andS. Tijs

9307 G. Imbens andA. Lancaster

9308 T. Ellingsenand K. WBrneryd

9309 H. Bester

9310 T'. Callan andA. van Soest

931 I M. Pradhan andA. van Scest

9312 Th. Nijman andE. Sentana

9313 K. W~meryd

9314 O.P.Attanasio andM. Browning

9315 F. C. Drost andB. J. M. Werker

9316 H. Hamers,P. Bonn andS. Tijs

CcntER Uiscussion Papcr Scries 1993

Title

Credibility and Duration of Political Contests and the Extent of

Rent Dissipation

Are Moral Objections to Free Riding Evolutionarily Stable?

Some Peculiarities of Group Decision Making in Teams

Euler Equations in Micro Data: Merging Data from Two Samples

A Simple Justification of Quantity Competition and the Coumot-Oligopoly Solution

The Consistency Principle For Games in Strategic Forrn

Case Control Studies with Contaminated Controls

Foreign Direct Investment and the Political Economy ofProtection

Price Commitment in Search Markets

Female Labour Supply in Farm Households: Farm andOff-Farm Participation

Formal and Informal Sector Employment in Urban Areas ofBolivia

Marginalization and Contemporaneous Aggregation inMultivariate GARCH Processes

Communication, Complexity, and Evolutionary Stability

Consumption over the Life Cycle and over the BusinessCycle

A Note on Robinson's Test of Independence

On Games Corresponding to Sequencing Situationswith Ready Times

9317 W. Guth On Ultimatum Bargaining Experiments - A Personal Review

Nu. Author~x) T'itlc

9318 M.J.G. van Eijs

9319 S.Ilurkens

9320 J.J.G. Lemmen andS.C.W. Eijffinger

9321 A.L. Bovenberg andS. Smulders

9322 K.-E. Wámeryd

9323 D. Talman,Y. Yamamoto andZ. Yang

On the Delcmiination of the Control Parameters of the Optimal Can-order Policy

Multi-sided Pre-play Communication by Buming Money

The Quanti[y Approach to Financial Integration: TheFeldstein-Horioka Criterion Revisited

Environmental Quality and Pollution-saving TechnologicalChange in a'Two-sector Endogenous Growth Model

The Will to Save Money: an Essay on Economic Psychology

The (2"'m'' - 2}Ray Algorithm: A New Variable DimensionSimplicial Algorithm For Computing Economic Equilibria onS"xRm

9324 H. Huizinga The Financing and'Iaxation of U.S. Direct Investment Abroad

9325 S.CW'. Eijffinger and Central Bank Independence: Theory and EvidenceE. Schaling

9326 T.C. Tu Infant Industry Protection with Leaming-by-Doing

9327 J.P.1.F. Scheepens Bankruptcy Litigation and Optimal Debt Contracts

9328 T.C. To Tariffs, Rent Extraction and Manipulation of Competition

9329 F. de 1ong, T. Nijman A Comparison of the Cost of Trading French Shares on theand A. Rcell Paris Bourse and on SEAQ Intemational

9330 H. Huizinga The Welfare Effects of Individual Retirement Accounts

9331 H. Huizinga Time Preference and International Tax Competition

9332 V. Peltkamp, Linear Production with Transport of Products, Resources andA. Koster, TechnologyA. van den Nouweland,P. Bortn and S. Tijs

9333 B. Lauterbach and Panic Behavior and the Perforrnance of Circuit Breakers:U. Ben-Zion Empirical Evidence

9334 B. Melenberg and Semi-parametric Estimation of the Sample Selection ModelA. van Scest

9335 A.L. Bovenberg and Green Policies and Public Finance in a Small Open EconomyF. van der Plceg

9336 E. Schaling On the Economic Independence of the Central Bank and thePersistence of Inflation

No. Author(s)

9337 G.-J.Otten

9338 M. Gradstein

9339 W. Guth andH. Kliemt

9340 f.C.l"o

9341 A. Demirgup-Kuntand II. liuizinga

9342 G.J. Almekinders

9343 E.R. van Dam andW.H. Haemers

9344 H. Carlsson andS. Dasgupta

9345 F. van der Plceg andA.L. Bovenberg

9346 1.P.C. Blanc andR.D. van der Mei

9347 J.P.C. Blanc

Title

Characterizations of a Game Theoretical Cost AllocationMethod

Provision of Public Goods With Incomplete Information:Decentralization vs. Central Planning

Competition or Co-operation

Export 5ubsidies and Gligopoly with Switching Costs

Barriers to Portfolio Investments in Emerging Stock Markets

Theories on the Scope for Foreign Exchange Market Intervention

Eigenvalues and the Diameter of Graphs

Noise-Proof Equilibria in Signaling Games

Environmental Policy, Public Goods and the Marginal Costof Public Funds

The Power-series Algorithm Applied to Polling Systems witha Dormant Server

Performance Analysis and Optimi7ation with the Power-seriesAlgorithm

9348 R.M.W.J. Beetsma and Intramarginal Interventions, Bands and the Pattern of EMSF. van der Plceg Exchange Rate Distributions

9349 A. Simonovits

9350 R.C. Douven andJ.C. Engwerda

9351 F. Vella andM. Verbeek

9352 C. Meghir andG. Weber

9353 V. Feltkamp

9354 R.J. de Groof andM.A. van Tuijl

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Is There Room for Convergence in the E.C.?

Estimating and Interpreting Models with EndogenousTreatment Effects: The Relationship Between Competing Estimatorsof the Union Impact on Wages

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9355 Z. Yang A Simplicial Algorithm for Computing Robust Stationary Points ofa Continuous Function on the Unit Simplex

9356 E. van Damme and Commitment Robust Equilibria and Endogenous TimingS. Hurkens

9357 W. Guth and B. Pcleg On Ring Forrnation In Auctions

9358 V. Bhaskar Neutral Stability In Asymmetric Evolutionary Games

9359 F. Vella and Estimating and Testing Simultaneous Equation Panel Data ModelsM. Verheek with Censored F,ndogenous Variables

9360 W.B. van den Hout The Power-Series Algorithm Extended to thc BMAP~PHII Queueand J.P.C. Blanc

9361 R. Heuts and1. de Klein

9362 K.-E. Wámeryd

9363 P.J.-J. Herings

9364 P.J.-J. Herings

9365 F. van der Plceg andA. L. Bovenberg

9366 M. Pradhan

9367 H.G. Bloemen andA. Kapteyn

9368 M.R. Baye,D. Kovenock andC.G. de Vries

9369 T. van de Klundertand S. Smulders

9370 G. van der Laan andD. Talman

9371 S. Muto

9372 S. Muto

An (s,q) Inventory Model with Stochastic and Interrelated LeadTimes

A Closer Look at Economic Psychology

On the Connectedness of the Set of Constrained Equilibria

A Note on "Macroeconomic Policy in a Two-Party SystemRepeated Game"

as a

Direct Crowding Out, Optimal Taxation and Pollution Abatement

Sector Participation in Labour Supply Models: Preferences orRationing?

The Estimation of Utility Consistent Labor Supply Models byMeans of Simulated Scores

The Solution to the Tullock Rent-Seeking Game When R~ 2:Mixed-Strategy Equilibria and Mean Dissipation Rates

The Welfare Consequences of Different Regimes of OligopolisticCompetition in a Growing Economy with Firm-Specific Knowledge

Intersection Theorems on the Simplotope

Altemating-Move Preplays and vN - M Stable Sets in Two PersonS[rategic Form Games

Voters' Power in Indirect Voting Systems with Political Parties: theSquare Root Effect

No. Author(s)

9373 ti. Smuldurs andR. Gradus

9374 C. Fcmandez,J. Osiewalski andM.FJ. Stecl

9375 E. van Damme

937G I'.M. Kurt

9377 A. L. Bovenbergand F. van der Plceg

9378 F. Thuijsman,B. Peleg, M. Amitaiand A. Shmida

9379 A. Ixjour andH. Verbon

9380 C. Femandez,J. Osiewalski andM. Stcel

Title

I'ullulion Ahatcmcnt and Lung-tcrni Gruwth

Marginal f:quivalence in v-Spherical Models

Evolutionary Game 'Theory

Pullutiun Cbnlrol and the Dynamics of the Firni: the l:ffècts ofMarket Based Instruments on Oplimal Firm Investments

Optimal Taxation, Public Goods and Environmental Policy withInvoluntary Unemployment

Automata, Matching and Foraging Behavior of Bees

Capital Mobility and Social Insurance in an Integra[ed Market

The Continuous Multivariate Location-Scale Model Revisited: A Taleof Robustness

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