Tilburg University

Efficiency comparisons of maximum likelihood-based estimators in garch models

Gonzalez-Rivera, G.; Drost, F.C.

Publication date:1998

Link to publication in Tilburg University Research Portal

Citation for published version (APA):Gonzalez-Rivera, G., & Drost, F. C. (1998). Efficiency comparisons of maximum likelihood-based estimators ingarch models. (CentER Discussion Paper; Vol. 1998-124). Unknown Publisher.

General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.

• Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal

Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Download date: 17. Dec. 2021

CBM ~ ~R D1 C 1s uss on84141998 for

nr.124 ~mic Research a era~~ ~~ ~ R ~ ~~ ~.~:''!. ~ -,n -- i,1~L,

~ ~r

t

~ ~(`~ X t~11,~,c ~ ry` r.~ .! ~: ~.. r ~ r c~ ~' rv '

I'1NIIIIIIIIIqNIIIIIIIV I,uIIIN~h~~l'~luTilburg University

,,,

Centerfor

Economic Research

No. 98124

EFFICIENCY COMPARISONS OF MAXIMUMLIKELIHOOD-BASED ESTIMATORS IN

GARCIi MODELS

By Gloria González-Rivera and Feike C. Drost

November 1998

ISSN 0924-7815

EFFICIENCY COMPARISONS OF MAXIMUM LIKELIHOOD-BASEDESTIMATORS IN GARCH MODELS'

Gloria González-RiveraUniveraity oF California, Riverside

DeparWnent of EconomicsRiverside, CA 92521

e-mail ggonzale~galaxy.ucr.eduand

Feike C. DrastTilburg University

Depaz~tment of Econometrics and CentERP.O. Box 90153, 5000 LE Tilburg

The Netherlandse-mail F.C.Droat~kub.nl

Abstract

In this paper, we investigate the loas of asymptotic efSciency of aemiparametric and quasi-maximum likelihood eatimators rclative tu maximuin likelihood rxtimat4FS in models withGeneralized Autoreg~easive Conditional Heteroacedasticity (GARCH). For a general time-varying location-scale model, the factors that contribute to differencea in efficiency amongthe estimators can be divided in two categoriea. One pertaina to the parametric apecificartions of the conditional mean and the conditional variance. The other corresponda to theahape characteristics of the conditiona! density of the atandardized errora, aummarized inthe coefficients of skewneas sud kurtoais together with the Fiaher information for locationand acale. The quantification of these factot~e has practical implicationa since it can helpto decide if the more complex semiparametric eatimator provides aufficient e~iciency gainawith respect to the simplest quasi-maximum likelihood eetimator. We alao prove that thereie no probability density function, with the exception of the normal, for which the asymp-totic efficiency of the three estimatoi~s is the same. Particular modela are also considered,for which the efficiency compaz~isons are greatly simpli8ed.

JEL Classification: C13, C14, C32Key Words: Fisher Information Matrix, MLE, QMLE, Semiparamettic eatimation.

' We wish to thank to the participauts of the 1995 Joint Meeting of the American Statis-tical Aasociation in Orlando, FL, the 19J7 Economettic Society Meetinga, New Orleans, LA,and seminara at UCLA and UCR. We are specially thankful to the editor, Richard Blun-dell, the associate editor, and two anonymous refereea, for their constructive commente.The firat suthor gratefully acknowledgea the financial aupport of the Academic Senate ofthe University of California, Riveireide. The usual caveat appliea.

1 Introduction

In this paper, we compare the efficiency properties of maximum-likelihood-based estima-tora in the context of Generalized Autoregressive Conditionally Heteroecedastic (GARCH)models. Asymptotically, consiatency ia both a desirable and required property of an esti-mator, but the property of maximal efficiency, though deairable, is not always attainable.Asymptotic efficiency is a function of the level of information available to the researcher.

We develop a atrategy for evaluating efficiency gains when eatimating several typea ofGARCH and GARCH-in-mean models, but the methodology will be extended to other time-varying location-scale models. The estiinatora we consider aze all based on the likelihoodprinciple. A likelihood function is constructed based on an assumed conditional probabilitydensity function. Depending on the amount of information available, we can eatimate amodel with a maximum likelihood (ML) eatimator where the conditional probability densityfunction ia fully known, a semipazametric (SP) eatimator where the density is cretimated witha data-based procedure, or a quasi-maximum likelihood (QML) eatimator where conditionalnotmality is assumed, though this assumption is likely to be false. The efficiency gains aredirectly proportional to the amount of information available; hence, ML ia more efficientthan SP, whicó is more efBcient than QML estimation.

The compazison of the asymptotic variance-covaziance matrices of the ML, SP, andQML eatimatore reveal that, in the general time-varying location-scale model, differenceain e~ciency are the result of the interaction between the epecified model and the shapecharacteristica of the conditional probabílity denaity function. These interactions can bediaentangled when we focua on interesting apecific location-scale models. Relevant factorathat contribute to differences in efficiency among the three estimatora are the Fisher in-formation for location, the Fisher information for scale, the coeeïcient of kurtoais and thecoefficient of skewness of the conditional density. Our reaults have practical implications.The empirical researcher can assess the 'closenesa' of the SP estimator to the ML and QMLestimators. When comparing a QML to an SP estimator, there is a practical trade-off be-tween simplicity in implementation and potential gains in eeïciency. If the SP estimator is`closer' to the QML than to the ML, the potential efficiency gains would be weighed againatthe costa of implementing the more complex SP estimator.

We distinguish among five typea of models, depending on the relation between theconditional mean and the conditional variance. The most general model consiats of a time-varying conditional mean and a time-vazying conditional vaziance, where mean and varianceare mutually dependent upon each other. The GARCH-in-mean models are examples of thiaclass. This general model can be particularized to provide four interesting specific modela.First, we consider a pure time-varying location model with homoscedastic errora, where theparameters of interest aze only the mean parameters. In thia group, we may include dy-namic specifications as ARMA modele and general regression modela. The second model isa pure time-varying scale model, where the pazameters of intereat are the conditional vari-

1

ance parametera. In this group, we tnay include any type of conditional heteroacedasticityspecification, such as the classic ARCH and GARCH, Exponential GARCH, Non-linearGARCH, Asymmetric GARCH. The tltird group of modela conaieta of a conditional meanand a conditional variance apecification such that the conditional mesn does not depend onthe parameters of the conditional variance, and the conditional variance does not dependon the parametere of the conditional mean. In this group, we may include regresaion modelswith heteroscedasticity other than GARCH, such as multiplicative heteroacedasticity, andARMA type models, where the conditional vaziance ia a function of past observationa. Thefourth group conaista of a conditional variance that dependa on the full set of inean andvariance parametera, but the conditional mean doea not depend on the conditional varianceparametera. In this group, we reatrict the conditional vaziance to be a symmetric functionof the errors as in the classical ARCH and GARCH models. In the third and fourth groups,we also impoae symmetry of the probability density function of the errora.

This paper proceeds as follows: In Section 2, we describe the general time-varyinglocation-ecale model and the varioua estimation methodologies. In Section 3, we considerefficiency comparisons among ML, QML, and SP estimatora for the general location-model.In Section 4, we examine the efficiency comparisons for four apecific location-scale modela.9CCtian 5 eoncludes the paper.

2 Location-Scale Models and Estimation Methodology

2.1 The General Location-Scale Model

Conaider a discrete time atochastic process {yt} paratneterized by a finite parameter vectorB. Conditioning on available information up to time t-1, the random variablea yt have condi-tional mean mt(Bo) and conditional variance ht(Bo), where Bo denotea the true but unknownparameter. The functiona mt(B) and ht(8) may be function of paat information, includinglagged exogenoua variablea xt,xt-l,xt-z,..., and of the parameter vector B, i.e. mt(B) -m(xt,zt-t,xe-z,...,yt-t,yt-z,...te), ht(B) - hÍxn y:-t,xr-z,...,yt-i,Ue-z,...;6). Suchprocesses are known as regression models with Generalized Autoregreasive ConditionallyHeteroacedasticity (GARCH), possibly including GARCH-in-mean behavior. We assumethat the starting valuea of the process are taken from the stationary distribution and,hence, the procesa, itself, is assumed stationazy. This implies that the scorea will be sta-tionary too. Alternatively, if the ataz~ting values are observed, Drost and Klaassen (1997)show that replacing the true non-stationary acores by the corresponding atationary onea hasno influence asymptotically, see also Koul and Schick ( 1995) for general conditions. Ourmodel will be

yt - me(eo) t ht(eo)un (I)

2

where {ut} is an i.i.d. aequence with zero mean, unit variance, finite fourth moments,and an absolutely continuous probability density function (pdf) g with derivative g', suchthat the Fisher information for location and the Fisher information for scale are finite.In general, the pdf g can be defined by additional parametere, say q, that are conaiderednuisance parameters, but may contain relevant information for the estimation of the vectorof parametera of interest, B.

This paper ia concerned with the specification of the conditional density function g andwith the e6'icíency properties of maximum likelihood-based eatimatora of the parametervector B. We assume that the models for mt(B) and ht(B) are correctly apecified. We uaethree estimation methods, maximum likelihood (ML), quasi-maximum Iikelihood (QML),and aemiparametric (SP) estimation, depending on how much information we have availableregarding the pdf. The iasue of efficiency ia directly related to knowledge eurrounding g.

The assumption of i.i.d {ut} innovations may be too restrictive for parametric maxi-mum likelihood methods; in fact, a weaker asaumption as {ut} being a martingale differ-ence aequence aufficea to render the QML eatitnator asymptotically normal (Bollerslev andWooldridge, 1tJ92). The i.i.d. assumption facilitatea the proofa of consiatency and aeymp-totic normality. Moreover, most of the current parametric and aemiparametric literatureadopt auch an assumption. Therefore, we retain the i.i.d ae~umption in order to comparethe three eatimation methods on equal grounda.

2.2 Notation

Let ut(B) - (yt-mt(B))~ ht B) denote calculated residuals. Inserting the true but unknownvalue of the parameter Bo yielda ut(Be) - ut. We define the vector function ~- (rGt,1G,)',baeed upon the location-acale acorea, by

Ti~1(ue) --9~(ur)

9(ut)and

~a(ut) - - (1 f ut9~(ut)l` 9Íut) I

In the special case of a standard normal density the function 1[i reducea to

F(u) - ~ uyu I ~ .

Define the matrix W as

x, -~tiy ~ 1 ant(9) 1 8ht(9) jr - te W.t - J .ht(B a8 ' 2ht(B) c~B

where 1 and a stands for location and scale reapectively.Finally, we will use the notation GX,Y7- E(XY') and IIXII2-CX,X~- E(XX').

3

2.3 Estimation Methodologies

Ficat, we conaider the case for which the pdF g is fully known to the researcher and theobject of interest ia the eatimation of B. Maximum likelihood estimation producea opti-mal estimatora under a set of regularity conditiona. MLE eetimators are consistent andasymptotically efficient since they achieve the Cramér-Rao lower bound.

For a eample of length T, the averaged log-likelihood function ia given by

GT(B) --2T ~ l09 heÍB) -F. T~l09 9(ui(B)). (Z)~ ~

The ML eatimator ia found by maximizing equation (2) with reapect to the vector ofparametera B. The score function ia given by

STI(B) - aGT(B)C7B

1 ~- 1 8m~ ( B) 9~(ut(B))- - T L he(B) ~ 9(ue(B))

1 8h~(B) q'(ui(B))l lf Zhi(B) ~ ~1 t ue(B)9(u~(B)) I 1

1 ~ We~e,T i (3)

where ~~ ia used as short-hand notation for ~(ui). The ML eatimator 8,,,~ eolvea the systemof equationa ST~(B) - 0. Since thia syatem ia nonlinear in B, the aolution is obtained vianumerical techniquea. Note that the two factora in the acore function, h~(B)-~~~(8qsi(B)~8B)and ~hi(B)-t(8h~(B)~8B), depend aolely on past information, and they rely on the epeci-fication of the conditional mean and the conditional variance equatione. The other twofactora, g'~g and (1 f u~f ~g), are functiona of u~ and depend on the ahape of the pdf g. Itia easy to ahow that the terma in (3) form a martingale difference sequence. The expecta-tion of the acore ia zero for any pdf since integration by parta results in E(g'~g) - 0 andE(urg'~g) - -1.

Proving conaiatency and asymptotic normality of the ML eatimator for (G)ARCH pro-cesaea is a non-trivial exerciee. Basawa, Feigin, and Heyde ( 1976) provide a set of aufScientconditiona for conaistency and asymptotic normality of estimators for dependent proceesea.Results are only available under the asaumption of conditional normality and only for a lim-ited clasa oF procesaea, mainly GARCH(1,1) and ARCH(p) ( see Weisa ( 1986), Lumsdaine(1996), and Lee and Hansen ( 1994)). Under a correct apecification of the variance equationand of the pdf g, the ergodic theorem and a central liinit theorem can be invoked to ahowthat

~(Bml -Bo) - ~ N(D,VmI), (4)

4

where V,;,tl - Bát is the expectation of the outer product of the ecore evaluated at the trueparameter vector Bo and is given by

Bot-E f7,(8GT(Bo)1 (a~r(Bo)l~l.l` eB J` aB J J (5)

Under the assumption of a con~ectly-speci8ed model, the information matrix equality holds;that is, Bp t - Apt, where the matrix Ap t is (minus) the expectation of the Hessian matrixgiven by

Amt - -E r~Gr(BO) I .o ` BB~B, J(6)

The second methodology that we consider for the estimation of the parameter vectorB is quasi-maximum likelihood estimation. In this case, the researcher does not have anyknowledge of the pdi that characterizes the standardized innovations ui and chooses thenormal pdf. The quasi-maximum likelihood estimator is the argument that maximizes thelikelihood function under the assumption of conditional normality, even though this maybe a false assumption. The score function corresponding to a qusei-maximum likelihoodfunction is

sy.~j(B) - i 1 Ómt(9)uf(B)-T ~ - hi B) 8B

f 2h~(B)~h~ )( i -ur(B)z)~

- 1 ~ 17, WiFi, (7)i

with Fi - F(u~). The acore function ~mt (B) preserves the martingale difference property.It is easy to see that the expectation of the score is equal to zero because u~ is a etandardizedinnovation, for which E(ut) - 0 and E ( ui)- 1. This property holds for any g and is thebasis for proving consistency and asymptotic normality of the QML estimator under a setof regularity conditions. These conditions are discussed in Wooldridge ( 1995), Lee andHansen ( 1994), Lumsdaine ( 19~J6), and Weiss ( 1986). The limiting distribution of the QMLestimator is

~ (Bqmt - Bo) ~ N (~r Vqm!) , (g)

where Vy,,,t - Ap t19m~iBpmtAp tt9mtl and where A9mt and B9mt are (minus) the expectationof the Hessian and the expectation of the outer product of the acore respectively calculatedunder conditional normality. This estimator is less efficient than the ML estimator, reAectingthe lack of information about the pdf. The finite-sample properties of QML and e~ciencylosses with respect to ML have been studied in several Monte Carlo simulations by Engle and

5

González-Rivera (1991) and by Bollerslev and Wooldridge (1992). Newey and Steigerwald(1997) have shown that a qua.gi-maximum likelihood approach with t-distributions is alsofeasible if an additional parameter is added. Although the derivationa fot this approach donot differ easentially from the ones in ordinazy (normal) QML, we will not include the exactexpressiona for these estimatora.

The third methodology that we consider is the semiparametric estimation of the pa-rameter vector B. In this eituation, the researcher doea not know the pdf of the standard-ized innovationa but assumea that it is sufficiently smooth (Hájek and Sidák (19G7)) to beapproximated by a nonparametric denaity eatimator. Semiparametric ARCH models wereintroduced by Engle and González-Rivera (1991), and their asymptotic propertiea were stud-ied by Linton (1993), Steigerwald (1994), Drost and Klaassen (1997). The aemiparametricestimator ia a two-etep estimator. In the Srst atep, conaistent eatimatea of the pazameteraof interest are obtained through, for example, quasi-maximum likelihood eatimation andare used to conatruct a nonparametric density of the atandardized innovationa. The aecondstep conaista of using this nonparametric denaity to adapt the initial estimator by a one-stepNewton-Raphson improvement. The goal is to recapture the seymptotic efficiency lossesdue to quasi-maximum likelihood estimation, which can be aubstantial when the departureof the tnir. pdf from normality is large. On efficiency grounda, the semiparametric eati-mator is an intermediate eatimator between the unattainable maximum likelihood and thequasi-maximum likelihood estimatore. The eemiparametric eatimator ia tecmed adaptive ifit happena to have the same asymptotic efAciency as the maximum likelihood estimator.The semiparametric efficiency bound depends on how informative the nuisance parameters,q, of the density aze for the eatimation of the parametera of interest B. Let S(q) be thepopulation score vector for the nuisance parameters and S(B) be the population acore vectorfor the parametera of interest. The vector of parametera q ia unknown and conaequently thesemiparametric eatimator of B cannot exploit the information contained in n. If q containaany information about B, the efHcient score for B is found by calculating the residual vectorR(B) from the projection of S(B) on the closure of the set of all lineaz combinationa ofS(rl), called the tangent set T. The tangent aet conaists of lineaa combinationa of S(n) and,becauae the u's are random variablea with mean zero and variance one, the elementa of thetangent set are orthogonal to the function vector (ui,ui - 1)'. Through the projection, allthe variation of S(B) due to S(q) ia removed (Newey (1990)). The residual vector R(B) iethe difference between S(B) and the projection and, by conatruction, ia orthogonal to this.Hence, R(B) is the eH'icient acore for B and the aemiparametric efficiency bound ia

~Vsy - ~T ~ E [Rt(eo)Rt(Bo)~~~- . (9)

We consider two cases: (i) the density g of the errora is completely unknown and (ii)the denaity g ie known to be symmetric. In (ii) the tangent aet contains only aymmetric

B

functions. In the rest of this article, expectations are implicitly taken under Bp and g.In the most general case (i), for a sample of length T, the sample average vector residual

is

RT(B) 1 ~ Rt(B)T tSTr(B)

E(Wrt) X ~, ~{~t~r(ut)- Gt~r. F~IIFII-2 F(ut)1 ~t

E(W,t) x T~{[~G,(ut)- G~e,F~IIFII-2 F(ut)~ }t

- STr(e)

- E(Wt) X T~{[~G(ut)- G~,F~IIFII-2 F(ut)]}. (IO)t

where STr(B) is given in ( 3). The derivation of equation ( 10) can be found along the linesin Bickel et al. (1993), see Drost et al. (1997) for the present time aeries set up. Note thatit is yuite easy ta verify that R~ (0) is indeed the effieient aeore. In the first plaee, R~(B)is orthogonal to the tangent set T and, secondly, the difference between STr(9) and RT(6)belongs to this tangent space.

In the case ( ii) where the error densities are symmetric, the tangent apace T consists ofsums of all symmetric functions orthogonal to F(ut). In this case, the difference STr(B) -RT(B) needs to be a symmetric function. We need to remove the non-symmetric residualof the projection of ~t onto F. Hence, in the symmetric case the sample average vectorresidual is given by

1~rvm (8) - ST`t (B)

- E(Wt) r 0`0 ~~ x T~{[~G(ut)- G~, F~IIFII-Z F(ut)] }. (II)

r

The conditions to show that R7.y~`(B) is the required efhcient scare are easily verified.

3 Efficiency Comparisons Among ML,QML,SP Estimators

In this section we present the results concerning the moat general time-varying location-scale model, and in the next section we particularize them for specific models. To simplifythe exposition and because we work with atationary acorea, we refer to one specific elementof the acore auch that Wt becomes W, Wjt becomes Wj, W,t becomes W„ and so on.

7

Observe that the expectations E(~F') and E(FF') can be explicitly calculated:

K-~~G, F7- ~ p 2~, L -IIFII2- I~ K~ 1~,

where ~- E(u~) and n. - E(u4). Furthermore `

M -11~GII2ie the Fiaher-information in the location acale model. Note that M- KL-~K ia poaitiveaemidefinite, aince

M - KL-~K -11~G - KL-IF'll2 .

We introduce some additional notation before stating our main resulta. Define

n-11WII2, A-E(W)E(W)~, E-II-A.

Moreover, define for some arbitrary symmetric positive semideSnite matrix A,

nA -11WA1~2112, AA - E(W)A E(W)~, ~A -~A - AA.

By conatruction these matrices are positive semidefinite.To facilitate the comparison of asyinptotic variance-covariance matricea we work with

the inverse of theae matricea, V,~~ , V9,~1, and V;pl. We focus in the absolute losaea ( gains).The relative losses are atraiglitforward to derive from the abaolute ones. Flirthermore,the construction ofthe asymptotic variance-covariance matrices relies on certain regularityconditions such as tliose in Bollerslev and Wooldridge ( 1992). Essentially, these conditionarequire the satisfaction of uniforcn weak lawa of large numbers and uniformly positive def-initeness for (minus) the expectation of the liessian, as well as for the expectation of theouter product of the score.

Uaing the acore function ( 3), and equationa (4) aad (5), for the maximum likelihoodeatimator we can write

vml -11W~~I~- ~M. (12)

For the quasi maximum likelihood estimator, uaing (7) and (8), we obtain

Vyml -~~CW~,WF)~IWF~I-2 WF 112- IIh-II~IIIx. (13)

For general error distributiona, using ( 9) and ( 10), the semiparametric information boundcan be written as

V;PI -~I W~G - E(W)~~G - KL-I F] 112- Eal-xt-~x f na~t-~x. (14)

8

The available information increases if the error diatributions are known to be aymmetric.The efficient score ia given in (11). Thus, the aemiparametric information bound ia

Vap,iym - ~~ W~Y - E(W )[tG - KL'~F] i" E(W)Z[~G - KL-~~ ~~2- ~M-AL-~A"-~IIh"L-~tii"AZ(M-AL-~K]Z, (15)

where Z is the indicator matrix Z - [1 0]~[1 0).Comparisona of expresaions ( 12), (13), and (14) yield the following reaults.

Result 1. For a general error distribution, V;pi - Vqm~ ia a positive aemidefinite matrix.

PROOF: Compaz~ing expressiona (13) and (14),

1 1 IVep - Vqm! -~M-AL-~I~" f inKL-~h- - IItiIIL~IIh'}. (1~)

To ahow that the aecond term on the right-hand eide is positive semideSnite, observe that

II~L-~~; - II~,'II~IIIh' -11WKL-1F- IIh'II~1WF~~2 .

This completea the proof. O

Reault 2. For a general error distribution, V,~~l - V;p1 ia a poaitive semidefinite matrix.

PROOF: Comparing expresaiona (12) and (14),

Vm~l -Vsp~ - AM-tiL-~K~ ~ (17)

The following Result 3 can be obtained directly from Resulta 1 and 2. For tranaparencyreaeons, we alao include a direct proof.

Result 3. For a general error distribution, Vm~ - Vq,~~ is a poaitive semidefinite matrix.

PROOF: Using the expressions for the asymptotic information matricea (12) and (13),we obtain by atraightforward calculations

t ~Vm~ -Vqm~-nM-F"L-~F f{IIF-L-~t; -IIhII~'IIF'}. (18)

Both terma on the right-hand aide are poaitive aemidefinite. ~

If the error probability density ia known to be symmetric, the efftcient score in theaemiparametric location-scale context ie sliglitly different from the acore for general errordistributions. Resulta 1 and 2 have to be modified to reflect the additional information.

H

We need to include an additional term Az[al-tiL-~ti]z. The detaile are left to the reader.Observe, however, that the aymmetric case is not a aubcase of Reaulta 1 and 2. Knowingthat the densities are symmetric yields essentially different scorea.

The joint implication of Results 1, 2, and 3 ia that, with the exception of the nor-mal density, there is not any other probability density function for which the asymptoticvariance-covariance matrices of the ML, QML, and SP estimators are equal. This is sum-marized in the following result.

Iiesult 4. For the general time-varying location-scale model, and asauming that i)E(W) has full rank, ii) dc E R~ (W - E(W))c ~ 0, then

vmll - vsp~ - Vyml (19)

if and only if the probability density function g(.) is normal. Furthermore, equality betweenany two variance-covariance inatrices impliea the equality of the three matrices.

PROOP: To prove sufficiency ia straightforward. If the density is normal, then K- L- M,and (19) followa. To prove necessity, consider the following. If (17) is equal to zero, and i)holds, then M- KL-~K. If (16) ia equal to zero and iij holds, then M- KL-~K andn~[G-~K - Rtin~~n~;. If (18) is equal to zero, then (16) and (17) are equal to zero. Theequality M- KL-~K yielda a pair of diffetential equationa for which the only solutionis the normal density. To find the aolution to this ayetem, proceed as in González-Rivera(1997). O

4 SpeciBc Models

In thia section, we discuss four interesting submodela of the general time-varying location-scale model presented in Section 3. Depending upon the relation between the parameters inthe mean and the parametens in the variance, we can have: i) A pure time-varying locationmodel. The parametere of interest are only location parameters. In thie group, we mayinclude dynamic apecificatione as ARMA models and general regreasion models. ii) A puretime-varying acale model. There is no conditional mean and the parametera of interestare only the variance parametera. In thia group we may include any type of conditionalheteroacedasticity specification, auch as the claseic ARCH (Engle (1982)) and GARCH(Bollcrslev (1986)), Exponential GARCH (Nelson (1991)), Non-linear GARCH (González-Rivera (1998)), Asymmetric GARCH (Ding et al. (1993)). iii) Block-diagonal models,where the parameter vector can be split in two groups. One group containa the parametersin the conditional mean, and the other contains the parametera in the conditional variance,auch that the conditional variance does not depend upon the parameters in the conditionalmean and the conditional mean doea not depend upon the parameters in the conditional

10

variance. In this group, we may include regression modela with heteroscedasticity otherthan GARCH, such as multiplicative heteroscedasticity, and ARMA type models, wherethe conditional variance is a function of past observationa. iv) Block-triangular models,where the conditional variance depends upon the full aet of inean and variance parametersbut the conditional mean does not depend upon the conditional variance parameters. Inthis group, we may include regression models and ARMA type models with conditionalheteroacedasticity, but we require the ARCH or GARCH pmcesa to be aymmetric as deSnedin Engle (1982), as well as symmetry of the pdf of the errora. For iii) and iv), we presentthe aimplified expressions only for aymmetric error diatributiona. In case of general errordiatributiona, the reaulting formulas are not esaentially aimpler than the general onea inSection 3.

4.1 Location models

In the clasa of pure location models, Results 1, 2, and 3 of Section 3 are greatly aimplified.This ia due to the fact that we do not have to differentiate the conditional variance withrespect to the parameter of interest. Thus the matrix W consiats of only one column,W~ -[Wj]. The mstrlcea II, A, and E contain excluaively location information. Thefunction vector F aimplifiea to F- u becauae the variance of the error distribution ia notreatricted to one in the location case. The matricea K, L, and M are reduced to numberak- 1, l- Eu2 - 02, and m- f(gt~g)2g. This impliea e.g. that expressiona like II,,,may be written as mII. Under a general error distribution, Resulta 1, 2, and 3 for the purelocation model are

Yspl - Yqml - [Tn - U-2]F.,

Ymtt - Vap~ - [m - ~-2]A~

Y,~rl - vy,~t - [m - o'ZjII.

Under the asaumption A~ 0 and E~ 0, the three estimators have the same asymptoticdistribution if and only if m- 0-2. Thia condition ia only satisfied by the class of normaldistributions. The case of E- 0 ie rather exceptional. It only happena when the derivativeof the conditional mean with respect to the mean parameters ia nonrandom, i.e. the i.i.d.location model. In this inatance, the aemiparametric estimator is as efficient as the quasimaximum likelihood estimator. The equality A- 0 can happen in several modele suchaa ARMA models without a conatant term and regresaion modela where the regreasorshave zero expectation. In theae madela, the aemiparametric eatimator is as efficient asthe maximum likelihood estimator for all error diatributiona. This property is known asadaptivity of the mean parametei~s.

If the error distribution ia restricted to the symmetric clasa of densitiea, we always haveadaptivity of the location parametera, independently of any restriction on the matrix A.

11

Resulta 1, 2, and 3 for symmetric densitiea aze

Vsyl - Vqmf - Ifn - O-2]II,

Vm~~ - V,P~ - O,

V,~~l - Vy,n~ - ( m - o-2]II.

4.2 Scale models

Reaulta I, 2, and 3 for the pure tiine-varying acale models aze alao greatly aimplified. Thematrix W consiate of only one column, Wi -~W,]. In contrast to the location model,the vector function F remaina unchanged becauae tlie error distribution haa two momentreatrictions. The matricea K and M reduce to the row-vector k-(0 2) and to the realnumber m - f(1 f ug~~g)zg respectively. The matrix L ia not affected. For the quaei-maximum likelihood eatimator, we find that ~m~ -~Bóm~` and that its asymptoticvariance covariance matrix simplifies to V9,~~ -~ll. For general ermr diatributiona,Resulta 1, 2, and 3 for the pure time-varying acale model are

Vsp~ - Vqml - Im -. 4 2]E f 4~Z y n,ti-I-~ (K-I)(K-I-( )~ 1 4

Vm~ - V'a - ~m - K - 1 - ~Z]A.

Vm! - Vqmf - Im - K4I]II.

Similaz to the pure location model, E- 0 ia only poasible in the i.i.d. scale model. Inthe pure acale models, A~ 0 happens in all practical econometric aituationa. Under theasaumption E~ 0 and A~ 0, adaptivity ia not possible in the ecale model with general errordiatributiona. However, V,;,~~ - V;P ~ if and only if the following condition holda m-~-~~C'González-Rivera ( 1997) has shown that thia condition is satisfied by a clasa of aymmetrized(~ - 0) square root chi-aquared distributions ( among which the normal density is a specialcase) and for a class of nonsymmetric distributions wíth ~~ 0.

If ~- 0, V,~,il - V;P1 - V9m~ for the set of distributiona deacribed in González-Rivera(1997). If ~~ 0, the aemipazametric estimator is always more efficient than the quasi-maximum likelihood eatimator. In other words, there is no density for which the asymptoticdiatributions of the three eatimators are the aame.

Note that adding a symmetry condition to the set of error diatributiona doea not increasethe information of the aemiparametric estimator in the pure ecale model. Consequently, theprevioua conclusiona remain for the symmetric case.

~a

4.3 Block-diagonal models

In these models, the parazneter vector is partitioned in two subsets, mean and vazianceparameters, such that the conditional mean doea not depend on the variance parametersand the conditional variance does not depend on the mean parameters. The matrix W ispartitioned as followa

W- ~ olm Wë J,where the superindexes 'm' and 'v' account for 'mean' and 'variance' respectively. Werestrict our attention to symmetric error distributions. Denote the diagonal elements of thediagonal matrix M by ml and m, (the off-diagonal elements are zero because of symmetry)and let the block-diagonal matrix II have a upper-left block III and a lower-right block II,.Results 1, 2, and 3 for block-diagonal models with symmetric error distributione are

V-t - V-t ~ntl - 1]III 0sp Qml - ~ ~m 4 E~-~J s

Vml -Vyt - 0 00 ~m, - ~]A, ) '

V-t - V9 t ~ml - 1JIIl 0mf ml - 0 Ims - K4IJ~s ~.

Note that the location parametet~s can be adaptively estimated, just as in the pure locationproblem with symmetric error distributions. Adaptivity of the ecale parameters will nothold. The asymptotic efficiency of the three estimators will be identical, i.e. V,~j - V;yt -Vq„~l, if and only if ml - 1 and m, - 4~(K - 1). These two conditions are joinUy satisfiedonly by the norma! distribution.

4.4 Block-triangular models

In these models, the conditional variance depends on the parameters of the conditional meanbut the conditional mean does not depend on the paz~ameters of the conditional variance.We restrict our attention to symmetric densities. The matrix W is of the following form

w-f Wlm W~ J .U W;

Fltrthermore, if the conditional variance is a symmetric process in the innovations of themean model, it can be shown that E(W; W;~) - 0(Theorem 4 in Engle, 1982). Exam-ples where this orthogonality condition is satisfied are the classical symmetric ARCH and

13

GARCH modela. In these casea, the matricea IIF and IIL are block-diagonal, but the equal-ity IIh~-~h - IIti.II~IIIh. - 0 doea not hold anymore. Neverthelesa, the lower-right blockof this matrix ia zero, and only for the acáe parametera do we obtain the same comparisonsas thoae of the block-diagonal models, i.e.

, -~ ~VsP~~~ - V9ml~~~ - m, -

V1~,~ - V,P~;I - ~ms -

i i ~Vml~~~ - V9ml~,~ - ms -

4.5 Numerical Efficiency Losses

For the particular modela of the previoua sections we calculate the relative el~ciency losa ofthe QML eatimator with reapect to the ML eatimator.

We consider a set of atandardized probability density functiona for which we compute thecoefficienta of akewnees and kurtosis, and the Fisher information of location and scale. For aataadardized Student-t with v degreea of freedom, we have that ~- 0, rc - 3(v-2)~(v-4),ml - v(v f 1)~((v - 2)(v -~ 3)), and m, - 2v~(v f 3). For a standardized Chi-square withv degrees of freedom, we óave that (- 2 2 v, ~: - 3(v f 4)~v, ml - v~(v - 4), andm, - 2v~(v - 4). With these expresaions, the efficiency lose of the QML eatimator withreapect to the ML estimator ia atraightforward to compute. In the following table we ahowsome examplea where the efficiency losa is quantified for the above mentioned probabilitydensity functiona.

Tàble IStandardized Shape Characteristics Efficiency Losa

DensitY (V ml.V'~ - 1)9bml m, ~ K Mean Par. Variance Par.

Normal 1 2 0 3 0 0Student-tv- 5 1.25 1.25 0 9.00 25 150v- 8 1.09 1.45 0 4.50 9 27v- 12 1.04 1.60 0 3.75 4 10Laplace 2 1 0 6 100 25Chi-Squarev- 10 1.67 3.33 0.89 4.20 67 167v- 15 1.36 2.73 0.73 3.80 36 91v- 20 1.25 2.50 0.63 3.60 25 63v- 30 1.15 2.31 0.52 3.40 15 38

14

With the exception of the Laplace diatribution, it can be aeen that the relative efHciencylosa ie larger for the variance parametera than for the mean parametera. Conaequently, theimplementation of a semiparametric eatimator has a lazger pay-off in thoae inatancea inwhich the variance pazameters are the parametera of intereat.

5 Conclusions

In thia paper we have quantified the asymptotic efficiency loasea (gains) of the ML, QML,and SP eatimatore in the context of GARCH modela. We have obtained a set of reaultsfor a general time-varying location-scale model. The factora that contribute to differenceain efficiency among the eatimatora can be divided in two categoriea. One pertaina to theparametric specificationa of the conditional mean and the conditional variance. The othercorreaponda to the ahape characteriatica of the conditional denaity of the atandazdized errors,summarized in the coefficient of skewneas and the coefficient of kurtoais together with theFiaher information for location and acale. We have proven that there ia no probabilitydensity function, with the exception of the normal, for which the asymptotic efficiency ofthe three eatimatora ie the same.

Out of the genernl location-ecale modrl, we have extiaCted four particulaz models. In apure time-varying location model, the coefficienta of akewnese and kurtoais, and the Fiaherinformation for acale do not play any role in explaining efficiency differencea. In a puretime-varying scale model, there is no need for the Fisher information for location, but theccefficient of akewnesa ia important in explaining differencea between the SP and QMLeatimatora, and between the SP and the ML eatimatora. Surpriaingly, however, akewnesais irrelevant in determining efficiency differencea between the ML and QML eatimatora. Inthe pure acale modela with akewnesa equal to zero, the three eatimatora can have equalaeymptotic efficiency for other denaitiea than the normal. Apart from pure location andpure acale modela, we have conaidered two more caaea, block-diagonal modele and block-triangular models, with symmetric denaity functiona. In the block-diagona) models, theasymptotic variance-covariance matrices are block diagonal between the mean and varianceparametera. Eeaentially, in these models, the efficiency comparisona reduce to those ofthe pure location and pure acale modela together. In the block-triangular modela, theasymptotic variance-covariance matrices are atill block diagonal between mean and varianceparametera, but it is only for the acale parametera where the efficiency compariaona reduceto those of the pure acale models.

These results have practical implicationa for the empirical reaearcher. A potential atrat-egy may be to etart the estimation procesa with a QML methodology. To recapture theefficiency loasea of the QML eatimator, we need to evaluate the matrix M, the cceE6cient ofskewness and the coefficient of kurtoais of the atandardized residuals. The matrix M can beeatimated by non-parametric methoda. The matrices II and A are already eatimated in the

15

QML estimation. Straightforward application of Results 1, 2, and 3 providea the efficiencyloss. Thie impliee that even with the most inefficient eatimator euch ae QML, the reeearchercan estimate the maximal ef6ciency bound provided by the unattainable ML estimator.

18

Bibliography

Basawa, I.V., P.D. Feigin, and C.C. Heyde. 1976. Asymptotic properties of maximumlikelihood estimators for stochastic processes. Sankhya: The Indian Journal of Statietica38:259-270.

Bickel, P.J., C.A.J. Klaassen, Y. Ritov, and J.A. Wellner. 1993. Ejj'icient and AdnptiveStatistica! Injerence for Semipammetric Modele. Johns Hopkina University Press.

Bollerslev, T. 1986. Generalized sutoregressive conditional heteroscedasticity. Journal ojEconometrica 31:307-327.

Bollerslev, T., and J.M. Wooldridge. 1992. Quasi-maximum likelihood estimation and infer-ence in dynamic models with time varying covariancea. Econometric Reviewa 11:143-72.

Ding, Z., R. F. Eng1e, and C.W.J. Granger. 1993. A long memory property of stock marketreturns and a new model. Journol of Empirica! Finnnce 1:83-106.

Drost, F.C., and C.A.J. Klaassen. 1997. Efficient estimation in semiparametric GARCHmodels. Journal of Econometrica 81:193-221.

Drost, F.C., C.A.J. Klaassen, and B.J.M. Werker. 1997. Adaptive estimation in time seriesmodels. The Annals oJ Statiatica 25:786-817.

Engle, R.F., and G. González-Rivera. 1991. Semiparametric ARCH models. Journal ojBueineas and Economic Statiatica 9:345-360.

Engle, R.F. 1982. Autoregressive conditional heteroscedasticity with estimates of the vari-ance of u.k. inAation. Econometrica 50:987-1008.

González-Rivera, G. 1997. A note on adaptation in GARCH models. Econometric Reviewa16:55~8.

González-Rivera, G. 1998. Smooth transitian GARCH modela. Studiea in Nonlinear Dy-namica and Econometrica vo1.3.2. forthcoming 1998.

Hájek, J., and Z. ~idák. 1967. Theory oj RaraR Teata. Academia, Prague.

Koul, H.L., and A. Shick. 1995. Efficient estimation in non-linear time series models. Work-ing Paper. Michigan State University and State University of New York, Binghamton.

Lee, S.W., and B.E. Hansen. 1994. Asymptotic theory for the GARCH(1,1) quasi-maximumlikelihood estimator. Econometric Theory 10:29-52.

Linton, O. 1993. Adaptive estimation in ARCH modela. Econometric Theorb 9:539-569.

17

Lumadaine, R.L. 1996. Consiatency and aaymptotic normality of the quasi-maximumlikelihood estimator in IGARCH(1,1) and covariance atationary GARCH(1,1) models.Econometrica 64:575-596.

Nelson, D.B. 1991. Conditional heteroskedasticity in asaet returna: A new approach.Econometricn 59:307-346.

Newey, W.K. 1990. Semiparametric efficiency bounda. Journa! oj Applied Econometrica5:99-135.

Newey, W.K., and D.G. Steigerwald. 1997. Asymptotic bias for quasi-maximum-likelihoodeatimatora in conditional heteroekedasticity modela. Econometrica 65:587-599.

Steigerwald, D.G. 1994. Efficient estimation of modela with conditional óeteroacedasticity.Working Paper. Department of Economics. Univeraity of California, Santa Barbara.

Weiss, A.A. 1986. Asymptotic theory for ARCH modela: Eetimation and testing. Econo-metric Theory 2:107-131.

Wooldridge, J.M. 1995. Estimation and inference for dependent processes. Handóook ojEconometrica 4.

18

No. Author(s) Title

9840 P. Bolton and X. Freixas A Dilution Cost Approach to Financial Intermediation andSecurities Markets

9841 A. Rustichini Minimizing Regret: The General Case

9842 1. Boone

9843 H.L.F. de Groot

9844 U. Gneery, W. Guth andF. Verboven

984~ A. Prat

9846 P. Borm and H. Hamers

9847 A.J. Hoogstrate and T. Osang

Competitive Pressure, Selection and Investments inDevelopment and Fundamental Research

Macroeconomic Conse~uences of Outsourcing. An Analysis ofGrowth, Welfare, and Product Variety

Presents or Investrna~ts? An Experimental Analysis

How Homogeneous Should a Team Be?

A Note on Games Corresponding to Sequencing Situations withDue Dates

Saving, Openness, and Growth

9848 H. Degryse and A. Irmen On the Incentives to Provide Fuel-Efficient Automobiles

9849 l. Bouckaert and H. Degryse

9850 J.R. ter Horst, Th. E. Nijmanand F.A. de Roon

9851 ].R. ter Horst, Th. E. Nijmanand F.A. de Roon

9852 F. Klaassen

9853 F.J.G.M. Klaassen andI.R. Magnus

9854 J. de Haan, F. Amtenbrinkand S.C.W. Eijffinger

9855 J.R, ter Horst, Th.E. Nijmanand M. Verbeek

9856 G.J. van den Berg, B. van derKlaauw and 1.C. van Ours

9857 U. Gneery and A. Rustichini

9858 C. Fershtman

9859 M. Kaneko

Price Competition Between an Expert and a Non-Expert

Style Analysis and Perforniance Evaluation of Dutch MutuelFunds

Performance Analysis of International Mutual Fundslncorporaáng Market FriMions

Improving GARCH Volatility Forecasts

On the I~epa~dence and Identical Dístribution of Points inTennis

Accountabiliry of Central Banks: Aspects and Quantification

Eliminating Biases in Evaluating Mutual Fund Performartcefrom a Survivorship Free Sample

Punitive Sanctions and the Transition Rate from Welfare toWork

Pay Enough-Or pon't Pay at All

A Note on Multi-Issue Two-Sided Bargaining: BilateralProcedures

Evolution of Thoughts: Dedudive Game TiKOries in theLiductive Game Situation. Part I

No. Author(s) Title

9860 M. Kaneko Evolution of Thoughts: Deductive Game Theories in theInductive Game Situation. Part II

9861 H. Huizinga and S.B. Nielsen

9862 M. Voomeveld and A. van denNouweland

9863 E.W. van Luijk and 1.C. vanOurs

9864 B.G.C. Dellaert and B.E. Kahn

9865 E.W. van Luijk and J.C. vanOurs

9866 G. van der Laan and R. vanden Brink

Is Coordination of Fiscal Deficits Necessary?

Cooperative Multicriteria Garnes with Public and PrivateCriteria; An Investigation of Core Concepts

On the Deterrninants of Opium Consumption; An EmpiricalAnalysis of Historical Data

How Tolerable is Delay? Consumers' Evaluations of InterndWeb Sites after Waiting

How Govemment Policy Affects the Consumption of HardDrugs: The Case of Opium in lava, 1873-1907

A Banzhaf Share Function for Cooperative Games in CoalitionStructure

9867 G. Kirchsteiger, M. Niederle The Endogenous Evolution of Markd Institutions Anand J. Potters Experimental Investigation

9868 E. van Damme and S. Hurkens Endogenous Price Leadership

9869 R. Piders and L. Warlop Visual Attention During Brand Choice: The Impact of TimePressure and Task Motivation

9870 I.P.C. Kleijnen and Short-Term Robustness of Production Management SystemsE.G.A. Gaury

9871 U. Hege Bank Dept and Publicly Traded Debt in Repeated Oligopolies

9872 L. Brcersma and 1.C. van Ours Job Searchers, Job Matches and the Elasticity of Matching

9873 M. Burda, W. Guth, Employment Duration and Resistance to Wage Reductions:G. Kirchsteiger and H. Uhlig Experimental Evidence

9874 J. Fidrmuc and J. Horváth Stability of Mondary Unions: Lessons from the Break-up ofCzechoslovakia

9875 P. Borm, D. Vem~eulen The Structure of the Sd of Equilibria for Two Person Muhi-and M. Voomeveld criteria Games

9876 l. Timmer, P. Borm and J. Suijs Linear Transformation of Products: Games and Economies

9877 T. Lensberg and E. van der A Cross-Cultural Study of Reciprocity, Trust and AltruísmHeijden in a Gift Exchange Experiment

9878 S.R. Mohan and A.J.J. Talman Refinement of Solutions to the Linear ComplementarityProblem

9879 l.l. Inman and M. Zeelenberg "Wow, I Could've Had a V8!": The Role of Regret in

No. Author(s) Title

Consumer Choice

9880 A. Konovalov Core Equivalence in Economies with Satiation

9881 R.M.W.J. Beetsma and The Optimality of a Monetary Union without a Fiscal UnionA.L. Bovenberg

9882 A. de Jong and R. van Dijk Detemtinants of Leverage and Agency Problems

9883 A. de Jong and C. Veld An Empirical Analysis of Incremental Capital StructureDecisions Under Managerial Entrenchment

9884 S. Schalk A Model Distinguishing Production and Consumption Bundles

9885 S. Eijffinger, E. Schaling and The Term StrucNre of Interest Rates and Inflation ForecastW. Verhagen Targeting

9886 E. Droste and J. Tuinstra Evolutionary Selection of Behavioral Rules in a CounwtModeL A Local Bifurcation Analysis

9887 U. Glunk and C.P.M. Wilderom High Performance on Multiple Domains: Operationalizing theStakeholder Approach to Evaluate Organizations

9888 B. van der Genugten A Weakened Form of Fictituous Play in Two-Person Zero-SumGames

9889 A.S. Kalwij Household Wealth, Female Labor Force Participation andFertility Decisions

9890 T. Leers, L. Meijdam and Ageing and Pension Reform in a Small Open Economy: theH. Verbon Role of Savings Incentives

9891 R.T. Frambach, J. Prabhu The [nfluence of Business Strategy on Market Orientation andand T.M.M. Verhallen New Product Activity

9892 H. Houba and G. van Lomwel Counter Intuitive Results in a Simple Model of WageNegotiations

9893 T.H.A. Bijmolt and R.G.M. GeneraliTations in Marketing Using Meta-Analysis withPieters Multiple Measurements

9894 E. van Damme and Evolution with Mutations Driven by Control CostsJ.W. Weibull

9895 A. Prat and A. Rustichini Sequential Common Agency

9896 J.H. Abbring, G.J. van den Berg Displaced Workers in the United States and the NetherlandsP.A. Gautier, A.G.C. vanLomwel and J.C. van Ours

9897 G J van den Berg, A.G.C. van Unemplovment Dynamics and AgeLomwel and 1.C. van Ours

No. Author(s)

9898 1. Fidrmuc

9899 R. Pieters, H. Baumgartner,J. Vermunt and T. Bijmolt

98100 A.L. Bovenberg andB.J. Heijdra

98101 F. Verboven

98102 O.J. Boxma, J.W. Cohen andQ.~B

Title

Political Support for Reforms: Economics of Voting inTransition Countries

importance, Cohesion, and Structural Equivalence in theEvolving Citation Network of the International Joumal ofResearch in Marketing

Environmental Abatement and Intergenerational Distribution

Gasoline or Diesel? Inferring Implicit Interest Rates fromAggregate Automobile Purchasing Data

Heavy-Traffic Analysis of the M~G~I Queue with PriorityClasse.c

98103 S.C.W. Eijffinger, A Theory of Central Bank AccountabilityM. Hoeberichts and E. Schaling

98104 G.1. van den Berg,P.A. Gautier, J.C. van Oursand G. Ridder

98105 Th. ten Raa and P. Mohnen

98106 M.P. Montero Garcia

98107 F. Palomino and A. Prat

98108 F. Palomino and A. Prat

98109 M. Wedel and T.H.A. Bijmolt

98110 A. Rustichini

981 I 1 E. Droste, M. Kosfeld andM. Voomeveld

98112 J.C. Engwerda

98113 ].C. Engwerda, B. van Aarleand J.E.J. Plasmans

Worker Turnover at the Firm Level and Crowding t~ut ofLower Educated Workers

Sources of Productivity Growth: Technology, Terms of Trade,ar~d Preference Shifts

A Bargaining Game with Coalition Formation

Dynamic [ncentives in the Money Management Toumament

Risk Taking and Optimal Contracts for Money Managers

Mixed Tree and Spatial Representation of DissimilarityJudgments

Sophisticated Players and Sophisticated Agents

A Myopic Adjustment Process leading to Best-Reply Matching

On the Scalar Feedback Nash Equilibria in the Infinite HorizonLQ-Game

Fiscal Policy Interadion in the EMU

98114 K.J.M. Huisman and P.M. Kort Strategic [nvestment in Technological Innovations

981 I S A. Cukierman and Y. Spiegel

98l l6 A. Cukierman and F. Lippi

When Do Representative and Direct Democracies Lead toSimilar Policy Choices?

Central Bank Independence, Centrali7ation of WageBargaining, Inflation and Unemployment -Theory ard Some

No. Author(s) Title

Evidcnce

9R I 17 E.G.A. Gaury, I.P.C. Kleijnen Customized Pull Systems for Single-Product Flow Linesand H. Pierreval

981 18 P.1.1. Herings, G. van der Laan Price-Quantiry Adjustment in a Keynesian Economyand D. Talman

981 19 R. Nahuis The Dynamics ofa General Purpose Technology in a Researchand Assimitation Model

98120 C. Dustmann and A. van Soest Language Fluency and Eamings: Estimation with MisclassifiedLanguage Indicators

98121 C.P.M. Wilderom and P.T. van A Test of the Leadership-Culture-Perfortnance Model Withinden Berg a Large, Dutch Fimtncial Organization

98122 M. Koster Multi-Service Serial Cost Sharing: An Incompatibility withSmoothness

98123 A. Prat Campaign Spending with Office-Seeking Politicians, RationalVoters, and Multiple Lobbies

98124 G. Gorvález-Rivera and F.C. Efficiency Comparisons of Maximum Likelihood-BascdDrost Estimators in GARCH Models

I I I I IÍ~Í I N INÍI IIÍ~NI IÍNIN II