Robust inference with GMM estimatorsdirectory.umm.ac.id/Data Elmu/jurnal/J-a/Journal of...

Journal of Econometrics 101 (2001) 37}69

Robust inference with GMM estimators

Elvezio Ronchetti!, Fabio Trojani",*!Department of Econometrics, University of Geneva, Blv. Carl Vogt 102, CH-1211 Geneva, Switzerland"Institute of Finance, University of Southern Switzerland, Via G. Buz 13, 6900 Lugano, Switzerland

Received 1 September 1999; received in revised form 1 April 2000; accepted 5 July 2000

Abstract

The local robustness properties of generalized method of moments (GMM) estimatorsand of a broad class of GMM based tests are investigated in a uni"ed framework.GMM statistics are shown to have bounded in#uence if and only if the function de"ningthe orthogonality restrictions imposed on the underlying model is bounded. Sincein many applications this function is unbounded, it is useful to have procedures thatmodify the starting orthogonality conditions in order to obtain a robust version ofa GMM estimator or test. We show how this can be obtained when a reference modelfor the data distribution can be assumed. We develop a #exible algorithm for construct-ing a robust GMM (RGMM) estimator leading to stable GMM test statistics. Theamount of robustness can be controlled by an appropriate tuning constant. We relateby an explicit formula the choice of this constant to the maximal admissible bias onthe level or (and) the power of a GMM test and the amount of contamination that onecan reasonably assume given some information on the data. Finally, we illustrate theRGMM methodology with some simulations of an application to RGMM testing forconditional heteroscedasticity in a simple linear autoregressive model. In this example we"nd a signi"cant instability of the size and the power of a classical GMM testingprocedure under a non-normal conditional error distribution. On the other side, theRGMM testing procedures can control the size and the power of the test undernon-standard conditions while maintaining a satisfactory power under an approxi-matively normal conditional error distribution. ( 2001 Elsevier Science S.A. All rightsreserved.

*Corresponding author. Tel.: #41-91-912-4723.E-mail addresses: [email protected] (E. Ronchetti), [email protected]

(F. Trojani).

0304-4076/01/$ - see front matter ( 2001 Elsevier Science S.A. All rights reserved.PII: S 0 3 0 4 - 4 0 7 6 ( 0 0 ) 0 0 0 7 3 - 7

MSC: C12; C13; C14

Keywords: ARCH models; GMM estimators and tests; In#uence function; Robust modelselection; Robustness of validity

1. Introduction

This paper analyzes the local robustness properties of estimators based on theGMM (cf. Hansen, 1982) and of test statistics based on a GMM estimator. Wecharacterize the local robustness of GMM estimators, of Hansen's speci"cationtest and of GMM-based tests that are GMM versions of the classical Wald,score, and likelihood-ratio test (cf. Newey and West, 1987a; Gourieroux andMonfort, 1989) by a single property: the boundedness of the underlying ortho-gonality function. Since many available econometric models are based on anunbounded orthogonality function, we propose a simple uni"ed setting forconstructing a robust GMM (RGMM) estimator yielding at once the localrobustness of all GMM-based tests.

The need for robust statistical procedures for estimation, testing and predic-tion has been stressed by many authors both in the statistical and econometricliterature; cf. for instance, Hampel (1974), Koenker and Bassett (1978), Huber(1981), Koenker (1982), Hampel et al. (1986), Peracchi (1990, 1991), Markatouand Ronchetti (1997), Krishnakumar and Ronchetti (1997). This paper focuseson locally robust GMM estimation and testing and contributes to the currentliterature in the following directions.

First of all, our results extend the application of robust instrumental variablesestimators proposed by Krasker and Welsch (1985), Krasker (1986) and Lucaset al. (1994) to a general GMM setting with nonlinear orthogonality conditionsand where some stationary ergodic dependence in the underlying data generat-ing process is admitted.

Secondly, the paper generalizes the robust testing framework developed byHeritier and Ronchetti (1994) to a general GMM setting. It uni"es and simpli"esthe theory by proposing a RGMM estimator leading to robust Wald, score andlikelihood-ratio type tests for general nonlinear parameter restrictions.

Finally, the paper provides some robust versions of Hansen's speci"cationtest. This yields RGMM model selection procedures that were not availablebefore.

RGMM analysis focuses implicitly on econometric models de"ned by somesets of orthogonality conditions that are at best ideal approximations to thereality. This task is accomplished by de"ning a reference distribution for thedata playing the role of a reference model for the underlying data generatingprocess. Of course, this entails a lack of generality compared to a standard

38 E. Ronchetti, F. Trojani / Journal of Econometrics 101 (2001) 37}69

nonparametric GMM situation. However, in many applications of the GMM,the reference model distribution is already implied by the problem underinvestigation (for instance in the case of normality of the error distribution).Furthermore, when no natural reference model is supplied we claim that it isoften useful to impose one in order to obtain GMM statistics that behavesu$ciently well at least over a restricted set of relevant model distributions. Theimplied orthogonality conditions are then approximate in the sense that theyshould be satis"ed by any model distribution &near' } in some appropriate sense} to the given reference model. When translating this argument in terms of theempirical distribution of the data, this means that in a RGMM frameworka small fraction of the observations can deviate from the rest of the samplewithout a!ecting the empirical moment conditions in a dramatic way. There-fore, the derived parameter estimates and statistics are representative for thestructure of the &majority' of the data. In other words, robust GMM procedurespay a small &insurance premium' in terms of e$ciency at the reference model inorder to be robust in a neighborhood of it.

At least in linear models with normal serially independent errors, the e!ects ofdi!erent kinds of distributional deviations from the assumptions are wellstudied and known to have an important impact on the asymptotic properties ofa GMM estimator; cf. Krasker and Welsch (1985), Krasker (1986) and Lucaset al. (1994). For the time-series context important work has been done byKuK nsch (1984) and Martin and Yohai (1986) within the framework of (linear)autoregressive models. Since there is a priori no reason to generally believe thatin a nonlinear model with stochastic time dependence these e!ects should be lessserious, a general RGMM framework can o!er a powerful complement to theclassical GMM in many applications.

In this paper we focus on locally robust GMM estimation and particularlyon GMM testing, that is on smooth GMM functionals that can be locallyapproximated by means of their in#uence function (IF); see Hampel (1968, 1974)and Hampel et al. (1986) for basic de"nitions and KuK nsch (1984) and Martin andYohai (1986) for the time-series context. Boundedness of the IF implies that ina neighborhood of the model the bias of an estimator cannot become arbitrarilylarge. In the testing framework this implies that in a neighborhood of the modelthe level of the test does not become arbitrarily close to 1 (robustness of validity)and the power does not become arbitrarily close to 0 (robustness of e$ciency).Hence, a bounded in#uence function is a desiderable local stability property ofa statistic.

Since in applications the IF of a GMM statistical functional is often un-bounded (some examples are listed in Section 2), we propose a robusti"edversion of a GMM estimator that is shown to induce at the same time(locally) robust GMM testing procedures. The RGMM estimator is constructedby applying a basic truncation argument of the theory of robust statisticsmodi"ed to the particular GMM situation. An important feature of the

E. Ronchetti, F. Trojani / Journal of Econometrics 101 (2001) 37}69 39

proposed estimator is that the amount of robustness imposed can be controlledby a tuning constant which is related by an explicit formula to both the maximallocal bias in the level and the power of a GMM test and to the magnitude of thegiven model deviation; see Section 4.

As an illustration of these general principles consider for instance a simpleAR(1) model with ARCH(1) (cf. Engle, 1982) errors for a random sequence(y

t)t|N

:

yt"b

0#b

1yt~1

#Jhtut, h

t"a

0#a

1u2t~1

, (1)

where (ut)t|N

is a standardized i.i.d. sequence with unknown distribution.A natural set of orthogonality conditions for a GMM estimation of the

parameters (a0, a

1, b

0, b

1) is given by

E[et]"0, E[e

tyt~1

]"0, E[gt!h

t]"0, E[g

tgt~1

]"0, (2)

where

et"y

t!b

0!b

1yt~1

, gt"e2

t. (3)

We will see that the unboundedness of the orthogonality function de"ning theseorthogonality conditions implies a lack of robustness of GMM estimators andtests. For this case we propose a small simulated application to testing fora conditional heteroscedasticity speci"cation in the linear autoregressive model.In this simple experiment we observe that for relevant sample sizes the classicalGMM procedure is unstable even under relatively small distributional devi-ations from the normality of the error distribution. Speci"cally, the GMMspeci"cation test often produces sizes that are higher than theoretically expectedand power curves that are already much #atter than under normality forconditional error distributions very near to the normal (for instance a t

9distri-

bution). On the other side, when introducing a normal reference model for theerror distribution it is possible to control for the empirical bias in the level andthe power of a RGMM test under non-standard situations. Of course, imposingmore robustness on a RGMM test has an impact on the power of the test at thereference model. However, in the proposed application it seems that the loss inpower of the RGMM test at the reference model is quickly compensated bya strong gain in power under non-normality of the error distribution.

One could argue that local robustness is not su$cient and global reliabilityshould be the target in order to guarantee resistance in the presence of a largeamount of contamination. This would require the development of high-break-down estimators, but it seems di$cult to extend the available results forregression models (see for instance Rousseeuw and Leroy, 1987) to a generalGMM setting. Indeed, the latter does not show the high degree of structurewhich is exploited in the de"nition of high-breakdown estimators for regressionmodels. Moreover, although we believe that high-breakdown estimators play


a useful role in the exploratory and estimation part of any data analysis, we feelthat small deviations from the model are more meaningful for inference.

The paper is structured as follows. In Section 2 we derive the in#uencefunction of a GMM estimator and show that GMM estimators have boundedIF if and only if the function inducing the natural orthogonality conditions ofthe model is bounded. We then give some examples of GMM estimators withunbounded IF. Section 3 is devoted to the de"nition and construction ofa RGMM estimator suited to induce stable GMM testing procedures. Section 4analyses the local robustness of tests constructed from a GMM estimator andderives some basic expansions for the power and level functionals of a GMMtest. These expansions provide a useful asymptotic bound for the asymptoticbias of level and power of a GMM test under small deviations from the modeldistribution. The bound is proportional to a particular supremum norm of theunderlying orthogonality function. Therefore, it can be used to obtain RGMMestimators that explicitly control the maximal bias of level and power ofa GMM test under deviations from the assumptions. It is this bound that allowsus to derive the explicit link between the &degree' of robustness of the RGMMestimator of Section 3, the amount of contamination that can be reasonablyassumed given some information on the data, and the maximal bias in level andpower of a RGMM test. Section 5 presents the results of our simulations ofa RGMM test for conditional heteroscedasticity in the errors of an autoregres-sive model and Section 6 concludes the paper with some summarizing remarksand suggestions for further research.

2. Robustness properties of GMM estimators

Let X:"(Xn)n|N

be a stationary ergodic sequence de"ned on an underlyingprobability space (X, F, P) and taking values in RN. Without loss of generality,we index the family P :"MPh , h3HN of distributions of RN by a parametervector h3H. h

0is the parameter vector corresponding to the model distribution

(the reference model) for X1. This notation is used in order to write several

GMM statistics used in the paper as functionals on a subset of P.The GMM consists in estimating indirectly some functional

a :PPA :"a(P)LRk of parameters of interest by introducing a functionh : RN]APRH enforcing a set of orthogonality conditions

Eh0 h(X1; a(Ph0 ))"0 (4)

on the structure of the underlying model. Let W:"(=n)n|N

be a sequence ofweighting symmetric positive-de"nite matrices converging a.s to a positivede"nite matrix =

0.


A generalized method of moments estimator (GMME) associated with W isa sequence (a8 (Phn ))n|N of (functional) solutions to the optimization problem

a8 (Phn )"arg mina|A

Ehn h?(X

1; a)=

nEhn h(X

1; a), n3N, (5)

where Phn :"1n+n

i/1dXi

is the empirical distribution of X1,2,X

n, and d

xde-

notes the point mass distribution at x3RN. This functional notation of theGMM minimization problem is useful for investigating the functional structureof a general GMM statistic later on.

Under appropriate regularity conditions (See Hansen, 1982) the GMMEexists, is strongly consistent and asymptotically normally distributed at themodel with an asymptotic covariance matrix given by

Rh0 (=0)"Sh0 (=0

)Eh0Lh?(X

1; a(Ph0 ))

La=

0<0=

0

]Eh0Lh(X

1; a(Ph0 ))

La?Sh0 (=0

),

where

<0

:"Eh0 [h(X1; a(Ph0 ))h

?(X1; a(Ph0 ))] (6)

(the covariance matrix of h(X1, a(Ph0 )) at the model Ph0 ) and

Sh0 (=0) :"CEh0

Lh?(X1; a(Ph0 ))

La=

0Eh0

Lh(X1; a(Ph0 ))

La? D~1

. (7)

The GMME associated to a sequence W such that

=0"<~1

0(8)

is asymptotically best in the sense of Hansen (1982) and yields a &smallest'asymptotic covariance matrix given by

Rh0 (=0)"CEh0

Lh?(X1; a(Ph0 ))

La<~1

0Eh0

Lh(X1; a (Ph0 ))

La? D~1

. (9)

We will adopt in the sequel the following shortened notation:

S0

:"Sh0 (=0), R

0:"Rh0 (=0

), a(h0) :"a(Ph0 ).

To analyze the asymptotic local stability properties of a GMME we consider theasymptotic optimality problem

mina|A

Eh0 h?(X

1; a)=

0Eh0 h(X

1; a) (10)


1 In the following we will always assume that the domain of the given statistical functional is anopen convex subset of P containing Ph0 and all empirical measures.

2 In the exactly identi"ed case (k"H) this expression simpli"es to:

IF(x; a8 , Ph0 )"!CEh0Lh?(X

1; a(h

0))

La D~1

h(x; a(h0))

the standard expression for an M-estimator de"ned by a score function h; cf. for instance Huber(1981).

corresponding to (5). Its unique solution is assumed to be a(Ph0 ) and to be in theinterior of A. The sequence of necessary (functional) equations

CEhnLh?(X

1; a8 (Phn ))

La D=n[Ehn h(X

1; a8 (Phn ))]"0 (11)

de"ning the GMME then converges a.s to the implicit (functional) equation

Eh0Lh?(X

1; a(Ph0 ))

La=

0Eh0 h(X

1; a(Ph0 ))"0. (12)

In order to describe the stability properties of a GMME in a neighborhood ofPh0 we introduce the following well-known concept from the theory of robuststatistics; cf. also Hampel et al. (1986).

Dexnition 1. The in#uence function IF( ) ; a8 , Ph0 ) of a statistical functional1 a8 isgiven by

IF(x; a8 , Ph0 )"lime?0

a8 ((1!e)Ph0#edx)!a8 (Ph0 )

e(13)

for all dx

such that this limit exists.

As a consequence, the in#uence function of a statistical functional describesthe linearized asymptotic bias of a statistic under single-point contaminationsdx

of the assumed model distribution Ph0 . An unbounded IF implies an un-bounded asymptotic bias of a statistic under single-point contaminations of themodel. Therefore, a natural robustness requirement on a statistical functional isthe boundedness of its in#uence function.

The in#uence function of a GMME is obtained by implicitly di!erentiatingthe necessary condition (12) in an arbitrary direction d

x. Straightforward calcu-

lations then yield:2

IF(x; a8 , Ph0 )"!S0Eh0

Lh?(X1; a (h

0))

La=

0h(x; a(h

0)). (14)


3This point is even more important for deriving robust GMM testing procedures; cf. Section 4.

Note that in deriving this expression we used condition (4) which is satis"ed byassumption at the model Ph0 . As a consequence, we can see that

f The IF of a GMME is linearly related to the orthogonality function of themodel h( ) ; a(Ph0 )).

f The IF of a GMME is bounded if and only if the orthogonality function of theunderlying model is bounded.

Expression (14) covers as special cases well-known situations where h is linear,as in Krasker and Welsch (1985), Krasker (1986) and in Lucas et al. (1994).

It is well-known that many econometric estimators can be interpreted asGMME, see Hansen (1982). Unfortunately, many of these turn out to benon-robust, because the corresponding function h is unbounded in the observa-tions. Well-known examples in the (linear) instrumental variables frameworkwere analyzed for instance in Krasker and Welsch (1985).

In addition to M-estimators that are de"ned through the roots of an implicitequation (these estimators can in fact be interpreted as particular GMME), thereis a broad class of nonlinear GMME where the given nonlinearity is in contrastwith the basic robustness principle of a bounded in#uence function.3 Someexamples are listed below.

Example 1. Nonlinear instrumental variables estimators (cf. Amemiya, 1974). Let(X

t)tw0

:"(X(1)t

, X(2)t

, X(3)t

)tw0

be a data generating process, with (for brevity)X(1)

ta scalar endogenous variable, X(2)

ta scalar exogenous variable and

X(3)t

some instrumental variable inducing the orthogonality restrictions

E[X(3)1

(X(1)1!m(X(2)

1; a))]"0

for some given nonlinear function m. Since the function

(x(1), x(2), x(3), a) C x(3)(x(1)!m(x(2); a))

is unbounded at least in x(1) and x(3) all these estimators have unbounded IF.Moreover, for di!erent nonlinear forms of m the robustness problems of a giveninstrumental variables estimator can be quite di!erent. For instance, di!erentpolynomial forms of m can induce very di!erent biases for the correspondingestimator under a slight single-point contamination of the underlying model. Itis then useful to have a general procedure for bounding this maximal biasindependently of the general form of m.

Example 2. GMM estimation of autoregressive models with conditionally hetero-scedastic errors. Let (y

t)t|N

be the ARCH process as de"ned in (1) with associated


orthogonality conditions (2). The function de"ning these orthogonality condi-tions is unbounded. Moreover, note that the observation y

t~1enters in the last

of these four orthogonality conditions as a polynomial of degree four. Therefore,for some choices of the model parameters the in#uence function of the impliedGMM estimator can be steep in some contamination directions. In Section 5 wewill apply the RGMM methodology to this particular example by derivinga RGMM testing procedure for conditional heteroscedasticity in some simula-tion experiments.

Example 3. GMM estimation of nonlinear empirical asset pricing models (cf. forexample Bansal et al. (1993)). Let a nonlinear pricing kernel (G

t)t|N

be de"ned by

Gt"G(R

f,t, R

M,t)"b

0#b

1R

f,t# +

j/1,3,5

bj,M

(RM,t

) j, (15)

where (Rf,t

)t|N

and (RM,t

)t|N

are some corresponding series of yields to maturityof the Treasury bill and of an aggregate equity index, respectively.

Given a set of instrumental variables (Zt)t|N

and a set of n contingent claimspay-o!s (x(1)

t,2, x(n)

t)t|N

, the natural orthogonality conditions implied by thegiven asset pricing equation are

E[Gt`1

(x(i)t`1

!1)Zt]"0, i"1,2, n. (16)

These orthogonality conditions are again unbounded. Moreover, RM,t`1

entersin all orthogonality conditions as a polynomial of degree "ve. Therefore, forsome choices of the model parameters the in#uence function of the impliedGMM estimator is again steep in some contamination directions.

3. Robust GMM estimation

It is not possible to construct robust GMME that are optimal in the sense ofHampel et al. (1986), because a best ML-estimator of a(Ph0 ) at the model is notgenerally available, even when its in#uence function is not required to bebounded. Speci"cally, the covariance matrices of the GMME induced by twodi!erent non-nested sets of orthogonality conditions are not generally rankable.

Instead, we require the bound on the IF to be satis"ed in a norm that isself-standardized with respect to the covariance matrix of the given GMME.This norm measures the in#uence of the estimator a8 relative to its variabilityexpressed by its covariance matrix. We will see in Section 4 that this is theappropriate norm for obtaining robust GMM testing procedures.

Formally, we look at GMME with a bounded self-standardized IF, that issatisfying

DDIF(x; a8 ,Ph0 )DDR~10

:"DDR~1@20

IF(x; a8 , Ph0 )DD)c, (17)


4 In the exactly identi"ed case (H"k) this inequality becomes an equality. In this situation thebound on the self-standardized IF of a GMM estimator provided by a bounded self-standardizednorm of h is exact and the following arguments in this section and in Section 4 still hold.

5 It is important to note that no further model assumptions are needed in order to perform thisconstruction.

where c is a given prespeci"ed positive constant. We can satisfy this conditionfor our RGMM estimator by bounding the self-standardized norm

DDh(x; a(h0))DD

V~10

:"DD<~1@20

h(x; a(h0))DD (18)

of h.Indeed, (17) is satis"ed when the self-standardized norm of h is bounded by

c because:4

DDIF(x; a8 , Ph0 )DD2R~10"h?(x; a(h

0))=

0Eh0

Lh(X1; a(h

0))

La?

]CEh0Lh?(X

1; a(h

0))

La=

0<0=

0Eh0

Lh(X1; a(h

0))

La? D~1

]Eh0Lh?(X

1; a(h

0))

La=

0h(x; a(h

0))

)DDh(x; a(h0))DD2

V~10

by the orthogonal projection property of the matrix

<1@20=

0Eh0

Lh(X1; a(h

0))

La? CEh0Lh?(X

1; a(h

0))

La=

0<0=

0Eh0

Lh(X1; a(h

0))

La? D~1

]Eh0Lh?(X

1; a(h

0))

La=

0<1@2

0.

To construct a GMME with self-standardized in#uence function bounded by c,we introduce the Huber function H

c: RHPRH; y C yw

c(y), de"ned by

wc(y) :"min(1, c/DDyDD) for yO0 and w

c(0) :"1, and a new mapping

hA,qc

: RN]APRH given by

hA,qc

(x; a) :"Hc(A[h(x; a)!q])"A[h(x; a)!q]w

c(A[h(x; a)!q]) (19)

for x3RN and a3A. The nonsingular matrix A3RHCH and the vector q3RH aredetermined by the implicit equations:5

Eh0 hA,qc

(X1; a(Ph0 ))"0 (20)

and

Eh0 hA,qc

(X1; a(Ph0 ))hA,q?

c(X

1; a(Ph0 ))"I. (21)


6Note that (by construction) the optimal asymptotic weighting matrix associated to this particu-lar GMME is the identity matrix.

a(h0) can be estimated by the sequence of "xed points of the algorithm described

by (19)}(21). Note that the bound imposed on the self-standardized in#uence of

our GMME cannot be chosen arbitrarily small. Indeed, c*JH, cf. Hampel etal. (1986), p. 228.

In some robust applications in the iid framework, the functional form impliedby (20) for the dependence of q on a and A can be determined explicitly. Forinstance, in linear regression models with normal errors, symmetry impliesq"0; cf. Hampel et al. (1986), chapter 6.

To apply the algorithm to a general GMM situation we propose to estimateA via the sequence of solutions to the empirical version of (21) and to determineq as the solution of (20) under the model probability Ph0 . In some models } as forinstance in the RGMM application presented in Section 5 } this will requirea simulation procedure.

Speci"cally, for a given bound c'JH on the self-standardized in#uence of a8 ,the computation of the robust GMME can be performed by the following foursteps:

f Fix a starting value a0

for a(h0) and initial values q

0:"0 and A

0such that

A?0A

0"[Ehn (h(X

1; a

0)h?(X

1; a

0))]~1.

f Compute new values q1

and A1

for q and A de"ned by

q1

:"Eh0 [h(X

1; a

0)w

c(A

0(h(X

1; a

0)!q

0))]

Eh0 wc(A

0(h(X

1; a

0)!q

0))

(22)

and

(A?1A

1)~1 :"Ehn [(h(X

1; a

0)!q

0)(h(X

1; a

0)!q

0)?

]w2c(A

0(h(X

1; a

0)!q

0))]. (23)

f Compute the optimal GMME a1

associated6 to the orthogonality functionhA1,q1c

.f Replace q

0and A

0by q

1and A

1, respectively, and iterate the second and

third step described above until convergence of the sequence of optimalGMME (a

n)n|N

associated with the sequence (hAn ,qnc

)n|N

of bounded ortho-gonality functions.

The robust GMME obtained in this way can be interpreted as the GMMEinduced by the truncated orthogonality conditions hA,q

cwhen satisfying the

orthogonality condition (20) for q and when simultaneously estimating Athrough the empirical version of (21).


hA,qc

is a truncated version of h. Because of the truncation, h must be shifted byq in order to satisfy the orthogonality condition (20). Moreover, (21) ensures thatc is an upper bound on the self-standardized in#uence function of the corre-sponding GMME, because } by construction } the self-standardized norm ofhA,qc

is equal to its euclidean norm which itself is bounded by the constant c.Existence and uniqueness of a solution (a(Ph0 ), q(Ph0 ) , A(Ph0 )) are implied by

the implicit function theorem and the FreH chet di!erentiability of the equationsystem de"ning the GMME of (a(Ph0 ), q(Ph0 ), A(Ph0 )) in a neighborhood of Ph0 ,which itself is implied by the boundedness of the function hA,q

c; cf. for example

Clarke (1986), Bednarski (1993). More speci"c conditions for a special model canbe found in Krasker and Welsch (1985). Regularity conditions for consistencyand asymptotic normality of a GMM estimator at the model Ph0 are provided inHansen (1982).

Whereas the original moment conditions h are usually dictated by economictheory, the truncated version hA,q

ctakes into account the realistic case that only

the &majority of the data' can reasonably "t the original moment conditions. Theweights w

c(A[h(x; a)!q]) assigned to each observation x can be used to detect

outlying points. The tuning constant c3(JH,R) controls the degree of robust-ness imposed on the procedure. It can be chosen by the analyst as a trade-o!between her theoretical moment conditions and those supported by the data.Some objective guidelines for the choice of c are presented in the next sectionwhere we directly focus on RGMM hypothesis testing. There we show that fora given amount of model contamination the constant c can be determined sothat the maximal bias in the level or the power of a GMM test remains belowa given bound.

As pointed out by a referee, one possible disadvantage of the RGMMestimator de"ned above could be the well-known poor performance insmall samples of GMM estimators when the asymptotic covariance matrix ofthe given orthogonality function is estimated; cf. for instance Koenker et al.(1994). Some protection in this respect should be supplied by the fact that theorthogonality function hA,q

cbehind our RGMM estimator is bounded; neverthe-

less we expect the issue of a covariance matrix estimation in our RGMMframework to be particularly important when the number of orthogonalityconditions is &high' (for example of an order higher than n1@3; cf. Koenker andMachado (1999)).

One possibility to improve the small sample performance of our RGMMestimator is to use an empirical likelihood version of GMM as pro-posed by Imbens et al. (1998). Speci"cally, one can use as estimating equationsfor the empirical likelihood those given by formula (5), p. 337 or (9), p. 339in their paper with (in their notation) t(z, h)"hA,q

c(z, h). The boundedness

of the function hA,qc

will preserve the robustness properties of the estimatorwhile the empirical likelihood version should improve the "nite sampleperformance.


7Only the results obtained for the likelihood ratio statistic are not available in the non-optimalGMM case since the corresponding statistic is then no longer asymptotically equivalent to a sym-metric functional form (see below).

4. Robust inference with GMM estimators

This section is devoted to the robustness properties of GMM-based teststatistics. The key idea in deriving RGMM procedures is to construct GMMestimators based on a bounded self-standardized norm of the given ortho-gonality function, as for instance in the case of the RGMM estimator de"ned inthe last section. For simplicity of notation we will derive all results for the case ofan optimal GMME (that is a weighting matrix=

0"<~1

0) based on a bounded

orthogonality function. Modi"cations to the general case are straightforward.7Several tests derived from a GMME can be constructed, for testing some

misspeci"cation of the model or some set of parameter restrictions on a. TheGMM speci"cation test proposed by Hansen (1982) is a test of the overidentify-ing restrictions implied by the null hypothesis given by (4), for the case whereH'k.

The asymptotic distribution of the statistic de"ning Hansen's test with respectto a sequence of local misspeci"cations is a noncentral s2(H!k, b) distribution.At the model b"0; see Newey (1985).

The GMM versions of the classical ML-tests are used to test a null hypothesis

g(a(h0))"0 (24)

for a smooth function g :APRr such that Lg?/La(a(h)) is of full column rank forall h3H.

The Wald, score and likelihood-ratio statistics induced from a best GMMEare all asymptotically equivalent under the null hypothesis (24) and with respectto a sequence of local alternatives to a(h

0). They are asymptotically noncentral

s2(r; b) distributed, with a noncentrality parameter b"0 at the model under thenull hypothesis (24).

We restrict our attention to GMM test statistics that can be written (at leastasymptotically) as simple quadratic forms of a functional ;. Speci"cally, weconsider functionals m de"ned asymptotically by a symmetric form

nm(Phn )"n;(Phn )?;(Phn ), n3N (25)

and consider the following test statistics:Hansen's test: Hansen's statistic (mM) is the symmetric form (25) with a func-

tional ; de"ned by

;M(Phn ) :"=1@20

Ehn h(X1; a8 (Phn )). (26)


A consistent estimator of =0

is given by the sequence W of positive de"niteestimators.

Wald-type test: The statistic of a GMM-based Wald-type test (mW) is of theform (25) with a functional ; de"ned by

;W(Phn ) :"CLgLa?

(a(h0))R

0Lg?

La(a(h

0))D

~1@2g(a8 (Phn )). (27)

Practically

LgLa?

(a(h0))R~1

0Lg?

La(a(h

0))

is approximated by estimating R0

with Rhn (=n) and a(h

0) with a8 (Phn ).

Score type tests: The statistic of a GMM-based score-type test (mS) is of theform (25) with a functional ; de"ned by

;S(Phn ) :"RK 1@20

EhnLh?(X

1; a( (Phn ))

La=

nEhn h(X

1; a( (Phn )), (28)

where a( (Phn ) is a solution to a constrained GMM minimization problem:

a( (Phn )"arg mina|A,g(a)/0

Ehn h?(X

1; a)=

nEhn h(X

1; a) (29)

and RK0

is the covariance matrix (9) evaluated at a"a( (Ph0 ). In applicationsRK0

can be consistently estimated by RK hn (=n).

Likelihood ratio-type test: The GMM likelihood ratio type-test is constructedwith a statistic mR that can be written asymptotically as a symmetric form. It isde"ned by

mR(Phn ) :"Ehn h?(X

1; a( (Phn ))=0

Ehn h(X1;a( (Phn ))

!Ehn h?(X

1;a8 (Phn ))=0

Ehn h(X1;a8 (Phn )).

Asymptotically one has

mR(Phn )";H(Phn )?;H(Phn )#o

1(1)

with a functional ;H that is explicitly given by (A.3) in the proof of Theorem 1.As mentioned in Section 1, the general goal of robust testing procedures is to

control the maximal bias on the level and the power of a test that can arisebecause of a slight distributional misspeci"cation of a null or an alternativehypothesis. This is called robustness of validity and e$ciency, respectively.

To analyze the asymptotic local stability properties of these tests we followthe general approach proposed by Heritier and Ronchetti (1994). In order toapply this methodology to the GMM setting we can assume the followinguniform convergence to normality of a robust GMME.


8The proofs of all theorems are given in the appendix.

Property 1. Let a bounded in#uence GMME a8 of a(Ph0 ) be given. It then follows:

Jn(a8 (Phn )!a8 (P0e,n,Q))PN(0, R0), nPR (30)

in distribution, uniformly over the sequence (Ue,n (Ph0 ))n|N of (e, n)-neighbor-hoods of Ph0 de"ned by

Ue,n (Ph0 ) :"GP0e,n,Q :"A1!e

JnBPh0 #e

JnQ DQ3dom(a8 )H, (31)

where the assumptions on dom(a8 ) are given in Footnote 1.

The neighborhood de"ned by (31) is probably the simplest way to formalizelocal perturbations of the model Ph0 . Note that d

K(P0e,n,Q , Ph0 ))e for all n3N

and Q3dom(a8 ), where dK

denotes the Kolmogoro! distance. Alternatively, onecould use more involved notions of distance between distributions.

Property 1 is stronger than the requirement of the existence of the in#uencefunction. Generally, one needs a stronger smoothness condition like FreH chetdi!erentiability in order to obtain uniform convergence; cf. Clarke (1986) andBednarski (1993). However, under appropriate regularity conditions (cf. Clarke,1986; Heritier and Ronchetti, 1994), bounded in#uence statistical functionalscan be shown to be FreH chet di!erentiable. As a particular case, the robustRGMM estimator proposed in the last section is FreH chet di!erentiable.

The next theorem provides a maximal asymptotic bias of the level of a GMMtest.8

Theorem 1. Let a8 be a GMME induced by a bounded orthogonality function h anddenote by a the level functional of the tests based on mM, mW, mS and mR, respectively.Let further (P0e, n, Q

)n|N

be a sequence of (e, n, Q)-contaminations of the underlyingnull distribution Ph0 , each of them belonging to a corresponding neighborhoodUe,n (Ph0 ), as dexned in (31).

Then

limn?=

a(P0e,n,Q)"a0#e2kKKPRN

IF(x; ;, Ph0 ) dQ(x)KK2#o(e2), eP0 (32)

for all Q3dom(a), where ;( ) ) is the ;-functional corresponding to each test,

k"!

LLb

Hr(g

1~a0 ; b)Db/0"

(1!a0)

2!

12

Hr`2

(g1~a0 ; 0),

where Hr( ) ; b) is the cumulative distribution function of a noncentral s2(r; b)

distribution with r degrees of freedom and noncentrality parameter b*0, g1~a0 is


the 1!a0

quantile of a s2(r; 0) distribution and a0"a(Ph0 ) is the nominal level of

the test. Moreover, the bias of a(P0e,n,Q) is uniformly bounded by the inequality

limn?=

Da(P0e,n,Q)!a0D)e2k sup

xDDh(x; a(h

0)) DD2

W0#o(e2).

As a consequence of the theorem, the maximal asymptotic bias of the level ofa GMM test that is derived from the robust GMME of the last section can bebounded by the inequality

limn?=

Da(P0e,n,Q)!a0

D)k(ec)2#o(e2). (33)

For robust testing purposes the asymptotic bound (33) can be used to choosec depending on both the maximal amount of contamination (e) expected by theresearcher } given some prior information on the nature of the data } and themaximal bias for the level (maxbias) he or she is willing to accept

c"1

eSmaxbias

k.

(34)

Table 1 presents the implied c values for e"5%, maxbias"$0.5% anda0"5%.By regressing logk vs. log r and for the case of a nominal level a

0"5% at the

model, one can obtain the simple approximation

c+3

er0.3(maxbias)1@2. (35)

We now come back to the robustness of e$ciency properties of a GMM test and"rst investigate the case of a GMM speci"cation test.

Let

(P!-5g,n )n|N :"AA1!g

JnBPh0#g

JnPh1B

n|N

(36)

be a sequence of local alternatives to Ph0 and

Ue,n (P!-5g,n ) :"GP1e,n,Q :"A1!e

JnBP!-5g,n#e

JnQ DQ3dom(a8 )H (37)

be the corresponding asymptotic neighborhood of P!-5g,n , for given n.A natural restriction on the magnitude of the contamination is DeD(DgD. This

allows us to distinguish the neighborhood Ue,n (P!-5g,n) of the local alternative fromthe given null hypothesis. On the other side, one could of course compare


a given neighborhood Ue0,n (Ph0 ) of the null hypothesis with a neighborhoodUe,n(P!-5g,n) of the local alternative. In this case a natural restriction will beDe0D#DeD(DgD.The next theorem is the &power' counterpart of Theorem 1 for the GMM

speci"cation test. Similarly to the case of the level, the theorem yields an explicitasymptotic bound by which the maximal asymptotic bias of the power can bebounded.

Theorem 2. Let a8 be a GMME induced by a bounded orthogonality function h anddenote by n the power functional of the test based on mM. Let further (P1e,n,Q)

n|Nbe

a sequence of (e, n, Q)-contaminations of the underlying local alternatives P!-5g,n eachof them belonging to a corresponding neighborhood Ue,n(P!-5g,n) as dexned in (37).

Then

limn?=

Dn(P1e,n,Q)!n(P!-5g,n )D"2kegPRN

IF?(x; ;M, P!-5g,n ) dQ(x)

]PRN

IF(x; ;M, Ph0 ) dPh1 (x)#o(g) (38)

with k dexned as in Theorem 1. Moreover, the bias of the asymptotic powerfunctional n is uniformly bounded by the inequality

limn?=

Dn(P1e,n,Q)!n(P!-5g,n)D)2keg maxMP

!-5g,n _Ph0Nsupx

DDh(x; a8 ( ) ))DD2W0

#o(g). (39)

Similarly to the case for the level, the maximal asymptotic bias of the power ofa GMM speci"cation test derived from the RGMM estimator of the last sectioncan be estimated by the inequality

limn?=

Dn(P1e,n,Q)!n(P!-5g,n)D)2kegc2#o(g). (40)

As in the case of the level (see (33)) this inequality can be used to relate the tuningconstant c of our RGMM estimator to the maximal bias in the power of theGMM speci"cation test, given a nominal level a

0at the model. For instance,

assuming H!k"1 and e"5%, a bound of 0.5% on the bias from a nominallevel of 5% implies c"4.18 and k"0.1145 (cf. Table 1). This yields an absolutemaximal bias in the power of a corresponding RGMM speci"cation test givenby 0.20g. For example for g"15% the implied maximal bias in the power isapproximatively 3%.

Theorem 2 illustrates the trade o! existing between power and robustnessof a GMM speci"cation test. Indeed, for a given maximal bias over thecontaminated neighborhood Ue,n (P!-5g,n) one cannot impose stronger robustnessrequirements on a RGMM estimator (that is a lower constant c) withoutsimultaneously looking at local alternatives that are more distant in the given


Table 1Values of the tuning constant c for bounding the maximal bias of the level of a GMM test!

r g1~a0 k c

1 3.84 0.1145 4.182 5.99 0.0749 5.173 7.81 0.0584 5.854 9.94 0.0490 6.395 11.07 0.0428 6.836 12.59 0.0383 7.227 14.07 0.0350 7.568 15.51 0.0323 7.879 16.92 0.0301 8.15

10 18.31 0.0283 8.41R 0 R

!The values of the tuning constant c are for a nominal level 5% at the model, for a maximal biasgiven by maxbias"$0.5% and for a model contamination e"5%. r is the number of degrees offreedom implied by the s2-test under scrutiny.

direction Ph1 (that is with a higher constant g). On the other side, imposingstronger robustness requirements by a lower constant c reduces the maximalbias from the power of the given local alternative P!-5g,n . However, for near localalternatives (and therefore low values of g) this will correspond to a low powerof the RGMM speci"cation test over the full contaminated neighborhoodUe,n(P!-5g,n).

We conclude this section by discussing the robustness of e$ciency propertiesof the GMM-based Wald, score, and likelihood ratio tests. Consider again theneighborhood de"ned by (31). For P0e,n,Q3Ue,n(Ph0 ) we de"ne a sequence ofparametric local alternatives to (24) by

gAa(h0)#

D

JnB"0 (41)

with a non-zero vector D3Rk.Similarly to the case of the GMM speci"cation test, a natural restriction on

the magnitude of the contamination is DeD(DDD. This allows us to distinguish theneighborhood Ue,n(Ph0 ) of the local alternative from the given null hypothesis.The next theorem is the power counterpart of Theorem 1 for the maximum-likelihood-type GMM tests. Similarly to Theorem 2 an explicit asymptoticbound for the maximal bias in the power of a parametric GMM test is provided.

Theorem 3. Let a8 be a GMME induced by a bounded orthogonality function h anddenote by n the power functional of the test based on mW, mS and mR, respectively.


Then the bias of n is uniformly bounded by the inequality

limn?=

Dn(P0e,n,Q)!n(Ph0 )D)2keDDDDDR~10

supx

DDh(x; a(h0))DDw

0#o(D), (42)

where k is dexned as in Theorem 1.

Similarly to the case for the power of a GMM speci"cation test, the maximalasymptotic bias of the power of a parametric GMM test derived from theRGMM estimator of the last section can be estimated by the inequality

limn?=


c#o(D). (43)

This bound can be used to relate the choice of c to the maximal bias in the powerof a parametric GMM test, given a nominal value a

0at the model.

5. An application to RGMM testing for conditional heteroscedasticity

In this section we consider a simple application of our RGMM methodologyto a test for ARCH structures in the errors of a linear autoregressive model. Thegoal is not to perform a full analysis of the robustness properties of ARCHtesting procedures but to outline the performance of the RGMM in a simpleapplication as well as the algorithm used to compute the RGMM estimator ofSection 3.

Let (yt)t|N

be the autoregressive process (1) with ARCH(1) error terms pres-ented in Example 2. Moreover, consider the orthogonality conditions given by(2) and (3).

A test for a constant conditional variance speci"cation of etcould be a Hansen

speci"cation test for the overidentifying orthogonality conditions implied by thenull hypothesis a

1"0 against an alternative hypothesis a

1'0.

Note that in the present formulation we treat all parameters that have to beestimated under the null hypothesis symmetrically. Of course, one could easilydevelop a two-stage RGMM testing procedure if (b

0, b

1) is treated as a vector of

nuisance parameters.To construct a GMM test for conditional heteroscedasticity behaving satis-

factorily under local deviations from normality, we consider as a referencemodel for y

tan autoregressive model with normally distributed errors u

tand

compute the RGMM estimator presented earlier for a given choice of the tuningconstant c (see below). By construction, the above RGMM test of induced bythis RGMM estimator maintains &good' level and power properties under localdeviations from the given reference model in a way that is formalized throughthe inequalities obtained in Theorems 1}3.


To compare the performance of the given GMM and RGMM tests for ARCHwe simulate the following distributions &near' the normal distribution as candi-date models of a possible data generating process for u

t.

1. Standard normal.In this experiment we compare the e$ciency of the RGMM and the

classical GMM testing procedures at the given reference model.2. Contaminated normal CN(e,K2).

F(x)"(1!e)U(x)#eUAx

KB, x3R, (44)

where U is the cumulative distribution function of a standard normal randomvariable. Here, we investigate the performance of the classical GMM and theRGMM under a known maximum distance e from the standard normalmodel and a given degree of contaminating variance K2. We simulate thiscase for a distance e"0.05 and a very high contaminating varianceK2"100. This choice is quite extreme. However, it allows us to compare theperformances of the RGMM and the GMM under dramatic symmetricdeviations from normality that could occur over a short time period in realdata.

3. Student tl with l degrees of freedom.We consider the cases l"5, 9 that allow for the existence of the fourth and

the eighth conditional moments of ut, respectively. Note that the t

9and

t5

distributions are already very near to the normal. As a consequence, in thisexample we can compare the numerical performance of the robust and theclassical GMM when very small deviations from normality are present.Moreover, in the t

5case we can investigate the impact of the non-existence of

some theoretical conditional moments of ut(assumed "nite by the GMM).

4. Double exponential DE.This distribution has a symmetric convex density. It is therefore qualitat-

ively di!erent from the normal already in the center of the distribution.Furthermore, it displays fat tails somewhere between the t

5and the

CN(0.05, 102) distribution.

All simulated error distributions were scaled in order to have variance 1. Thissmall simulation design covers a good spectrum of tail behaviors for distribu-tions of u

tthat have heavier tails than the normal and still satisfy minimal

moment requirements. Indeed, the tail indices (cf. Gasko and Rosenberger, 1983,p. 322) of these distributions are 1 for the standard normal distribution, 1.16 and1.34 for the Student t

9and t

5distributions, respectively, 1.63 for the double

exponential, and 3.42 for the contaminated normal. For comparison, a standardCauchy distribution has a tail index of 9.22.


9All QQ plots are based on simulated samples of 5000 observations.

Fig. 1. QQ Plot of the unconditional distribution of (yt) (sample size 5000 observations) under

standard normal, t5, double exponential and contaminated normal (e"0.05, K"10) errors. The

ARCH parameter was set to a1"0.

We simulate (1) for the parameter choice (b0, b

1, a

0)"(0.4, 0.3, 0.25) and for

di!erent values of a1, ranging from 0 to 0.3, under the di!erent distributions

for utpresented above and for sample sizes ¹"250, 500, 1000. Note that for

a1'1

3the fourth unconditional moments of u

tdo not exist even under normal-

ity of ut.

As an illustration, some QQ plots9 of the unconditional distribution ofa process (y

t) without and with ARCH e!ects (a

1"0 and a

1"0.2, respectively)

for some of the distributions considered above are presented in Figs. 1 and 2.From these graphs one can see that the e!ects on the unconditional distribu-

tion of ytof a &slight' modi"cation of the conditional distribution of u

tcan be

quite important, when ARCH e!ects are present. This is particularly true for thetails of the induced distributions. As expected, a given tail index of a conditional


Fig. 2. QQ Plot of the unconditional distribution of (yt) (sample size 5000 observations) under

standard normal, t5, double exponential and contaminated normal (e"0.05, K"10) errors. The

ARCH parameter was set to a1"0.2.

distribution for ut

induce fatter tails in the unconditional distributions ofytwhen ARCH structures are present.Each model is simulated 1000 times. The corresponding empirical rejection

frequency for the RGMM and the GMM Hansen's test is calculated for a "xednominal level of 5%. The estimated standard error of the empirical rejectionfrequency p( is given by (using the binomial distribution) ((p( (1!p( ))/1000)1@2. It is0.7%, 1.0% and 1.5% for p("5%, 20%, 50%, respectively.

The tuning constant for the RGMME was set at c"2.09. This allows toobtain a maximal bias of $0.5% in the level of the RGMM test also forcontaminations e"10% (cf. Table 1 above) of the unconditional distribution ofyt. We imposed such a strong robustness restriction on our RGMME because

the unconditional distribution of ytshows even fatter tails than the conditional

distribution of ut

when ARCH e!ects are present, a fact that can make thedistance between the induced unconditional distributions of y

tlarger than the

distance between the assumed conditional distributions of ut.


Table 2GMM and RGMM simulation results under u

t&N(0, 1)!

Rejection frequency GMM Rejection frequency RGMM

a1

¹"250 ¹"500 ¹"1000 ¹"250 ¹"500 ¹"1000

0.00 0.08 0.08 0.05 0.02 0.02 0.020.05 0.05 0.09 0.19 0.02 0.06 0.070.10 0.09 0.28 0.62 0.06 0.14 0.290.15 0.20 0.52 0.90 0.12 0.31 0.620.20 0.32 0.74 0.97 0.21 0.51 0.870.25 0.45 0.84 0.98 0.35 0.71 0.950.30 0.56 0.89 0.98 0.49 0.86 0.99

!Each entry in the table corresponds to the empirical rejection frequency of the hypothesis a1"0

obtained using 5% critical values for the s2 test. The constant c for the RGMM test was set toc"2.09.

For each simulation run we used a0"(0.2, 0.2, 0.2) as a starting point for the

algorithm and always obtained convergence.In the "rst step of the algorithm we set q"0 and updated the matrix A

after having estimated the covariance matrix of h(X1,a

0) with a Newey and

West (1987b) covariance matrix estimator. In the second step of the algorithmwe simulated an ARCH process corresponding to the parameter choice a

0and computed the expectations needed to solve (22) and thereby obtain q

1.

A1

is obtained after having estimated the covariance matrix of(h(X

1, a

0)!q

0)w

c(A

0(h(X

1, a

0)!q

0)) with a Newey and West (1987b)

covariance matrix estimator. Note that at this stage of the algorithm an autocor-relation robust covariance matrix estimator is necessary even when (h(X

t, a

0))t|N

is conditionally uncorrelated because this does not generally imply that(hA0 ,q0

c(X

t, a

0))t|N

is uncorrelated. In the third step we computed the GMMEa1

associated to the orthogonality function hA1 ,q1c

. The second and third stepabove are then iterated until convergence of the sequence (a

n)n|N

of GMMEassociated to the sequence (hAn ,qn

c)n|NH of bounded orthogonality functions.

The results are presented in Tables 2}6. Although the goal of this simulation isnot to perform a full analysis of the robustness properties of ARCH testingprocedures, some of the features obtained are worth noting.

First of all, the RGMM test yields empirical sizes that are very stable acrossall simulated distributions (generally between 0.02 and 0.03). On the other side,the empirical sizes of the classical GMM tests are far less stable ranging between0.05 (for the normal distribution and for a sample size ¹"1000) and 0.11 (insome experiments with the DE and t

5distributions) for distributions that are

not &too far' from the normal. In the case of heavier tails (the contaminatednormal case) the classical GMM test breaks down. Although this simulation is



t&DE!


a1

¹"250 ¹"500 ¹"1000 ¹"250 ¹"500 ¹"1000

0.00 0.11 0.10 0.09 0.03 0.03 0.030.05 0.04 0.04 0.09 0.03 0.06 0.120.10 0.04 0.12 0.31 0.07 0.14 0.320.15 0.06 0.23 0.54 0.11 0.26 0.580.20 0.10 0.33 0.71 0.18 0.41 0.780.25 0.16 0.43 0.78 0.26 0.57 0.910.30 0.21 0.50 0.79 0.34 0.70 0.96




t&t

9!


a1

¹"250 ¹"500 ¹"1000 ¹"250 ¹"500 ¹"1000

0.00 0.09 0.09 0.07 0.02 0.02 0.020.05 0.05 0.05 0.11 0.04 0.06 0.100.10 0.05 0.16 0.42 0.09 0.14 0.300.15 0.12 0.35 0.69 0.13 0.31 0.620.20 0.21 0.54 0.83 0.23 0.50 0.840.25 0.30 0.65 0.87 0.35 0.83 0.950.30 0.38 0.73 0.88 0.46 0.83 0.99



too limited to draw "nal conclusions, we observe empirical sizes re#ectinga rather &conservative' behavior of the RGMM test and a drastic liberal behav-ior of the classical GMM test. Indeed, already under normality the empiricalsizes of the classical GMM test are often higher than the given nominal level of5% (for sample sizes ¹"250, 500).

Secondly, the RGMM test yields empirical power curves that are fairly stablefor almost all the simulated distributions. In particular, for distributions that arenot &too far' from the normal (the DE, the t

9and the t

5distributions) the

empirical asymptotic power when ¹"1000 deviates from that obtained under



t&t

5!


a1

¹"250 ¹"500 ¹"1000 ¹"250 ¹"500 ¹"1000

0.00 0.10 0.11 0.11 0.02 0.02 0.030.05 0.05 0.05 0.06 0.03 0.07 0.110.10 0.06 0.10 0.24 0.05 0.14 0.330.15 0.11 0.18 0.43 0.11 0.28 0.610.20 0.15 0.29 0.59 0.17 0.46 0.820.25 0.21 0.40 0.67 0.29 0.64 0.930.30 0.27 0.48 0.71 0.40 0.78 0.97




t&CN(0.05, 10)!


a1

¹"250 ¹"500 ¹"1000 ¹"250 ¹"500 ¹"1000

0.00 0.35 0.51 0.48 0.02 0.01 0.020.05 0.16 0.19 0.17 0.02 0.03 0.060.10 0.09 0.08 0.05 0.03 0.06 0.140.15 0.06 0.04 0.02 0.06 0.11 0.240.20 0.04 0.03 0.03 0.07 0.16 0.360.25 0.04 0.03 0.06 0.10 0.22 0.480.30 0.04 0.04 0.11 0.13 0.28 0.60



normality by no more than $0.09 (with a maximal absolute deviation of 0.09obtained for the DE case when a

1"0.2). In the contaminated normal case

di!erences are larger. However, note that in this case the classical GMM testdoes not even produce a monotonically increasing power curve.

The stability of the RGMM is paid for through a loss in power undernormality, compared to the classical GMM test. For instance, when a

1"0.1 the

power of the robust GMM test under normality is half that of the classical one.Somehow surprisingly however, this clear power advantage is already lost forvery small deviations from normality. Indeed, for the di!erent sample sizes the


classical and robust power curves in the t9

experiment are quite comparable,with some small advantages for the GMM (RGMM) for large (small) samplesizes. For larger deviations, (the t

5, the DE and the CN case) the power of the

RGMM is clearly higher than that of the classical test. This suggests that in realdata applications already very small contaminations of the underlying modelcould a!ect the e$ciency of classical GMM testing procedures. On the otherside, a RGMM procedure could then be helpful in maintaining this e$ciencyloss below a given bound.

6. Conclusions

We derived a RGMM estimator that generates robust tests for a broad classof GMM test statistics. Special cases are Hansen's speci"cation test and likeli-hood-type GMM tests like the Wald, the score and the likelihood ratio test.

We presented an algorithm to compute our RGMM estimator, in which thedegree of robustness required by a researcher can be controlled through thechoice of an appropriate tuning constant c.

We explicitly related the choice of this tuning constant to two key variables:the amount of contamination that one can realistically assume with respect toa given data set and to the available data information, and the maximal bias oflevel and (or) power of a GMM test that one is ready to admit for the given test.

In some simulated experiments we presented evidence that the optimalperformance of a GMM test at the model can be strongly worsened even whensmall deviations are present. In these experiments the RGMM testing procedurebehaves well in controlling small distributional deviations from the assump-tions. Moreover, the e$ciency loss at the model of the RGMM procedure seemsto be reasonable when considering its performance under small model mis-speci"cations.

Further research on RGMM testing includes the study of its performanceunder more general model structures and model deviations (for instance asym-metric deviations) than those presented above. Applications to more complexmacroeconomic and "nancial models where a reference model for the datadistribution can be assumed could produce interesting robust results that can becompared with those obtained with classical methodologies. Finally, a furtherissue is the small sample behavior of RGMM statistics.

Acknowledgements

The authors thank the Co-Editor, the Associate Editor and three referees forvery valuable comments that improved the presentation of the paper.


10See Clarke (1986), Bednarski (1993) and Heritier and Ronchetti (1994).

Appendix A

A.1. Proof of Theorem 1

We prove the statement of the theorem only for the score and likelihood ratiostatistics. Those for mM and mW can be proved by similar arguments. As notedafter Property 1 a8 is FreH chet di!erentiable.10 This implies the FreH chet di!erenti-ability of ;S. A "rst-order Von Mises (1947) expansion of ;S then gives up toterms of order o(e)

Jn(;S(Phn )!;S(P0e,n,Q))PN(0, Ir), nPR (A.1)

in distribution uniformly for all Q3dom(;S), using (30). As shown by Heritierand Ronchetti (1994) the asymptotic level under contamination of the corre-sponding symmetric test functional induced by;S can be then approximated bythe second-order expansion given by (32) with ;";S. Note that the equalityfor k in the statement of the theorem is obtained by a result of Johnson and Kotz(1991), Chapter 28, p. 132, formula 1. Then, by (4) and using the hypothesisg(a(Ph0 ))"0, we obtain

IF(x; ;S, Ph0 )"R1@20

Eh0Lh?(X

1; a(h

0))

La=

0

]AEh0Lh(X

1; a(h

0))

La?IF(x; a( , Ph0 )#h(x; a(h

0))B.

The constrained GMM estimator (a( (Phn ), jK (Phn )) is de"ned by the system of"rst-order conditions

EhnLh?(X

1; a( (Phn ))

La=

nEhn h(X

1; a( (Phn ))!

Lg?

La(a( (Phn ))jK (Phn )"0

g(a( (Phn ))"0,

where jK : dom(jK )PRr is the corresponding statistical functional of Lagrangemultipliers. Di!erentiating implicitly the limit version of these necessary condi-tions in direction d

x3dom(a( ) } while imposing (4) and jK (Ph0 )"0 } and solving

the corresponding system of implicit equations gives

IF(x; a( , Ph0 )"(I!R0M

0)IF(x; a8 , Ph0 ), (A.2)


where

M0"

Lg?

La(a(h

0))A

Lg

La?(a(h

0))R

0

Lg?

La(a(h

0))B

~1 Lg

La?(a(h

0)).

Inserting this result in (A.2) and using (14) with =0"<~1

0yields

DDIF(x; ;S, Ph0 )DD2"KKM0R0Eh0

Lh?(X1; a(h

0))

La=

0h(x; a(h

0))KK

2

R0

.

Moreover, by the orthogonal projection property of M0R

0(with respect to the

scalar product induced by R0):

DDIF(x; ;S, Ph0 )DD2)KKEh0Lh?(X

1; a(h

0))

La=

0h(x; a(h

0))KK

2

R0

)DDh(x; a(h0))DD2

W0.

This proves the theorem for aS.To apply the approximation (32) to the likelihood-ratio statistic remember

that a8 is FreH chet di!erentiable. A second-order von Mises (1947) expansion ofmR under the hypotheses (4) and g(a(Ph0 ))"0 then gives up to terms of ordero(e2):

mR(P0e,n,Q)"e2n CPRNCEh0

Lh(X1; a(h

0))

La?IF(x; a( ,Ph0 )#h(x; a(h

0))D

?

dQ(x)=0

]PRNCEh0Lh(X

1; a(h

0))

La?IF(x; a( , Ph0 )#h(x; a(h

0))DdQ(x)

!PRNCEh0Lh(X

1; a(h

0))

La?IF(x; a8 , Ph0 )#h(x; a(h

0))D

?

dQ(x)=0

]PRNCEh0Lh(X

1; a(h

0))

La?IF(x; a8 , Ph0 )#h(x; a(h

0))DdQ(x)D.

This expression can be simpli"ed by using (A.2), (14) with =0"<~1

0, and the

orthogonal projection property of M0R0

(with respect to the scalar productinduced by R

0), to obtain

mR(P0e,n,Q)"e2n CPRN

(IF(x; a( , Ph0 )!IF(x; a8 , Ph0 ))?dQ(x)R~1

0

]PRN

(IF(x; a( , Ph0 )!IF(x; a8 , Ph0 )) dQ(x)D#o(e2), eP0.


The expression on the right-hand side of this formula is the second-ordervon Mises expansion under the hypotheses (4) and g(a(Ph0 ))"0 of aHausman functional mH : dom(mH)PR`, de"ned by the symmetric formmH(Phn ) :";H(Phn )

?;H(Phn ), where

;H(Phn ) :"R~1@20

[a( (Phn )!a8 (Phn )] (A.3)

(cf. Hausman, 1978; Holly, 1982). As a consequence the di!erence between thelevels under contamination of the likelihood ratio test and a Hausman testde"ned by the critical region MnmH(Phn )*g

1~a0 N, where g1~a0 is the 1!a

0quantile of a s2(r, 0) distribution, is of order o(e2). Hence, the asymptotic biasunder a given P0e,n,Q-contamination of the level of the likelihood ratio test can beequivalently investigated by analyzing that of the Hausman test. Similar argu-ments to those developed above for ;S can be now applied to ;H. The IF ofUH is given by

IF(x; ;H, Ph0 )"!R1@2

0M

0IF(x; a8 , Ph0 )

"R1@2

0M

0R0Eh0

Lh?(X1; a(h

0))

La=

0h(x; a(h

0))

using (A.2) and (14). Furthermore, again by the properties of orthogonal projec-tions, this quantity can be bounded by the self-standardized norm of h asfollows:

DDIF(x; ;H, Ph0 )DD2"KKM0R

0Eh0

Lh?(X1; a(h

0))

La=

0h(x; a(h

0))KK

2

R0

)KKEh0Lh?(X

1; a(h

0))

La=

0h(x; a(h

0))KK

2

R0

)DDh(x; a(h0))DD2

W0.

This proves the theorem for the level functional aR of the likelihood-ratio typetest. h


By the FreH chet di!erentiability of a8 , nmM(Phn ) is asymptotically uniformlys2(r, b(e)) distributed with b( ) )"nDD;M(P1

> ,n,Q)DD2. Moreover, up to order O(1/n)

we have n(P1> ,n,Q

)"1!Hr(g

1~a0 ; b( ) )). A "rst-order Taylor expansion thenyields

n(P1e,n,Q)!n(P!-5g,n)"Ln(P1

> ,n,Q)

Le Ke/0

e#o(e)

up to terms of order O(1/n).


Some calculations then yield

Ln(P1> ,n,Q

)

Le Ke/0

"kLbLe Ke/0

"2knL;M(P1

> ,n,Q)?

Le Ke/0

;M (P!-5g,n)

"2kJnCPRN

IF?(x; ;M, P!-5g,n ) dQ(x)D;M (P!-5g,n) (A.4)

up to terms of order O(1/n). Writing

;M(P!-5g,n)";M(Ph0 )#g

JnPRN

IF(x; ;M, Ph0 ) dPh1 (x)#o(g) (A.5)

we obtain

Dn(P1e,n,Q)!n(P!-5g,n)D)2keg maxMP

!-5g,n _Ph0 Nsupx

DDIF(x; ;M, ) )DD2#o(g)

)2keg maxMP

!-5g,n _Ph0 Nsupx

DDh(x, a8 ( ) ))DD2W0

#o(g) (A.6)

using ;M(Ph0 )"0. This concludes the proof of the theorem. h


The functional ;W is asymptotically equivalent to ;S and ;R at the modelunder the local alternatives given by (41); cf. Gourieroux and Monfort (1989).Moreover the FreH chet di!erentiability of a8 implies that this equivalenceis uniform. It is therefore su$cient to prove the theorem for the functional;W. The statistic nmW(Phn ) is asymptotically uniformly s2(r, b(e)) distributedwith b( ) )"nDD;W(P0

>,n,Q)DD2. Again, up to order O(1/n) we have n(P0

>,n,Q)"1

!Hr(g

1~a0 ; b( ) )). As in the proof of Theorem 2 a "rst-order Taylor expansionthen yields

n(P0e,n,Q)!n(Ph0 )"Ln(P0

>,n,Q)

Le Ke/0

e#o(e) (A.7)

up to terms of order O(1/n).Similar calculations to those in the proof of Theorem 2 then give

Ln(P0> ,n,Q

)

Le Ke/0

"kLbLe Ke/0

"2knL;W(P0

> ,n,Q)?

Le Ke/0

;W(Ph0 )

"2kJn CPRN

IF?(x; ;W, Ph0 ) dQ(x)D;W(Ph0 )

up to terms of order O(1/n).


Expanding ;W(Ph0 ) with respect of D we have

;W(Ph0 )"!CLg

La?(a(h

0))R

0

Lg?

La(a(h

0))D

~1@2 Lg

La?(a(h

0))

D

Jn#oA

D

JnB.(A.8)

Since

IF(x; ;W, Ph0 )"!CLg

La?(a(h

0))R

0

Lg?

La(a(h

0))D

~1@2

]Lg

La?(a(h

0))R

0Eh0

Lh?(X1; a(h

0))

La=

0h(x; a(h

0)) (A.9)

we obtain up to terms of order o(D):

Ln(P0> ,n,Q

)

Le Ke/0

"2kCPRN

h?(x; a(h0)) dQ(x)D=0

Eh0Lh(X

1; a(h

0))

La?R

0

]Lg?

La(a(h

0))C

Lg

La?(a(h

0))R

0

Lg?

La(a(h

0))D

~1 Lg

La?(a(h

0))D.

The Cauchy}Schwarz inequality then gives

KALn(P0

>,n,Q)

Le Ke/0BK)2k KKPRN

h(x; a(h0)) dQ(x)KK

W0

]KKLg?

La(a(h

0))C

Lg

La?(a(h

0))R

0

Lg?

La(a(h

0))D

~1

]Lg

La?(a(h

0))DKKR

0

#o(D).

Using again the properties of orthogonal projections we obtain

KALn(P0

>,n,Q)

Le Ke/0BK)2ksup

xDDh(x; a(h

0))DD

W0DDDDDR~1

0#o(D). (A.10)

Finally, by inserting this expression in the Taylor expansion (A.7) we get


supx

DDh(x; a(h0)DD

W0#o(D). (A.11)

This concludes the proof of the Theorem. h


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