The ability of six alternative bootstrap methods to reduce the bias of GMM parameter estimates is examined in an instrumental variable framework using Monte Carlo analysis. Promising results were found for the two bootstrap estimators suggested in the paper

Ramalho, Joaquim J.S.

Repositório Científico da Universidade de Évora

 UNIVERSIDADE DE ÉVORA  DEPARTAMENTO DE ECONOMIA     DOCUMENTO DE TRABALHO Nº 2005/10 June     Bootstrap bias-adjusted GMM estimators   Joaquim J. S. Ramalho* Universidade de Évora, Departamento de Economia, CEMAPRE                 * The author gratefully acknowledges partial financial support from Fundação para a Ciência e Tecnologia, program POCTI, partially funded by FEDER. Address for correspondence: Joaquim J. S. Ramalho, Universidade de Évora, Departmento de Economia, 7000-803 ÉVORA, Portugal. Tel.: +351-266740894. Fax: +351-266740809. E-mail: jsr@uevora.pt.       UNIVERSIDADE DE ÉVORA DEPARTAMENTO DE ECONOMIA Largo dos Colegiais, 2 – 7000-803 Évora – Portugal Tel.: +351 266 740 894  Fax: +351 266 742 494 www.decon.uevora.pt    wp.economia@uevora.pt    Abstract:  The ability of six alternative bootstrap methods to reduce the bias of GMM parameter estimates is examined in an instrumental variable framework using Monte Carlo analysis. Promising results were found for the two bootstrap estimators suggested in the paper.    Palavras-chave/Keywords: GMM, Bootstrap, Empirical Likelihood, Instrumental Variables, Monte Carlo  Classificação JEL/JEL Classification: C13, C14                      1 IntroductionIt is now widely recognized that the eﬃcient two-step generalized method of moments (GMM) es-timator may have large biases for the sample sizes typically encountered in economic applications;see, for example, the several Monte Carlo studies that appeared in the July 1996 special issue ofthe Journal of Business & Economic Statistics. In this paper we analyze the ability of six alter-native bootstrap procedures to reduce the finite sample bias of GMM parameter estimates. Threeof those alternatives were already proposed by other authors: the standard, nonparametric (NP)bootstrap; Hall and Horowitz’s (1996) recentered nonparametric (RNP) bootstrap; and Brownand Newey’s (2002) constrained empirical likelihood (CEL) bootstrap. Monte Carlo evidence byHorowitz (1998) and Ramalho (2005) shows that application of these bootstrap methods reducesthe bias of the GMM estimator but does not completely eliminate it. Therefore, in this paperwe suggest two alternative bootstrap techniques, both of which use the empirical likelihood (EL)distribution function (see Qin and Lawless, 1994) to generate the bootstrap samples. The finitesample bias of all the corresponding bootstrap bias-corrected GMM estimators are examined inan instrumental variable framework through a Monte Carlo analysis.2 GMM estimationLet yi, i = 1, ..., n, be independent and identically distributed observations on a data vector y, θa k-dimensional vector of parameters of interest, and g (y, θ) an s-dimensional vector of functionsof the observed variables and parameters of interest. Throughout, we assume that s > k and thatthe true parameter vector θ0 uniquely satisfies the moment conditionsEF [g (y, θ0)] = 0, (1)where EF [·] denotes expectation taken with respect to the unknown distribution function F (y).Define gi (θ) ≡ g (yi, θ), i = 1, ..., n, and gn (θ) ≡ n−1Pni=1 gi (θ). Regularity conditions areassumed such that gn (θ)p→ EF [g (y, θ)] and√ngn (θ0)d→ N (0, V ), where the asymptotic variancematrix V ≡ EF£gi (θ0) gi (θ0)0¤is positive definite andp→ and d→ denote convergence in probabilityand convergence in distribution, respectively.The eﬃcient GMM estimator θˆGMM is obtained from minimization of the optimal quadraticform of the sample moment indicatorsQn = gn (θ)0hVn³θ˜´i−1gn (θ) , (2)2where θ˜ is a preliminary consistent estimator for θ0 and V˜n ≡ Vn³θ˜´is a consistent estimatorfor V . Let G ≡ EFh∂gi(θ0)∂θ0i. Under suitable regularity conditions, see Newey and McFad-den (1994), θˆGMM is a consistent, asymptotically normal estimator of θ0,√n³θˆGMM − θ0´d→Nh0, (G0V −1G)−1i, and is asymptotically eﬃcient among all estimators based only on (1).3 The empirical likelihood distribution functionConsider again the moment conditions (1). Implicitly, by giving the same weight (n−1) to eachobservation, GMM uses the empirical distribution function Fn (y) ≡ n−1Pni=1 1 (yi ≤ y) as esti-mate for F (y), where the indicator function 1 (yi ≤ y) is equal to 1 if yi ≤ y and 0 otherwise.However, since the moment conditions (1) are assumed to be satisfied in the population, thisinformation can be exploited in order to obtain a more eﬃcient estimator of F (y). Actually, wemay obtain an alternative estimator for θ in (1) by choosing the estimator θˆ that minimizes thedistance, relatively to some metric, between Fn (y) and a distribution function Fp (y) satisfyingthe moment conditions (1). The distribution Fp (y) is, hence, the member of the class F (θ) of alldistribution functions that satisfy (1), F (θ) ≡ ©Fp : EFp [g (y, θ0)] = 0ª, that is closest to Fn.In the selection of a particular probability measure in F (θ), diﬀerent metrics for the closenessbetween Fp (y) and Fn (y) may be used. The most common choices for the metricM (Fn, Fp) areparticular cases of the Cressie-Read (1984) power-divergence statistic, namelyM (·) =P dFn (y)ln [dFn (y) /dFp (y)] which produces the so-called EL estimator. Thus, the EL estimator θˆEL canbe described as the solution to the programmaxθnXi=1ln pELi subject to pELi ≥ 0,nXi=1pELi = 1 andnXi=1pELi g (yi, θ) = 0, (3)where pELi ≡ dFp (yi), i = 1, ..., n, and the last restriction is an empirical measure counterpart tothe moment conditions (1), imposing them numerically in the sample; for an alternative motiva-tion of EL estimators, see Newey and Smith (2004).From optimization of (3), it is straightforward to show thatpˆELi ≡ pELi³θˆEL, λˆEL´=1nh1 + λˆ0ELg³yi, θˆEL´i , i = 1, ..., n, (4)where θˆEL and λˆEL, the s-vector of Lagrange multipliers associated to the last restriction of (3),result from unconstrained optimization of the saddle function n−1Pni=1 ln [1 + λ0gi (θ)] . Thus,3the EL distribution function Fp (y) is given byFp (y) =nXi=1pˆELi 1 (yi ≤ y) . (5)See Qin and Lawless (1994) for details.4 Alternative bootstrapping for the GMM estimatorAssume that a random sample S of size n is collected from a population whose (unknown)distribution function is F (y). Bootstrap samples are generated by randomly sampling S withreplacement. This resampling is based on a certain distribution function, F ∗ (y), which assignseach observation a given probability of being sampled. In general, using the bootstrap, the biasof the GMM estimator θˆGMM can be estimated as follows: 1) compute θˆGMM by minimizing(2) based on S; 2) generate B bootstrap samples S∗j , j = 1, ..., B, of size n accordingly withthe chosen F ∗ (y): S∗j =©y∗j1, ..., y∗jnª, where y∗ji, i = 1, ..., n, denotes the observations includedin the bootstrap sample S∗j ; 3) for each bootstrap sample calculate the GMM estimator θˆ∗j ≡argminθg∗jn (θ) Vˆ ∗−1jn g∗jn (θ), j = 1, ..., B, where g∗jn (θ) = n−1Pni=1 g¡y∗ji, θ¢and Vˆ ∗−1jn uses a pre-liminary consistent estimator for θ0 based on the bootstrap sample S∗j ; 4) average the B GMMestimators calculated in the preceding step: θ¯∗ = 1BPBj=1 θˆ∗j ; 5) estimate the bias of the GMMestimator θˆ by calculating:bˆ = θ¯∗ − θˆGMM . (6)Subtracting the bias (6) from the GMM estimator θˆGMM , it is then possible to obtain the bias-corrected GMM estimatorθˆBCGMM = 2θˆGMM − θ¯∗. (7)As discussed next, these general procedures may be used to reduce the finite sample bias ofGMM parameter estimates in several distinct forms.4.1 Nonparametric bootstrapThe NP bootstrap is probably the most commonly applied bootstrap technique in econometrics.In this case, the bootstrap samples are generated using the empirical distribution function Fn (y),so each observation has equal probability n−1 of being drawn. However, direct application ofthe NP bootstrap in the GMM framework seems to be unsatisfactory in many cases. Indeed,when the model is overidentified, while the population moment conditions EF [g (y, θ)] = 0 are4satisfied at θ = θ0, the estimated sample moments are typically non-zero, that is, there is no θsuch that EFn [g (y, θ)] = 0 is met, except in very special cases. Therefore, Fn (y) may be a poorapproximation to the true underlying distribution of the data and, hence, the NP bootstrap maynot yield a substantial improvement over first-order asymptotic theory in standard applicationsof GMM.4.2 Recentered nonparametric bootstrapIn order to guarantee that the moment conditions exploited by GMM estimators hold exactlyin each replication of the bootstrap, Hall and Horowitz (1996) suggested using the recenteredmoment indicatorsgc¡y∗j , θ¢= g¡y∗j , θ¢− 1nnXi=1g³yi, θˆGMM´, (8)since EFn£gc¡y∗j , θ¢¤= 0. To implement this RNP bootstrap method some adaptations must bemade to the general procedures described earlier. Namely, in step 1 we have to calculate alsogn³θˆGMM´and in step 3 GMM estimation is now based on the recentered moment indicators(8), with the weight matrix specified accordingly.4.3 Constrained empirical likelihood bootstrapInstead of recentering the moment conditions and keeping Fn (y) as resampling distribution,Brown and Newey (2002) suggested generating the bootstrap samples using a diﬀerent distribu-tion, say F1 (y), such that EF1hg³y, θˆGMM´i= 0. Namely, they proposed the employment of aconstrained version of the EL distribution function (5), which is given byF cp (y) =nXi=1pˆCELi 1 (yi ≤ y) , (9)wherepˆCELi =1nh1 + λˆ0CELg³yi, θˆGMM´i , i = 1, ..., n, (10)and λˆCEL results from maximization of n−1Pni=1 lnh1 + λ0gi³θˆGMM´i; in other words, F cp (y)results from solving the program (3) conditional on θ = θˆGMM . SincePni=1 pˆCELi gi³θˆGMM´=0 is the first-order condition characterizing λˆCEL, this CEL bootstrap imposes, in eﬀect, themoment conditions, evaluated at θˆGMM , on the sample: EFCELhg³y, θˆGMM´i= 0.Brown and Newey (2002) proved that the CEL bootstrap is asymptotically eﬃcient relativeto the NP and RNP methods, since F cp (y) is a more eﬃcient estimator of F (y) than Fn (y).54.4 Recentered empirical likelihood bootstrapThe two bootstrap methods that we propose in this paper are based on the EL distributionFp (y) given in (5). Although they are not expected to be more eﬃcient than the CEL bootstrap,the fact that Fp (y) is used instead of F cp (y) may lead to better results in finite samples for tworeasons: first, the former distribution do not result from an optimization conditional on θ =θˆGMM as the latter; second, there are some Monte Carlo evidence suggesting that θˆEL displaysless bias than θˆGMM in small samples; see inter alia Ramalho (2005).As before, some correction seems to be necessary to apply this EL bootstrap to the GMMestimator, sincePni=1 pˆELi g³yi, θˆGMM´6= 0 in general. Analogously to Hall and Horowitz (1996),we suggest using the recentered moment indicatorsgc¡y∗j , θ¢= g¡y∗j , θ¢−nXi=1pˆELi g³yi, θˆGMM´, (11)since EFEL£gc¡y∗j , θ¢¤= 0. This recentered EL (REL) bootstrap can be implemented applyingsimilar procedures to those described for the RNP method, with only two (obvious) alterations:Fp (y) is used instead of Fn (y) and (11) instead of (8).4.5 Post-hoc empirical likelihood bootstrapThe expected failure of the EL bootstrap in providing significantly less biased GMM estimatorscan be also explained as follows. Let pEL =¡pˆEL1 , ..., pˆELn¢be the n-dimensional resampling vectorthat assigns each observation a given probability of being sampled in the EL bootstrap. By usingthis resampling vector and estimating the bias utilizing the formula given in (6), we are notadequately estimating the bias of the GMM estimator that we intended to correct. Actually, inthe calculation of (6), we are comparing GMM estimators that can be based on quite distinctsamples: while θˆGMM results from the minimization of the quadratic form (2), θ¯∗ is the averageof the standard GMM estimators θˆj, j = 1, ..., B, each of which, due to the way the bootstrapsamples are constructed, can be interpreted as minimizing also (2) but with gn (θ) replaced bygp (θ) ≡Pni=1 pˆELi g (yi, θ), which, in small samples, can be rather diﬀerent. Based on thesearguments, we suggest below the post-hoc EL (PHEL) bootstrap, which uses a post-samplingadjustment to the EL bootstrap GMM estimator.1Define paj ≡¡paj1, ..., pajn¢as the actual or post-resampling vector calculated from the bootstrapsample S∗j , that is paji = #©y∗ji = yiª±n is the proportion of times that the i-th original data1For other applications of post-sampling adjustments, see Efron (1990).6point appeared in the bootstrap sample S∗j . Define also the average post-resampling vector p¯a ≡(p¯a1, ..., p¯an) = B−1PBj=1 paj . In this framework, the j-th bootstrap estimator θ¯∗j can be expressedas a function of the j-th post-resampling vector: θ¯∗j = θ¡paj¢. Similarly, we have for the originalGMM estimator θˆGMM = θ (p0), where p0 = (n−1, ..., n−1). Define also θˆa= θ (p¯a) as the GMMestimator resultant from the application of the average post-sampling probabilities p¯a, i.e. basedon g¯a (θ) =Pni=1 p¯ai g (yi, θ).Instead of using bˆ = θ¯∗− θ (p0), we propose the calculation of the bias of the GMM estimatoras:b¯ = θ¯∗ − θ (p¯a) . (12)The intuition behind this is the following. Although the theoretical expectation of the resamplingvector pEL is p0, its actual average is p¯a. Thus, using θ (p¯a) instead of θ (p0) in the estimationof the bias, we might be able to correct for this discrepancy. In fact, in (12), we are eﬀectivelycomparing GMM estimators based on similar samples, in opposition to what was happeningbefore. The bias-corrected GMM estimator is then found by calculating:θˆBCGMM = θˆGMM − θ¯∗+ θˆa. (13)When both n and B go to infinity, θˆa will converge to θˆGMM , so asymptotically this method willproduce the same results as the other bootstrap techniques discussed in the previous sections.Note that we could have also opted for estimating the bias by b¯ = θ¯∗ − θ¡pˆEL¢, since p¯a ' pˆEL.However, the utilization of the post-resampling probabilities are expected to provide a slightfurther improvement.In terms of procedures, the algorithm presented earlier must be modified as follows. In step3, for each bootstrap sample, in addition to the GMM estimator θ¯∗j , we calculate also paj . In step4, the average post-resampling vector p¯a needs also to be calculated. In the final step, we needto obtain θˆa and, instead of (6), the bias is calculated according to (12).5 Monte Carlo simulationConsider the linear instrumental variable model described by equationsYi = θ0 ·Xi + i,Xi =sXj=1π · Zij + ui,7where Yi and Xi denote the dependent variable and an exogenous regressor, respectively. Allthe instruments Zij are i.i.d. N (0, Is) variables, while (i, ui)0 is N (0,Ω), where Ω is a (2× 2)-matrix with diagonal and oﬀ diagonal elements 1 and ρ, respectively. We considered three diﬀerentdegrees of non-orthogonality betweenXi and i, ρ = (0.25, 0.50, 0.75). Let R2f = sπ2/ (sπ2 + 1) bethe theoretical R2 of the first stage regression, which measures the overall fit of the instrumentsto the endogenous regressor Xi. We fix s = 10 and set the value of π in such a way thatR2f = (0.15, 0.30) in all the experiments. The value of θ0 was fixed in order to keep constant theoverall fit of Yi to Xi in the structural equation (R2 = 0.5). For each one of the 6 parametercombinations of s, R2f , and ρ we generated 5000 Monte Carlo samples of size n = 200.In Table 1 we report for each estimator the mean and median bias, the median absolute error(MAE), and the standard error (SE) across replications. As expected, the bias of the GMMestimator increases with the endogeneity of the model and decreases with the strength of theinstruments. The same pattern can be observed for all the bootstrap GMM estimators. Theutilization of any one of the bootstrap methods allows the bias of the GMM estimator to besubstantially reduced, although at the expense of an increment in its dispersion. Clearly, the twoestimators suggested in this paper display the best performances in terms of mean and medianbias, particularly the PHEL bootstrap, which produces the only estimator which is approximatelymean unbiased in all cases. Conversely, the Monte Carlo distribution of this estimator is slightlymore disperse. Overall, these results suggest that the estimators developed in this paper will beuseful, at least, in settings similar to those replicated in this Monte Carlo study.8ReferencesCressie, N. and T.R.C. Read, 1984, Multinomial goodness-of-fit tests, Journal of the RoyalStatistical Society Series B 46(3), 440-464.Brown, B.W. and W.K. Newey, 2002, Generalised Method of Moments, Eﬃcient Bootstrapping,and Improved Inference, Journal of Business and Economic Statistics 20, 507-517.Efron, B., 1990, More eﬃcient bootstrap computations, Journal of the American StatisticalAssociation 85(409), 79-89.Hall, P. and J.L. Horowitz, 1996, Bootstrap critical values for tests based on generalised-method-of-moments estimators, Econometrica 64(4), 891-916.Horowitz, J.L., 1998, Bootstrap methods for covariance structures, Journal of Human Resources33(1), 39-61.Newey, W.K. and D. McFadden, 1994, Large sample estimation and hypothesis testing, in:R.F. Engle and D.L. McFadden, eds., Handbook of Econometrics, Vol. 4 (North-Holland,Amsterdam) 2111-2245.Newey, W.K. and R.J. Smith, 2004, Higher order properties of GMM and generalized empiricallikelihood estimators, Econometrica 72(1), 219-255.Qin, J. and J. Lawless, 1994, Empirical likelihood and general estimating equations, Annals ofStatistics 22(1), 300-325.Ramalho, J.J.S., 2005, Small sample bias of alternative estimation methods for moment conditionmodels: Monte Carlo evidence for covariance structures, Studies in Nonlinear Dynamics andEconometrics, forthcoming.9Table 1: Monte Carlo results (5000 replications; n = 200; s = 10)Estimator Bias MAE SEmean medianρ = 0.25, R2f = 0.15GMM .050 .054 .112 .158NP .019 .027 .123 .189RNP .019 .027 .121 .187CEL .019 .026 .122 .186REL .016 .023 .122 .191PHEL .004 .017 .127 .214ρ = 0.25, R2f = 0.3GMM .023 .026 .074 .108NP .006 .009 .078 .119RNP .006 .009 .077 .117CEL .006 .009 .077 .117REL .005 .008 .078 .118PHEL .004 .007 .078 .119ρ = 0.5, R2f = 0.15GMM .099 .106 .131 .153NP .036 .049 .127 .187RNP .036 .050 .125 .185CEL .037 .049 .124 .183REL .026 .041 .126 .191PHEL .004 .027 .129 .214ρ = 0.5, R2f = 0.3GMM .045 .049 .080 .106NP .011 .018 .078 .119RNP .011 .017 .078 .118CEL .010 .016 .077 .117REL .008 .014 .078 .119PHEL .005 .012 .078 .120ρ = 0.75, R2f = 0.15GMM .147 .157 .165 .142NP .052 .070 .132 .184RNP .053 .072 .131 .181CEL .057 .076 .131 .177REL .033 .054 .129 .190PHEL .001 .031 .132 .214ρ = 0.75, R2f = 0.3GMM .067 .073 .090 .102NP .016 .025 .081 .119RNP .016 .026 .079 .118CEL .016 .024 .079 .117REL .009 .018 .079 .120PHEL .005 .015 .080 .12210

Bootstrap bias-adjusted GMM estimators

Abstract

Similar works

Full text

Available Versions

Repositório Científico da Universidade de Évora