GMM with Many Moment Conditions

This paper provides a first order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak or uninformed) instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter, and conditions for consistent GMM estimation are given. A general framework for the GMM limit distribution theory is developed based on epiconvergence methods. Some illustrations are provided, including consistent GMM estimation of a panel model with time varying individual effects, consistent limited information maximum likelihood estimation as a continuously updated GMM estimator, and consistent IV structural estimation using large numbers of weak or irrelevant instruments. Some simulations are reported.     

GMM,GEL, Serial Correlation,and Asymptotic Bias

For stationary time series models with serial correlation, we consider generalized method of moments (GMM) estimators that use heteroskedasticity and autocorrelation consistent (HAC) positive definite weight matrices and generalized empirical likelihood (GEL) estimators based on smoothed moment conditions. Following the analysis of Newey and Smith (2004) for independent observations, we derive second order asymptotic biases of these estimators. The inspection of bias expressions reveals that the use of smoothed GEL, in contrast to GMM, removes the bias component associated with the correlation between the moment function and its derivative, while the bias component associated with third moments depends on the employed kernel function. We also analyze the case of no serial correlation, and find that the seemingly unnecessary smoothing and HAC estimation can reduce the bias for some of the estimators.     

Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators

In an effort to improve the small sample properties of generalized method of moments (GMM) estimators, a number of alternative estimators have been suggested. These include empirical likelihood (EL), continuous updating, and exponential tilting estimators. We show that these estimators share a common structure, being members of a class of generalized empirical likelihood (GEL) estimators. We use this structure to compare their higher order asymptotic properties. We find that GEL has no asymptotic bias due to correlation of the moment functions with their Jacobian, eliminating an important source of bias for GMM in models with endogeneity. We also find that EL has no asymptotic bias from estimating the optimal weight matrix, eliminating a further important source of bias for GMM in panel data models. We give bias corrected GMM and GEL estimators. We also show that bias corrected EL inherits the higher order property of maximum likelihood, that it is higher order asymptotically efficient relative to the other bias corrected estimators.     

Fixed‐Smoothing Asymptotics in a Two‐Step Generalized Method of Moments Framework

This paper develops the fixed‐smoothing asymptotics in a two‐step generalized method of moments (GMM) framework. Under this type of asymptotics, the weighting matrix in the second‐step GMM criterion function converges weakly to a random matrix and the two‐step GMM estimator is asymptotically mixed normal. Nevertheless, the Wald statistic, the GMM criterion function statistic, and the Lagrange multiplier statistic remain asymptotically pivotal. It is shown that critical values from the fixed‐smoothing asymptotic distribution are high order correct under the conventional increasing‐smoothing asymptotics. When an orthonormal series covariance estimator is used, the critical values can be approximated very well by the quantiles of a noncentral F distribution. A simulation study shows that statistical tests based on the new fixed‐smoothing approximation are much more accurate in size than existing tests.     

GMM Estimation of Autoregressive Roots Near Unity with Panel Data

This paper investigates a generalized method of moments (GMM) approach to the estimation of autoregressive roots near unity with panel data and incidental deterministic trends. Such models arise in empirical econometric studies of firm size and in dynamic panel data modeling with weak instruments. The two moment conditions in the GMM approach are obtained by constructing bias corrections to the score functions under OLS and GLS detrending, respectively. It is shown that the moment condition under GLS detrending corresponds to taking the projected score on the Bhattacharya basis, linking the approach to recent work on projected score methods for models with infinite numbers of nuisance parameters (Waterman and Lindsay (1998)). Assuming that the localizing parameter takes a nonpositive value, we establish consistency of the GMM estimator and find its limiting distribution. A notable new finding is that the GMM estimator has convergence rate , slower than , when the true localizing parameter is zero (i.e., when there is a panel unit root) and the deterministic trends in the panel are linear. These results, which rely on boundary point asymptotics, point to the continued difficulty of distinguishing unit roots from local alternatives, even when there is an infinity of additional data.     

Consistent Estimation of Models Defined by Conditional Moment Restrictions

In econometrics, models stated as conditional moment restrictions are typically estimated by means of the generalized method of moments (GMM). The GMM estimation procedure can render inconsistent estimates since the number of arbitrarily chosen instruments is finite. In fact, consistency of the GMM estimators relies on additional assumptions that imply unclear restrictions on the data generating process. This article introduces a new, simple and consistent estimation procedure for these models that is directly based on the definition of the conditional moments. The main feature of our procedure is its simplicity, since its implementation does not require the selection of any user‐chosen number, and statistical inference is straightforward since the proposed estimator is asymptotically normal. In addition, we suggest an asymptotically efficient estimator constructed by carrying out one Newton–Raphson step in the direction of the efficient GMM estimator.     

Testing Parameters in GMM Without Assuming that They Are Identified

We propose a generalized method of moments (GMM) Lagrange multiplier statistic, i.e., the K statistic, that uses a Jacobian estimator based on the continuous updating estimator that is asymptotically uncorrelated with the sample average of the moments. Its asymptotic χ² distribution therefore holds under a wider set of circumstances, like weak instruments, than the standard full rank case for the expected Jacobian under which the asymptotic χ² distributions of the traditional statistics are valid. The behavior of the K statistic can be spurious around inflection points and maxima of the objective function. This inadequacy is overcome by combining the K statistic with a statistic that tests the validity of the moment equations and by an extension of Moreira's (2003) conditional likelihood ratio statistic toward GMM. We conduct a power comparison to test for the risk aversion parameter in a stochastic discount factor model and construct its confidence set for observed consumption growth and asset return series.     

A Conditional Likelihood Ratio Test for Structural Models

This paper develops a general method for constructing exactly similar tests based on the conditional distribution of nonpivotal statistics in a simultaneous equations model with normal errors and known reduced‐form covariance matrix. These tests are shown to be similar under weak‐instrument asymptotics when the reduced‐form covariance matrix is estimated and the errors are non‐normal. The conditional test based on the likelihood ratio statistic is particularly simple and has good power properties. Like the score test, it is optimal under the usual local‐to‐null asymptotics, but it has better power when identification is weak.     

Generalized Method of Integrated Moments for High‐Frequency Data

We propose a semiparametric two‐step inference procedure for a finite‐dimensional parameter based on moment conditions constructed from high‐frequency data. The population moment conditions take the form of temporally integrated functionals of state‐variable processes that include the latent stochastic volatility process of an asset. In the first step, we nonparametrically recover the volatility path from high‐frequency asset returns. The nonparametric volatility estimator is then used to form sample moment functions in the second‐step GMM estimation, which requires the correction of a high‐order nonlinearity bias from the first step. We show that the proposed estimator is consistent and asymptotically mixed Gaussian and propose a consistent estimator for the conditional asymptotic variance. We also construct a Bierens‐type consistent specification test. These infill asymptotic results are based on a novel empirical‐process‐type theory for general integrated functionals of noisy semimartingale processes.     

Efficient Derivative Pricing by the Extended Method of Moments

In this paper, we introduce the extended method of moments (XMM) estimator. This estimator accommodates a more general set of moment restrictions than the standard generalized method of moments (GMM) estimator. More specifically, the XMM differs from the GMM in that it can handle not only uniform conditional moment restrictions (i.e., valid for any value of the conditioning variable), but also local conditional moment restrictions valid for a given fixed value of the conditioning variable. The local conditional moment restrictions are of special relevance in derivative pricing to reconstruct the pricing operator on a given day by using the information in a few cross sections of observed traded derivative prices and a time series of underlying asset returns. The estimated derivative prices are consistent for a large time series dimension, but a fixed number of cross sectionally observed derivative prices. The asymptotic properties of the XMM estimator are nonstandard, since the combination of uniform and local conditional moment restrictions induces different rates of convergence (parametric and nonparametric) for the parameters.     

Random Effects Estimators with many Instrumental Variables

In this paper we propose a new estimator for a model with one endogenous regressor and many instrumental variables. Our motivation comes from the recent literature on the poor properties of standard instrumental variables estimators when the instrumental variables are weakly correlated with the endogenous regressor. Our proposed estimator puts a random coefficients structure on the relation between the endogenous regressor and the instruments. The variance of the random coefficients is modelled as an unknown parameter. In addition to proposing a new estimator, our analysis yields new insights into the properties of the standard two‐stage least squares (TSLS) and limited‐information maximum likelihood (LIML) estimators in the case with many weak instruments. We show that in some interesting cases, TSLS and LIML can be approximated by maximizing the random effects likelihood subject to particular constraints. We show that statistics based on comparisons of the unconstrained estimates of these parameters to the implicit TSLS and LIML restrictions can be used to identify settings when standard large sample approximations to the distributions of TSLS and LIML are likely to perform poorly. We also show that with many weak instruments, LIML confidence intervals are likely to have under‐coverage, even though its finite sample distribution is approximately centered at the true value of the parameter. In an application with real data and simulations around this data set, the proposed estimator performs markedly better than TSLS and LIML, both in terms of coverage rate and in terms of risk.     

Distortions of Asymptotic Confidence Size in Locally Misspecified Moment Inequality Models

This paper studies the behavior, under local misspecification, of several confidence sets (CSs) commonly used in the literature on inference in moment (in)equality models. We propose the amount of asymptotic confidence size distortion as a criterion to choose among competing inference methods. This criterion is then applied to compare across test statistics and critical values employed in the construction of CSs. We find two important results under weak assumptions. First, we show that CSs based on subsampling and generalized moment selection (Andrews and Soares (2010)) suffer from the same degree of asymptotic confidence size distortion, despite the fact that asymptotically the latter can lead to CSs with strictly smaller expected volume under correct model specification. Second, we show that the asymptotic confidence size of CSs based on the quasi‐likelihood ratio test statistic can be an arbitrary small fraction of the asymptotic confidence size of CSs based on the modified method of moments test statistic.     

Consistent Estimation with a Large Number of Weak Instruments

This paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables (IV) regression when the available instruments are weak and the number of instruments, K_n, goes to infinity with the sample size. We show that consistent estimation depends importantly on the strength of the instruments as measured by r_n, the rate of growth of the so‐called concentration parameter, and also on K_n. In particular, when K_n→∞, the concentration parameter can grow, even if each individual instrument is only weakly correlated with the endogenous explanatory variables, and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature. Hence, the use of many weak instruments may actually improve the performance of certain point estimators. More specifically, we find that the limited information maximum likelihood (LIML) estimator and the bias‐corrected two‐stage least squares (B2SLS) estimator are consistent when , while the two‐stage least squares (2SLS) estimator is consistent only if K_n/r_n→0 as n→∞. These consistency results suggest that LIML and B2SLS are more robust to instrument weakness than 2SLS.     

矩约束模型的最优矩条件选取方法

大量经济、金融以及企业管理等领域研究对象的行为特征可以通过矩约束模型来刻画。然而,该模型中参数的估计对矩条件的选取非常敏感。如何选取最优的矩条件,进而得到更准确的参数估计和更精确的统计推断,是实证研究面临的重要问题。本文从估计量均方误差（MSE）最小的角度,研究了一般矩约束模型两步有效广义矩（GMM）估计的最优矩条件选取方法。首先,利用迭代的方法,推导出两步有效GMM估计的高阶MSE,然后通过Nagar分解,求出了两步有效GMM估计量的近似MSE。接着,基于近似MSE表达式,给出了两步有效GMM估计矩条件选取准则的一般理论,即定义了最优的矩条件,提出了两步有效GMM估计的最优矩条件选取准则,并证明了选取准则的渐近有效性。模拟结果表明,本文提出的矩条件选取方法能够很好地改善两步有效GMM估计量的有限样本表现,降低估计量的有效样本偏差。本研究为实证研究中面临的矩条件选择问题提供了理论依据。     

Empirical Likelihood‐Based Inference in Conditional Moment Restriction Models

This paper proposes an asymptotically efficient method for estimating models with conditional moment restrictions. Our estimator generalizes the maximum empirical likelihood estimator (MELE) of Qin and Lawless (1994). Using a kernel smoothing method, we efficiently incorporate the information implied by the conditional moment restrictions into our empirical likelihood‐based procedure. This yields a one‐step estimator which avoids estimating optimal instruments. Our likelihood ratio‐type statistic for parametric restrictions does not require the estimation of variance, and achieves asymptotic pivotalness implicitly. The estimation and testing procedures we propose are normalization invariant. Simulation results suggest that our new estimator works remarkably well in finite samples.     

Functional Differencing

In nonlinear panel data models, the incidental parameter problem remains a challenge to econometricians. Available solutions are often based on ingenious, model‐specific methods. In this paper, we propose a systematic approach to construct moment restrictions on common parameters that are free from the individual fixed effects. This is done by an orthogonal projection that differences out the unknown distribution function of individual effects. Our method applies generally in likelihood models with continuous dependent variables where a condition of non‐surjectivity holds. The resulting method‐of‐moments estimators are root‐N consistent (for fixed T) and asymptotically normal, under regularity conditions that we spell out. Several examples and a small‐scale simulation exercise complete the paper.     

GMM with Weak Identification

This paper develops asymptotic distribution theory for GMM estimators and test statistics when some or all of the parameters are weakly identified. General results are obtained and are specialized to two important cases: linear instrumental variables regression and Euler equations estimation of the CCAPM. Numerical results for the CCAPM demonstrate that weak‐identification asymptotics explains the breakdown of conventional GMM procedures documented in previous Monte Carlo studies. Confidence sets immune to weak identification are proposed. We use these results to inform an empirical investigation of various CCAPM specifications; the substantive conclusions reached differ from those obtained using conventional methods.     

The Time Series and Cross‐Section Asymptotics of Dynamic Panel Data Estimators

In this paper we derive the asymptotic properties of within groups (WG), GMM, and LIML estimators for an autoregressive model with random effects when both T and N tend to infinity. GMM and LIML are consistent and asymptotically equivalent to the WG estimator. When T/N→ 0 the fixed T results for GMM and LIML remain valid, but WG, although consistent, has an asymptotic bias in its asymptotic distribution. When T/N tends to a positive constant, the WG, GMM, and LIML estimators exhibit negative asymptotic biases of order 1/T, 1/N, and 1/(2N−T), respectively. In addition, the crude GMM estimator that neglects the autocorrelation in first differenced errors is inconsistent as T/N→c>0, despite being consistent for fixed T. Finally, we discuss the properties of a random effects pseudo MLE with unrestricted initial conditions when both T and N tend to infinity.     

Firm‐level Evidence on Gender Wage Discrimination in the Belgian Private Economy

In this paper we explore a matched employer–employee data set to investigate the presence of gender wage discrimination in the Belgian private economy labour market. Contrary to many existing papers, we analyse gender wage discrimination using an independent productivity measure. Using firm‐level data, we are able to compare direct estimates of a gender productivity differential with those of a gender wage differential. We take advantage of the panel structure to identify gender‐related differences from within‐firm variation. Moreover, inspired by recent developments in the production function estimation literature, we address the problem of endogeneity of the gender mix using a structural production function estimator alongside instrumental variable‐general method of moments (IV‐GMM) methods where lagged value of labour inputs are used as instruments. Our results suggest that there is no gender wage discrimination inside private firms located in Belgium, on the contrary.     

Conditional Linear Combination Tests for Weakly Identified Models

We introduce the class of conditional linear combination tests, which reject null hypotheses concerning model parameters when a data‐dependent convex combination of two identification‐robust statistics is large. These tests control size under weak identification and have a number of optimality properties in a conditional problem. We show that the conditional likelihood ratio test of Moreira, 2003 is a conditional linear combination test in models with one endogenous regressor, and that the class of conditional linear combination tests is equivalent to a class of quasi‐conditional likelihood ratio tests. We suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter. These plug‐in tests perform well in simulation and have optimal power in many strongly identified models, thus allowing powerful identification‐robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong.