Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

# Further Results on the Limiting Distribution of GMM Sample Moment Conditions

Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

# Further Results on the Limiting Distribution of GMM Sample Moment Conditions

## Article excerpt

Working Paper 2010-11 July 2010

Abstract: In this paper, we extend the results in Hansen (1982) regarding the asymptotic distribution of generalized method of moments (GMM) sample moment conditions. In particular, we show that the part of the scaled sample moment conditions that gives rise to degeneracy in the asymptotic normal distribution is T-consistent and has a nonstandard limiting distribution. We derive the asymptotic distribution for a given linear combination of the sample moment conditions and show how to conduct statistical inference. We demonstrate the finite-sample properties of the proposed asymptotic approximation using simulation.

JEL classification: C13, C32, G12

Key words: GMM

1. INTRODUCTION

Over the past thirty years, the generalized method of moments (GMM) has established itself as arguably the most popular method for estimating economic models defined by a set of moment conditions. In his seminal paper, Hansen (1982) developed the asymptotic distributions of the GMM estimator, sample moment conditions, and test of over-identifying restrictions for possibly nonlinear models with sufficiently general dependence structure. This large sample theory proved to cover a large class of models and estimators that are of interest to researchers in economics and finance.

There are cases, however, in which the root-T convergence and asymptotic normality of the GMM sample moment conditions and estimators based on these moment conditions do not accurately characterize their limiting behavior. For example, Gospodinov, Kan, and Robotti (2010) demonstrate that some GMM estimators, which are functions of the sample moment conditions, are proportional to the GMM objective function and, hence, cannot be root-T consistent and asymptotically normally distributed for correctly specified models. This situation is directly related to the results in Lemma 4.1 and its subsequent discussion in Hansen (1982) which correctly point out that the covariance matrix of the sample moment conditions is singular.

In this paper, we study the case that gives rise to degeneracy in the asymptotic approximation in Lemma 4.1 of Hansen (1982) and establish the appropriate limiting theory. Interestingly, we show that in this case, the scaled sample moment conditions evaluated at the GMM estimator are characterized by a non-standard asymptotic behavior. In particular, we demonstrate that the estimated GMM moment conditions converge to zero (the value implied by the population moment conditions) at rate T and are asymptotically distributed as a product of jointly normally distributed random vectors.

The rest of this paper is organized as follows. Section 2 introduces the general framework and notation and discusses some motivating examples that illustrate the discontinuity in the asymptotic approximation of the sample moment conditions. This section also provides the main theoretical results on the limiting behavior of linear combinations of sample moment conditions and presents an easy-to-implement rank test that determines which asymptotic approximation should be used. Section 3 reports simulation results based on a problem in empirical asset pricing and Section 4 concludes.

2. ASYMPTOTICS FOR GMM SAMPLE MOMENT CONDITIONS

2.1. NOTATION AND ANALYTICAL FRAMEWORK

Let [theta] [member of] [THETA] denote a p x 1 parameter vector of interest with true value [[theta].sub.0] that lies in the interior of the parameter space [THETA] and [g.sub.t] ([theta]) be a known function {g : [R.sub.p] [right arrow] [R.sup.m], m > p} of the data and [theta] that satisfies the set of population orthogonality conditions

E[[g.sub.t] ([[theta].sub.o]) = [o.sub.m]. (1)

The GMM estimator of [[theta].sub.o] is defined as

[??] = [argmin.sub.[theta][member of][THETA][bar.g]T]'[W.sub.T][bar.g]T([theta]), (2)

where [W.sub.T] is an m x m positive-definite weight matrix and

[bar. …

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