Empirical likelihood estimation and inference

Abstract

Introduction Since Hansen's (1982) seminal paper, the generalised method of moments (GMM) has become an increasingly important method of estimation in econometrics. Given assumed population moment conditions, the GMM estimation method minimises a quadratic form in the sample counter-parts of these moment conditions. The quadratic form is constructed using a positive definite metric. If this metric is chosen as the inverse of a positive semi-definite consistent estimator for the asymptotic variance matrix of the sample moments, then the resultant GMM estimator is asymptotically efficient. Estimation using GMM is semi-parametric and, therefore, a particular advantage of GMM is that it imposes less stringent assumptions than, for example, the method of maximum like-lihood (ML). Although consequently more robust, GMM is of course generally less efficient than ML. There is some Monte Carlo evidence indicating that GMM estimation may be biased in small samples where the bias seems to arise mainly from the metric used. For example, Altonji and Segal (1996) suggest that GMM estimators using an identity matrix metric may perform better in finite samples than an efficient GMM estimator obtained using the inverse of the estimated asymptotic variance matrix of the sample moments. When the variables of interest in the data-generation process (DGP) are independent and identically distributed, an important recent paper (Qin and Lawless (1994)) shows that it is possible to embed the components of the sample moments used in GMM estimation in a non-parametric likelihood function; namely, an empirical likelihood (EL) framework. The resultant maximum EL estimator (MELE) shares the same first-order properties as Hansen's (1982) efficient GMM estimator.