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Empirical Likelihood Lecture

# Empirical Likelihood Lecture - Imbens/Wooldridge Lecture...

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Imbens/Wooldridge, Lecture Notes 15, Summer ’07 1 What’s New in Econometrics NBER, Summer 2007 Lecture 15, Wednesday, Aug 1st, 4.30-5.30pm Generalized Method of Moments and Empirical Likelihood 1. Introduction Generalized Method of Moments (henceforth GMM) estimation has become an important unifying framework for inference in econometrics in the last twenty years. It can be thought of as nesting almost all the common estimation methods such as maximum likelihood, or- dinary least squares, instrumental variables and two–stage–least–squares and nowadays it is an important part of all advanced econometrics text books (Gallant, 1987; Davidson and McKinnon, 1993; Hamilton, 1994; Hayashi, 2000; Mittelhammer, Judge, and Miller, 2000; Ruud, 2000; Wooldridge, 2002). Its formalization by Hansen (1982) centers on the presence of known functions, labelled “moment functions”, of observable random variables and un- known parameters that have expectation zero when evaluated at the true parameter values. The method generalizes the “standard” method of moments where expectations of known functions of observable random variables are equal to known functions of the unknown pa- rameters. The “standard” method of moments can thus be thought of as a special case of the general method with the unknown parameters and observed random variables entering additively separable. The GMM approach links nicely to economic theory where orthogonal- ity conditions that can serve as such moment functions often arise from optimizing behavior of agents. For example, if agents make rational predictions with squared error loss, their prediction errors should be orthogonal to elements of the information set. In the GMM framework the unknown parameters are estimated by setting the sample averages of these moment functions, the “estimating equations,” as close to zero as possible. The framework is sufficiently general to deal with the case where the number of moment functions is equal to the number of unknown parameters, the so–called “just–identified case”, as well as the case where the number of moments exceeding the number of parameters to be estimated, the “over–identified case.” The latter has special importance in economics where

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Imbens/Wooldridge, Lecture Notes 15, Summer ’07 2 the moment functions often come from the orthogonality of potentially many elements of the information set and prediction errors. In the just-identified case it is typically possible to estimate the parameter by setting the sample average of the moments exactly equal to zero. In the over-identified case this is not feasible. The solution proposed by Hansen (1982) for this case, following similar approaches in linear models such as two– and three– stage–least–squares, is to set a linear combination of the sample average of the moment functions equal to zero, with the dimension of the linear combination equal to the number of unknown parameters. The optimal linear combination of the moments depends on the
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Empirical Likelihood Lecture - Imbens/Wooldridge Lecture...

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