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Unformatted text preview: Notes on Generalized Method of Moments (GMM) 1 Je f Thurk University of Texas at Austin Introduction This note is intended to introduce the reader to estimation using generalized method of moments (GMM), providing basic intuition into why and how the estimation procedure works. GMM provides the researcher with a f exible platform for estimation on a broad array of topics, data sets, and hypotheses. Further, one may view many of the estimation procedures you learn in the econometrics sequence (OLS, IV, ML) as GMM estimators, therefore, understanding this procedure provides a solid econometric foundation. Framework What is a ”Moment”? By now you’ve probably heard the word ”moment” thrown around in a number of di f erent circumstances, seemingly at whim. You can think of a moment as a descriptive statistic of any distribution of data (e.g., mean, variance, skew). For our purposes here, we require that the researcher construct a moment that has mean zero, which is not much of restriction since we can often make any data series center around zero by shifting, detrending, etc. De f nition of the GMM Estimator Suppose we have a data set x i , where i = 1 , ..., n drawn from an unknown probability distribution P and we know that the parameter vector θ ∈ Θ satis F es the following moment condition E [ ψ ( x i , θ )] = 0 (1) for some known function ψ ( · ). Note that we’ve placed little restriction on ψ ( · ) and θ , and these can be vectors of di f erent degrees. The basic idea behind GMM is to construct ψ ( · ) to form a valid moment condition and use the sample data x i to form a sample analog of E [ · ] = 0 using the law of large numbers. The researcher then chooses parameters ˆ θ to solve 1 n n X i =1 ψ ( x i , θ ) = 0 (2) If ψ ( · ) is q x 1 vector then we have q moment conditions, requiring that our parameter vector θ (a p x 1 vector ) solve (2) for each. Heuristically, you can think of (2) identifying a system of q equations and p unknowns. If q=p, then we call the system ”just identi F ed” and there may be a unique solution. If q > p, then we call the system ”overidenti...
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This note was uploaded on 08/06/2008 for the course ECON 387 taught by Professor Corbae during the Spring '07 term at University of Texas.
- Spring '07