EC2610 Fall 2004 GMM Notes for EC2610 1 Introduction These notes povide an introduction to GMM estimation. Their primary purpose is to make the reader familiar enough with gmm to be able to solve problem set assignments. For the more theoretical foundations, properties and extensions of GMM, or to better understand its workings, interested reader should consult any Hayashi, Hamilton, etc, as well as the original GMM article by Hansen(1982) Available lecture notes for graduate econometrics courses, e. g. by Chamberlain (Ec 2140), by Pakes and Porter(Ec 2144), also contain very useful reviews of GMM Generalized Method of Moments provides asymptotic properties for estima- tors and is general enough to include many other commonly used techniques like Ols and ML Having such an umbrella to encompass many of the estimators is very useful, as one doesn't have to derive each estimator property separately With such a wide range, it is not surprising to see GMM used extensively, but one should also be careful when it is appropriate to apply. Since GMm deals with asymptotic properties, it works well for large samples, but does not pro- vide an answer when the samply size is small, or what is "largeenough sample ize. Also, when applying GMm, one may forgo certain desirable properties like efiiciency. 2 GMM Framework 2.1 Definition of GMM Estimator Let i, i=1,., n be i i.d. random draws from the unknown population distri- bution P For a known function y/, the parameter Bo Ee(usually also in the interior of e) is known to satisfy the key moment condition E(ai, 80)=0 This equation provides the core of the GMM estimation. The appropriate function y and the parameter Bo are usually derived from a theoretical model Both yl and Bo can be vector valued and not necessarily of the same size. Let the size of wl be q, and the size of 0 be p. The mean is 0 only at the true parameter value Bo, which is assumed to be unique over some neighborhood around Bo Along with equation (1), one also imposes certain boundary conditions for the 2nd order moment and partial derivative one E[v(x;60)v(x;,60)≡更<∞ 0 0e sm(a)EC2610 Fall 2004 GMM Notes for EC2610 1 Introduction These notes povide an introduction to GMM estimation. Their primary purpose is to make the reader familiar enough with GMM to be able to solve problem set assignments. For the more theoretical foundations, properties and extensions of GMM, or to better understand its workings, interested reader should consult any of the standard graduate econometrics textbooks, e.g., by Greene, Wooldridge, Hayashi, Hamilton, etc., as well as the original GMM article by Hansen (1982). Available lecture notes for graduate econometrics courses, e.g. by Chamberlain (Ec 2140), by Pakes and Porter (Ec 2144), also contain very useful reviews of GMM. Generalized Method of Moments provides asymptotic properties for estima￾tors and is general enough to include many other commonly used techniques, like OLS and ML. Having such an umbrella to encompass many of the estimators is very useful, as one doesnít have to derive each estimator property separately. With such a wide range, it is not surprising to see GMM used extensively, but one should also be careful when it is appropriate to apply. Since GMM deals with asymptotic properties, it works well for large samples, but does not pro￾vide an answer when the samply size is small, or what is "large" enough sample size. Also, when applying GMM, one may forgo certain desirable properties, like eÖiciency. 2 GMM Framework 2.1 DeÖnition of GMM Estimator Let xi ; i = 1; :::; n be i.i.d. random draws from the unknown population distri￾bution P. For a known function ; the parameter 0 2 (usually also in the interior of ) is known to satisfy the key moment condition: E [ (xi ; 0)] = 0 (1) This equation provides the core of the GMM estimation. The appropriate function and the parameter 0 are usually derived from a theoretical model. Both and 0 can be vector valued and not necessarily of the same size. Let the size of be q, and the size of  be p. The mean is 0 only at the true parameter value 0, which is assumed to be unique over some neighborhood around 0: Along with equation (1), one also imposes certain boundary conditions for the 2nd order moment and partial derivative one: E (xi ; 0) 0 (xi ; 0)   < 1 and      @ 2 j (x; ) @k@l       m(x) 1