ECONOMETRICS phần 8

Định lý Phân phối của GMM Ước tính phi tuyến Theo các điều kiện đều đặn chung, p ở đâu và Thuật ngữ mô tả mà người sử dụng thường khá gây hiểu lầm. Khi nói về thuyền trưởng của ngành công nghiệp hiện đại và các nhà lãnh đạo doanh nghiệp lớn, ví dụ, | CHAPTER 11. GENERALIZED METHOD OF MOMENTS 197 GMM The General Case In its most general form GMM applies whenever an economic or statistical model implies the X 1 moment condition E gf ft 0. Often this is all that is known. Identification requires l k dim ft . The GMM estimator minimizes J ft n gn ft Wn gn ft where 9n ft n X gi ft n i 1 and 1 Wn 1 X gig i - gn9 n nn i 1 with gi gi ft constructed using a preliminary consistent estimator ft perhaps obtained by first setting Wn I. Since the GMM estimator depends upon the first-stage estimator often the weight matrix Wn is updated and then ft recomputed. This estimator can be iterated if needed. Theorem Distribution of Nonlinear GMM Estimator Under general regularity conditions pn ft - ft - n 0 G Q 1G where n E gfg and G E @ft gi ft . The variance of ft may be estimated by Vp g n G 1 where n n-1 X g g i and G n-1 X @ft gi ft . i The general theory of GMM estimation and testing was exposited by L. Hansen 1982 . Over-Identification Test Overidentified models k are special in the sense that there may not be a parameter value ft such that the moment condition CHAPTER 11. GENERALIZED METHOD OF MOMENTS 198 Eg yi Xi Zi @ 0 holds. Thus the model - the overidentifying restrictions - are testable. For example take the linear model yi @iXii @2x2i ei with E xiiei 0 and E x2iSj 0. It is possible that @2 0 so that the linear equation may be written as yi @ixii ej. However it is possible that @2 0 and in this case it would be impossible to find a value of @1 so that both E xii yi X0ii@i 0 and E X2i yi X0ii@i 0 hold simultaneously. In this sense an exclusion restriction can be seen as an overidentifying restriction. Note that gn Egi and thus gn can be used to assess whether or not the hypothesis that Egi 0 is true or not. The criterion function at the parameter estimates is Jn n g n Wg n2g n g g - n9ngn i gn. is a quadratic form in gn and is thus a natural test statistic for H0 Egi 0. Theorem Sargan-Hansen . .