Exercise 13.4 In the linear model estimated by GMM with general weight matrix W the asymptotic vari- ance of Bgmm is v=(
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- Exercise 13 4 In The Linear Model Estimated By Gmm With General Weight Matrix W The Asymptotic Vari Ance Of Bgmm Is V 1 (35.02 KiB) Viewed 29 times
legiblly. This topic is about generalized method of moments.
Exercise 13.4 In the linear model estimated by GMM with general weight matrix W the asymptotic vari- ance of Bgmm is v=(QwQ) ¹QwowQ (QwQ)-¹. (a) Let V₁ be this matrix when W = -¹. Show that V₁ = (Q'-¹)-¹. (b) We want to show that for any W, V-Vo is positive semi-definite (for then Vo is the smaller possible covariance matrix and W=Q2-¹ is the efficient weight matrix). To do this start by finding matrices A and B such that V = A'QA and V₁ = B'QB. (c) Show that B'A= B'OB and therefore that B' (A-B) = 0. (d) Use the expressions V = A'DA, A = B + (A-B), and B' (A-B) = 0 to show that V≥ Vo.