tailieunhanh - Lecture Advanced Econometrics (Part II) - Chapter 13: Generalized method of moments (GMM)

Lecture "Advanced Econometrics (Part II) - Chapter 13: Generalized method of moments (GMM)" presentation of content: Orthogonality condition, method of moments, generalized method of moments, GMM and other estimators in the linear models, the advantages of GMM estimator, GMM estimation procedure. | Advanced Econometrics Chapter 13 Generalized Method of Moments Chapter 13 GENERALIZED METHOD OF MOMENTS GMM I. ORTHOGONALITY CONDITION The classical model Xn X p 8 1 E 1X 0 2 E ss X a1 3 X and 8 are generated independently. If E 8i Xf 0 then for equation i Yi X p 8i i Ịxk k x1 E Xi i Ex- E XA X Exi E 8 X Xi J 0 MOAK0 Orthogonality condition. Note Cov X 8i E X -E X -E E Xi -E X E X i -E X E Ci E X 8 1 0 if E X 0 y kx1 1x1 J kx1 So for the classical model E X 8 i i y kx1 1x1 0 kx1 Nam T. Hoang UNE Business School 1 University of New England Advanced Econometrics Chapter 13 Generalized Method of Moments II. METHOD OF MOMENTS Method of moments involves replacing the population moments by the sample moment. Example 1 For the classical model Population E X . 0 E X Y- X 3 0 y L___________ z k x1 population moment Sample moment of this 1 1 1 X Y- Xi 3 1 X Y - X 3 n i i kx1 1x1 n kxn nx1 _______v______z kx1 ị Moment function A function that depends on observable random variables and unknown parameters and that has zero expectation in the population when evaluated at the true parameter. m 3 - moment function - can be linear or non-linear. 3 is a vector of unknown parameters. kx1 ị E m 3 0 population moment. ị Method of moments involves replacing the population moments by the sample moment. Example 1 For the classical linear regression model The moment function m 3 The population function E X-S 0 E m 3 E -X X3 0 The sample moment of E Xị et is n I I 1 X Y-X 3 1 X Y - X 3 n i 1 kx1 1x1 n kxn nx1 _____v_____z kx1 Replacing sample moments for population moments Nam T. Hoang UNE Business School 2 University of New England Advanced Econometrics Chapter 13 Generalized Method of Moments i X-n - y - X p X Y - X X p 0 X X p X Y 0 PMOM X X -1 X Y POLS Example 2 If Xt are endogenous Cov X s 0 1xk Suppose Zị ZXi Z2i ZLi is a vector of instrumental variables for Xt 1xL ixk Zi satisfies E ei Zi 0 E Z isi 0 and Cov Z isi 0 ixL Lx1 Lx1 We have E ZL 8 L Lx1 1x1 Lx1 E Z y - X p Lxl .