Lecture33-2010 - Generalized Method of Moments Estimator:...

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Generalized Method of Moments Estimator: Lecture XXXIII Charles B. Moss November 30, 2010 I. Basic Derivation of the Linear Estimator A. Starting with the basic linear model y t = x 0 t θ 0 + u t (1) where y t is the dependent variable, x t is the vector of independent variables, θ 0 is the parameter vector, and u t is the residual. In addition to these variables we will introduce the notion of a vector of instrumental variables denoted z t . 1. Reworking the original formulation slightly, we can express the residual as a function of the parameter vector u t ( θ 0 )= y t x 0 t θ 0 (2) 2. Based on this expression, estimation follows from the popu- lation moment condition E[ z t u t ( θ 0 )] = 0 (3) Or more speciFcally, we select the vector of parameters so that the residuals are orthogonal to the set of instruments. a) Note the similarity between these conditions and the or- thogonality conditions implied by the linear projection space P c = X ( X 0 X ) 1 X 0 (4) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXXIII Fall 2010 3. Further developing the orthogonality condition, note that if a single θ 0 solves the orthogonality conditions, or that θ 0 is unique that E[ z t u t ( θ )] = 0 if and only if θ = θ 0 (5) Alternatively z t u t ( θ )] 6 =0if θ 6 = θ 0 (6) a) Going back to the original formulation z t u t ( θ )] = E [ z t ( y t x 0 t θ )] (7) b) Taking the ±rst-order Taylor series expansion z t ( y t x 0 t θ )] = E [ z t ( y t x 0 t θ 0 )] z t x 0 t ]( θ θ 0 ) ∂θ ( y t x 0 t θ )= x 0 t (8)
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture33-2010 - Generalized Method of Moments Estimator:...

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