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Lecture 32-2007

# Lecture 32-2007 - Generalized Method of Moments Estimator...

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1 Generalized Method of Moments Estimator Lecture XXXII I. Basic Derivation of the Linear Estimator A. Starting with the basic linear model 0 t t t y x u where t y is the dependent variable, t x is the vector of independent variables, 0 is the parameter vector, and t u is the residual. In addition to these variables we will introduce the notion of a vector of instrumental variables denoted t z . 1. Reworking the original formulation slightly, we can express the residual as a function of the parameter vector 0 0 t t t u y x . 2. Based on this expression, estimation follows from the population moment condition 0 0 t E z u Or more specifically, 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 orthogonality conditions implied by the linear projection space: 1 ' ' c P X X X X 3. Further developing the orthogonality condition, note that if a single 0 solves the orthogonality conditions, or that 0 is unique that 0 0if and only if t t E z u Alternatively, 0 0 if t t E z u a) Going back to the original formulation t t t t t E z u E z y x b) Taking the first-order Taylor series expansion 0 0 t t t t t t t t

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Lecture 32-2007 - Generalized Method of Moments Estimator...

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