Slides33-2010 - Generalized Method of Moments Estimator:...

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Generalized Method of Moments Estimator: Lecture XXXIII Charles B. Moss November 30, 2010 Charles B. Moss () Generalized Method of Moments November 30, 2010 1 / 18
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1 Basic Derivation of Linear Estimator 2 The Limiting Distribution 3 DiFerenital Demand Model Charles B. Moss () Generalized Method of Moments November 30, 2010 2 / 18
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Basic Derivation of the Linear Estimator 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 . I 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) Charles B. Moss () Generalized Method of Moments November 30, 2010 3 / 18
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Continued 2. Based on this expression, estimation follows from the population 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. I Note the similarity between these conditions and the orthogonality conditions implied by the linear projection space P c = X ( X 0 X ) 1 X 0 (4) Charles B. Moss () Generalized Method of Moments November 30, 2010 4 / 18
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Continued 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 E [ z t u t ( θ )] 6 =0 if θ 6 = θ 0 (6) Charles B. Moss () Generalized Method of Moments November 30, 2010 5 / 18
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Continued I Going back to the original formulation E [ z t u t ( θ )] = E [ z t ( y t x 0 t θ )] (7) I
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Slides33-2010 - Generalized Method of Moments Estimator:...

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