TAILIEUCHUNG - Class Notes in Statistics and Econometrics Part 30

CHAPTER 59 Generalized Method of Moments Estimators. This follows mainly [DM93, Chapter 17]. A good and accessible treatment is [M´9]. The textbook [Hay00] uses GMM as the organizing principle for all 9 estimation methods except maximum likelihood. | CHAPTER 59 Generalized Method of Moments Estimators This follows mainly DM93 Chapter 17 . A good and accessible treatment is . The textbook Hay0 uses GMM as the organizing principle for all estimation methods except maximum likelihood. A moment y of a random variable y is the expected value of some function of y. Such a moment is therefore defined by the equation E g y - y 0. The same parameter-defining function g y y defines the method of moments estimator y of y if one replaces the expected value in with the sample mean of the elements of an observation vector y consisting of independent observations of y. In other words y y is that value which satisfies n 52n i ff yj y 0. 1287 1288 59. GENERALIZED METHOD OF MOMENTS ESTIMATORS The generalized method of moments estimator extends this rule in several respects the yi no longer have to be . the parameter-defining equations may be a system of equations defining more than one paramter at a time there may be more parameter-defining functions than parameters overidentification and not only unconditional but also conditional moments are considered. Under this definition the OLS estimator is a GMM estimator. To show this we will write the linear model y Xft e row by row as yi xj ft i where xi is as in various earlier cases the ith row of X written as a column vector. The basic property which makes least squares consistent is that the following conditional expectation is zero E yi xirft xi 0. This is more information than just knowing that the unconditional expectation is zero. How can this additional information be used to define an estimator From follows that the unconditional expectation of the product E xi yi x ft o. Replacing the expected value by the sample mean gives 1 Xi yi x7 3 o n i 1 59. GENERALIZED METHOD OF MOMENTS ESTIMATORS 1289 which can also be written as xi xn ïi - xjft . . . -r- L n - xnßJ 1XT y - X 3 o. n 1 n These are exactly the OLS Normal .

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