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Econometrics I
Professor William Greene
Notes 23. The Generalized Method of Moments
I. Essential Results for the Method of Moments Estimator
Moment Equation: Defines a sample statistic that mimics a population expectation:
The population expectation:
E[ mi (?) ] = 0. Subscript i indicates it depends on data vector indexed by i (or t for a
time series setting)
Use nonlinear instrumental variables regression as an example: There are K parameters, ?
yi = f(xi,?) + ?i. There exists a set of K instrumental variables, zi such that E[zi ?i] = 0.
The sample counterpart is the moment equation
(1/n)?i zi ?i = (1/n)?i zi [yi - f(xi,?)] = (1/n)?i mi (?) = (?) = 0.
The method of moments estimator is the solution to the moment equation(s).
(How the solution is obtained is not always obvious, and varies from problem to problem. This one can
be solved by solving the optimization problem
? = the solution to Min wrt ? [(?)]?[(?)] = (1/n2)[??ZZ??]
F.O.C. = (2/n2) GZ Z? where G is the n?K matrix of derivatives of the regression functions.
Note that the solution has 2(1/n)GZ ? (1/n)Z? which is what we looked for.
Example: Linear regression, f(xi,?) = xi??, then G = X, as usual.
Variance of the method of moments estimator, based on the Slutsky Theorem and the Delta Method:
Let V = The covariance matrix of (?). This is a sample mean, so usually, V = (1/n)? for some matrix ?. (Stated without proof). The covariance matrix of the estimator is
Var[bMM] = (G-1) V (G-1)? where G = ?(?)/???
For the nonlinear least squares, IV problem, suppose Var[?i] = ?2
mi = zi?i, so the variance of mi = ?2zizi?. With independent observations,
Var[(?)] = (1/n2)?i ?2zizi? = ?2/n2 (Z?Z). The derivative matrix is the sum of the terms,
G = (1/n) ?i[-zi xi0] = -(1/n)Z?X0. Combining terms, then,
(G-1) V (G-1)? = [-(1/n)Z?X0]-1 {?2/n2 (Z?Z)} [-(1/n)Z?X0]-1? = ?2(Z?X0)-1Z?Z(Z?X0)-1?.
For the linear regression function, this is the
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