Generalized method of moments
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- GMM may also mean Gaussian mixture model.
The generalized method of moments is a very general statistical method for obtaining estimates of parameters of statistical models. It is a generalization, developed by Lars Peter Hansen, of the method of moments.
The term GMM is very popular among econometricians but is hardly used at all outside of economics, where the slightly more general term estimating equations is preferred. The method is also closely related to the classical theory of minimum chi-square estimation.
[edit] Description
The idea of the generalized method of moments is to use moment conditions that can be found from the problem with little effort. We assume that the data are a stochastic process <math>(Y_1, Y_2, \ldots ).</math> In mathematical language, we start out with a (vector valued) function <math>f</math> that depends both on the parameter and a single observation and that has mean zero for the true value of the parameter, <math>\theta = \theta_0,</math> i.e.
- <math>E[f(Y_i,\theta_0)] = 0.\,</math>
To turn this function into a parameter estimate, we minimize the associated chi-square statistic
- <math>\ \left(\sum_1^N f(Y_i,\theta)\right)^TA^{-1}\left(\sum_1^N f(Y_i,\theta)\right)</math>
where superscript <math>T</math> is used for transpose, and <math>A</math> is a matrix. <math>A</math> may be known a priori or estimated from the sample.
[edit] See also
- Instrumental variable, subclass of this method
- Method of moments
[edit] Further reading
- Lars Peter Hansen (1982). "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica 50, 1029-1054.
- Kirby Faciane (2006). Statistics for Empirical and Quantitative Finance. H.C. Baird: Philadelphia. ISBN 0978820894.
- Alastair R. Hall (2005). Generalized Method of Moments (Advanced Texts in Econometrics). Oxford University Press. ISBN 0198775202.de:Momentenmethode
