# Generalized method of moments

Generalized method of moments

:"GMM may also mean Gaussian mixture model.":"For the Thai entertainment company, see GMM Grammy."

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.

Description

The idea of the generalized method of moments is to use moment conditionsthat can be found from the problem with little effort. We assume thatthe data are a stochastic process $\left(Y_1, Y_2, ldots \right).$ In mathematical language, we start out with a (vector valued) function $f$ that depends both on the parameter and a single observation and that has mean zero for the true value of the parameter, $heta = heta_0,$ i.e.

:$E \left[f\left(Y_i, heta_0\right)\right] = 0.,$

To turn this function into a parameter estimate, we minimize the associated chi-square statistic

:$hat\left\{ heta\right\} = ext\left\{arg\right\} min_\left\{ heta\right\} left\left(sum_\left\{i=1\right\}^N f\left(Y_i, heta\right) ight\right)^TAleft\left(sum_\left\{i=1\right\}^N f\left(Y_i, heta\right) ight\right)$

where superscript $T$ denotes transpose, and $A$ is a positive-definite "weighting" matrix. $A$ may be known a priori or estimated from the sample.

* Instrumental variable, subclass of this method
* Method of moments
* Method of simulated moments, which is used when the moment functions must be approximated by simulations

* Lars Peter Hansen (1982): "Large Sample Properties of Generalized Method of Moments Estimators", "Econometrica" 50, 1029-1054.
* Lars Peter Hansen (2002): "Method of Moments" in International Encyclopedia of the Social and Behavior Sciences, N. J. Smelser and P. B. Bates (editors), Pergamon: Oxford.
* Kirby Faciane (2006): "Statistics for Empirical and Quantitative Finance". H.C. Baird: Philadelphia. ISBN 0-9788208-9-4.
* Alastair R. Hall (2005). "Generalized Method of Moments (Advanced Texts in Econometrics)". Oxford University Press. ISBN 0-19-877520-2.

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