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 conditions that can be found from the problem with little effort. We assume that the data are a stochastic process $$(Y_1, Y_2, \ldots ).$$ 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, $$\theta = \theta_0,$$ i.e.


 * $$ E[f(Y_i,\theta_0)] = 0.\,$$

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


 * $$\ \hat{\theta} = \text{arg} \min_{\theta} \left(\sum_{i=1}^N f(Y_i,\theta)\right)^TA\left(\sum_{i=1}^N f(Y_i,\theta)\right)$$

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