Generalized Method of Moments

In econometrics and statistics, the generalized method of moments (GMM) is a generic method for estimating parameters in statistical models. Usually it is applied in the context of semiparametric models, where the parameter of interest is finite-dimensional, whereas the full shape of the distribution function of the data may not be known, and therefore maximum likelihood estimation is not applicable.

The method requires that a certain number of moment conditions were specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the true values of the parameters. The GMM method then minimizes a certain norm of the sample averages of the moment conditions.

The GMM estimators are known to be consistent, asymptotically normal, and efficient in the class of all estimators that do not use any extra information aside from that contained in the moment conditions.

GMM was developed by Lars Peter Hansen in 1982 as a generalization of the method of moments, which was introduced by Karl Pearson in 1894. Hansen shared the 2013 Nobel Prize in Economics in part for this work.

Suppose the available data consists of T observations {Y_t }_{t = 1,...,T}, where each observation Y_t is an n-dimensional multivariate random variable. We assume that the data come from a certain statistical model, defined up to an unknown parameter θ ∈ Θ. The goal of the estimation problem is to find the “true” value of this parameter, θ₀, or at least a reasonably close estimate.

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