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 data's distribution function may not be known, and therefore maximum likelihood estimation is not applicable. The method requires that a certain number of moment conditions be specified for the model. These moment conditions are functions of the model parameters and the data, such that their expectation is zero at the parameters' true values. The GMM method then minimizes a certain norm of the sample averages of the moment conditions, and can therefore be thought of as a special case of minimum-distance estimation. The GMM estimators are known to be consistent, asymptotically normal, and most efficient in the class of all estimators that do not use any extra information aside from that contained in the moment conditions. GMM were advocated by Lars Peter Hansen in 1982 as a generalization of the method of moments, introduced by Karl Pearson in 1894. However, these estimators are mathematically equivalent to those based on "orthogonality conditions" (Sargan, 1958, 1959) or "unbiased estimating equations" (Huber, 1967; Wang et al., 1997). Suppose the available data consists of T observations {Yt } t = 1,...,T, where each observation Yt 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. A general assumption of GMM is that the data Yt be generated by a weakly stationary ergodic stochastic process. (The case of independent and identically distributed (iid) variables Yt is a special case of this condition.) In order to apply GMM, we need to have "moment conditions", that is, we need to know a vector-valued function g(Y,θ) such that where E denotes expectation, and Yt is a generic observation.