Generalized Method of Moments

Identification and Overidentification

For a given set of moment functions g(Xt,θ), the true parameter sets the expected moment functions to zero by definition. When Eg(Xt,θ) = 0 at only the true parameter, we say that the true parameter vector is identified by the moment conditions. A necessary condition for the identification of the true parameter is that the number of moment conditions should be at least as large as the number of parameters. When the number of moment conditions is exactly equal to the number of parameters (and when the true parameter is identified), we say that the true parameter is exactly identified. On the other hand, if there are more moment conditions than necessary, we say that the true parameter is overidentified.

Generalized Method of Moments

When there is a set of moment conditions that exactly identifies a parameter vector, method of moments estimation is widely used. As the true parameter sets the population moments to zero, the method of moments estimator sets the sample moments to zero. More precisely, when the true parameter is exactly identified by Eg(Xt,θ) = 0, the method of moments estimator θ∘ satisfies T−1ΣTt=1 g(Xt, θ∘) = 0.

If the true parameter is overidentified, that is, if there are more moment conditions than are necessary to identify θ, then it is usually impossible to set the sample moment vector to zero (because there are more equations than parameters). The generalized method of moments (GMM) was introduced by Lars Peter Hansen in 1982 in order to handle this case. Let g¯(θ) = T−1ΣTt=1g(Xt,θ) for notational simplicity. Instead of setting the sample moment functions simultaneously to zero (which is usually impossible), Hansen proposed to minimize the quadratic distance of the sample moment vector from zero, that is, to minimize g¯(θ)′g¯(θ) with respect to θ over the parameter space. The minimizer is called the generalized method of moments (GMM) estimator.

The GMM estimator is consistent and asymptotically normal. In addition, the GMM procedure contains method of moments estimation as a special case. The method of moments estimator sets g¯(γ∘) = 0, in which case the criterion function g¯(θ)′g¯(θ) attains the minimal value zero at θ = θ∘.

Weighted GMM and Optimal GMM

A symmetric and positive definite constant matrix W can be used in the criterion function to form a weighted criterion function g¯(θ)′Wg¯(θ), whose minimizer is called the weighted GMM estimator using the matrix W as weights. Because any symmetric and positive definite matrix can be decomposed into A′A for some nonsingular matrix A (e.g., by a Cholesky decomposition), we observe that any weighted criterion function can be regarded as the (unweighted) quadratic distance of the transformed sample moment vector Ag¯(θ) = T−1ΣTt=1Ag(Xt,θ) from [Page 391]zero. Because W is a constant matrix, A is also a constant matrix, and therefore the transformed moment conditions are also valid, because E[Ag(Xt,θ)] = AEg(Xt,θ) = 0. This obvious fact shows that any weighted GMM estimator (using a symmetric and positive definite weight matrix) is also consistent and asymptotically normal.

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