ABSTRACT
The generalized method of moments (GMM) provides a computationally convenient method for inference on the structural parameters of economic models. The method has been applied in many areas of economics but it was in empirical finance that the power of the method was first illustrated. Hansen (1982) introduced GMM and presented its fundamental statistical theory. Hansen and Hodrick (1980) and Hansen and Singleton (1982) showed the potential of the GMM approach to testing economic theories through their empirical analyzes of, respectively, foreign exchange markets and asset pricing. In such contexts, the cornerstone of GMM inference is a set of conditional moment restrictions. More generally, GMM is well suited for the test of an economic theory every time the theory can be encapsulated in the postulated unpredictability of some error term u(Yt, ) given as a known function of p
of Empirical
unknown parameters ∈ ⊆Rp and a vector of observed random variables Yt. Then, the testability of the theory of interest is akin to the testability of a set of conditional moment restrictions,
Et[u(Yt+1, )] = 0, (2.1) where the operator Et[.] denotes the conditional expectation given available information at time t. Moreover, under the null hypothesis that the theory summarized by the restrictions (Equation 2.1) is true, these restrictions are supposed to uniquely identify the true unknown value 0 of the parameters. Then, GMM considers a set of H instruments zt assumed to belong to the available information at time t and to summarize the testable implications of Equation 2.1 by the implied unconditional moment restrictions:
E[t()] = 0 where t() = zt ⊗ u(Yt+1, ). (2.2) The recent literature on weak instruments (see the seminal work by Stock and Wright 2000) has stressed that the standard asymptotic theory of GMM inference may be misleading because of the insufficient correlation between some instruments zt and some components of the local explanatory variables of [∂u(Yt+1, )/∂]. In this case, some of the moment conditions (Equation 2.2) are not only zero at 0 but rather flat and close to zero in a neighborhood of 0.
