Generalization of GMM to a continuum of moment conditions

This paper proposes a version of the generalized method of moments procedur e that handles both the case where the number of moment conditions is finit e and the case where there is a continuum of moment conditions. Typically, the moment conditions are indexed by an index parameter that takes its valu es in an interval, The objective function to minimize is then the norm of t he moment conditions in a Hilbert space. The estimator is shown to be consi stent and asymptotically normal. The optimal estimator is obtained by minim izing the norm of the moment conditions in the reproducing kernel Hilbert s pace associated with the covariance, We show an easy way to calculate this estimator, Finally, we study properties of a specification test using overi dentifying restrictions. Results of this paper are useful in many instances where a continuum of moment conditions arises. Examples include efficient estimation of continuous time regression models, cross-sectional models tha t satisfy conditional moment restrictions, and scalar diffusion processes.