Generalization of GMM to a Continuum of Moment Conditions

This paper proposes a version of the generalized method of moments procedure
that handles both the case where the number of moment conditions is finite and
the case where there is a continuum of moment conditions. Typically, the moment
conditions are indexed by an index parameter that takes its values in an
interval. The objective function to minimize is then the norm of the moment conditions
in a Hilbert space. The estimator is shown to be consistent and asymptotically
normal. The optimal estimator is obtained by minimizing the norm of the
moment conditions in the reproducing kernel Hilbert space associated with the
covariance. We show an easy way to calculate this estimator. Finally, we study
properties of a specification test using overidentifying 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, crosssectional
models that satisfy conditional moment restrictions, and scalar diffusion
processes.

1. INTRODUCTION
In his seminal paper, Hansen (1982) has extended the method of moments to
overidentified models, i.e., models in which the number of moment conditions
is greater than the number of parameters. This method is now very popular,
and its properties are well established (for a survey, see Hall, 1993; or Ogaki,
1993).