Generalization of GMM to a Continuum of Moment Conditions

Abstract

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, cross-sectional models that satisfy conditional moment restrictions, and scalar diffusion processes.

Instrumental variable models for discrete outcomes

“…An unobserved scalar random variable U is continuously distributed, normalised Uniform on (0; 1) and restricted to be distributed independently of instrumental variables Z. The instrumental variables are, by restriction, excluded from the threshold crossing function p and vary in a set of 3 Full exploitation of the restrictions of the model yields a continuum of moment inequalities on which there are few research results at this time although inference with point identi…cation induced by a continuum of moment equalities is quite well understood, see for example Carrasco and Florens (2000). 4 In Chesher (2007) there are additional examples covering cases in which the outcome has a binomial or a Poisson structural function.…”

Nonparametric estimation for compound Poisson process via variational analysis on measures

“…In the Lévy process setting, the most straightforward approach for estimating the distribution F(x) = P(W h ≤ x) is the moments fitting, see Feuerverger and McDunnough (1981b) and Carrasco and Florens (2000). Estimates of Λ can be obtained by maximising the likelihood ratio (see e.g.…”

Empirical Characteristic Function Estimation and its Applications

“…Since gðrÞ is a continuous function, the procedure (2.2) basically matches the ECF and CF continuously over an interval and hence can be viewed as a special class of GMM on a continuum of moment conditions given by Carrasco and Florens (2000). To see this, consider the objective function of the GMM procedure based on a continuum of moment conditions defined in Carrasco and Florens (2000), Z Z h n ðr; hÞg n ðr; sÞh n ðs; hÞdrds; ð2:4Þ where h h is the conjugate of h. If we choose g n ðr; sÞ ¼ gðrÞIðr À sÞ; h n ðr; hÞ ¼ ð1=nÞ P hðr; X j ; hÞ, (2.4) is equivalent to (2.2).…”

“…To see this, consider the objective function of the GMM procedure based on a continuum of moment conditions defined in Carrasco and Florens (2000), Z Z h n ðr; hÞg n ðr; sÞh n ðs; hÞdrds; ð2:4Þ where h h is the conjugate of h. If we choose g n ðr; sÞ ¼ gðrÞIðr À sÞ; h n ðr; hÞ ¼ ð1=nÞ P hðr; X j ; hÞ, (2.4) is equivalent to (2.2). The above continuous ECF procedure has been used in Press (1972), Paulson et al (1975), Thorton and Paulson (1977), and more recently in .…”