Trinity tests of functions for conditional moment models

Abstract

This paper considers conditional moment models where the parameters of interest include both finite-dimensional parameters and unknown functions. We propose sup-Wald, sup-quasi-likelihood ratio and sup-Lagrange multiplier statistics for testing functionals restrictions uniformly over the support for both finite and infinite dimensional components of the parameters. The trinity of three statistics holds because they are asymptotically equivalent and can be strongly approximated by a sequence of chi-squared processes. Based on these results, we can, for instance, construct confidence intervals and uniform confidence bands for the unknown functions, the partial derivatives of the unknown functions and the functional combinations of the two parameters.