Testing subspace restrictions in the presence of high dimensional nuisance parameters

Abstract

A framework for hypothesis testing of functional restrictions against general alternatives is proposed. The parameter space is a subset of a reproducing kernel Hilbert space (RKHS). The null hypothesis does not necessarily define a parametric model. The test allows us to deal with possibly infinite dimensional nuisance parameters. The methodology is based on a moment equation similar in spirit to the construction of the efficient score in semiparametric statistics. The feasible version of such moment equation requires to consistently estimate projections in the space of RKHS. A tractable asymptotic theory is established for this problem. Simulation results show that the finite sample performance of the test is consistent with the asymptotic results and that ignoring the effect of nuisance parameters highly distorts the size of the tests.