Inference for Additive Models in the Presence of Possibly Infinite Dimensional Nuisance Parameters

TL;DR

This paper develops a new inference framework for additive models within RKHS, enabling hypothesis testing with infinite-dimensional nuisance parameters, and demonstrates its effectiveness through theoretical analysis and simulations.

Contribution

It introduces a novel testing methodology for functional restrictions in RKHS that handles infinite-dimensional nuisance parameters, with consistent estimation and tractable asymptotics.

Findings

The proposed test accurately controls size in finite samples.

Ignoring nuisance parameters leads to distorted test size.

Simulation results confirm the theoretical properties.

Abstract

A framework for estimation and hypothesis testing of functional restrictions against general alternatives is proposed. The parameter space is a reproducing kernel Hilbert space (RKHS). The null hypothesis does not necessarily define a parametric model. The test allows us to deal with 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 and it is shown that this is possible using the proposed approach. This allows us to derive some tractable asymptotic theory and critical values by fast simulation. Simulation results show that the finite sample performance of the test is consistent with the asymptotics and that ignoring the effect of nuisance parameters highly distorts the size of the tests.