Asymptotic Theory for Kernel Estimators under Moderate Deviations from a Unit Root, with an Application to the Asymptotic Size of Nonparametric Tests

TL;DR

This paper develops new asymptotic theory for kernel density estimators applied to autoregressive processes with moderate deviations from a unit root, ensuring valid nonparametric inference even with highly persistent regressors.

Contribution

It introduces a novel asymptotic framework for kernel estimators under moderate deviations from a unit root, extending the existing literature to highly persistent processes.

Findings

Null rejection probability of nonparametric t test is uniformly controlled.

Validates usual nonparametric inference for highly persistent regressors.

Fills a theoretical gap for nearly integrated autoregressive processes.

Abstract

We provide new asymptotic theory for kernel density estimators, when these are applied to autoregressive processes exhibiting moderate deviations from a unit root. This fills a gap in the existing literature, which has to date considered only nearly integrated and stationary autoregressive processes. These results have applications to nonparametric predictive regression models. In particular, we show that the null rejection probability of a nonparametric t test is controlled uniformly in the degree of persistence of the regressor. This provides a rigorous justification for the validity of the usual nonparametric inferential procedures, even in cases where regressors may be highly persistent.