A specification test for nonlinear nonstationary models

TL;DR

This paper develops a new limit theory for kernel smoothed U-statistics used in specification testing of nonlinear, nonstationary time series models, accommodating endogenous regressors with unit roots.

Contribution

It introduces a novel weak convergence result for partial sums of nonstationary time series, linking the test's limit distribution to the intersection local time of a Gaussian process.

Findings

The proposed test performs well in finite samples.

The limit distribution involves the intersection local time of Gaussian processes.

The theory applies to models with near unit roots and endogenous regressors.

Abstract

We provide a limit theory for a general class of kernel smoothed U-statistics that may be used for specification testing in time series regression with nonstationary data. The test framework allows for linear and nonlinear models with endogenous regressors that have autoregressive unit roots or near unit roots. The limit theory for the specification test depends on the self-intersection local time of a Gaussian process. A new weak convergence result is developed for certain partial sums of functions involving nonstationary time series that converges to the intersection local time process. This result is of independent interest and is useful in other applications. Simulations examine the finite sample performance of the test.