Pivotal Test Statistic for Nonparametric Cointegrating Regression Functions

Abstract

This article focuses on cointegrating regression models in which covariate processes exhibit long range or semi-long range memory behaviors, and may involve endogeneity in which covariate and response error terms are not independent. We assume semilong range memory is produced in the covariate process by tempering of random shock coefficients. The fundamental properties of long memory processes are thus retained in the covariate process. We modify a test statistic proposed for the long memory case by Wang and Phillips (2016) to be suitable in the semi-long range memory setting.The limiting distribution is derived for this modified statistic and shown to depend only on the local memory process of standard Brownian motion. Because, unlike the original statistic of Wang and Phillips (2016), the limit distribution is independent of the differencing parameter of fractional Brownian motion, it is pivotal. Through simulation we investigate properties of nonparametric function estimation for semilong range memory cointegrating models, and consider behavior of both the modified test statistic under semi-long range memory and the original statistic under long range memory. We also provide a brief empirical example.