STRUCTURAL CHANGE IN NONSTATIONARY AR(1) MODELS

Abstract

This article revisits the asymptotic inference for nonstationary AR(1) models of Phillips and Magdalinos (2007a) by incorporating a structural change in the AR parameter at an unknown time k 0 . Consider the model ${y_t} = {\beta _1}{y_{t - 1}}I\{ t \le {k_0}\} + {\beta _2}{y_{t - 1}}I\{ t > {k_0}\} + {\varepsilon _t},t = 1,2, \ldots ,T$ , where I {·} denotes the indicator function, one of ${\beta _1}$ and ${\beta _2}$ depends on the sample size T , and the other is equal to one. We examine four cases: Case (I): ${\beta _1} = {\beta _{1T}} = 1 - c/{k_T}$ , ${\beta _2} = 1$ ; (II): ${\beta _1} = 1$ , ${\beta _2} = {\beta _{2T}} = 1 - c/{k_T}$ ; (III): ${\beta _1} = 1$ , ${\beta _2} = {\beta _{2T}} = 1 + c/{k_T}$ ; and case (IV): ${\beta _1} = {\beta _{1T}} = 1 + c/{k_T}$ , ${\beta _2} = 1$ , where c is a fixed positive constant, and k T is a sequence of positive constants increasing to ∞ such that k T = o ( T ). We derive the limiting distributions of the t -ratios of ${\beta _1}$ and ${\beta _2}$ and the least squares estimator of the change point for the cases above under some mild conditions. Monte Carlo simulations are conducted to examine the finite-sample properties of the estimators. Our theoretical findings are supported by the Monte Carlo simulations.