STRUCTURAL CHANGE IN NONSTATIONARY AR(1) MODELS

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 timek0. Consider the model${y_t} = {\beta _1}{y_{t - 1}}I\{ t \le {k_0}\} + {\beta _2}{y_{t - 1}}I\{ t > {k_0}\} + {\varepsilon _t},t = 1,2, \ldots ,T$, whereI{·} denotes the indicator function, one of${\beta _1}$and${\beta _2}$depends on the sample sizeT, 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):${\beta _1} = {\beta _{1T}} = 1 + c/{k_T}$,${\beta _2} = 1$, wherecis a fixed positive constant, andkTis a sequence of positive constants increasing to ∞ such thatkT=o(T). We derive the limiting distributions of thet-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.