Testing for Nonstationary Parameter Specifications in Seasonal Time Series Models

Abstract

Let $Y_t$ be an autoregressive process satisfying $Y_t = \alpha_1 Y_{t - 1} + \alpha_2 Y_{t - d} + \alpha_3 Y_{t - d - 1} + e_t$, where $\{e_t\}^\infty_{t = 0}$ is a sequence of $\operatorname{iid}(0, \sigma^2)$ random variables and $d \geq 2$. Such processes have been used as parametric models for seasonal time series. Typical values of $d$ are 2, 4, and 12 corresponding to time series observed semi-annually, quarterly, and monthly, respectively. If $\alpha_1 = 1, \alpha_2 = 1, \alpha_3 = - 1$ then $\Delta_1\Delta_d Y_t = e_t$, where $\Delta_r Y_t$ denotes $Y_t - Y_{t - r}$. If $(\alpha_1, \alpha_2, \alpha_3) = (1, 1, - 1)$ the process is nonstationary and the theory for stationary autoregressive processes does not apply. A methodology for testing the hypothesis $(\alpha_1, \alpha_2, \alpha_3) = (1, 1, - 1)$ is presented and percentiles for test statistics are obtained. Extensions are presented for multiplicative processes, for higher order processes, and for processes containing deterministic trend and seasonal components.