Testing for Nonstationary Parameter Specifications in Seasonal Time Series Models

Let Yₜ be an autoregressive process satisfying Yₜ = α₁ Yt - 1 + α₂ Yt - d + α₃ Yt - d - 1 + eₜ, where ₜ\^∞t = 0 is a sequence of iid(0, σ²) random variables and d ≥ 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 α₁ = 1, α₂ = 1, α₃ = - 1 then Δ₁Δd Yₜ = eₜ, where Δᵣ Yₜ denotes Yₜ - Yt - r. If (α₁, α₂, α₃) = (1, 1, - 1) the process is nonstationary and the theory for stationary autoregressive processes does not apply. A methodology for testing the hypothesis (α₁, α₂, α₃) = (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.