Estimation for Autoregressive Processes with Unit Roots

Let Yₜ satisfy the stochastic difference equation Yₜ = ∑ᵖj = 1ηⱼYt - j + eₜ for t = 1, 2, ⋯, where the eₜ are independent identically distributed (0, σ²) random variables and the initial conditions (Y-p + 1, Y-p + 2, ⋯, Y₀) are fixed constants. It is assumed the true, but unknown, roots m₁, m₂, ⋯, mₚ of mᵖ - ∑ᵖj = 1ηⱼmp - j = 0 satisfy m₁ = m₂ = 1 and |mⱼ| < 1 for j = 3, 4, ⋯, p. Let η̂ denote the least squares estimator of η = (η₁, η₂, ⋯, ηₚ)' obtained by the least squares regression of Yₜ on Yt - 1, Yt - 2, ⋯, Yt - p for t = 1, 2, ⋯, n. The asymptotic distributions of η̂ and of a test statistic designed to test the hypothesis that m₁ = m₂ = 1 are characterized. Analogous distributional results are obtained for models containing time trend and intercept terms. Estimated percentiles for these distributions are obtained by the Monte Carlo method.