Estimation for Autoregressive Processes with Unit Roots

Abstract

Let $Y_t$ satisfy the stochastic difference equation $Y_t = \\sum^p_{j = 1}\\eta_jY_{t - j} + e_t$ for $t = 1, 2, \\cdots$, where the $e_t$ are independent identically distributed $(0, \\sigma^2)$ random variables and the initial conditions $(Y_{-p + 1}, Y_{-p + 2}, \\cdots, Y_0)$ are fixed constants. It is assumed the true, but unknown, roots $m_1, m_2, \\cdots, m_p$ of $m^p - \\sum^p_{j = 1}\\eta_jm^{p - j} = 0$ satisfy $m_1 = m_2 = 1$ and $|m_j| < 1$ for $j = 3, 4, \\cdots, p$. Let $\\hat{\\mathbf{\\eta}}$ denote the least squares estimator of $\\mathbf{\\eta} = (\\eta_1, \\eta_2, \\cdots, \\eta_p)'$ obtained by the least squares regression of $Y_t$ on $Y_{t - 1}, Y_{t - 2}, \\cdots, Y_{t - p}$ for $t = 1, 2, \\cdots, n$. The asymptotic distributions of $\\hat{\\mathbf{\\eta}}$ and of a test statistic designed to test the hypothesis that $m_1 = m_2 = 1$ 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.