Asymptotic Inference for Nearly Nonstationary AR(1) Processes

Abstract

A first-order autoregressive process, $Y_t = \\beta Y_{t - 1} + \\epsilon_t$, is said to be nearly nonstationary when $\\beta$ is close to one. The limiting distribution of the least-squares estimate $b_n$ for $\\beta$ is studied when $Y_t$ is nearly nonstationary. By reparameterizing $\\beta$ to be $1 - \\gamma/n, \\gamma$ being a fixed constant, it is shown that the limiting distribution of $\\tau_n = (\\sum^n_{t = 1}Y^2_{t - 1})^{1/2}(b_n - \\beta)$ converges to $\\mathscr{L}(\\gamma)$ which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of $\\tau_n$ proposed by Ahtola and Tiao (1984).