Asymptotic Inference for Nearly Nonstationary AR(1) Processes

A first-order autoregressive process, Yₜ = β Yt - 1 + εₜ, is said to be nearly nonstationary when β is close to one. The limiting distribution of the least-squares estimate bₙ for β is studied when Yₜ is nearly nonstationary. By reparameterizing β to be 1 - γ/n, γ being a fixed constant, it is shown that the limiting distribution of τₙ = (∑ⁿt = 1Y²t - 1)1/2(bₙ - β) converges to L(γ) which is a quotient of stochastic integrals of standard Brownian motion. This provides a reasonable alternative to the approximation of the distribution of τₙ proposed by Ahtola and Tiao (1984).