Maximum Likelihood Type Estimation for Nearly Nonstationary Autoregressive Time Series

The nearly nonstationary first-order autoregression is a sequence of autoregressive processes yₙ(k + 1) = φₙyₙ(k) + ε(k + 1), 0 ≤ k ≤ n, where the ε(k)'s are iid mean zero shocks and the autoregressive coefficient φₙ = 1 - β/n for some β > 0, so that φₙ → 1 as n → ∞. We consider a class of maximum likelihood type estimators called M estimators, which are not necessarily robust. The estimates are obtained as the solution φ̂ₙ of an equation of the form ∑n - 1k = 0yₙ(k)ψ(yₙ(k + 1) - φ yₙ(k)) = 0, where ψ is a given "score" function. Assuming the shocks have 2 + δ moments and that ψ satisfies some regularity conditions, it is shown that the limiting distribution of n(φ̂ₙ - φₙ) is given by the ratio of two stochastic integrals. For a given shock density f satisfying regularity conditions, it is shown that the optimal ψ function for minimizing asymptotic mean squared error is not the maximum likelihood score in general, but a linear combination of the maximum likelihood score and least squares score. However, numerical calculations under the constraint yₙ(0) = 0 show that the maximum likelihood score has asymptotic efficiency no lower than 40%.