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) = ₙyₙ (k) + (k + 1), 0 k n, where the (k) 's are iid mean zero shocks and the autoregressive coefficient ₙ = 1 - /n for some > 0, so that ₙ 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 ₙ of an equation of the form ^n - 1₊ = ₀yₙ (k) (yₙ (k + 1) - yₙ (k) ) = 0, where is a given "score" function. Assuming the shocks have 2 + moments and that satisfies some regularity conditions, it is shown that the limiting distribution of n (ₙ - ₙ) is given by the ratio of two stochastic integrals. For a given shock density f satisfying regularity conditions, it is shown that the optimal 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\%.