Maximum Likelihood Type Estimation for Nearly Nonstationary Autoregressive Time Series

Abstract

The nearly nonstationary first-order autoregression is a sequence of autoregressive processes $y_n(k + 1) = \phi_ny_n(k) + \varepsilon(k + 1), 0 \leq k \leq n$, where the $\varepsilon(k)$'s are iid mean zero shocks and the autoregressive coefficient $\phi_n = 1 - \beta/n$ for some $\beta > 0$, so that $\phi_n \rightarrow 1$ as $n \rightarrow \infty$. We consider a class of maximum likelihood type estimators called $M$ estimators, which are not necessarily robust. The estimates are obtained as the solution $\hat{\phi}_n$ of an equation of the form $\sum^{n - 1}_{k = 0}y_n(k)\psi(y_n(k + 1) - \phi y_n(k)) = 0,$ where $\psi$ is a given "score" function. Assuming the shocks have $2 + \delta$ moments and that $\psi$ satisfies some regularity conditions, it is shown that the limiting distribution of $n(\hat{\phi}_n - \phi_n)$ is given by the ratio of two stochastic integrals. For a given shock density $f$ satisfying regularity conditions, it is shown that the optimal $\psi$ 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 $y_n(0) = 0$ show that the maximum likelihood score has asymptotic efficiency no lower than 40${\tt\%}$.