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September, 1991 Maximum Likelihood Type Estimation for Nearly Nonstationary Autoregressive Time Series
Dennis D. Cox, Isabel Llatas
Ann. Statist. 19(3): 1109-1128 (September, 1991). DOI: 10.1214/aos/1176348240

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${\tt\%}$.

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Dennis D. Cox. Isabel Llatas. "Maximum Likelihood Type Estimation for Nearly Nonstationary Autoregressive Time Series." Ann. Statist. 19 (3) 1109 - 1128, September, 1991. https://doi.org/10.1214/aos/1176348240

Information

Published: September, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0742.62084
MathSciNet: MR1126316
Digital Object Identifier: 10.1214/aos/1176348240

Subjects:
Primary: 62M10
Secondary: 60F17 , 62E20 , 62F12

Keywords: Asymptotic efficiency , autoregressive processes , maximum likelihood estimation , Non-Gaussian time series

Rights: Copyright © 1991 Institute of Mathematical Statistics

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Vol.19 • No. 3 • September, 1991
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