Maximum likelihood estimation for α-stable autoregressive processes

Abstract

We consider maximum likelihood estimation for both causal and noncausal autoregressive time series processes with non-Gaussian αstable noise. A nondegenerate limiting distribution is given for maximum likelihood estimators of the parameters of the autoregressive model equation and the parameters of the stable noise distribution. The estimators for the autoregressive parameters are n 1/α -consistent and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. The bootstrap is asymptotically valid under general conditions. The estimators for the parameters of the stable noise distribution have the traditional n 1/2 rate of convergence and are asymptotically normal. The behavior of the estimators for finite samples is studied via simulation, and we use maximum likelihood estimation to fit a noncausal autoregressive model to the natural logarithms of volumes of Wal-Mart stock traded daily on the New York Stock Exchange. . This reprint differs from the original in pagination and typographic detail. 1 2 B. ANDREWS, M. CALDER AND R. A. DAVIS Rachev [28]), signal processing (Nikias and Shao [29]) and teletraffic engineering (Resnick [32]). The focus of this paper is maximum likelihood (ML) estimation for the parameters of autoregressive (AR) time series processes with non-Gaussian stable noise. Specific applications for heavy-tailed AR models include fitting network interarrival times (Resnick [32]), sea surface temperatures (Gallagher [20]) and stock market log-returns (Ling [24]). Causality (all roots of the AR polynomial are outside the unit circle in the complex plane) is a common assumption in the time series literature since causal and noncausal models are indistinguishable in the case of Gaussian noise. However, noncausal AR models are identifiable in the case of non-Gaussian noise, and these models are frequently used in deconvolution problems (Blass and Halsey [3], Chien, Yang and Chi [10], Donoho [16] and Scargle [36]) and have also appeared for modeling stock market trading volume data (Breidt, Davis and Trindade [5]). We, therefore, consider parameter estimation for both causal and noncausal AR models. We assume the parameters of the AR model equation and the parameters of the stable noise distribution are unknown, and we maximize the likelihood function with respect to all parameters. Since most stable density functions do not have a closed-form expression, the likelihood function is evaluated by inversion of the stable characteristic function. We show that ML estimators of the AR parameters are n 1/α -consistent (n represents sample size) and converge in distribution to the maximizer of a random function. The form of this limiting distribution is intractable, but the shape of the distribution for these estimators can be examined using the bootstrap procedure. We show the bootstrap procedure is asymptotically valid provided the bootstrap sample s...

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