The Annals of Statistics

Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors

Shiqing Ling and W. K. Li

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Abstract

This paper investigates the maximum likelihood estimator (MLE) for unstable autoregressive moving-average (ARMA) time series with the noise sequence satisfying a general autoregressive heteroscedastic (GARCH) process. Under some mild conditions, it is shown that the MLE satisfying the likelihood equation exists and is consistent. The limiting distribution of the MLE is derived in a unified manner for all types of characteristic roots on or outside the unit circle and is expressed as a functional of stochastic integrals in terms of Brownian motions. For various types of unit roots, the limiting distribution of the MLE does not depend on the parameters in the moving-average component and hence, when the GARCH innovations reduce to usual white noises with a constant conditional variance, they are the same as those for the least squares estimators (LSE) for unstable autoregressive models given by Chan and Wei (1988). In the presence of the GARCH innovations, the limiting distribution will involve a sequence of independent bivariate Brownian motions with correlated components. These results are different from those already known in the literature and, in this case, the MLE of unit roots will be much more efficient than the ordinary least squares estimation.

Article information

Source
Ann. Statist., Volume 26, Number 1 (1998), 84-125.

Dates
First available in Project Euclid: 28 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1030563979

Digital Object Identifier
doi:10.1214/aos/1030563979

Mathematical Reviews number (MathSciNet)
MR1611800

Zentralblatt MATH identifier
0932.62103

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62E20: Asymptotic distribution theory
Secondary: 60F17: Functional limit theorems; invariance principles

Keywords
Bivariate Brownian motion GARCH models limiting distribution maximum likelihood estimation stochastic integral unstable ARMA models

Citation

Ling, Shiqing; Li, W. K. Limiting distributions of maximum likelihood estimators for unstable autoregressive moving-average time series with general autoregressive heteroscedastic errors. Ann. Statist. 26 (1998), no. 1, 84--125. doi:10.1214/aos/1030563979. https://projecteuclid.org/euclid.aos/1030563979


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