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Statistica Sinica 20 (2010), 1363-1393



Ngai Hang Chan and Rong-Mao Zhang

Chinese University of Hong Kong and Zhejiang University

Abstract: Random walk models driven by GARCH errors are widely applicable in diverse areas in finance and econometrics. For a first-order autoregressive model driven by GARCH errors, let $\widehat{\phi}_{n}$ be the least squares estimate of the autoregressive coefficient. The asymptotic distribution of $\widehat{\phi}_{n}$ is given in Ling and Li (2003) when the GARCH errors have finite variances. In this paper, the limit distribution of $\widehat{\phi}_{n}$ is established as functionals of a stable process when the GARCH errors are heavy-tailed with infinite variances. An estimate of the tail index of the limiting stable process is proposed and its asymptotic properties are derived. Furthermore, it is shown that the least absolute deviations procedure works well under the unit-root and heavy-tailed GARCH setting. This research provides a relatively broad treatment of unit-root GARCH models that includes the commonly entertained unit-root IGARCH scenario.

Key words and phrases: Autoregressive process, GARCH, heavy-tailed, IGARCH, stable processes and unit-root.

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