Bernoulli

Limit experiments of GARCH

Boris Buchmann and Gernot Müller

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Abstract

GARCH is one of the most prominent nonlinear time series models, both widely applied and thoroughly studied. Recently, it has been shown that the COGARCH model (which was introduced a few years ago by Klüppelberg, Lindner and Maller) and Nelson’s diffusion limit are the only functional continuous-time limits of GARCH in distribution. In contrast to Nelson’s diffusion limit, COGARCH reproduces most of the stylized facts of financial time series. Since it has been proven that Nelson’s diffusion is not asymptotically equivalent to GARCH in deficiency, in the present paper, we investigate the relation between GARCH and COGARCH in Le Cam’s framework of statistical equivalence. We show that GARCH converges generically to COGARCH, even in deficiency, provided that the volatility processes are observed. Hence, from a theoretical point of view, COGARCH can indeed be considered as a continuous-time equivalent to GARCH. Otherwise, when the observations are incomplete, GARCH still has a limiting experiment, which we call MCOGARCH, which is not equivalent, but nevertheless quite similar, to COGARCH. In the COGARCH model, the jump times can be more random than for the MCOGARCH, a fact practitioners may see as an advantage of COGARCH.

Article information

Source
Bernoulli, Volume 18, Number 1 (2012), 64-99.

Dates
First available in Project Euclid: 20 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.bj/1327068618

Digital Object Identifier
doi:10.3150/10-BEJ328

Mathematical Reviews number (MathSciNet)
MR2888699

Zentralblatt MATH identifier
1291.62162

Keywords
COGARCH Le Cam’s deficiency distance random thinning statistical equivalence time series

Citation

Buchmann, Boris; Müller, Gernot. Limit experiments of GARCH. Bernoulli 18 (2012), no. 1, 64--99. doi:10.3150/10-BEJ328. https://projecteuclid.org/euclid.bj/1327068618


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