Open Access
February 2012 Limit experiments of GARCH
Boris Buchmann, Gernot Müller
Bernoulli 18(1): 64-99 (February 2012). DOI: 10.3150/10-BEJ328


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.


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Boris Buchmann. Gernot Müller. "Limit experiments of GARCH." Bernoulli 18 (1) 64 - 99, February 2012.


Published: February 2012
First available in Project Euclid: 20 January 2012

zbMATH: 1291.62162
MathSciNet: MR2888699
Digital Object Identifier: 10.3150/10-BEJ328

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

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability

Vol.18 • No. 1 • February 2012
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