Limit experiments of GARCH

TL;DR

This paper investigates the theoretical relationship between GARCH and COGARCH models, showing that GARCH converges to COGARCH under certain conditions and introducing MCOGARCH as a limiting experiment when observations are incomplete.

Contribution

It establishes the convergence of GARCH to COGARCH in Le Cam's framework and introduces MCOGARCH as a new limiting experiment for incomplete observations.

Findings

GARCH converges to COGARCH when volatility is observed.

MCOGARCH is a limiting experiment for GARCH with incomplete data.

COGARCH's jump times are more random, offering practical advantages.

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 Kl\"{u}ppelberg, 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.