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Abstract: The discrete-time GARCH methodology which has had such a profound influenceon the modelling of heteroscedasticity in time series is intuitively wellmotivated in capturing many `stylized facts- concerning financial series, andis now almost routinely used in a wide range of situations, often includingsome where the data are not observed at equally spaced intervals of time.However, such data is more appropriately analyzed with a continuous-time modelwhich preserves the essential features of the successful GARCH paradigm. Onepossible such extension is the diffusion limit of Nelson, but this isproblematic in that the discrete-time GARCH model and its continuous-timediffusion limit are not statistically equivalent. As an alternative,Kl\-{u}ppelberg et al. recently introduced a continuous-time version of theGARCH the `COGARCH- process which is constructed directly from a backgrounddriving L\-{e}vy process. The present paper shows how to fit this model toirregularly spaced time series data using discrete-time GARCH methodology, byapproximating the COGARCH with an embedded sequence of discrete-time GARCHseries which converges to the continuous-time model in a strong sense inprobability, in the Skorokhod metric, as the discrete approximating grid growsfiner. This property is also especially useful in certain other applications,such as options pricing. The way is then open to using, for the COGARCH,similar statistical techniques to those already worked out for GARCH models andto illustrate this, an empirical investigation using stock index data iscarried out.



Author: Ross A. Maller, Gernot Müller, Alex Szimayer

Source: https://arxiv.org/







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