GARCH modelling in continuous time for irregularly spaced time series data

Abstract

The discrete-time GARCH methodology which hits had such a profound influence on the modelling of heteroscedasticity in time series is intuitively Well motivated in capturing many 'stylized facts' concerning financial series, and is now almost routinely used in a wide range of situations, often including some where the data are not observed at equally spaced intervals of time. However, such data is more appropriately analyzed with a continuous-time model which preserves the essential features of the successful GARCH paradigm. One possible such extension is the diffusion limit of Nelson, but this is problematic in that the discrete-time GARCH model and its continuous-time diffusion limit tire not statistically equivalent. As till alternative, Kluppelberg et al. recently introduced a continuous.-time version of the GARCH (the 'COGARCH' process) which is constructed directly from a background driving Levy process, The present paper how to fit this model to irregularly spaced time series data using discrete-time GARCH methodology, by approximating the COGARCH with an embedded sequence of discrete-time, GARCH series which converges to the continuous-time model in a strong sense (in probability, in the Skorokhod metric), as the discrete/approximating grid grows finer. 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 and to illustrate this, an empirical investigation using stock index data is carried out

Data cloning estimation of GARCH and COGARCH models

“…Maller et al (2008) proved that this model can be expressed as a continuous time limit of a sequence of GARCH models. As the COGARCH model may be approximated by an appropriate set of GARCH processes, the parameters of the COGARCH model can be estimated using the relation between them.…”

“…Then, the process (G n (t), σ n (t)) can be considered as an approximation to the COGARCH model (G(t), σ(t)) for n large enough. Maller et al (2008) used this approximation to fit the model to unequally spaced time data, by deriving a pseudo-maximum likelihood function and numerically maximizing it in order to estimate the corresponding parameters.…”

“…But, in this case, as the procedure is based on a MCMC algorithm, it is not necessary to numerically maximize the pseudolikelihood function as in Maller et al (2008).…”

“…On the other hand, estimators based in pseudo-maximum likelihood were studied by Maller et al (2008), who approximated the COGARCH model with an embedded sequence of discrete-time GARCH series. Moreover, Müller (2010) used a Markov chain Monte Carlo (MCMC) based estimation procedure for the COGARCH model, driven by a compound Poisson process.…”

“…In order to deal with the approximation to the COGARCH model, Maller et al (2008) fixed an interval [0, T ], where T > 0, and took the sequence of integer numbers (N n ) n≥1 such that lim n→∞ N n = ∞ and a sequence of partitions 0 = t 0 (n) < . .…”