GARCH models

Chemical processes can be modelled in detail as a bunch of equations and differential equations based on chemical and physical laws. These laws rely on static assumptions about the environment where the corresponding processes take place. In practice, all components are infiuenced by stochastic factors that can influence the static process behaviour and/or the dynamic process characteristics. Incorporating continuous stochastic processes to a differential equation leads to stochastic differential equations. Linear stochastic differential equations (SDEs) are usually formulated in the following general form using the Wiener process W t)  

It can be shown that for a linear SDE with coefficient functions Cj(f) (which are continuous in f) an explicit, unique solution always exists and the SDE has finite first and second-order moments.  

Time series models and SDEs deal with the same sort of stochastic process. Both differ only in the domain of variables, which are either discrete or continuous. For instance, chemical processes are continuous by nature. In practice, however, the condition of a 

It can be shown that ARCH and GARCH models are able to approximate stochastic differential processes if the latter fulfil certain properties. Albeit the goodness of fit is limited, both types of methods are related and can be converted into each other. Moreover, simple stochastic processes show quite simple auto-correlation structures similar to basic ARMA models. For instance, the Ornstein-Uhlenbeck process can be seen as the continuous equivalent of the AR(1) process. In other words, an Ornstein-Uhlenbeck process measured in discrete intervals can be interpreted/modeUed as an AR(1) process (see also (2.23), (2.60), and (2.61)).  

This subsection briefly introduces GARCH models as discrete counterpart of continnous stochastic processes. In contrast to ARMA models, the basic idea is that the variance/ volatility in time is no longer deterministic and constant bnt depends on previous errors and volatility, i.e. 

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The 8 period ARCF1 model produces quite a substantial change in the estimates. Once again, this probably results from the restrictive assumption about the lag weights in the ARCH model. The GARCH model follows. 

The remainder of this section introduces the relevant notation with an additional focus on the extension to heteroscedastic models (so-called (G)ARCH and ARMA-GARCH models) as these can be seen as the discrete-time counterpart of continuous stochastic processes formulated in terms of SDEs. 

E.g. see Fornari and Mele (2001) for an application of GARCH models as diffusion approximations fitted to financial data sets. 

E.g. see Bauwene et al. (2006) for a review of recent multivariate GARCH models and Bollerslev (2008), Hansen and Lunde (2005) or Degiannakis and Xekalaki (2004) for a review of univariate (G)ARCH... 

It has to be noted that the estimation of ARMA-GARCH models requires some more sophisticated methods compared to simple ARMA models, see Francq and Zakoian (2004). 

Bauwens, L., Laurent, S., and Rombouts, J. Multivariate GARCH models a survey. Journal of Applied Econometrics, 21(1) 79-109, 2006. 

Wang, Y. Asymptotic nonequivalence of GARCH models and diffusions. The Annals of Statistics, 30(3) 754-783, 2002. 

Various forms of General Auto-Regressive Conditional Heteroske-dastic (GARCH) models have been used to estimate return volatility. Such models express current volatility as a function of previous returns and forecasts. For instance, the GARCH(1,1) model takes the form ... 

Ausin, M. C. Galeano, P. 2007. Bayesian estimation of the Gaussian mixture GARCH model. Computational Statistics Data Analysis. 51(5) 2636-2652. 

Bauwens, L. Lubrano, M. 1998. Bayesian inference on GARCH models using the Gibbs sampler. Econometrics Journal, 1(1) C23-C46. 

Wei, S. X. 2002. A censored-GARCH model of asset returns with price limits. Journal of Empirical Finance, 9(2) 197-223. 

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