Consider the time series
where . Here, zt is an independent and identically distributed series of standardized random variables. Econometrics Toolbox™ supports standardized Gaussian and standardized Student's t innovation distributions. The constant term, , is a mean offset.
A conditional variance model specifies the dynamic evolution of the innovation variance,
where Ht–1 is the history of the process. The history includes:
Conditional variance models are appropriate for time series that do not exhibit significant autocorrelation, but are serially dependent. The innovation series is uncorrelated, because:
E(εt) = 0.
E(εtεt–h) = 0 for all t and
However, if depends on , for example, then εt depends on εt–1, even though they are uncorrelated. This kind of dependence exhibits itself as autocorrelation in the squared innovation series,
Two characteristics of financial time series that conditional variance models address are:
Volatility clustering. Volatility is the conditional standard deviation of a time series. Autocorrelation in the conditional variance process results in volatility clustering. The GARCH model and its variants model autoregression in the variance series.
Leverage effects. The volatility of some time series responds more to large decreases than to large increases. This asymmetric clustering behavior is known as the leverage effect. The EGARCH and GJR models have leverage terms to model this asymmetry.