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# Documentation

## Conditional Variance Models

Consider the time series

${y}_{t}=\mu +{\epsilon }_{t},$

where ${\epsilon }_{t}={\sigma }_{t}{z}_{t}$. 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, $\mu$, is a mean offset.

A conditional variance model specifies the dynamic evolution of the innovation variance,

${\sigma }_{t}^{2}=Var\left({\epsilon }_{t}|{H}_{t-1}\right),$

where Ht–1 is the history of the process. The history includes:

• Past variances, ${\sigma }_{1}^{2},{\sigma }_{2}^{2},\dots ,{\sigma }_{t-1}^{2}$

• Past innovations, ${\epsilon }_{1},{\epsilon }_{2},\dots ,{\epsilon }_{t-1}$

Conditional variance models are appropriate for time series that do not exhibit significant autocorrelation, but are serially dependent. The innovation series ${\epsilon }_{t}={\sigma }_{t}{z}_{t}$ is uncorrelated, because:

• E(εt) = 0.

• E(εtεt–h) = 0 for all t and $h\ne 0.$

However, if ${\sigma }_{t}^{2}$ depends on ${\sigma }_{t-1}^{2}$, 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, ${\epsilon }_{t}^{2}.$

 Tip   For modeling time series that are both autocorrelated and serially dependent, you can consider using a composite conditional mean and variance model.

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.