## Presample Data for Conditional Variance Model Estimation

Presample data is data from time points before the beginning of the observation period. In Econometrics Toolbox™, you can specify your own presample data or use automatically generated presample data.

In a conditional variance model, the current value of the innovation conditional variance, ${\sigma }_{t}^{2},$ depends on historical information. Historical information includes past conditional variances, ${\sigma }_{1}^{2},{\sigma }_{2}^{2},\dots ,{\sigma }_{t-1}^{2},$ and past innovations, ${\epsilon }_{1},{\epsilon }_{2},\dots ,{\epsilon }_{t-1}.$

The number of past variances and innovations that a current conditional variance depends on is determined by the degree of the conditional variance model. For example, in a GARCH(1,1) model, each conditional variance depends on one lagged variance and one lagged squared innovation,

`${\sigma }_{t}^{2}=\kappa +{\gamma }_{1}{\sigma }_{t-1}^{2}+{\alpha }_{1}{\epsilon }_{t-1}^{2}.$`

In general, difficulties arise at the beginning of the series because the likelihood contribution of the first few innovations is conditional on historical information that is not observed. In the GARCH(1,1) example, ${\sigma }_{1}^{2}$ depends on ${\sigma }_{0}^{2}$ and ${\epsilon }_{0}.$ These values are not observed.

For the GARCH(P,Q) and GJR(P,Q) models, P presample variances and Q presample innovations are needed to initialize the variance equation. For an EGARCH(P,Q) model, max(P,Q) presample variances and Q presample innovations are needed to initialize the variance equation.

If you want to specify your own presample variances and innovations to `estimate`, use the name-value arguments `V0` and `E0`, respectively.

By default, `estimate` generates automatic presample data as follows. For GARCH and GJR models:

• Presample innovations are set to an estimate of the unconditional standard deviation of the innovation series. If there is a mean offset term, presample innovations are specified as the sample standard deviation of the offset-adjusted series. If there is no mean offset, presample innovations are specified as the square root of the sample mean of the squared response series.

• Presample variances are set to an estimate of the unconditional variance of the innovation series. If there is a mean offset term, the presample innovations are specified as the sample mean of the squared offset-adjusted series. If there is no mean offset, presample variances are specified as the sample mean of the squared response series.

For EGARCH models:

• Presample variances are computed as for GARCH and GJR models.

• Presample innovations are set to zero.