## 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.