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,

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.

To supply your own presample variances and innovations, use the appropriate name-value arguments for the data type matching the in-sample response data:

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

For EGARCH models:

See Also

Objects

• garch | egarch | gjr

Functions

• estimate

Related Topics

• Maximum Likelihood Estimation for Conditional Variance Models
• Conditional Variance Model Estimation with Equality Constraints
• Initial Values for Conditional Variance Model Estimation
• Optimization Settings for Conditional Variance Model Estimation