For conditional variance models, the innovation process is where zt follows a
standardized Gaussian or Student’s t distribution with degrees of freedom. Specify your distribution choice in the model
The innovation variance, can follow a GARCH, EGARCH, or GJR conditional variance process.
If the model includes a mean offset term, then
estimate function for
gjr models estimates
parameters using maximum likelihood estimation.
fitted values for any parameters in the input model equal to
estimate honors any equality constraints in the input model,
and does not return estimates for parameters with equality constraints.
Given the history of a process, innovations are conditionally independent. Let Ht denote the history of a process available at time t, t = 1,...,N. The likelihood function for the innovation series is given by
where f is a standardized Gaussian or t density function.
The exact form of the loglikelihood objective function depends on the parametric form of the innovation distribution.
If zt has a standard Gaussian distribution, then the loglikelihood function is
If zt has a standardized Student’s t distribution with degrees of freedom, then the loglikelihood function is
estimate performs covariance matrix estimation for
maximum likelihood estimates using the outer product of gradients
 Bollerslev, T. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. Vol. 31, 1986, pp. 307–327.
 Bollerslev, T. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics. Vol. 69, 1987, pp. 542–547.
 Engle, R. F. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. Vol. 50, 1982, pp. 987–1007.
 Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.