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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 object property Distribution.
The innovation variance, can follow a GARCH, EGARCH, or GJR conditional variance process.
If the model includes a mean offset term, then
The estimate function for garch, egarch, and gjr model objects estimates parameters using maximum likelihood estimation. estimate returns fitted values for any parameters in the input model object equal to NaN. estimate honors any equality constraints in the input model object, 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 (OPG) method.