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# Documentation

## Maximum Likelihood Estimation for Conditional Variance Models

### Innovation Distribution

For conditional variance models, the innovation process is ${\epsilon }_{t}={\sigma }_{t}{z}_{t},$ where zt follows a standardized Gaussian or Student's t distribution with $\nu >2$ degrees of freedom. Specify your distribution choice in the model property Distribution.

The innovation variance, ${\sigma }_{t}^{2},$ can follow a GARCH, EGARCH, or GJR conditional variance process.

If the model includes a mean offset term, then

${\epsilon }_{t}={y}_{t}-\mu .$

The estimate function for garch, egarch, and gjr models estimates parameters using maximum likelihood estimation. estimate returns fitted values for any parameters in the input model equal to NaN. estimate honors any equality constraints in the input model, and does not return estimates for parameters with equality constraints.

### Loglikelihood Functions

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

$f\left({\epsilon }_{1},{\epsilon }_{2},\dots ,{\epsilon }_{N}|{H}_{N-1}\right)=\prod _{t=1}^{N}f\left({\epsilon }_{t}|{H}_{t-1}\right),$

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

$LLF=-\frac{N}{2}\mathrm{log}\left(2\pi \right)-\frac{1}{2}\sum _{t=1}^{N}\mathrm{log}{\sigma }_{t}^{2}-\frac{1}{2}\sum _{t=1}^{N}\frac{{\epsilon }_{t}^{2}}{{\sigma }_{t}^{2}}.$

• If zt has a standardized Student's t distribution with $\nu >2$ degrees of freedom, then the loglikelihood function is

$LLF=N\mathrm{log}\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\pi \left(\nu -2\right)}\Gamma \left(\frac{\nu }{2}\right)}\right]-\frac{1}{2}\sum _{t=1}^{N}\mathrm{log}{\sigma }_{t}^{2}-\frac{\nu +1}{2}\sum _{t=1}^{N}\mathrm{log}\left[1+\frac{{\epsilon }_{t}^{2}}{{\sigma }_{t}^{2}\left(\nu -2\right)}\right].$

estimate performs covariance matrix estimation for maximum likelihood estimates using the outer product of gradients (OPG) method.