## 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.

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