## 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.

## References

[1] Bollerslev, T. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics. Vol. 31, 1986, pp. 307–327.

[2] 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.

[3] Engle, R. F. “Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica. Vol. 50, 1982, pp. 987–1007.

[4] Glosten, L. R., R. Jagannathan, and D. E. Runkle. “On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks.” The Journal of Finance. Vol. 48, No. 5, 1993, pp. 1779–1801.

[5] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.