Maximum Likelihood Estimation for Conditional Variance Models

Innovation Distribution

For conditional variance models, the innovation process is $${\epsilon }_t={\sigma }_t{z}_t,$$ where z_t 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 H_t denote the history of a process available at time t, t = 1,...,N. The likelihood function for the innovation series is given by

$$f({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_N|H_{N-1})={\displaystyle \prod _{t=1}^{N}f(}{\epsilon }_t|H_{t-1}),$$

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 z_t has a standard Gaussian distribution, then the loglikelihood function is

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

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

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

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

References

[1] Bollerslev, Tim. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (April 1986): 307–27. https://doi.org/10.1016/0304-4076(86)90063-1.

[2] Bollerslev, Tim. “A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return.” The Review of Economics and Statistics 69 (August 1987): 542–47. https://doi.org/10.2307/1925546.

[3] Engle, Robert. F. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica 50 (July 1982): 987–1007. https://doi.org/10.2307/1912773.

[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, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.