Maximum Likelihood Estimation for Conditional Variance Models

Innovation Distribution

For conditional variance models, the innovation process is $ε_{t} = σ_{t} z_{t},$ wherez_tfollows a standardized Gaussian or Student’stdistribution with $ν > 2$ degrees of freedom. Specify your distribution choice in the model propertyDistribution.

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

If the model includes a mean offset term, then

$ε_{t} = y_{t} - μ .$

Theestimatefunction forgarch,egarch, andgjrmodels estimates parameters using maximum likelihood estimation.estimatereturns fitted values for any parameters in the input model equal toNaN.estimatehonors 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. LetH_tdenote the history of a process available at timet,t= 1,...,N. The likelihood function for the innovation series is given by

$f (ε_{1}, ε_{2}, \dots, ε_{N} | H_{N - 1}) = \prod_{t = 1}^{N} f (ε_{t} | H_{t - 1}),$

wherefis a standardized Gaussian ortdensity function.

The exact form of the loglikelihood objective function depends on the parametric form of the innovation distribution.

Ifz_t有一个标准的高斯分布,然后呢loglikelihood function is

$L L F = - \frac{N}{2} \log (2 π) - \frac{1}{2} \sum_{t = 1}^{N} \log σ_{t}^{2} - \frac{1}{2} \sum_{t = 1}^{N} \frac{ε_{t}^{2}}{σ_{t}^{2}} .$
Ifz_thas a standardized Student’stdistribution with $ν > 2$ degrees of freedom, then the loglikelihood function is

$L L F = N \log [\frac{Γ (\frac{ν + 1}{2})}{\sqrt{π (ν - 2)} Γ (\frac{ν}{2})}] - \frac{1}{2} \sum_{t = 1}^{N} \log σ_{t}^{2} - \frac{ν + 1}{2} \sum_{t = 1}^{N} \log [1 + \frac{ε_{t}^{2}}{σ_{t}^{2} (ν - 2)}] .$

estimateperformscovariance matrix estimationfor maximum likelihood estimates using the outer product of gradients (OPG) method.

参考文献

[1]Bollerslev, Tim. “Generalized Autoregressive Conditional Heteroskedasticity.”Journal of Econometrics31 (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 Statistics69 (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.”Econometrica50 (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.