VaR : Student-t GARCH

Question

I have a question on the VaR estimation via the student t GARCH model.

Under this framework, the one day ahead VaR estimate is calculated by the following formula:

$$VaR_{p}=\mu_{t+1}+\sigma_{t+1}\sqrt{\frac{\nu-2}{\nu}}z_{p}$$

Where $z_{p}$ is the unconditional student-t quantile of the estimated innovations.

As you know, for the parameters estimation of the Student-t GARCH model the corresponding (Student-t) log likelihood function should be maximised (Maximum Likelihood methodology). When this estimation is conducted the $\nu$ (degrees of freedom) is also estimated since it is one of the parameters of the log likelihood function.

The question is if this is the actual $\nu$ (degrees of freedom) that we need to use in the above formula or I have to re-estimate it by fitting the Student-t pdf to the estimated innovations (residuals)? If this is the case can this fitting (parameters estimation) be done via the fitdist MATLAB function (Student-t is not included in the provided range of distributions)?

Answer 1

I think you are mixing the residuals versus the standardized residuals ( 0 mean and unit variance residuals) and/or the student distribution vs the standardized student distribution.

The degree of freedom you obtain from the MLE estimation is the one you should use in the Var formula.

The standardized student Garch model is:

$Y_{t} = \mu_{t} + \sigma_{t} z_{t} \qquad z_{t} \sim t(0,1,v)\qquad, v>2$

where $z_{t}$ are the standardized student innovations (see Bollerslev original formulation). You always need to have standardized innovations since the conditional variance of residuals $ (\epsilon_{t}= \sigma_{t} z_{t})$ must be equal to $\sigma_{t}^{2}$ : $Var(\epsilon_{t}) = \sigma_{t}^{2}$ only if $Var(z_{t})=1$

When you fit your model you obtain the residuals : $ \epsilon_{t}= \sigma_{t} z_{t}$

Then the standardized t-innovations are recovered by :

$z_{t} = \epsilon_{t} \sigma_{t}^{-1} $

MLE estimate is based on the fact that those innovations are IID. The loglikelihood estimation returns arch+garch parameters and parameters of the distribution of the error term : in this case $v$

Finally the VaR is given by :

$Var_{t} = \mu_{t} + \sigma_{t} st_{\alpha,v} $

with $st_{\alpha,v} $ being the left quantile at $ \alpha $% for the standardized t-distribution with (estimated) number of degrees of freedom ($v$).

However if you use a non-standardized t-distribution to express the VaR , and since you know that the variance of the t-distribution is : $ Variance(f_{tdist})=\sigma_{tdist}^{2}= \frac{v}{v-2} $ you can expressed the VaR by rescaling the student distribution as:

\begin{equation} \begin{split} Var_{t} & = \mu_{t} + \sigma_{t} st_{\alpha,v} \\ & = \mu_{t} + \sigma_{t} \left( t_{\alpha,v } \sigma_{tdist}^{-1} \right)\\ & = \mu_{t} + \sigma_{t} \left( t_{\alpha,v } \left[\sqrt\frac{v}{v-2} \right]^{-1} \right)\\ & = \mu_{t} + \sigma_{t} t_{\alpha,v } \sqrt\frac{v-2}{v} \\ \end{split} \end{equation}

Note that $st_{\alpha,v}$ , the standardized t-distribution is different of $t_{\alpha,v}$ the non-standardized t-distribution

The MLE estimation is based on $st_{\alpha,v}$ , from the estimation you get $v$ and then the factor $ \sqrt\frac{v-2}{v} $ is just a rescaling operation which does not participate to the MLE estimation.

In Matlab you can use the tLocationScaleDistribution with $\mu =0$ and $\sigma = 1 $.