Using the Estimated Student-t Degrees of Freedom in GARCH VaR

# VaR : Student-t GARCH


# VaR : Student-t GARCH












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 by Malick (score 2)

https://quant.stackexchange.com/a/36408

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 $.