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I am wondering about the correct formula of the standardized Student's t distribution. In the rugarch package on page 15 it is given as:

Substituting $\frac{(\nu-2)}{\nu}$ into 49 we obtain the standardized Student's distribution: $$ f\left(\frac{x-\mu}{\sigma}\right)=\frac{1}{\sigma}f(z)=\frac{1}{\sigma}\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{(\nu-2)\pi}\Gamma\left(\frac{\nu}{2}\right)}\left(1+\frac{z^2}{(\nu-2)}\right)^{-\left(\frac{\nu+1}{2}\right)} $$

whereas from a book (Carol Alexander Market Risk Analysis, Quantiative Methods in Finance, Vol. 1 page 99) I have the following formula:

Setting $\mu=0$ and $\beta\nu=\nu-2$ in (I.3.55) given the density for the standardized Student t distribution, i.i. the Student t distribution with zero expercation and unit variance. This that the density function $$ f_\nu(x)=((\nu-2)\pi)^{-1/2}\Gamma\left(\frac{\nu}{2}\right)^{-1}\Gamma\left(\frac{\nu+1}{2}\right)\left(1+\frac{x^2}{\nu-2}\right)^{-\left(\frac{\nu+1}{2}\right)} $$

Now I don't know which of them is the correct one? Why is there an additional $\frac{1}{\sigma}$ in the first formula?

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The $\frac{1}{\sigma}$ comes from the garch volatility. So this should not be relevant in this context.

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