# Parameters for numerically fitting t-distribution to log-returns

I am fitting the t-distribution to log-returns numerically (not using R, MATLAB, Stata, etc.), but rather using general programming. Assuming the log-return values are $r_t$, and the $t$-variates are $x_t$, then given $r_t=ax_t + b$, and knowing the variance of t-distribution, $\nu/(\nu-2)$, the variance of $r_t$ becomes $\sigma^2(r_t)=a^2(\nu/(\nu-2))$. Would the parameters to fit for the $r_t$ then be $b$, the location, and $a=\sqrt{\sigma^2(r_t)/(\nu/(\nu-2))}$, where $\sigma(r_t)$ is the measured standard deviation of the observed log-return?

The solution would be to first initialize $a=1$ and $b=0$ and then calculate

$$\hat{r_t}=x_t=\frac{r_t-b}{a}=\frac{r_t-b}{\sqrt{\frac{\sigma^2(r_t)}{\frac{\nu}{\nu-2}}}}$$

with minimization of $MSE=\frac{1}{T}\sum_t r_t-\hat{r}_t$ ?

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For MM: from wikipedia you get the mean and the variance. In your notation you can fit $b = \bar{r}$ so $b$ equals the empirical mean. The excess kurtosis is defined here and you can solve for the parameter $\nu$ from the estimated kurtosis of $r_t$ without estimating its volatility (if I am not mistaken this cancels out due to the definition). Then you go back and estimate your $a$ from the sample variance and your fitted $\nu$.
For the MLE you would need to write down the density of $r_t$.
Another comment to your proposed method. It is not that far from what I propose. But you don't use higher moments in order to estimate $\nu$. Using the density (MLE) or the kurtosis (MM) you go beyond modelling mean and variance only.
The excess kurtosis is defined here and you can solve for the parameter $\nu$ from the estimated kurtosis without estimating its volatility. Beware that excess kurtosis is only defined and finite for $\nu>4$, so for the common range of $\nu\in[3, 4.5]$ encountered in finance the sample kurtosis is at best unrelieble. –  Quartz Feb 19 at 12:03