Fitting a Student t Distribution to Log Returns
Summary
The document considers how to estimate location, scale, and degrees of freedom when fitting a Student t distribution to observed log returns. It notes that scaling a t variate changes its variance according to the distribution’s degrees of freedom, and asks whether location and scale can be recovered from the sample mean and standard deviation. It also questions a proposed approach based on rescaling observations and minimizing squared error.
The response recommends method of moments or maximum likelihood. Under method of moments, the sample mean estimates location, excess kurtosis provides information about degrees of freedom, and the sample variance then helps determine scale. Maximum likelihood instead requires specifying the density of the transformed returns. The central caveat is that fitting only mean and variance leaves the tail parameter unidentified; a higher moment or the full likelihood is needed. The discussion offers estimation routes, but gives no numerical example or assessment of their finite-sample performance.
Key ideas
- The sample mean can estimate the location parameter under the method of moments.
- Excess kurtosis can inform the degrees-of-freedom estimate for a t distribution.
- Once degrees of freedom are estimated, sample variance can be used to estimate scale.
- Maximum likelihood requires the density for the transformed return observations.
- Matching mean and variance alone does not capture the tail shape.
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# Parameters for numerically fitting t-distribution to log-returns
# 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 $MSE=\frac{1}{T}\sum_t (r_t-\hat{r}_t)^2$ ?
## Answer by Richi Wa (score 2)
https://quant.stackexchange.com/a/9419
It could be much more simple: if you use the method of moments (MM) then you estimate the mean and the variance and for example the kurtosis of your sample. Then you fit the parameters to these statistics. Alternatively you use maximum-likelihood (MLE).
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.