1. The Student t distribution
So if the normal distribution isn't a good model for log-returns over shorter horizons, then what is?
2. The Student t distribution
One distribution that often fits better is the Student t distribution. The Student t distribution in its general has three parameters: a location parameter mu, a scaling parameter sigma, and a degree-of-freedom parameter nu.
The probability density takes the form shown on the slide.
The smaller the value of nu, the more heavy-tailed or leptokurtic the distribution. The larger the value of nu, the closer it gets to a normal distribution.
So for one additional parameter, you can get a family that is much more flexible than the normal and contains normal as the limiting case as nu goes to infinity.
3. Fitting the Student t distribution
The best way to fit the distribution is by the method of maximum likelihood and that has been implemented in the function fit (dot) st in the QRM package.
Here is an example where the Student t distribution is fitted to the FTSE log-returns for 2008-2009.
The data are passed to the fit.st() function and the fitted model is assigned to the object tfit. tfit is a list with a component called par ests. This component is assigned to tpars and displayed and then the values nu, mu and sigma are stored for later use.
As you can see the value of nu in the first component of tpars is very small, about two point nine five. This indicates a very heavy-tailed t distribution. In fact for nu less or equal to 4 the kurtosis is actually infinite.
4. Displaying the fitted Student t distribution
How does it look when the fitted curve is superimposed on the data?
In the code the histogram of the ftse returns is plotted and the fitted normal is again superimposed in red.
The fitted t density is computed and assigned to the object yvals. It then superimposed on the picture in blue.
The calculation of the tdensity uses the standard R function dt(). Because this function only takes the degree of freedom argument nu and does not take arguments for mu and sigma, you have to do a little bit more work to calculate the density. Before passing the data to the dt() function, mu is subtracted from ftse and then the difference is divided by sigma. Then the density is scaled by 1 over sigma to get the calculation right.
Clearly the t distribution fits much better.
In the remaining exercises you'll try fitting a t distribution to data yourself and you'll also test other kinds of data like foreign-exchange returns and interest-rate returns for normality.
5. Let's practice!
So over to you for some practice.