Student T Fit
Looks at Student T distribution fit to the S&P 500 log return series.
This notebook uses same time-series of log returns for the S&P 500 as the MarketReturnDistribution notebook. That notebook contains a number of plots which will not be shown again here.
Start ... Tuesday 3 January 1950
End ... Friday 25 October 2013
Unlike the stable distribution, the Student T distribution can fit a wide range of tail behavior, extending from very heavy tails to the normal tails as the parameter, ν → ∞. The Student T tail behavior tends to occur more smoothly and more centrally, where as in the stable distribution as α → 2, the tail behavior appears abruptly further and further from the center of the distribution. Below is the histogram, the smoothed empirical density and the Student T distribution fit for the series of log returns. The fit is pretty good for the center of the distribution.
The tail fit is also pretty good, with the tail exponent, α or ν, of 3.1 being closer to the actual tail behavior of the data. This is a symmetric distribution so it cannot fit the tails separately; in this case the left tail has a better fit than the right tail (red).
© Copyright 2013 Robert H. Rimmer, Jr. Sun 27 Oct 2013
