As shown in a previous post; i.e. TASI’s Normality Assumption, TASI violates the normality assumption due mainly to the presence of negative skewness (i.e. -0.8486) and high positive kurtosis (i.e. 9.903). But why then assume normality? Because estimation of the probability density function (PDF) for TASI log return suggests that TASI’s PDF is a normal density. This is very convenient to financial modelers because using a parametric statistical model such as the normal density function would simplify their understanding of the financial markets.
In this post, I’ll try to estimate the density of TASI log returns using the histogram and the kernel density estimation (KDE) then I’ll compare them with the normal density.
The histogram is a simple and well-known estimator of probability density. The figure below is a histogram of TASI index log returns from 7-Jan-2007 to 31-Dec-2015 using 30 cells. The vertical dashes at the bottom represents the data points. Notice the outliers around -.10. Notice that the histogram roughly resembles a normal density. Notice also the high density of returns in the middle causing the histogram to appear more like a city skyline than a density curve.
The histogram is a crude estimator of density. A better estimator is the kernel density estimator (KDE). In the figure blow, KDE is mush closer in estimating the true density of TASI log returns which resembles a normal density curve. Using KDE requires some thought about the variance-bias trade off. Considering that, I am using a bandwidth of 1.5 in the plot below so that the KDE curve doesn’t overfit nor underfit the data.
Finally, in the figure below I plot both the histogram and the KDE compared to a normal density curve having the same mean and standard deviation.
Having the normal density curve overlaid on the same plot, it is easy to notice the high peak corresponding to the high positive kurtosis and the negative skewness.
Notice that although the histogram and the KDE resemble a normal density curve, the fact is, both are not a normal density curve. Thus, the distribution of TASI returns is not normal although both the histogram and the KDE suggest it.