Is the Density of TASI Unimodal?

I established in my previous posts that the normality assumption for the log returns of TASI index and its sectors is false. In this post, I’ll use the Normal probability plot to investigate the distributions of the data. In particular, I want to know if the log returns of the TASI index is unimodal or not. The short answer is TASI index log returns is not unimodal. But then what?

Below are the density plot and the normal probability plot for TASI index log returns.

R_Lab_Ch4_Fig21
Density (left) and Normal probability (right) plots for TASI index log returns

The normal plot of a random sample coming from a normal distribution must be close enough to a straight line. But as you can see, the normal plot of TASI index log returns is far from being linear (see the plotted normal line). This confirms of course that the data is not coming from a normal distribution but also reveals other informations.

The convexity and concavity in the normal plot worth checking. The alteration between concavity and convexity in the normal plot for TASI index log returns indicates a complex behavior which can be investigated by the density plot (left). Studying the normal plot (right) shows that convexity is changed three times; concave to convex to concave to convex. Further studying the density plot (left) shows that TASI log returns has a multimodal distribution.

One last point worth mentioning, The Shapiro-Wilks test uses the normality plot to test normality. For TASI index log returns, the Shapiro-Wilk test rejects the null hypothesis of normality with a p-value less than 2.2e-16; W is 0.84611.

Thank you.

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Author: Thamir K. AlHashemi

Practitioner in the fields of quantitative finance and risk management. A lifetime learner. Reading is the only hobby that I haven't failed in ;)

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