Variance Ratio plots are not enough!

This paper is just 8 pages long but conveys an important point about random walk tests. The paper analyzes the use of variance and absolute variation as measures of volatility while testing a series for random walk. The paper suggest the following plot :

for different values of zeta. For zeta=1 ,one ends up using absolute variation and for zeta=2, one ends up using variance. If the time series has fat tails, it might happen that variance ratio plots do not show anything fishy. However in such cases, absolute variation plots have a higher probability of highlighting fat tails.

The takeaway from the paper is : Always us absolute variation plots in conjunction with variance ratio plots.