# Does variance predict skewness?

I am trying to get some ideas on how to test for an implicit relationship, if any, between variance and skewness. That is, given a very large data set (e.g 90 years of monthly returns), is there a way to generally test if skewness is likely to increase with increasing variance or vice versa?

I will be very grateful for any ideas or suggestions regarding this topic.

• In general there can be no such relations. Consider the following two examples. Example 1: Let $X = \pm n$ with probability 1/2 each. What is the variance? (It will vary with $n$.) What is the skewness? (It will be constant.) Example 2: Let $X = -n$ with probability $p = 1 / 3n^2$, $X = 2n$ with probability $p/2$ and $X = 0$ otherwise. What is the variance? (It will be constant.) What is the skewness? (It will vary with $n$.) Conclusion? Commented Oct 9, 2012 at 23:49
• @cardinal That also depends on how skewness is defined.
– user10525
Commented Oct 19, 2012 at 13:19
• I`m performing a downside risk capm research for the Brazilian market. (50 IBOVESPA COMPANIES, daily returns, 7 years period) After providing proofs that returns distribution is not normally distributed (komolgorov-smirnov test), I found a statistical and significant relationship between variance a skewness of returns. Further control test are required, but it could mean that the relationship between returns and skewness ( I tested it too) may be partly explained by variance. Any comments would be appreciated. GD
– user37331
Commented Jan 15, 2014 at 7:26