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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.

Bloodline
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    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? – cardinal Oct 09 '12 at 23:49
  • @cardinal That also depends on how skewness [is defined](http://en.wikipedia.org/wiki/Skewness#Other_measures_of_skewness). –  Oct 19 '12 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 –  Jan 15 '14 at 07:26

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In general it cannot. Take normal distribution as an example. Variance can take any positive value but skewness is still zero.

And there are other similar questions:

Does skewness predict variance? Or does variance predict kurtosis (since variance is second and kurtosis is fourth moment)? Do answers depend on type of distribution?

sitems
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    While variance and skewness are unrelated in the normal distribution by definition, that does not mean this is always the case... – analystic Oct 10 '12 at 06:21