RE: st: Testing for differences in skewness and kurtosis?

[Cameron McIntosh <cnm100@hotmail.com>]

A bootstrap approach might indeed be palatable here.  On that note, I would suggest perhaps looking into the literature on skewness persistence (in which robust measures of both skewness and kurtosis have been developed):

Ergun, A.T. (June 13, 2011). Skewness and Kurtosis Persistence: Conventional vs. Robust Measures. Midwest Finance Association 2012 Annual Meetings Paper. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1857653

Muralidhar, K. (1993). The Bootstrap Approach for Testing Skewness Persistence. Management Science, 39(4), 487-491.

Nath, R. (1996). A Note on Testing for Skewness Persistence. Management Science, 42(1), 138-141.

Sun, Q., & Yan, Y. (2003). Skewness persistence with optimal portfolio selection. Journal of Banking & Finance, 27(6), 1111–1121.

Adcock, C.J., &  Shutes, K. (2005). An analysis of skewness and skewness persistence in three emerging markets. Emerging Markets Review, 6(4), 396–418.

Cam

> Date: Sat, 5 May 2012 17:05:16 +0100
> Subject: Re: st: Testing for differences in skewness and kurtosis?
> From: njcoxstata@gmail.com
> To: statalist@hsphsun2.harvard.edu
> 
> I'll add a
> 
> 6. You could also end up showing that skewness and kurtosis are
> similar but both imply non-normality. That would also undermine your
> tests based on normality. Only one possible outcome of the tests, that
> skewness and kurtosis are similar and results for both imply
> approximate normality would be entirely good news for you.
> 
> 
> On Sat, May 5, 2012 at 4:33 PM, Nick Cox <njcoxstata@gmail.com> wrote:
> > This question provokes various comments from me. They are all
> > variations on a single theme, that this is much more problematic than
> > you imply.
> >
> > 1. Large-sample standard errors for skewness and kurtosis _if_ the
> > parent is normal are given in many mathematical statistics texts and
> > could be the basis for a test. But if the normality of the
> > distributions is even slightly in doubt I have the impression that
> > they aren't worth much. Even if it is not in doubt, the large-sample
> > results do not kick in quickly. (There is at least one command in
> > official Stata that is wildly cavalier about this point! Some
> > economists use the so-called Jarque-Bera test based on asymptotic
> > standard errors as a test for normality.)
> >
> > 2. This is linked to the fact that skewness and kurtosis, as dependent
> > on third and fourth powers of deviations from the mean, can be very
> > sensitive to departures of any kind. I don't know in consequence what
> > a robust test of skewness and kurtosis would look like; that sounds a
> > contradictory request to make. If you are interested in robust
> > comparisons, skewness and kurtosis are not where you start.
> >
> > 3. You sound very confident that your distributions are normal but
> > with real data that is at best an approximation. It is an
> > approximation that you can get away frequently with tests on means,
> > less frequently with tests on variances, and even less frequently with
> > higher moments. Box in Biometrika 1953 remains pertinent.
> >
> > 4. Let's imagine that you applied such a test. As parent normals both
> > have skewness 0 and kurtosis 3, a difference in skewness or kurtosis
> > between two samples would be likely to arise if at least one
> > distribution was really not normal. This is just a restatement of the
> > fact that if two quantities are different, they can't be equal, and so
> > not equal to any particular constant. So, you would need to consider
> > that possibility of non-normality directly. If you concluded that that
> > was so, you could end up undermining the results of your previous
> > tests.  So, I don't think you can ignore the question of whether the
> > samples come from the same distribution. It's an assumption behind
> > what you are doing, even if it is not independently interesting.
> >
> > 5. In some ways the most direct way to compare skewness and kurtosis
> > would be to bootstrap, but that would ignore the question of whether
> > the distributions are normal, which is a crucial assumption so far as
> > you are concerned. Perhaps better is to simulate normal samples with
> > the same sizes, means and SDs.
> >
> > When you have two distributions, a -qqplot- is a restatement of _all_
> > the information on those distributions. It would throw light on
> > whether apparent non-normal skewness or kurtosis arises from
> > individual outliers or something more systematic.
> >
> > Nick
> >
> > Box, G.E.P. 1953. Non-normality and tests on variances. Biometrika 40: 318-335.
> >
> > On Sat, May 5, 2012 at 3:34 PM, George Murray <george.murray16@gmail.com> wrote:
> >
> >> I am currently working with a very simple dataset, with two variables,
> >> V0 and V1 (around 150 obs each), each normally distributed, and the
> >> difference of the means of the distribution of the variables are
> >> (statistically) different, but the standard deviations are equal. I
> >> would like to test whether there exists any significant difference in
> >> the skewness of these two variables. Can this be done through
> >> hypothesis testing, or is this only possible through some simulation
> >> technique (bootstrapping?) Is there a test that is robust to the
> >> aforementioned conditions? Is there an equivalent test for kurtosis?
> >> Is anyone aware of how this can be calculated with Stata? (And no, I
> >> am not trying to test whether they come from the same distribution)
> >>
> 
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