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Gujarati-Skewness/Kurtois Calc formula & clalrification

Thread starter #1
Hi David,
Could you please provide some guidance on the following:

1) Sample Skewness and Kurtosis Formulas

I have seen 2 differnt versions,can you please clarify which is correct:

Version#1: Sample Skewness

Sum of Third moment about the mean / cube of Std Deviation * 1/ N

Version#2:
[ [ Sum of Third moment about the mean] / (n-1) ] / cube of Std Deviation

Which one is correct?

2) In Gujarati 3.8, can you please provide additional details concerning the use/purpose of the Alternative Skew and Kurtosis calculation that is provided.

Thank you
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#2
Hi Paul,

If by "Sum of Third moment about the mean" you mean "Sum of [X - mean]^3" then your #1 appears to me to be the population skew:
[Sum of [X-mean]^3 / n ] * 1/StdDev^3 = E[(x-mean)^3] / StdDev^3 = Population third moment / StdDev^3
i.e, that looks to match the skew when n = entire population size
or, note Gujarati's 3.10: the six-sided die is fully characterized by the uniform PDF. That's not a sample, it's an expectation, so there is he is right to use population (sum and divided 6). Put another way, your #1 is fine for a population (unrealistic if historical) or if, ex ante, we have a distribution that fully characterizes *expectation* the random variable (e.g., our expectations for a 6 sided die are not a sample)

While #2 is Gujarati's sample skew:
Sample third moment / StdDev^3
i.e., this for when we only have a sample (any realistic exercise where we use historical data is effectively a sample)
this is analogous to using Sum of [x - mean]^2 / (n - 1) for sample variance instead of divide by (n) for population.
the net effect is a slightly larger sample skew.

but caution: it's not "wrong" to use #1 on a sample, like it's not wrong to use (n) in denominator. These are estimators. There can be more than one. Using #1 is "incorrect" to Gujrati's sample but it is still an *estimator* of skew...

Here is the learning XLS on skew/kurt: http://www.bionicturtle.com/premium/spreadsheet/2.a.2._skew_kurtosis/

please note i have 2 sample skew/kurt cacs:
1. Gujarati's which matches your #2
2. The excel sample skew which is different than either (yes, it's a 3rd variation!) which adjusts for small sample bias. Just another estimator (this one "unbiased"). Note how similar they are...

for the actual data on Google returns, i get .107 sample skew (your #2, our "proper" sample skew) versus .108 "unbiased" sample skew (not your #1, which would be smaller, this is larger). The 1/100th difference is precision not justified by the reality of the distribution, we are lucky if we are in the rough neighborhood.

Hope this helps, David
 
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