st: RE: Standard Errors for Skewness & Kurtosis

[Nick Cox <n.j.cox@durham.ac.uk>]

This received various good answers from Stas Kolenikov and Steve Samuels. 

The most optimistic answer was given by neither. In very large samples, and especially in very close to normal distributions, the standard errors depend on sample size alone. See also the documentation for -varnorm-. Otherwise put, some normality testing routines in Stata pivot on using large-sample results for possibly rather small samples. In particular, some economists know such procedures as Jarque-Bera tests although the basic idea belongs to neither Jarque nor Bera. 

Stas and Steve's comments were appropriately different but both more guarded.  

Some simulations alone show that even for "good" situations, i.e. sample sizes not too small and distributions close to normal, the sampling distributions of skewness or kurtosis are not even symmetric and uncorrelated. Some sample code: 

program mysim, rclass
        version 11.1
        syntax [, obs(integer 1) mu(real 0) sigma(real 1) ]
        drop _all
        set obs `obs'
        tempvar z
        gen `z' = rnormal(`mu',`sigma')
        summarize `z', detail 
        return scalar kurt = r(kurtosis)
        return scalar skew = r(skewness)
end

simulate kurt=r(kurt) skew=r(skew), reps(100000) nodots: mysim, obs(100)
rcspline kurt skew

where for the last command using -rcspline- from SSC you may substitute your favourite scatterplot smoother. 

The trace is not flat! 

A yet further problem is discussed in the next Speaking Stata column in Stata Journal 10(3). 

Nick 
n.j.cox@durham.ac.uk 

Philip Burgess

I can't find any ado that calculates the standard errors for skewness

Do you know of such or how I might estimate these?


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