Re: st: Jarque-Bera test

[Nick Cox <njcoxstata@gmail.com>]

A qualification: the Jarque-Bera test uses standard errors for
sampling distributions that depend on sample size, but the formulas
are approximations for large n. The sampling distributions being
Gaussian is an asymptotic result.

Nick

On Thu, Sep 27, 2012 at 2:44 AM, Nick Cox <njcoxstata@gmail.com> wrote:
> I commented on this matter in a thread started by Rodrigo Briceño yesterday.
>
> The main premise of the Jarque-Bera test is that skewness and kurtosis
> for samples from a Gaussian (normal) themselves have sampling
> distributions that are Gaussian. With other assumptions this allows a
> portmanteau test for normality using a chi-square statistic. The roots
> of the idea go back at least as far as Fisher, R.A. 1925. Statistical
> methods for research workers. Edinburgh: Oliver and Boyd.
>
> Here is a simulation which I think is suggestive. Stata 12 users won't
> need to -set memory-.
>
> clear
> set memory 100M
> set obs 1000000
> set seed 280352
> gen Gaussian = rnormal()
> gen block = ceil(_n/100)
> statsby skewness=r(skewness) kurtosis=r(kurtosis), nodots by(block)
> clear  : su Gau, d
> scatter kurtosis skewness, ms(oh) msize(*0.5) yla(, ang(h)) ///
> xli(0) yli(3) subtitle(10000 samples of size 100 from Gaussian)
> name(skewkurt, replace)
> qnorm skewness, name(skew, replace)
> qnorm kurtosis, name(kurt, replace)
>
> The sampling distribution for skewness is dodgy in the tails and that
> for kurtosis is way off Gaussian.
>
> An exercise for anyone seriously using Jarque-Bera is to follow
> through to check the consequences for Jarque-Bera, naturally using
> sample sizes of interest to them.
>
> I have done no literature search to see whether this comment is
> standard in some of the literature but I see enough uses of
> Jarque-Bera to suppose that knowledge of this problem is not
> widespread (or not widespread enough).
>
> The essence of the matter is that Jarque-Bera uses asymptotic results
> regardless of sample size for a problem in which convergence to those
> results is very slow. This approach is decades out of date and I am
> surprised that StataCorp support the test without a warning. The
> Doornik-Hansen test, for example, looks much more satisfactory. This
> was added in Stata 11 after Jarque-Bera was added. See
> http://www.stata.com/help.cgi?mvtest_normality
> Users of Stata 7..10 without access to Stata 11..12 and so the
> official implementation of the Doornik-Hansen test can download
> -omninorm- from SSC.
>
> (A separate point is that I regard any test as inferior as a means of
> checking normality to -qnorm- but that is secondary here.)
>
> Nick
>
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