st: Jarque-Bera test

[Nick Cox <njcoxstata@gmail.com>]

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 This approach is decades out of date and I am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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