1
$\begingroup$

What exactly does the Jarque-Bera test statistic represent?

It seems like the p-value is of more relevance when interpreting the data.

I would assume that as the JB statistic tests for normality, a value of 0 would mean the data is perfectly normal. Does this mean that all we look for in a JB test statistic is its proximity to 0?

$\endgroup$
2
$\begingroup$

Firstly note that failure to reject a null doesn't mean the null is true -- so just because a goodness of fit test came out perfectly consistent with a normal, that doesn't imply that the data were drawn from a normal distribution.

However, in the case of this particular test the connection is even more tenuous -- even if the population skewness and kurtosis were the same as for a normal, it still doesn't imply normality. There's an infinity of distributions that have skewness 0 and kurtosis 3. A number of examples can be found on site here with a bit of searching.

The closer the Jarque-Bera test statistic* is to zero, the closer the sample skewness and kurtosis are to the values 0 and 3. Rejection indicates inconsistency with those values (and hence with normality), failure to reject doesn't imply normality.

You could perhaps interpret it as a weighted sum-of-squared deviations from the expected cumulants under normality. (At least that would be the interpretation in very large samples - the values are asymptotic, but the approach of the joint distribution to the asymptotic distribution comes in quite slowly; the distribution shows clear signs of dependence even for samples of size 300 for example)

*(that name for the test is popular among econometricians but it is wrongly named, since the statistic was proposed, used and discussed decades before they came along).

$\endgroup$
  • $\begingroup$ Regarding the footnote: is the Jarque-Bera test know to others under a different name? $\endgroup$ – snoram Nov 12 '17 at 20:35
  • 1
    $\begingroup$ Well, it's such an old test that it has had a few names. For example Bowman and Shenton wrote a paper explaining about how the asymptotics kick in very slowly and looked at how to improve the test in 1975 -- it was already an old test by then. To my recollection the joint distribution is discussed in Kendall & Stuart, though I don't remember who first raised it as a test. I think it may have been Pearson the younger or maybe even Pearson the elder. I'll try to find out for sure. [I could have told you in the mid 80s when I was investigating goodness of fit, but that was a long time ago]... $\endgroup$ – Glen_b -Reinstate Monica Nov 12 '17 at 20:41
  • 1
    $\begingroup$ I recall reading a discussion (may have been in Kendall & Stuart) comparing the asymptotic test to the rectangle test (testing each term on its own at say the 2.5% level), for example. What it should be called and what it gets called nowadays are different things. $\endgroup$ – Glen_b -Reinstate Monica Nov 12 '17 at 20:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.