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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?

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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).

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  • $\begingroup$ Regarding the footnote: is the Jarque-Bera test know to others under a different name? $\endgroup$ – snoram Nov 12 '17 at 20:35
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    $\begingroup$ 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 Nov 12 '17 at 20:41
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    $\begingroup$ 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 Nov 12 '17 at 20:53

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