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If I'm failing the Jarque-Bera test but the residuals still appear to be normally on a qq plot and histogram, is it acceptable to say that my residuals are approximately normally distributed? Asked another way, how reliable are those normality tests and do you need to meet them to pass the normality assumption.

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If I'm failing the Jarque-Bera test but the residuals still appear to be normally on a qq plot and histogram, is it acceptable to say that my residuals are approximately normally distributed?

No, it is appropriate to state they are bell-shaped after transformations and normalizations.

Asked another way, how reliable are those normality tests and do you need to meet them to pass the normality assumption.

Your question depends on the characteristics of the sample. Restated, your question is "does the Jarque-Bera have sufficient power to test my hypothesis?"

That is, unfortunately, a complex question that doesn't generate a simple automatic answer for this topic. Failing Jarque-Bera after the transformations in your comments is a pretty good statement that your data is not normally distributed. Assuming your sample size is large enough to have power over the hypothesis, then Frequentist decision theory says you should behave as if the data were not normally distributed.

Given the seventy years of data on heavy tails, Jarque-Bera is ideally suited to detect them while other tests look for other types of departure. Alternatively, the Shapiro-Wilk test will test disturbances in the position of order statistics. A rejection by Shapiro-Wilk implies that not only are your tails wrong, but everything is in the wrong place. If your sample is large enough to have power, then these would be the most powerful tests for your problem.

Based on your results, you should reject normality. You also should not go test hunting in the hopes of finding a result that fits what you want. Equity data, in particular, is notoriously non-normal. Think of all the transformations you had to do just to get it close.

The importance of the normal distribution for most applications is the existence of a covariance matrix. Not all distributions have them. If it does not exist, you are in a different class of problems.

The question you should be asking is "if the normality assumption does not hold, what does this imply for the specific problem that I am facing?"

Predictive work is sensitive to the distributions involved. If these are equities and given your transformation $\log(y)$, then the distribution should be the hyperbolic secant distribution, but that won't necessarily hold for an index. That is also ignoring the impact of liquidity costs and constraints.

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Your question is very general and it's hard to answer specifically without more details from you

  • what are you trying to use this data for ?
  • How big is your dataset ?
  • Are you interested in the tails of the disrtribution of your sample or the bulk ?
  • Have you cleaned-up your data to remove outliers ?
  • The qq plot is only visual test whereas Jarque-Berra is statistical so hard to compare the 2
  • Have you tried other normality tests ? Kolmogorov-Smirnov is not sensitive to tails as much as JB

For more infos you can look a lot of material online. See this nice presentation about normality tests

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  • $\begingroup$ 1) Ultimate goal is a predictive mode. 2) Interested in the bulk. 3) Reluctant to remove outliers but have done normalizing transformations. e.g., log (y) 4) I have tried the shipiro test, which also fails. 5) Data set of about 2,400 $\endgroup$ – Ajk Feb 3 at 18:08
  • $\begingroup$ Thanks for extra details, please update those into your question for better quality. Have you tried KS test ? $\endgroup$ – Ezy Feb 3 at 18:30
  • $\begingroup$ Haven't, but will. How about the sample size? In the source you referenced,it said that as the sample size increases, the tests become more restrictive and less important. Might that be the case here? $\endgroup$ – Ajk Feb 3 at 20:59
  • $\begingroup$ @Ajk it is hard to make any statement absent any numerical data $\endgroup$ – Ezy Feb 3 at 21:11
  • $\begingroup$ What numerical data do you need? $\endgroup$ – Ajk Feb 3 at 21:18

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