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I am analyzing buy-and-hold abnormal returns of stocks (dependent variable) using OLS regression. These returns, however, tend to be positively skewed (and are so in my case). The residuals obtained by OLS are slightly skewed (skewness of 0.921 and kurtosis of 5.073).

Although the histogram of residuals looks quite normal, I am concerned about the heavy tails in the qq-plot. Is it valid to assume that the residuals are approximately normal or is the normality assumption violated in this case?

I already tried to transform the dependent variable using log-modulus and cube roots (as there are negative values) in order to get a smoother qq-plot, but did not get improved results.

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marked as duplicate by mdewey, Community Feb 20 '18 at 13:33

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  • $\begingroup$ did you try skewed Student t? it often works pretty well for daily data. normality assumption is very suspicious on return series, though people keep using it, btw, your returns do not look like normal $\endgroup$ – Aksakal Feb 13 '18 at 17:47
  • $\begingroup$ Normally distributed errors are not an assumption for linear regression. Search the site for references and comments on that point. Is your question "do my residuals follow a normal distribution" or "do my residuals have the skewness and kurtosis of a normal distribution"? The latter seems irrelevant because, for other procedures that do depend on normal dist-n, I can generate data satisfying 0-skew, 3 kurt moments but are strongly nonnormal. $\endgroup$ – AdamO Feb 13 '18 at 17:49
  • $\begingroup$ @AdamO, yes we say that normality assumption is not important usually in OLS etc., however, he's working with financial data. That gives me a pause. Depending on what he's going to do with data, normality assumption could be a difference between bankruptcy and success. that's why I'd be cautious in this case unless you know exactly what OP is going to do with the data. asset returns are often not normal, have fat tails etc. So, in risk management area the normality is a very dangerous assumption, it can underestimate the risk to a great degree $\endgroup$ – Aksakal Feb 13 '18 at 17:53
  • $\begingroup$ @Aksakal There is no inherent difference to the nature of the question simply because of its application. $\endgroup$ – AdamO Feb 13 '18 at 17:58
  • $\begingroup$ I'd be most concerned if it was clear the model was missing systematic structure -- I'd want to smooth the residual versus fitted plot as just one check, but it's not evident here -- and/or there were signs of relying literally on the normality assumption using inferential tools after fitting the model. (As a perhaps trivial point, talking about "ideal conditions" rather than "assumptions" I find clarifying.) $\endgroup$ – Nick Cox Feb 13 '18 at 17:59
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Economists do not worry about normality in papers, as far as I noticed. Your residuals are not normal, but it doesn't really matter despite the fact that you're conducting inference study.

However, you could use bootstrapping instead of t-stats, that way you'd avoid whole normality discussion altogether. You seem to have enough observations for a respectable bootstrapping analysis.

If you really need to test normality then look at the popular tests such as Jarque-Bera or Anderson-Darling. They don't just look at critical values of skew and kurtosis in isolation. Some of the tests have a critical value for a combo of skew and kurtosis.

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  • $\begingroup$ Jarque-Bera, Doornik-Hansen are some pairs of economists showing interest in testing for normality. Their tests are widely used. $\endgroup$ – Nick Cox Feb 13 '18 at 20:25

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