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I am trying to perform an OLS on time series for a project for college. The professor told me that I need my regressors to be normal in order to justify the use of a linear regression. His argument was that only if the joint distribution of dependent and independent variables were normal (and thus every marginal distribution were normal too) the expected values of the dependent conditioning for the values of our regressors would line up on a line, estimated by the OLS. Though I found an entirely different opinion on a reliable forum. This opinion stated that normality of the regressors does not matter. It doesn't cause any problem in the use of OLS. Can anyone confirm one or the other view? Can you please give me a link to a paper I could make reference ?

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  • $\begingroup$ The thread at stats.stackexchange.com/questions/16381 is a detailed discussion of all the OLS assumptions. The answers contain links and references. However, it does not explicitly address the claim that "only if the joint distribution of dependent and independent variables were normal (and thus every marginal distribution were normal too) the expected values of the dependent conditioning for the values of our regressors would line up on a line." Counterexamples are abundant and easy to construct: let $Y|X$ have any distribution with mean $\alpha+\beta X$. $\endgroup$ – whuber May 30 '15 at 18:33
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In regression we condition on X so its distribution is irrelevant. Only the conditional distribution of Y (or equivalently, the residuals) is relevant. Although there may be special considerations in time series (e.g., how are you handling autocorrelation?) your professor is incorrect.

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  • $\begingroup$ What if we assume a multivariate normal distribution for the regressors and the response? The conditional mean then is linear. I suspect this is what it is meant here. $\endgroup$ – JohnK May 30 '15 at 14:07
  • $\begingroup$ Yes, I think that's what the professor was talking about. So what's the implication of this condition on OLS? Frank Harrel says it is not necessary at all. Any ideas on how to make sense of my professor's words? Was he just plain wrong or he meant something....? $\endgroup$ – Paolo May 30 '15 at 14:40
  • $\begingroup$ If he was speaking in general and not about a specific time series model (which wouldn't be OLS anyway) then he is wrong. $\endgroup$ – Frank Harrell May 30 '15 at 16:50

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