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In a college course I try to measure the abnormal returns (the returns that are below or over the returns of the market) of a companies stock after a specific event based on linear OLS regression. When interpreting the results with t-statistic and corresponding p-value, the professor of the course told me that these abnormal returns I observe have to be normally distributed. Otherwise the t-test would lead to miss specified results. If I am not mistaken, OLS assumes the normal distribution of the error terms. Unfortunately I don’t really get why the effect I would like to observe has to be normally distributed in order to get unbiased results. What Problems arise when the data i observe are not normally distriuted? Do I have to work with approach in this case (not OLS regression)? Thanks in advance

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what you care about is the distribution of the sample mean of the errors. This is where the central limit theorem helps, because even if your individual errors are not normally distributed, the mean of say 100 (ie the number of datapoints) of them will be.

the p-value is calculated based on an assumption that the (sample mean) of the underlying data is normally distributed, if its not, then you would have to use a different formula to calculate probability. [however, as I say, I don't believe it should be a problem, assuming your t-statistic is calculated from a large enough sample]

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  • $\begingroup$ Thanks for your answer. So basically, even when the abnormal returns in my sample are not normally distributed, i can assume that the abnormal returns of the overall population will be? How exactly do i know whether my sample is large enough to calculate a useful t statistic? $\endgroup$ – Stelios Mar 2 '19 at 17:04
  • $\begingroup$ so I am perhaps confused about your test. are you looking at a single data point (eg company), and deciding if that has an abnormal return, because then the sample size is 1, and CLT is useless and you have to assume that your general population has normally distributed 'abnomal' returns; or are you looking at an average of multiple data points, where you can appeal to CLT see eg 195.134.76.37/applets/AppletCentralLimit/… $\endgroup$ – seanv507 Mar 2 '19 at 17:44
  • $\begingroup$ So the actual research question is, whether an event x has an impact of firms stock prices (effect measured as abnormal returns). I have a sample of 10 firms and calculated the abnormal returns for every day within a specific period of time (e.g. five days before the event accurs until 1 day after the event). After that i cumulate the abnormal returns for every day and firm. In a last step i average all the cumulated returns over the number of firms (10) so that i end up with a single cumulated average abnormal return. All my test were not significant which is somehow confusing. Thanks a lot. $\endgroup$ – Stelios Mar 2 '19 at 20:06
  • $\begingroup$ So then the clt will be helpful, and as you can see in the demo I linked,even 10 make it close to normal. It's hard to say anymore without diving into the data. Why don't you attach your analysis in the question. E.g. As r script $\endgroup$ – seanv507 Mar 2 '19 at 21:08
  • $\begingroup$ Thanks again for your help, I really appreciate it! If you want you can download the r file with the code as well as the data input here: ufile.io/p4tyq What does confuse me is the fact, that none of the Abnormal returns or the Cumulative average abnormal returns are significant, since there seems to be a clear decrease in the stock price of the firms when i look at their stock price charts for the specific event date. $\endgroup$ – Stelios Mar 3 '19 at 8:54

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