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I am trying to fit a multiple linear regression (OLS) model with IPO underpricing as dependent variable. As part of my master thesis I would like to analyze the effect of venture capital certification (dummy) on IPO underpricing (raw returns - not transformed). My sample consists of 199 IPOs.

Up to now I got the following plots. However, I am concerned about the residuals at the upper end of the qq-plot. Is the assumption of normality in the residuals violated in this case? I would report robust standard errors in order to control for the observed heteroscedasticity.

Further, I am not sure how to deal with the observation at the upper right in the qq-plot (represents first-day return of 63% vs. 51% the next smaller). Could this be an outlier or may it be simply the result of a misspecified model?. The studentized residual for this particular observation is 5.36 (all others below 3) and it has a Cook's D of 0.16. I run the regression twice: with and without this particular observation. It turns out that, if removed, the coefficients are getting larger in absolute values. The variables of interest (VC reputation and VC type) turned from weakly significant (at 10% level) into significant (at the 5% level). Running a robust regression, both the coefficients as well as the p-values are even more significant. However, the robust regression excludes 4 observations. Hence, should I simply stick with OLS and remove this potential outlier (and mention that I did so)? Or would it be better to report both results (with and without the observation)?

Thank you!

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  • $\begingroup$ The worst of all solutions is to remove the outlier just because it is awkward. You need to report what you've reported here. Saying as much as you can about the substantive context of the outlier is likely to be expected. You don't say which kind of robust regression you used. That's essential. It's likely that it downweighted 4 observations to zero weight; that's not quite the same as excluding them. Their values were considered! $\endgroup$ – Nick Cox Jan 28 '18 at 10:59
  • $\begingroup$ Running a robust regression, both the coefficients as well as the p-values are even more significant. Which other regression analysis you have conducted ? $\endgroup$ – Subhash C. Davar Jan 28 '18 at 15:40
  • $\begingroup$ @NickCox Sorry, I should be clearer. I used the robust regression implemented in stata (rreg command) with M-estimators (combination of Huber-weights and Tukey's biweights). Indeed, no observation were excluded, 4 received a weight of 0. Compared to OLS, the coefficients do not change dramatically, only the SE decrease which makes the main variable of interest highly significant (1% level). Following your suggestions, I won't exclude any values but will report the circumstances, instead. So, am I fine proceeding with the OLS regression (regarding normality)? $\endgroup$ – dtribus Jan 28 '18 at 20:09
  • $\begingroup$ @subhashc.davar Sorry for not clearly specifying. I meant robust regression with M-estimators. Thus, weighting resiuals according to their size. $\endgroup$ – dtribus Jan 28 '18 at 20:15
  • $\begingroup$ rreg in Stata is arguably best ignored. See statalist.org/forums/forum/general-stata-discussion/general/… and its references for arguments why. $\endgroup$ – Nick Cox Jan 28 '18 at 21:20
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You have a large sample size and there is only this one value that seems to be an issue, while the rest seems very close to a normal distribution. Do you have any reason to exclude this point (bad measure? possible typing error?)? Excluding points is usually a bad solution when they are measures similar to the others that were not due to obvious experimental mistakes, but removing it and comparing the regression with and without it can give you useful information on the importance of this outlier on your conclusions. See Whether to delete cases that are flagged as outliers by statistical software when performing multiple regression? and http://www.rapidinsightinc.com/handle-outliers/

Try to do the analysis after removing this point, if you get the same conclusions then it will give you support to argue that this point did not influence your results, and you can report the results of the analysis that include this point while mentioning this issue and that removing this point did not change your conclusions.

If you get different conclusions when removing this point, then you may have a problem and you may need to use a robust regression.

This issue is very similar to the one reported here: What to do if residuals are not normally distributed?

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  • $\begingroup$ Thank you for your answer. I have checked the point and I don't find plausible reasons to exclude the observation. Without the point the standard errors reduce about 10%, the coefficients do not change dramatically. Thus, I only observe that the variables of interest get statistically significant, which is of course favorable and in line with the robust regression. So would you still recommend to report the results with the point included? $\endgroup$ – dtribus Jan 28 '18 at 20:10
  • $\begingroup$ I updated my answer to include questions that were similar to yours. The most transparent thing to do would be to report both analyses and to discuss the differences, but this may not be accepted by certain journals if you are trying to publish these results. For a master thesis, it should be welcomed by the examiners that you provided honest data and explored different way to analyze them, but don't spend too much time discussing it. $\endgroup$ – Nakx Jan 29 '18 at 2:24

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