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QQ plot of standardised residuals

I am required to build a OLS model. Currently, My model of log(response) against a number of predictors have fulfilled homogeneity assumption (constant variance) and low multicollinearity (based on VIF and eigensystem analysis). However, I am stubbed at failing the normality test of my standardised residuals (failed Shapiro-Wilk's test for normality). I have looked at the recommended Box-Tidwell's transformation for my predictors but they suggested values are about 0.8, suggesting no transformation is needed. In terms of the outlier analysis, I was able to identify a number of leverage points and there are no influential points (based on cook's distance). My prof has also told us not to remove any data points.

How can I improve on my model to meet the normality assumption? Any help will be greatly appreciated.

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    $\begingroup$ Are you sure you need the normality assumption? The OLS estimator has a ton of desirable properties without the assumption. If your sample is quite large, you probably do not need normality. $\endgroup$ Apr 9, 2021 at 7:57
  • $\begingroup$ Thanks for the reply. Yes the data is large (654 data points), but fulfilling the normality assumption is a marking point to this project, and based on my discussion with the professor, she mentioned that it is important for the model to be adequate (fulfil all assumptions) prior to any further use of the model. $\endgroup$
    – Darren
    Apr 9, 2021 at 8:00
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    $\begingroup$ It might be one of the cases where the professor needs to learn some statistics... Pick up any basic econometrics or statistics textbook and find the chapter on regression assumptions and what they yield in terms of properties of the OLS estimator. Perhaps that will help in discussing with your professor. $\endgroup$ Apr 9, 2021 at 9:18

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The Shapiro-Wilk's test is a statistical test based on Monte-Carlo sampling, or with other words, there is some randomness involved. You could also try to rerun it a few times to make sure the result is consistent and it was not a statistical fluke (that you got very unlucky Monte Carlo samples). You should also have a look at the p-value to see how far you are off from normality (according to the test), or if it's possible, have a look at the histogram of values generated by the Monte-Carlo simulation.

Furthermore, you should be aware of the following:

From: https://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test

Like most statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis (i.e., although there may be some statistically significant effect, it may be too small to be of any practical significance); thus, additional investigation of the effect size is typically advisable, e.g., a Q–Q plot in this case.[5]

So, you are exactly going the right way about it: is the deviation from normality large enough for it to be meaningful? Do you think the deviation in the QQ-plot is practically meaningful? I think it is hard to judge, but perhaps it is sufficiently normal already. To answer this question, we really should know what is in the end the purpose of checking the normality assumption.

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