# Ran multivariate linear regression, checked normal probability plot, residuals are not normal. What can I do?

One of the required assumptions for multiple linear regression is that residuals are normally distributed, correct?

After running my regression, my normal probability plot is showing the typical 'heavy tail' S shape.

Does this inability to satisfy the assumption deem my whole model useless? Is there anyway I can get normal residuals?

My dependent variable is VaR, and my independent variables are Average Return, Log of Market value, dummy variable 1 and dummy variable 2.

Edit: I've tried transforming the independent variable (VaR)(Square root, Log, reciprocal), but it doesn't seem to make sufficient difference

This means that a linear regression is not the best model for your data. You may want to try a regularized regression (LASSO/Ridge) to see if penalizing the coefficients will help.

Regression analysis, as a minimization of the sum of squared errors, does not require normality of the error term.

The requirements are that errors are homoscedastic and uncorrelated. And these are the fundamental assumptions (together with exogeneity). Then estimators are unbiased, optimal (exhibit the minimum variance within the class of unbiased estimators) and consistent (the variance also goes to zero with sample size). Normality is not required.

If errors are normal, then we can also say something about the standard errors of these estimates and build confidence intervals. However, there are ways to build these confidence intervals even if the errors are not Gaussian, for example by bootstrapping them. Therefore I would focus on the other assumptions which are more material.

• My apologies if I've not understood what you're saying, I've only been studying financial modelling daily for 2-3 weeks. 1)My hypothesis relies on the 95% confidence level, so does this mean that I have to bootstrap? As my current confidence level isn't correct (because errors aren't normal)? 2) What are the implications of having heavy tailed qq plot? 3)Is there anything I can do to get a normal qq plot? – Harry Jan 5 '15 at 0:33
• Heavy tailed errors usually imply wider confidence intervals to the Gaussian ones. You can bootstrap. However, in my opinion, the other assuptions are more important to be tested: (1) Are errors serially correlated? (2) Is the volatility of the errors serially correlated? (3) In a multiple regression, are the various regressors correlated between them? If the answer to these three is 'no' then you should be fine. – Kiwiakos Jan 5 '15 at 0:46
• Ok thanks, I understand. Can I test these three assumptions in Excel? I'm not sure how to get errors, and what I need to correlate them with. – Harry Jan 5 '15 at 1:01
• @Harry: If you really want to do serious stuff in this area you should move to R! :-) – vonjd Jan 5 '15 at 9:11
• I did spend a few days using it, but struggled finding and adapting the codes I needed. I managed to run multiple regresssion in R, but couldn't do anything else like structual break test (chow) etc. – Harry Jan 5 '15 at 10:19

If your errors are non-normal and your sample is large non-normality is not important. You can rely on the Central Limit Theorem which implies that the test statistics (t and F statistics) have approximately the same distribution as in the normal case. Standard errors, in your case will be larger because of the fat tails. For a good textbook treatment of this treatment see, for example, Wooldridge (2013), Introductory Econometrics, Fifth Edition, South Western.

You could do your tests in Excel but you would have to do a lot of work and I would not recommend it. You should have a look at gretl http://gretl.sourceforge.net/ which is an easy to use econometrics package and is free.

If you wish to do some serious econometrics you need to obtain a better understanding of the underlying theory. You could start with Wooldridge or any of the other excellent test books available. Many introductory texts will give the normality requirement in an early section and then generalize this in a later section.

• I will have a look for this book. Many thanks for your response – Harry Jan 7 '15 at 21:52