# Heavy-tailed residuals for OLS regression with large n. Implications?

I am trying to fit a multiple regression on a dataset with n=8619.

First of all, using an untransformed Y as the response variable (ie Y = aX + bX +..) resulted in a residual plot with increasing error variance.

I then tried transforming Y to sqrt(Y) which made the residual plot look better.

However, the residuals still exhibit a wide-tailed distribution (see QQ plot below).

My question is - to what extent does this affect the validity of the model? I am aware that non-normal residuals and variances will result in inaccurate p-values/standard errors, but if I recall correctly, the inaccuracy is much more pronounced with smaller samples.

With my sample size (n=8619), is it large enough to be resistant to such a wide-tailed residual distribution?

Thanks.

OLS doesn't require normal errors to estimate the coefficients, as you noted. In large samples you can apply CLT (central limit theorem) to obtain the p-values.

The problem with fat tails is that they may be coming from a distribution which will not let you apply CLT. For instance, there's a family of distributions called stable. Usually, to apply CLT when you add random variables their sum converges to normal distribution. The stable random variables add up to a stable distribution regardless of the sample size, whether it's 30 or 8,000. They have other nasty properties, e.g. some of these distributions do not have mean or variance, which will make coefficient variance-covariance calculation "interesting".

So, unfortunately, with heavy tails I can't tell not to worry because your sample is large. You should look into your errors closer in this case.

You should not worry about non normality of the error term, since all it ensures is that the distribution of $\hat{\beta}_k$ is exactly normal - such that the simple t-statistic has an exact t-distribution when the variance is estimated. If you do not have normality, then t-distribution is only an approximation. But for n larger than 8000 you should be fine.

I don’t know what you are fitting, but to me $\sqrt{Y}$ does not seem like a naturel interpretation? But perhaps it makes sense, given your context?

If you are worried about heteroskedasticity, you can always just compute robust standard errors - package “sandwich” in R does this - the so called Huber White standard errors (note that they have only asymptotic justification, but again I would think that 8000 units is fine). Using these, in place of the usual OLS errors, does not change inference.

What you should be worried about, in term of validity (of the estimates of the parameters), is whether or not you model is (1) correctly specified and (2) if you have included all relevant variables. Otherwise your estimates might very well be biased and inconsistent. Here I am assuming that you do not do an actual random experiment (in which case nevermind).

• Thanks! I am trying to fit a model of taxi rides given time and weather effects. Using Y would be more natural than sqrt(Y), but sqrt(Y) fixes the iffy error variances a little and improves the Rsquared marginally (still deciding which one to use though). As for including all terms, I'm still working on that, although based on preliminary BIC estimates this seems to be the best I can do with the variables I have! Commented May 2, 2015 at 20:59
• Well you could maybe try using $ln(y)$ instead of the square root - then you get a semi elastic model. Also $r^2$ should not be your main focus (but I understand why you would like that), getting casual estimates is (in my oppion) more important. Getting estimates which you interpret is even more important. Commented May 2, 2015 at 21:06