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I am running a linear regression (just a single IV) and have selected the robust error option (vce robust) in Stata due to heteroscedasticity (and because it is sometimes recommended to do so anyway). However, try as I might, I cannot find any advice on whether I should be testing for normality after I have selected this robust option or whether running the robust option negates the need to do so. Any help on whether normality should be tested with this option checked would be greatly appreciated.

BASED ON ANSWERS: My main focus is on understanding the regression model, so I will be looking at the (slope) coefficient and its 95% CI as well as statistical significance (I have a continuous IV). In another linear regression I had hoped to make predictions (with CI and, hopefully, PI) also. I was OK with checking the assumptions of a regression analysis until I reached the option to use robust standard errors. From the answers received am I correct in saying that asymptotic normality is needed, but not readily/easily tested for (and is rarely tested in practice)? So I could run the regression with robust errors and not test for normality. I assume that other assumptions (e.g., unusual points) still hold. I checked Stata and it does seem that it predicts when robust errors are used. Is it correct to use these predictions?

Peter S
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  • See a related question and answer [here](http://stats.stackexchange.com/questions/2492/is-normality-testing-essentially-useless). – AdamO May 17 '16 at 17:11

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To make things simple, suppose you have 3 observations. Robust standard errors allow for a variance-covariance matrix of the errors to look like this: $$\Sigma = \begin{bmatrix} \sigma_{1} & 0 & 0\\ 0 & \sigma_{2} & 0 \\ 0 & 0 & \sigma_{3} \end{bmatrix} $$

The diagonal terms are the variances of the errors for each of the 3 observations. The covariance terms are all zero because we still assume that the errors are uncorrelated across observations. If you want to relax that, you will need cluster-robust errors.

Ordinary, non-robust errors assume that $\sigma_1=\sigma_2=\sigma_2=\sigma$: all observations have the same (unknown) error variance. Neither the robust nor the non-robust VCE make any assumptions about the distribution of the error, such as normality. They only make assumptions about the variance being the same (homoskedasticity) or different across observations (heteroskedasticity).

Thus, depending on what you are doing, you may still need to test for normality of the errors. Alternatively, you may be able to bootstrap or rely on asymptotics for inference.

dimitriy
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  • I disagree with your answer. And can you propose a method of testing for normality in a sample of $r_i \sim \mathcal{N}(0, \sigma_i^2)$ with each residual having a different variance? – AdamO May 17 '16 at 17:18
  • @AdamO I am afraid you are correct that a general test does not exist. I should have written "worry" rather than "test". I would look at the distribution of residuals using `qnorm` separately for each group (say foreign and domestic cars or high income versus low income or whatever the theoretically relevant dimension for the heteroskedasticity is) or perform one of the diagnostic tests like Jarque-Bera within each group. – dimitriy May 17 '16 at 17:47
  • It is certainly possible to create some hybrid approaches. The simplest would be a binning approach, hoping that adjacent observations are homoscedastic. Alternately, fit a linear, polynomial, or spline model to the squared residuals, estimate the $\sigma_i^2$s and their standard errors, then transforming the residuals to a consistent scale and apply any desired test. Also an interesting read here: https://arxiv.org/pdf/1101.1402.pdf – AdamO May 17 '16 at 18:10
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If your goal is to use a fitted model to do prediction, then you cannot use robust standard errors because you need to make the assumption of normality as well as homoscedasticity. This is not how most people use linear models. Most use linear models to test for statistical significance of model parameters.

In this case, robust standard errors give you a general test of significance that depends neither on normality nor homoscedasticity. Furthermore, it has very comparable power to the classical (non-robust) standard errors when those assumptions happen to be met. For that reason, I agree with Stata and would advocate using robust standard errors for most analyses. You would only not use robust standard errors if your $n<40$ because it behaves poorly in small samples.

It is important to note that even with classic standard errors, there is no normality assumption. I honestly don't know where this assumption came about, though I see it frequently mentioned in introductory texts. Even Gauss, in his seminal paper on the least squares estimation, specifically noted that normality is not a necessary assumption.

It's the Lindeberg Feller central limit theorem ensures that, under a minimal number of assumptions the normal approximation to the sampling distribution of model parameters is very good. Regardless of whether robust or non-robust standard errors are used.

Nonetheless the answer to your question is that no, normality assumptions are not important with robust standard errors.

AdamO
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    I think if you care the variance of each prediction, you cannot use robust standard errors. But you can still predict, since the coefficients will be the same regardless of which standard errors were used. Also, Stata uses a finite sample correction with robust errors that may ameliorate poor small sample behavior. Alternatively, hc2 or hc3 VCEs may work even better with small samples and heteroskedasticity. – dimitriy May 17 '16 at 18:07
  • On the origins, *exact* inference based on $t$ and $F$ requires normality, though approximation is often quite good. Cutoffs like $n=30$ cannot be sufficient of all possible distributions, and the quality of the approximation will also depend on the degrees of freedom, not just the sample size. – dimitriy May 17 '16 at 18:10
  • @DimitriyV.Masterov is exact inference even possible with sandwich estimators? I assumed it boiled down to the [Behrens-Fisher](https://en.wikipedia.org/wiki/Behrens%E2%80%93Fisher_problem) problem where good estimators are certainly available, but nothing provides exact inference. – AdamO May 17 '16 at 18:13
  • @DimitriyV.Masterov interestingly, I think the $n=30$ or $n=40$ cutoff is a consideration *not* for normal approximation but for degrees of freedom, the sandwich tends toward anticonservative estimates. I may be mistaken. You correctly mention the small sample size corrected sandwich estimators (I rarely use these as I usually work with larger datasets) and agree these perform well. – AdamO May 17 '16 at 18:21
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    When you add the normality assumption to the Gauss-Markov assumptions OLS has i) finite sample properties ii) OLS is the minimum variance unbiased estimator. This is why (in stats books) you add the assumption. See my answer – Repmat May 17 '16 at 19:38
  • @Repmat the statement and proof of GM that was presented to me was for asymptotic tests and (possibly) non-normal errors. It's true that WLS will be the BLUE estimator in finite samples for normal errors, but it's also asymptotically BLUE. – AdamO May 17 '16 at 20:34
  • I am not talking about BLUE (linear estimators) I am saying that OLS under normality and GM has the minimum variance amongst all unbiased estimators (not just the linear ones) – Repmat May 17 '16 at 20:40
  • @Repmat I think that is not a result of Gauss Markov but actually Neyman Pearson since it is maximum likelihood in that case. – AdamO May 17 '16 at 20:44
  • Would you please read my comments, GM AND normality. Again I emphasize, GM AND normality. – Repmat May 17 '16 at 20:49
  • @Repmat I don't disagree with what you claim to have written. GM and RSEs have different implications for use in analyses. GM says you *know* the precision and can use it to reweight observations for more efficient analyses. RSEs say you don't know the precision but use empirical plug-in estimates of variance to obtain asymptotically correct tests. – AdamO May 18 '16 at 17:13
  • @AdamO Many thanks for all your responses. Please see my clarification to my original question (I don't seem capable of @ all of you). – Peter S May 19 '16 at 11:49
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To answer your question: Should I test for normality, when I have used robust standard errors in my analysis?

The answer is: no.

Why? The assumption of normality gives OLS finite sample propperties when you also assume the Gauss-Markov (GM) assumptions to be true. When you use robust standard errors, you do not assume GM because homoscedlasticity is one of these assumptions.

Buttom line, the robust errors only have asymptotic justification and adding normality in this case gives you exactly nothing. OLS is already asymptotically normal regardless of the underlying distribution.

EDIT: Note that when you add normality to the GM assumptions you are in the classical linear model setup. In this case OLS has i) finite sample properties and is the minimum variance unbiased estimator (it reaches the Cramer Rao lower bound). That is, you can fint no other (unbiased) estimator that gives you a better result.

Repmat
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