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So I'm slowly going through the Stock and Watson book and I'm a bit confused on how to deal with the issue of homoscedacity/heteroscedacity. Specifically, it is mentioned that economic theory tells us that there's no reason for us to assume that errors will be homoscedastic, so their advice is that we assume heteroscedasticity and always use the heteroscedastic robust standard errors when performing our regression analysis. The way I'm being taught this material, in STATA for example, is that we just run the reg command, always sure to include r for robust standard error.

My question(s) is this: if our default position is to assume heteroscedacticity, then is it also correct that OLS is no longer the best unbiased linear estimator as one of the Gauss-Markov assumptions is violated? And if this is the case, is it also correct that GLS would be the BLUE estimator? Lastly, if both of these assumptions are correct, why would we not just run GLS regressions as our default and not OLS?

Thanks.

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    $\begingroup$ It would be good to get some clarification about your meaning of "GLS." My understanding of GLS is that you have to provide specific information about the error variances. What do you have in mind doing in the general (and by far most common) case when there is no such information directly available? $\endgroup$ – whuber Nov 26 '18 at 15:42
  • $\begingroup$ Hi, I'm not sure what to clarify. I'm not very familiar with the Generalized Least Squares method, but in your comment, is that the answer? Namely, that even assuming that our errors are heteroscedastic, we do an OLS regression anyways, because to do a GLS we need information on the error terms that we don't have? Sorry, this is all quite new to me, and I'm sure I've not phrased the question well. Thanks to all for their comments. $\endgroup$ – anguyen1210 Nov 26 '18 at 15:49
  • $\begingroup$ GLS as a method has more pedagogic value than practical. One almost never sees GLS used in papers because researchers typically don't know $\operatorname{Cov}[ \epsilon \mid X] = \Omega$. You could assume some structure on $\Omega$ and estimate the rest, but such a procedure (i.e. FGLS) can have big problems with robustness! Use a poor $\Omega$ and your estimates will be worse than OLS. To me, GLS is interesting mostly from the standpoint of developing a deeper understanding of linear algebra and OLS. $\endgroup$ – Matthew Gunn Nov 27 '18 at 5:50
  • $\begingroup$ @MatthewGunn but GLS with a compound symmetry correlation structure (assuming exchangeability of responses within cases) is identical to a random intercept multilevel model. So in this sense, an identical model is regularly used. Also, in the case where you are analyzing reasonably normal data from a randomized control trial with not small n, I see no reason not to use it since you can simultaneously model heterogeneous variances by the groups. I think the reason it is not common has a lot to do with researchers ignoring substantive questions about variances. $\endgroup$ – Heteroskedastic Jim Nov 28 '18 at 1:02
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Because GLS is BLUE if you know the form of heteroskedasticity (and correlated errors). If you misspecify the form of heteroscedasticity, GLS estimates will lose their nice properties.

Under heteroscedasticity, OLS remains unbiased and consistent, but you lose efficiency.

So unless you're certain of the form of heteroscedasticity, it makes sense to stick with unbiased and consistent estimates from OLS. Then adjust inference for heteroskedasticity using robust standard errors which are valid asymptotically if you don't know the form of heteroscedasticity.

A hybrid approach is to do your best at specifying the form of heteroskedasticity but still apply robust standard errors for inference. See Resurrecting weighted least squares (PDF).

Modeling is all about tradeoffs and resources. If you are convinced there is nothing to be learned from modeling the form of heteroscedasticity, then specifying its form is a waste of time. I would argue that there is usually something to be learned in empirical applications. But tradition/convention nudges us away from studying heteroscedasticity, looking at the variances, since all they are is "error".

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OLS is still unbiased when the data are correlated (provided the mean model is true). The net effect of heteroscedasticity is that it offsets the errors, so that the 95% CI for the regression is, at times, too tight and at other times too wide. Even still, you can correct the standard errors by using the sandwich or heteroscedasticity consistent standard error (HC) estimator. Technically, this is not "ordinary least squares" but it results in the same effect summary measures: a slope, interecept, and 95% CIs for their values, just no global test, or F-tests, and no validity to prediction intervals.

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