01:220:322 Chap.5 Heteroskedasticity and Robust Standard Errors
Heteroskedasticity and Robust Standard Errors
What heteroskedasticity is
Homoskedasticity, assumption MLR.5, says the variance of the error is the same whatever the values of the regressors.
Heteroskedasticity is its failure, and it is common: spending varies more among high earners and wages vary more among experienced workers.
What it does not break
Unbiasedness of OLS was shown without MLR.5, so heteroskedasticity does not bias the coefficients.
Answers that call the slopes biased because of a fan-shaped residual plot lose points on this course's exams.
What it does break
Without MLR.5, OLS is no longer best among linear unbiased estimators, and more importantly the usual standard errors are invalid, so the usual t statistics, confidence intervals and F statistics are unreliable even in large samples.
The fix
Keep the OLS coefficients and replace the standard errors with heteroskedasticity-robust ones, which remain valid whether or not the variance is constant.
In Stata the option is robust, or r, added after the regression command. From the inference lecture onward, nearly every regression in the notes is run this way.
A common myth
Robust standard errors are not always larger than the usual ones.
They can be smaller for one coefficient and larger for another in the same regression, so a conclusion can survive or reverse only after the robust numbers are checked.
Recognising heteroskedasticity
A scatter of residuals against a regressor that fans out or narrows as the regressor grows is the classic picture.
Binary outcomes guarantee it: in a linear probability model the error variance equals the probability times one minus the probability, which changes with every regressor.
Group averages, firm sizes and incomes are other frequent sources.
Robust inference in the rest of the course
From the inference lecture onward, regressions are run with the robust option by default, and joint F tests are computed by Stata in their robust form.
The habit is simple: add the robust option, then read t statistics, p-values, intervals and F tests exactly as before.
Efficiency, briefly
Because OLS is no longer best under heteroskedasticity, more efficient estimators exist in principle. The course's emphasis is practical: keep OLS and make the inference valid, rather than change the estimator.
What this chapter covers
- 01
MLR.5 homoskedasticity
- 02
Consequences for bias, efficiency and standard errors
- 03
Heteroskedasticity-robust standard errors
- 04
Robust t tests and intervals in Stata
Worked example · free
A conclusion that reverses
- 2Usual t = 0.30/0.12 = 2.50, above 1.96, so the usual test rejects.
- 2Robust t = 0.30/0.19 = 1.58, below 1.96, so the robust test does not reject; report the robust result.
Key terms
- Homoskedasticity
- Constant error variance across all values of the regressors, assumption MLR.5.
- Heteroskedasticity
- Error variance that changes with the values of the regressors.
- Robust Standard Error
- A standard error that stays valid whether or not the error variance is constant.
- Efficiency
- Having the smallest variance within a class of estimators, such as linear unbiased estimators.
- Gauss-Markov Assumptions
- MLR.1 to MLR.5, the conditions under which OLS is the best linear unbiased estimator.
Heteroskedasticity and Robust Standard Errors FAQ
Does heteroskedasticity make OLS biased?
No. Unbiasedness relies on MLR.1 to MLR.4 only. What fails is the usual variance formula, so the reported standard errors and the tests built on them cannot be trusted.
Are robust standard errors always larger than the usual ones?
No. Depending on where the large residuals fall relative to the mean of each regressor, a robust standard error can be larger or smaller, and different coefficients can move in opposite directions.
Should I use robust standard errors even if the variance looks constant?
They are valid under homoskedasticity too, so using them costs little in large samples and protects the inference if the variance does change with the regressors.
Can heteroskedasticity be ignored in a large sample?
No. A large sample does not repair the usual standard errors, because their formula is wrong rather than noisy. The coefficients become more precise, but only robust standard errors give tests and intervals that can be trusted.
How does the robust option change a Stata printout?
The coefficient column is identical. The standard error column changes, and so do the t statistics, p-values, confidence intervals and the overall F statistic that are built from it; the R-squared is unchanged.
Exam move
For each regression printout you meet, recompute one t statistic with the usual and the robust standard error and say whether the conclusion changes.
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