Questions: Robust Standard Errors

Question 1

A researcher runs an OLS regression and finds evidence of heteroskedasticity. She switches from classical OLS standard errors to robust (Huber-White) standard errors. What does this change fix?

• A. The bias in the OLS coefficient estimates caused by heteroskedasticity
• B. The efficiency loss in the OLS estimates — robust SEs make OLS as efficient as GLS
• C. The validity of standard errors, confidence intervals, and hypothesis tests
• D. Both the coefficient estimates and the inference, producing a fully corrected regression

Answer: (C)

Heteroskedasticity does not bias OLS coefficient estimates — they remain consistent. What fails is the classical standard error formula, which assumes constant error variance and produces incorrect SEs when that assumption fails. Robust standard errors correct only the inference side: the SEs, confidence intervals, and p-values become valid even without knowing the form of the heteroskedasticity. The coefficient estimates are unchanged. This is the central point: robust SEs fix your hypothesis tests, not your estimates.

Question 2

A researcher studies the effect of a job training program assigned randomly at the county level, with outcome data measured at the individual worker level. At what level should she cluster her standard errors?

• A. The individual worker level — more clusters produce more precise standard errors
• B. The firm level — workers in the same firm share economic environment
• C. The county level — this is where the policy assignment variation occurs
• D. No clustering is needed because the assignment was random

Answer: (C)

The rule is to cluster at the level where the treatment assignment occurs. The program was assigned at the county level, meaning all workers in the same county got the same treatment or control condition — their error terms are likely correlated through shared local economic conditions and the common assignment. Clustering at the individual level ignores this correlation (under-clustering). Clustering at the county level accounts for within-county correlation, producing valid standard errors. Random assignment justifies the causal interpretation of the coefficient but does not eliminate within-cluster correlation in the errors.

Question 3

Robust (Huber-White) standard errors are generally larger than the classical OLS standard errors they replace.

• T. True
• F. False

Answer: False

Robust standard errors can be larger or smaller than classical OLS SEs in any given regression. On average they tend to be larger when heteroskedasticity is present (because the OLS formula was underestimating variance), but this is not guaranteed. In samples with particular variance patterns, robust SEs can be smaller. The value of robust SEs is not that they give a more conservative answer — it is that they give the correct answer under a broader set of conditions.

Question 4

Using clustered standard errors makes OLS coefficient estimates less biased and more efficient, in addition to correcting the inference.

• T. True
• F. False

Answer: False

Clustered standard errors only correct the inference — the standard errors, confidence intervals, and p-values. The OLS coefficient estimates themselves are unchanged: they remain unbiased (under the usual OLS assumptions) but may be inefficient compared to GLS or FGLS if within-cluster correlation is severe. The correction is entirely on the variance-estimation side, not the coefficient side. If efficiency matters and the within-cluster correlation structure can be modeled correctly, alternatives like FGLS would be used instead.

Question 5

Why should you cluster standard errors at the level of policy assignment rather than at the individual level, even if your data is measured at the individual level?

Model answer: When a policy is assigned at a group level (e.g., state, county, school), all individuals in the same group share the same treatment status. This creates correlation among their error terms: they face common unobserved shocks, common local economic conditions, and common institutional factors. Treating them as independent observations overstates the effective sample size — you have far fewer independent pieces of information than you have individuals. Clustering at the assignment level accounts for this within-group correlation, producing standard errors that reflect how much independent variation you actually have.

The intuition is that 50 workers in the same county receiving the same policy treatment are not 50 independent data points about the policy's effect — they share a common cause and common context. If you estimate the effect with data on 1,000 workers across 20 counties (50 per county), you effectively have 20 quasi-independent observations about the policy, not 1,000. Standard errors clustered at the county level reflect this reality; individual-level SEs do not, leading to spuriously small p-values and overconfident inference.