A researcher runs OLS and discovers that error variance increases with the level of x (heteroskedasticity). What is the primary consequence for the OLS estimates?
AThe coefficient estimates β̂ are biased and inconsistent
BThe coefficient estimates β̂ are still unbiased, but standard errors are biased, making inference unreliable
CBoth coefficients and standard errors are unbiased; only efficiency is lost
DThe model must be re-estimated using a different technique because OLS cannot be applied
Heteroskedasticity violates the efficiency assumption (homoskedasticity) but does not affect unbiasedness or consistency of the coefficient estimates. The coefficients β̂ are still correct on average. However, OLS standard errors assume constant variance; when variance is non-constant, the computed standard errors are wrong, which invalidates t-tests and confidence intervals. The fix is to use heteroskedasticity-robust standard errors — not to discard OLS.
Question 2 True / False
If the OLS assumption E[u|x] = 0 is violated due to an omitted variable, the coefficient estimates are still unbiased as long as the sample is large enough.
TTrue
FFalse
Answer: False
E[u|x] = 0 (exogeneity) is required for unbiasedness and consistency. When an omitted variable is correlated with x, it enters the error term u, making u correlated with x and violating this assumption. The resulting omitted variable bias does not shrink as the sample grows — it is a persistent, systematic error. A larger sample just estimates the wrong parameter more precisely. No amount of data can fix a violation of exogeneity.
Question 3 Short Answer
The Gauss-Markov theorem says OLS is BLUE. What does each of those four letters mean, and why does the 'unbiased' part depend on a different assumption than the 'best' part?
Think about your answer, then reveal below.
Model answer: BLUE stands for Best Linear Unbiased Estimator. 'Unbiased' (E[β̂] = β) requires E[u|x] = 0 — the exogeneity assumption. 'Best' (minimum variance among linear unbiased estimators) requires homoskedasticity and no serial correlation. These are governed by separate assumptions, so it is possible to have an unbiased but inefficient estimator (when homoskedasticity fails) or a biased but precise one.
The distinction between unbiasedness and efficiency maps directly onto which assumptions are load-bearing for each property. E[u|x] = 0 is the hardest assumption to satisfy — it rules out omitted variable bias, measurement error in x, and simultaneity — and its failure destroys the fundamental validity of OLS. Homoskedasticity and no serial correlation govern efficiency only: their failure means OLS is no longer the minimum-variance estimator, but coefficients remain interpretable. This is why heteroskedasticity-robust standard errors are a fix that preserves the coefficient estimates while correcting the inference.