Questions: Endogenous Regressors: Bias and Consequences

Question 1

A researcher regresses crime rates on police deployment across cities and finds a positive coefficient — more police, more crime. The most likely explanation is:

• A. Measurement error in the crime rate variable attenuates the true negative coefficient
• B. Simultaneous causality: high-crime cities deploy more police, so police deployment is correlated with the error term
• C. The model is correctly specified; police presence genuinely increases crime
• D. Omitting income as a control variable causes the police coefficient to flip sign

Answer: (B)

This is a classic simultaneous causality problem. Crime causes police deployment (cities respond to crime), while police may also affect crime — both causal arrows run simultaneously. OLS cannot distinguish them and produces a biased estimate. The positive coefficient likely reflects the reverse causation (high crime → more police) dominating the estimate, not evidence that police cause crime. More data won't fix this; the regressor is endogenous by design.

Question 2

What distinguishes endogeneity bias from ordinary sampling variance?

• A. Endogeneity bias affects only small samples; it disappears with larger datasets
• B. Endogeneity produces wider confidence intervals but an unbiased point estimate
• C. Endogeneity makes OLS inconsistent — the estimator converges to the wrong value even with unlimited data
• D. Endogeneity bias is larger in magnitude but still correctable through heteroskedasticity-robust standard errors

Answer: (C)

This is the key distinction. Sampling variance shrinks as n grows — more data averages out random noise. Endogeneity causes inconsistency: the OLS estimator converges to a value that is systematically wrong (not the true causal coefficient), because Cov(X, u) ≠ 0 is a structural feature of the model, not a sampling accident. Collecting more data makes you more confidently wrong. This is why endogeneity requires a fix at the design level — instrumental variables, randomization, or natural experiments — not just a larger sample.

Question 3

In a wage regression that omits individual ability, the OLS coefficient on education will be biased upward because ability is positively correlated with both education and wages.

• T. True
• F. False

Answer: True

The direction-of-bias formula confirms this: the omitted variable bias is approximately Cov(education, ability) / Var(education) × (effect of ability on wages). Both terms are positive — more-able workers tend to get more education, and ability directly raises wages — so the bias is positive. OLS attributes to education some of the wage premium that actually belongs to ability, inflating the education coefficient.

Question 4

Classical measurement error in the dependent variable Y (rather than in a regressor X) causes endogeneity bias in the OLS coefficient estimates.

• T. True
• F. False

Answer: False

Measurement error in Y adds noise to the error term but does not correlate with the regressors X (under classical assumptions), so OLS remains unbiased — just less precise. Endogeneity bias arises from measurement error in a REGRESSOR X, not in Y. When the measured X* = X + ε, the true X and the measurement error ε both enter the model, making the observed regressor correlated with the error term. This causes attenuation bias — coefficients are biased toward zero.

Question 5

Why does endogeneity make OLS inconsistent rather than merely imprecise, and why does this distinction matter practically?

Model answer: Inconsistency means the estimator does not converge to the true parameter as sample size grows — it converges to a wrong value. This happens because Cov(X, u) ≠ 0 is a systematic feature: OLS attributes variation in Y to X even when that variation actually comes from the correlated error. More data amplifies the false signal rather than averaging it away. Practically, this means you cannot fix endogeneity by gathering more observations; you must address the source of correlation — through instrumental variables that isolate exogenous variation in X, through randomization that breaks the X-u correlation, or through a research design that eliminates the confounding path.

The contrast with imprecision is crucial for research strategy. An imprecise estimate is still correct on average — you need more data to narrow the confidence interval. An inconsistent estimate is wrong on average, and gathering more data only makes you more confident in the wrong answer. This motivates the entire instrumental variables literature: find a variable that moves X but is uncorrelated with u, so it extracts only the clean, exogenous variation in X that OLS cannot isolate on its own.