Returns to Schooling: OLS, Ability Bias and Twin Estimates
A labour economist uses a cross-section of 2,000 urban workers to estimate ln(monthly wage) = α + β years of schooling + γ experience + ε. OLS gives β̂ = 0.09 with a standard error of 0.01. A senior colleague notes that parental income and cognitive ability are omitted; both raise wages and are positively correlated with schooling. A separate study of identical twins finds β̂ = 0.05 with a standard error of 0.02, where differences in schooling within twin pairs are used. The government is considering a subsidy that would raise average schooling by one year and cites the 0.09 estimate to justify a 9% wage gain. You are asked to assess whether 0.09 is a credible causal return, to reconcile it with the twin estimate, and to advise on the subsidy appraisal. Assume classical measurement error in self-reported schooling and that twins share family background.
1. Using the omitted-variable-bias formula, assess the direction and likely size of the bias in the OLS estimate of 0.09. Suppose the ability index has a wage coefficient of 0.05 and a schooling coefficient of 0.8.
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The omitted variable bias formula is plim β̂ = β + γδ, where γ is the effect of the omitted ability index Z on wages and δ is the coefficient from regressing schooling on Z after controlling for experience.
Both γ and δ are positive: ability raises wages and is positively correlated with schooling. Therefore the bias γδ is positive and OLS overstates the causal return to schooling.
With γ = 0.05 and δ = 0.8, the bias is 0.05 × 0.8 = 0.04. So the OLS estimate of 0.09 decomposes approximately as 0.09 = β + 0.04, implying β ≈ 0.05. That is exactly the twin estimate.
The calculation is illustrative because Z is unobserved and δ and γ are not directly estimated. But the sign is unambiguous: family background and ability make the cross-sectional 0.09 an upper bound on the causal return, unless other biases work in the opposite direction.
2. Reconcile the cross-sectional estimate of 0.09 with the twin fixed-effects estimate of 0.05. Which is more credible for policy?
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The twin fixed-effects estimate differences wages and schooling within identical twin pairs. This removes all shared family background, including parental income, neighbourhood, and shared genetic ability, so the ability bias identified above is largely eliminated. That explains why 0.05 is lower than 0.09.
However, 0.05 is not automatically the true causal effect. Classical measurement error in self-reported schooling attenuates the OLS coefficient toward zero, and differencing within twins can amplify this attenuation, so 0.05 may be a lower bound. The twin sample is also small, with a standard error of 0.02, and twins may not be representative of the population. If twins differ in unobserved ability, some upward bias remains.
The two estimates bracket the plausible causal return. The cross-section is likely biased upward by ability and family background; the twin estimate is likely closer to causal but may be biased downward by measurement error. A defensible range is roughly 0.05 to 0.07, with 0.05 as the more credible central estimate for policy.
3. The government proposes a subsidy that would raise average schooling by one year, citing the 9% wage gain. How should the appraisal change?
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The appraisal should not use 0.09 as the causal return. If the credible return is about 0.05, a one-year increase in schooling raises wages by about 5%, not 9%. Using 0.09 would overstate the benefit by roughly 80% and could justify a subsidy that fails a cost-benefit test.
The appraisal should also account for general equilibrium effects. A large subsidy expands the supply of educated workers, which may compress the schooling premium. The marginal recipients of the subsidy may have lower returns than the average worker in the cross-section. There may be deadweight loss from the taxation used to finance the subsidy, and the wage gain may take years to materialise.
A prudent appraisal uses a sensitivity range, for example 0.03 to 0.07, with 0.05 as the central case. If the subsidy still passes a cost-benefit test at the lower bound, it is robust. If it passes only at 0.09, the case is fragile and should not be based on the OLS estimate alone.