Angrist, J. D., & Krueger, A. B. (1991). Does compulsory school attendance affect schooling and earnings? Quarterly Journal of Economics, 106(4), 979-1014.
Does an extra year of schooling actually raise earnings, or does the OLS correlation between education and wages just reflect the fact that more able, more motivated people both stay in school longer and earn more? Angrist and Krueger estimate the impact of years of schooling on log weekly wages using 1970 and 1980 Census data.
\[ \log(Wage_i) = \beta_0 + \beta_1 Educ_i + \beta_2 X_i + \varepsilon_i \]
Log weekly wages for person \(i\), regressed on years of schooling, plus controls like age and region.
This is a classic omitted-variable problem rather than reverse causation. Unmeasured traits like innate ability are likely correlated with both how much schooling someone gets and how much they earn, so \(Educ_i\) is correlated with \(\varepsilon_i\) and OLS overstates the return to schooling.
Angrist and Krueger use quarter of birth to construct their instrument, applying two-stage least squares.
First stage:
\[ Educ_i = \pi_0 + \pi_1 QOB_i + \pi_2 X_i + v_i \]
Second stage:
\[ \log(Wage_i) = \alpha_0 + \alpha_1 \widehat{Educ}_i + \alpha_2 X_i + e_i \]
Kids born earlier in the calendar year start school older and hit the legal dropout age after completing fewer years of schooling than kids born later in the year, since compulsory schooling laws are tied to age, not grade level. So quarter of birth ends up mechanically correlated with completed years of schooling for people near that cutoff.
Since a person’s birthday can’t plausibly be correlated with other personal traits that affect earnings, the variation in schooling driven by birth timing is treated as exogenous. Nobody chooses their season of birth, so it shouldn’t be picking up ability or family background.
If birth season is systematically linked to something else that affects earnings independent of schooling, like health at birth or parental income timing, the exclusion restriction fails. This isn’t just theoretical either. One study using genetic data found some evidence that factors influencing education aren’t randomly distributed across birth quarters, and separately, later work by Bound, Jaeger, and Baker showed the quarter-of-birth instrument is quite weak, correlating only weakly with schooling, which amplifies bias in 2SLS rather than eliminating it.
Kind of a fun irony that the paper that helped launch the modern IV era in economics is now mostly cited as the textbook example of what a weak instrument looks like.