Econometric Methods for Establishing Causality
1. RCTs Versus Quasi-Experimental Methods
A randomized controlled trial bakes causality into the data: random assignment severs every systematic link between treatment status and unobserved factors. Consequently, the simple difference in average outcomes is unbiased, provided compliance and sample integrity hold. The four quasi-experimental tools imitate this causal magic only by leaning on single-point assumptions:
- Ordinary Least Squares (OLS): Assumes no omitted variable is correlated with the regressor.
- Differences-in-Differences (DiD): Relies on parallel pre-treatment trends.
- Instrumental Variables (IV): Hinges on an exclusion restriction—the instrument affects the outcome only through the endogenous regressor.
- Regression Discontinuity (RD): Trusts that units cannot precisely manipulate the cutoff score.
If any assumption fails, bias enters the estimation. RCTs build the firewall; the others retrofit it.
2. Differences-in-Differences (DiD) and Causal Inference
Differences-in-Differences (DiD) treats time as a second control group. It compares the before-after change in a treated cohort to the contemporaneous change in an untreated cohort. When the pre-intervention slopes match, any post-intervention divergence is attributed to the treatment.
In Professor Angrist’s banking graphic, the 6th and 8th Federal Reserve Districts track nearly parallel declines until the 1930 Caldwell collapse strikes only the 6th. Afterward, the 6th’s bank count plunges; the 8th’s merely drifts. The vertical gap that opens—net of the 8th’s own erosion—is the DiD estimator. The inference is credible only if you believe the counterfactual: absent the collapse, the yellow 6th-District line would have continued shadowing the blue 8th-District line.
3. Instrumental Variables (IV): Stages and Exclusion
Instrumental Variables (IV) are required when the regressor of interest is endogenous—correlated with unobserved determinants of the outcome—rendering OLS inconsistent. The IV method proceeds in two stages:
- Stage 1 (First Stage): Projects the endogenous regressor on an external instrument plus controls, isolating the exogenous component.
- Stage 2 (Second Stage): Regresses the outcome on those fitted values, converting the instrument-induced shock into a causal estimate.
In Angrist and Evans (1998), a twin birth raises the probability of a third child by approximately 60 percentage points, while a same-sex pair raises it by approximately 6 percentage points. Dividing each instrument’s impact on maternal labor supply by its impact on family size yields nearly identical “per-child” effects. This serves as a crucial robustness check, confirming that the exclusion restriction holds and the Local Average Treatment Effect (LATE) is well identified.
4. Interpreting the NLSY Returns-to-Schooling Table
The progression of estimates in the NLSY Returns-to-Schooling Table dramatically illustrates the impact of omitted-variable bias:
- Column (1): Shows a raw 13% wage premium per additional year of schooling.
- Column (2): Adding age dummies barely moves the needle, confirming that cohort composition is not driving the result.
- Column (3): Incorporating family-background controls trims the premium to 11%, revealing modest upward bias from socioeconomic factors.
- Column (4): Introducing the AFQT ability score cuts the estimate to 8.7%; cognitive skill had been masquerading as schooling’s payoff.
- Column (5): Finally, occupation dummies push the return down to 6.6%, indicating that part of the earlier premium simply captured better occupational sorting.
This progression justifies the shift toward IV or RD designs when precision regarding the causal return to schooling matters.