1. Regression Discontinuity
Liang Dai

2. Illustration
Suppose we are interested in the cohort effect of elite high school A on the college entrance exam scores (高考) of graduates of middle school M.
One can regress the score of M graduates on a dummy of A attendance
Problem: selection bias
Most talented graduates from M get into A

3. Way out
Suppose high school enrollment is completely determined by entrance exam scores (中考)
And the graduating class of M is not large enough to affect the cutoff
Di=1 if score Xi>=x0, and Di=0 if Xi<x0
Then we can overcome the selection bias by comparing X graduates with score RIGHT above and below x0.
Key assumption: all variables other than treatment, e.g. talent, have to be continuous at the cutoff.

4. Example 1 Angrist & Lavy, QJE 99
Does smaller class size enhance scholastic achievement?
Hot policy debate, interesting labor economics question
Challenge: ?

5. Example 1 Angrist & Lavy, QJE 99
Does smaller class size enhance scholastic achievement?
Hot policy debate, interesting labor economics question
Challenge: class size correlated with unobservable factors, e.g. student quality, teacher’s experience, etc.

6. Regression Discontinuity
Maimonides’ rule on Israeli primary school enrollment: class size can’t exceed 40.
E.g. total enrollment=80 => average class size=40;
Total enrollment=81 => average class size=27.
Predicted class size:
f=e/[int((e-1)/40)+1]

10. Example 2
Raul, JF 06
Does a firm’s investment depend on its internal cash holdings, controlling for its investment opportunities (Tobin’s Q)?
Evidence for financial frictions
w/o financial constraints, external capital is as cheap as internal capital (Recall MM), so internal cash holdings don’t matter
Challenge: ?

11. Example 2
Raul, JF 06
Does a firm’s investment depend on its internal cash holdings, controlling for its investment opportunities (Tobin’s Q)?
Evidence for financial frictions
w/o financial constraints, external capital is as cheap as internal capital (Recall MM), so internal cash holdings don’t matter
Challenge: cash holdings are endogenous
Persistence of profit
Measurement error in Q
First large-sample test

12. Regression Discontinuity
Treatment D: Cash
Running variable X: defined pension plan balance
Discontinuity: when defined pension plan is in deficit, the firm has to contribute its cash to it. Not otherwise
Exogeneity of cutoff: funding status mostly determined by value of the asset side of pension fund, independent of firm’s investment opportunity
Analogy: Whether you are just below or above cutoff of exam scores must be independent of talent

13. Firms contribute mostly when funding status <0

14. Investment correlated with funding status only to the left of 0