Mostly Harmless Econometrics: An Empiricist’s Companion
Appendix: Derivation of the average derivative formula
Begin with the regression of Yi on Si :
Cov(Yj, Si) _ E[h(Si)(Si - E[Si])] V(Si) _ E[Si(Si - E[Si])] '
Let К-ж = lim h (t). By the fundamental theorem of calculus, we have:
t—» — OO
s
h (si) = к_ж + / h' (t) dt.
Substituting for h(Si), the numerator becomes
|
/ |
+ 1 ps
/ h' (t) (s - E[Si)g(s)dtd.
- OO J — OO
where g(s) is the density of si at s. Reversing the order of integration, we have
|
/ |
+i p+i
h' (tW (s - E[Si])g(s)dsdt.
-OO J t
The inner integral is easily seen to be equal to g, t = fE[si|si > t] — E[si|si < t]}{P(si > t)[1 — P(si > t)},
which is clearly non-negative. Setting si =Yi, the denominator can similarly be shown to be the integral of these weights. We therefore have a weighted average derivative representation of the bivariate regression coefficient, CoV((Yf ^, equation (3.3.8) in the text. A similar formula for a regression with covariates, Xi, is derived in the appendix to Angrist and Krueger (1999).
