# 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).