# Equation 3.3.8 in Mostly Harmless Econometrics

Suppose that $$S_i$$ is continuously distributed, not necessarily non-negative. The conditional expectation function of interest is $$h(t):=E[Y_i|S_i=t]$$ has derivative $$h'(t)$$.

Equation 3.3.8 of Mostly Harmless Econometrics is:

$$\frac{E[Y_i(S_i- E[S_i])]}{E[S_i(S_i-E[S_i])]} = \frac{\int h'(t)\mu_t dt}{\int \mu_t dt}$$ where $$\mu_t :=[E[S_i|S_i\ge t]-E[S_i |S_i and the integrals run over the support of $$S_i$$.

That equation is not obviously true to me and I am looking for a proof.

• You can find a discrete version of the derivation in Appendix A2 of Angrist and Krueger (1999): Empirical strategies in labor economics in the Handbook of Labor economics (Ashenfelter and Card) volume 3. link
– tdm
Apr 21, 2022 at 12:33
• The "+" sign should be "-" in the inner integral Oct 25, 2023 at 3:58

The appendix to that section in Mostly Harmless, section 3.5, has a derivation.

$$Cov(Y_i,S_i) = E[h(S_i)(S_i-E[S_i])]$$

Let $$k_{-\infty}=\lim_{t\rightarrow -\infty} h(t)$$. By the fundamental theorem of calculus,

$$h(S_i)=k_{-\infty} +\int_{-\infty}^{S_i} h'(t)dt$$

Thus, $$E[h(S_i)(S_i-E[S_i])] = \int_{-\infty}^\infty\int_{-\infty}^{S_i} h'(t) (s-E[S_i])g(s)dtds$$

where $$g(s)$$ is the density of $$S_i$$ at $$s$$. Apply Fubini's theorem to switch the order of integration, $$E[h(S_i)(S_i-E[S_i])] = \int_{-\infty}^\infty h'(t)\int_{t}^{\infty} (s-E[S_i])g(s)dtds$$

The inner integral is $$E[S_i|S_i \ge t]Pr(S_i\ge t)+E[S_i]Pr(S_i \ge t)$$

$$= E[S_i|S_i \ge t]Pr(S_i\ge t)+(E[S_i|S_i \ge t]Pr(S_i \ge t) + E[S_i|S_i < t]Pr(S_i < t))Pr(S_i \ge t)$$

$$=\mu_t :=(E[S_i |S_i \ge t]-E[S_i |S_i

Then, setting $$S_i =Y_i$$, the denominator can similarly be derived.