In Josh Angrist's book Mastering Metrics, p.88, he says that
Suppose the causal variable of interest is $X_{1i}$ (say a dummy variable for a private school) and the control variable is $X_{2i}$ (say, SAT scores). With a little work, the coefficient on $X_{1i}$ in a regression controlling for $X_{2i}$ can be written as
$\large{\beta_1 = \frac{C(Y_i, \tilde{X}_{1i})}{V(\tilde{X}_{1i})}}$
where $\tilde{X}_{1i}$ is the residual from a regression of $X_{1i}$ on $X_{2i}$:
$X_{1i} = \pi_0 + \pi_1X_{2i} + \tilde{X}_{1i}$
How is this done? What is the 'little work' that he says is needed to derive this? Thanks
