# instrumental variables covariance

I'm got stuck in the following book on page 86 (ftp://nozdr.ru/biblio/kolxo3/G/GL/Angrist%20J.D.,%20Pischke%20J.-S.%20Mostly%20Harmless%20Econometrics%20(PUP,%202008)(ISBN%20069112034X)(O)(290s)GL.pdf).

They explain how instrumental variables work and conclude that from the regression formula given in 4.1.2 it follows that the the regression coefficient of interest p = Cov(y,z)/Cov(s,z), where z is the instrumental variable and s is the variable of interest. I do not see how this follows from simply observing the regression formula in 4.1.2 (y = a + ps + Ay + v), given that z is uncorrelated with y or v. I would be grateful for help.

• Please make your question self-contained by writing down the key parts of the source in your question. Also, the source link seems rather dodgy. Please remove it if it is not a legal source. – Christoph Hanck Jan 4 '18 at 8:33

The "exclusion restriction" of the validity of the instrument $z$ says that $z$ is uncorrelated with the "compound" error term $\eta=A\gamma+v$, which consists of the "structural" error $v$ and the influence of the omitted variable $A$ (ability in their example, which one often does not observe). That is, $Cov(z,\eta)=0$.
If you apply the covariance operator with $z$ to either side of $$y=\alpha+\rho s+\eta$$ you get $$Cov(y,z)=Cov(\alpha+\rho s+\eta,z)$$ Using linearity of covariances, $Cov(z,\eta)=0$ and that covariance with a constant is of course zero, this simplifies to $$Cov(y,z)=Cov(\alpha,z)+\rho Cov(s,z)+Cov(\eta,z)=\rho Cov(s,z)$$ Solving for $\rho$ gives the desired result.