Consider TSLS estimation with a single included endogenous variable and a single instrument. Then the predicted value from the first-stage regression is \(\hat{X}_{i}=\hat{\pi}_{0}+\hat{\pi}_{1} Z_{i}\). Use the definition of the sample variance and covariance to show that \(s_{\hat{X} Y}=\hat{\pi}_{1} s_{Z Y}\) and \(s_{\hat{X}}^{2}=\hat{\pi}_{1}^{2} s_{Z}^{2}\). Use this result to fill in the steps of the derivation of Equation (12.4) in Appendix 12.2.

Equation (12.4)

TSLS SZY Szx*