In the regression model \(y=\beta_{1}+\beta_{2} x+e\), assume \(x\) is endogenous and that \(z\) is a valid instrument. In Section 10.3.5, we saw that \(\beta_{2}=\operatorname{cov}(z, y) / \operatorname{cov}(z, x)\).

a. Divide the denominator of \(\beta_{2}=\operatorname{cov}(z, y) / \operatorname{cov}(z, x)\) by \(\operatorname{var}(z)\). Show that \(\operatorname{cov}(z, x) / \operatorname{var}(z)\) is the coefficient of the simple regression with dependent variable \(x\) and explanatory variable \(z\), \(x=\gamma_{1}+\theta_{1} z+v\). Note that this is the first-stage equation in two-stage least squares.

b. Divide the numerator of \(\beta_{2}=\operatorname{cov}(z, y) / \operatorname{cov}(z, x)\) by \(\operatorname{var}(z)\). Show that \(\operatorname{cov}(z, y) / \operatorname{var}(z)\) is the coefficient of a simple regression with dependent variable \(y\) and explanatory variable \(z\), \(y=\pi_{0}+\pi_{1} z+u\). 

c. In the model \(y=\beta_{1}+\beta_{2} x+e\), substitute for \(x\) using \(x=\gamma_{1}+\theta_{1} z+v\) and simplify to obtain \(y=\pi_{0}+\pi_{1} z+u\). What are \(\pi_{0}, \pi_{1}\), and \(u\) in terms of the regression model parameters and error and the first-stage parameters and error? The regression you have obtained is a reduced-form equation.

d. Show that \(\beta_{2}=\pi_{1} / \theta_{1}\).

e. If \(\hat{\pi}_{1}\) and \(\hat{\theta}_{1}\) are the OLS estimators of \(\pi_{1}\) and \(\theta_{1}\), show that \(\hat{\beta}_{2}=\hat{\pi}_{1} / \hat{\theta}_{1}\) is a consistent estimator of \(\beta_{2}=\pi_{1} / \theta_{1}\). The estimator \(\hat{\beta}_{2}=\hat{\pi}_{1} / \hat{\theta}_{1}\) is an indirect least squares estimator.