Suppose that \(x\) is endogenous in the regression \(y_{i}=\beta_{1}+\beta_{2} x_{i}+e_{i}\). Suppose that \(z_{i}\) is an instrumental variable that takes two values, one and zero; it is an indicator variable. Make the assumption \(E\left(e_{i} \mid z_{i}\right)=0\).

a. Show that \(E\left(y_{i} \mid z_{i}\right)=\beta_{1}+\beta_{2} E\left(x_{i} \mid z_{i}\right)\).

b. Assume \(E\left(x_{i} \mid z_{i}\right) eq 0\). Does \(z_{i}\) satisfy conditions IV1-IV3? Explain.

c. Write out the conditional expectation in (a) for the two cases with \(z_{i}=1\) and \(z_{i}=0\). Solve the two resulting equations for \(\beta_{2}\).

d. Suppose we have a random sample \(\left(y_{i}, x_{i}, z_{i}\right), i=1, \ldots, N\). Give an intuitive argument that a consistent estimator of \(E\left(y_{i} \mid z_{i}=1\right)\) is the sample average of the \(y_{i}\) values for the subset of observations for which \(z_{i}=1\), which we might call \(\bar{y}_{1}\).

e. Following the strategy in part (d) form \(\bar{y}_{1}, \bar{y}_{0}, \bar{x}_{1}\), and \(\bar{x}_{0}\). Show that the empirical implementation of the expression in (c) is \(\hat{\beta}_{\text {WALD }}=\left(\bar{y}_{1}-\bar{y}_{0}\right) /\left(\bar{x}_{1}-\bar{x}_{0}\right)\), which is the Wald Estimator, in honor of Abraham Wald.

f. Explain how \(E\left(x_{i} \mid z_{i}=1\right)-E\left(x_{i} \mid z_{i}=0\right)\) might be viewed as a measure of the strength of the instrumental variable \(z_{i}\).