Suppose that \(x_{i}\) 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.

a. Let \(N_{1}=\sum z_{i}\) be the number of \(z_{i}\) values such that \(z_{i}=1\). Show that \(\sum z_{i} x_{i}=N_{1} \bar{x}_{1}\) where \(\bar{x}_{1}\) is the sample average of the \(x_{i}\) values corresponding to \(z_{i}=1\).

b. Let \(N_{0}=N-\sum z_{i}=N-N_{1}\) be the number of \(z_{i}\) values such that \(z_{i}=0\). Show that \(\sum x_{i}=N_{1} \bar{x}_{1}+\) \(N_{0} \bar{x}_{0}\) where \(\bar{x}_{0}\) is the sample average of the \(x_{i}\) values corresponding to \(z_{i}=0\).

c. Show that \(N \sum x_{i} z_{i}-\sum z_{i} \sum x_{i}=N_{1} N_{0}\left(\bar{x}_{1}-\bar{x}_{0}\right)\)

d. Show that \(N \sum y_{i} z_{i}-\sum z_{i} \sum y_{i}=N_{1} N_{0}\left(\bar{y}_{1}-\bar{y}_{0}\right)\)

e. Use the results in (c) and (d) to show that the IV estimator of \(\beta_{2}\) in (10.17) reduces to \(\hat{\beta}_{2}=\left(\bar{y}_{1}-\bar{y}_{0}\right) /\left(\bar{x}_{1}-\bar{x}_{0}\right)\).

f. The estimated variance of the IV estimator is given in (10.18a). Show that \(\sum\left(z_{i}-\bar{z}\right)\left(x_{i}-\bar{x}\right)=\) \(\sum z_{i} x_{i}-N \bar{z} \bar{x}=N_{1} N_{0}\left(\bar{x}_{1}-\bar{x}_{0}\right)\).
g. Using the result in part (f), suppose \(\left(\bar{x}_{1}-\bar{x}_{0}\right) \simeq 0\). How does this indicate that the IV \(z_{i}\) is weak?
h. \(\sum\left(z_{i}-\bar{z}\right)\left(x_{i}-\bar{x}\right) / \sum\left(z_{i}-\bar{z}\right)^{2}\) is the OLS estimate of the slope coefficient from a regression of \(x_{i}\) on \(z_{i}\). True or False? How does this value relate to the weak instrument discussion in part (g)? If this coefficient is small, with a low \(t\)-value, does it imply that \(z_{i}\) is a weak IV? Explain.