Angrist and Krueger (1991) use quarter of birth as an instrumental variable to estimate the returns to schooling, using a sample of 327,509 from the 1980 census. The model of interest is \(\ln (W A G E)=\) \(\beta_{1}+\beta_{2} E D U C+e\).

a. Let \(\overline{\ln (W A G E)}\) denote the average of the natural log of weekly wage. For men born in the first quarter of the year the average is 5.8916, and for men born in the fourth quarter of the year the average is 5.9027 . What is the approximate percentage difference in wages for the two groups of men?

b. The standard error of the difference in means from part (a) is 0.00274 . Is the difference in \(\overline{\ln (W A G E)}\) statistically significant? What is the two-tail \(p\)-value?

c. Let \(\overline{E D U C}\) denote the average years of schooling. For men born in the first quarter of the year the average is 12.6881, and for men born in the fourth quarter of the year the average is 12.7969. What is the approximate percentage difference in years of schooling for the two groups of men? Is there a reason why men born in the fourth quarter have higher average schooling than men born in the first quarter?

d. The standard error of the difference in means from part (c) is 0.0132 . Is the difference in \(\overline{E D U C}\) statistically significant? What is the two-tail \(p\)-value.

e. Compute the Wald estimate of the return to schooling, \(\hat{\beta}_{2, \text { WALD }}\) using the results above. What is the instrumental variable \(z\) being used in this case? The Wald estimator is introduced in Exercise 10.4.

f. Explain why the result in (d) is important to the success of the Wald estimator.

Data From Exercise 10.4:-

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 \(z_{i}\).