How much does education affect wage rates? The data file cps5_small contains 1200 observations on hourly wage rates, education, and other variables from the 2013 Current Population Survey (CPS). 

a. Create the variable \(L W A G E=\ln (W A G E)\). Construct a histogram and calculate detailed summary statistics. Does the histogram appear bell shaped and normally distributed? A normal distribution is symmetrical with no skewness, skewness \(=0\). The tails of the normal distribution have a certain "thickness." A measure of the tail thickness is kurtosis, discussed in Appendix C.4.2. For a normal distribution, the kurtosis \(=3\), discussed in Appendix C.7.4. How close are the measures of skewness and kurtosis for LWAGE to 0 and 3, respectively?

b. Obtain the OLS estimates from the log-linear regression model \(\ln (W A G E)=\beta_{1}+\beta_{2} E D U C+e\) and interpret the estimated value of \(\beta_{2}\).

c. Obtain the predicted wage, \(\widehat{W A G E}=\exp \left(b_{1}+b_{2} E D U C\right)\), for a person with 12 years of education and for a person with 16 years of education.

d. What is the marginal effect of additional education for a person with 12 years of education and for a person with 16 years of education? [Hint: This is the slope of the fitted model at those two points.]

e. Plot the fitted values \(\widehat{W A G E}=\exp \left(b_{1}+b_{2} E D U C\right)\) versus \(E D U C\) in a graph. Also include in the graph the fitted linear relationship. Based on the graph, which model seems to fit the data better, the linear or log-linear model?

f. Using the fitted values from the log-linear model, compute \(\Sigma(W A G E-\widehat{W A G E})^{2}\). Compare this value to the sum of squared residuals from the estimated linear relationship. Using this as a basis of comparison, which model fits the data better?