Use the data in HTV to answer this question. The data set includes information on wages, education,

parents' education, and several other variables for $1,230$ working men in $1991 .$

(i) What is the range of the $e d u c$ variable in the sample? What percentage of men completed

twelfth grade but no higher grade? Do the men or their parents have, on average, higher levels

of education?

(ii) Estimate the regression model

$$e d u c=\beta_{0}+\beta_{1} \text { motheduc }+\beta_{2} \text { fatheduc }+u$$

by OLS and report the results in the usual form. How much sample variation in $e d u c$ is ex-

plained by parents' education? Interpret the coefficient on motheduc.

(iii) Add the variable $a b i l($ a measure of cognitive ability) to the regression from part (ii), and report the results in equation form. Does "ability" help to explain variations in education, even after

controlling for parents' education? Explain.

(iv) (Requires calculus) Now estimate an equation where abil appears in quadratic form:

$$e d u c=\beta_{0}+\beta_{1} \text { motheduc }+\beta_{2} \text { fatheduc }+\beta_{3} a b i l+\beta_{4} a b i l^{2}+u$$

Using the estimates $\hat{\beta}_{3}$ and $\hat{\beta}_{4}$ use calculus to find the value of abil, call it abil, where educ is minimized. (The other coefficients and values of parents' education variables have no effect; we are holding parents' education fixed.) Notice that abil is measured so that negative values are permissible. You might also verify that the second derivative is positive so that you do indeed have a minimum.

(v) Argue that only a small fraction of men in the sample have "ability" less than the value calculated in part (iv). Why is this important?

(vi) If you have access to a statistical program that includes graphing capabilities, use the estimates

in part (iv) to graph the relationship between the predicted education and abil. Set motheduc and

fatheduc at their average values in the sample, 12.18 and $12.45,$ respectively.