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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 completedtwelfth grade but no higher grade? Do the men or their parents have, on average, higher levelsof 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 aftercontrolling 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 estimatesin part (iv) to graph the relationship between the predicted education and abil. Set motheduc andfatheduc at their average values in the sample, 12.18 and $12.45,$ respectively.

(i) Min 6, Max 20, Mean 13. 512 men or 41.6 percent . No (ii) See video (iii) Yes (iv) $abil^*=-3.93$ (v) (vi) Plot takes a J curve form. See video.

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Chapter 3

Multiple Regression Analysis: Estimation

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Scott S.

October 7, 2021

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Yes, Part one. The range of the education variable in the sample is running from six Yeah to 20 with the mean of 13 in the sample. There are 512 men completing 12th grade, but no higher grade. The simple side is 12 30. So the number of men with education equals 12. Mix up 41.6% of the sample. The men and their parents do not have higher levels of education on average. And you would know that by looking at the summary statistic of education, Mother's education and Father's education? Yeah, or two. We will estimate the regression model by old L s. This is the regression result to decide how much sample variation in education is explained by parents education. We need to look at the value of our square. Our square is the measure of the goodness of fit. Of the model in this model are square is 0.249 That means almost 25% of the simple variation of education can be explained by Father's and mother's education. Yeah, the estimate on Mother education is 0.304 and this estimate is significant at the 1% level. This value implies that on increase of one unit in the mother's highest education would increase yes, her child's highest great by 0.3 unique on average, all else equal are three. We will add another explanatory variable. Ah, measure of ability. This is through the estimation result. There's ability help to explain variations in education after controlling for parents education. The answer is yes. You can see that model fit improves. We have a higher are square. Our square in this new model is 0.4, and before we have another square of 0.25 Ability also has a positive and highly significant coefficient. So I will write down the two reasons Part four. We add to the right hand side of the equation, the quadratic term of ability. This is the regression result. We would use the estimate Beta three and beta four, which is the estimates on ability and Ability square to find the value of ability that minimized education. You can do that by taking partial derivative of education with respect to ability and said that to zero now the partial derivative of education will respect you. Ability is the sum of Okay, The estimates of beta three plus the estimate of beta four. Okay, terms to and times the value of ability. We know the value of beta three head and beta forehead with a three head is 30.401 and beta forehead is we have to wait. A forehead is 0.51 And as a problem say's the second derivative is positive. So we actually have a minimum of education. We win. Set this to be equal to zero. That's how we find ability. Star. The value of ability star is minus 0.401 divided by 0.102 and you can get minus 3.93 This is the value of ability that minimized education. Okay, Part five, Ali. A small fraction of men in the simple have ability less than this value it is. So because if there is a large fraction of men with ability less than this value, it means there are many cases where men with low ability have a good education. If this is true, then we would not be able to find a significant and positive influence of ability on education. It means that our measure of ability might be flawed. So that's why this is important. And if you, um, count the number of cases where man has ability lower than minus 3.93 you can find that on Lee. 15 men in the sample with such low ability measures Heart. Six. Yeah, we win predict education based on the estimates in Part four, and we fixed mother education and further education at their average values in the sample, which is 12.18 and 12.45 respectively. Thes numbers are the estimates from the regression in Part four, So if you plot the predicted education against ability, you wouldn't have a J curve. Plot ability ranges from almost minus 626 and education wind range from, as I said before in part one from sixth to 20. So here I win roughly draw off the plot. You make it, you wouldn't get so. The value of ability that minimized education is almost minus four. We will get a plot like this. Okay, this is ability stars value we found in Part four and a smaller or larger value of ability. It's associate ID with a higher education level right. This should be very should be quite flat. And then they start going up like this. The actual plot is much smoother than what I had here. But you get the idea.

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