HW 1

1. Write the meaning of each of the following terms without referring to the book (or your notes), and compare your definition with the version in the text for each:

1. constant or intercept (p. 7)
2. cross-sectional (p. 21)
3. dependent variable (p. 5)
4. estimated regression equation (p. 14)
5. expected value (p. 9)
6. independent (or explanatory) variable (p. 5)
7. linear (p. 8)
8. multivariate regression model (p. 12)
9. regression analysis (p. 5)
10. residual (p. 15)
11. slope coefficient (p. 7)

1. stochastic error term (p. 8)

3. Not all regression coefficients have positive expected signs. For example, a Sports Illustrated article by Jaime Diaz reported on a study of golfing puns of various lengths on the Professional Golfers' Association (PGA) Tour.11 The article included data on the percentage of putts made ( Pi) as a function of the length of the putt in feet (Li). Since the longer the putt, the less likely even a professional is to make it, we'd expect Li to have a negative coefficient in an equation explaining Pi. Sure enough, if you estimate an equation on the data in the article, you obtain:

\({{\hat{P}}_{i}}=83.6-4.1{{L}_{i}}\) (1.22)

1. Carefully write out the exact meaning of the coefficient of \({{L}_{i}}\).
2. Suppose someone else took the data from the article and estimated:
   \({{P}_{i}}=83.6-4.1{{L}_{i}}+{{e}_{i}}\)
   Is this the same result as that of Equation 1.22? If so, what definition do you need to use to convert this equation back to Equation 1.22?
3. Use Equation 1.22 to determine the percent of the time you'd expect a PGA golfer to make a 10-foot putt. Does this seem realistic? How about a 1-foot putt or a 25-foot putt? Do these seem as realistic?
4. Your answer to part c should suggest that there's a problem in applying a linear regression to these data. What is that problem?

5. If an equation has more than one independent variable, we have to be careful when we interpret the regression coefficients of that equation. Think, for example, about how you might build an equation to explain the amount of money that different states spend per pupil on public education. The more income a state has, the more they probably spend on public schools, but the faster enrollment is growing, the less there would be to spend on each pupil. Thus, a reasonable equation for per pupil spending would include at least two variables: income and enrollment growth:

\({{S}_{i}}={{\beta }_{0}}+{{\beta }_{1}}{{Y}_{i}}+{{\beta }_{2}}{{G}_{i}}+{{\varepsilon }_{i}}\) (1.24)

where: \({{S}_{i}}\) = educational dollars spent per public school student in the i ^th state

\({{Y}_{i}}\) = per capita income in the i ^th state (in dollars)

\({{G}_{i}}\) = the percent growth of public school enrollment in the i ^th state

1. State the economic meaning of the coefficients of Y and G. (Hint: Remember to hold the impact of the other variable constant.)
2. If we were to estimate Equation 1.24, what signs would you expect the coefficients of Y and G to have? Why?
3. Silva and Sonstelie estimated a cross-sectional model of per student spending by state that is very similar to Equation 1.24:12
   \({{\hat{S}}_{i}}=-183+0.1422{{Y}_{i}}-5926{{G}_{i}}\) (1.25)
   N = 49
   Do these estimated coefficients correspond to your expectations? Explain Equation 1.25 in common sense terms.
4. The authors measured G as a decimal, so if a state had a 10 percent growth in enrollment, then G equaled .10. What would Equation 1.25 have looked like if the authors had measured G in percentage points, so that if a state had 10 percent growth, then G would have equaled 10? (Hint: Write out the actual numbers for the estimated coefficients.)

7. Let's return to the wage determination example of Section 1.2. In that example, we built a model of the wage of the i ^th worker in a particular field as a function of the work experience, education, and gender of that worker:

where:

\({{Y}_{i}}\) = \[WAG{{E}_{i}}\] = the wage of the i ^th worker

\({{X}_{1i}}\) = \[EX{{P}_{i}}\] = the years of work experience of the i ^th worker

\({{X}_{2i}}\) = \[ED{{U}_{i}}\] = the years of education beyond high school of the i ^th worker

\({{X}_{3i}}\) = \(GEN{{D}_{i}}\) = the gender of the i ^th worker (1 = male and 0 = female)

1. What is the real-world meaning of \({{\beta }_{2}}\) ? (Hint: If you're unsure where to start, review Section 1.2.)
2. What is the real-world meaning of 133? (Hint: Remember that GEND is a dummy variable.)

8. Have you heard of "RateMyProfessors.com"? On this website, students evaluate a professor's overall teaching ability and a variety of other attributes. The website then summarizes these student-submitted ratings for the benefit of any student considering taking a class from the professor.

Two of the most interesting attributes that the website tracks are how "easy" the professor is (in terms of workload and grading), and how "hot" the professor is (presumably in terms of physical attractiveness). An article by Otto and colleagues indicates that being "hot" improves a professor's rating more than being "easy." To investigate these ideas ourselves, we created the following equation for RateMyProfessors.com:

\(\text { RATING }_{\mathrm{i}}=\beta_{0}+\beta_{1} \text { EASE }_{\mathrm{i}}+\beta_{2} \mathrm{HOT}_{\mathrm{i}}+\epsilon_{\mathrm{i}}\)

where: \(\quad \operatorname{RATING}_{\mathrm{i}}=\) the overall rating (5= best ) of the ith professor \(\operatorname{EASE}_{\mathrm{i}}=\) the easiness rating (5= easiest ) of the ith professor

\(\mathrm{HOT}_{\mathrm{i}}=1\) if the ith professor is considered "hot," 0 otherwise

To estimate Equation 1.27, we need data, and Table 1.2 contains data for these variables from 25 randomly chosen professors on RateMyProfessors.com. If we estimate Equation $1.27$ with the data in Table 1.2, we obtain:

1. Take a look at Equation 1.28. Do the estimated coefficients support our expectations? Explain.
2. See if you can reproduce the results in Equation 1.28 on your own. To do this, take the data in Table 1.2 and use Stata or your own regression program to estimate the coefficients from these data. If you do everything correctly, you should be able to verify the estimates in Equation 1.28. (If you're not sure how to get started on this question, either take a look at the answer to Exercise 2 in Appendix A or read Appendix 1.7.)
3. This model includes two independent variables. Does it make sense to think that the teaching rating of a professor depends on just these two variables? What other variable(s) do you think might be important?
4. Suppose that you were able to add your suggested variable(s) to Equation $1.28$. What do you think would happen to the coefficients of EASE and HOT when you added the variable(s)? Would you expect them to change? Would you expect them to remain the same? Explain.
5. (optional) Go to the RateMyProfessors.com website, choose 25 observations at random, and estimate your own version of Equation 1.27. Now compare your regression results to those in Equation 1.28. Do your estimated coefficients have the same signs as those in Equation 1.28? Are your estimated coefficients exactly the same as those in Equation 1.28? Why or why not?

(d) I would expect the coefficients to change, but not by much.

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