Introductory Econometrics

Jeffrey M. Wooldridge

Chapter 4

Multiple Regression Analysis: Inference - all with Video Answers

Educators

Chapter Questions

Problem 1

The following model can be used to study whether campaign expenditures affect election outcomes:
voteA$=\beta_{0}+\beta_{1} \log (e x p e n d A)+\beta_{2} \log (e x p e n d B)+\beta_{3} p r t y s t r A+u$
where voteA is the percentage of the vote received by Candidate A, expendA and expendB are campaign expenditures by Candidates $A$ and $B$ , and prystrA is a measure of party strength for Candidate A (the percentage of the most recent presidential vote that went to A' party).
(i) What is the interpretation of $\beta_{1} ?$
(ii) In terms of the parameters, state the null hypothesis that a 1$\%$ increase in A's expenditures is offset by a 1$\%$ increase in B's expenditures.
(iii) Estimate the given model using the data in VOTE1 and report the results in usual form. Do A's expenditures affect the outcome? What about B's expenditures? Can you use these results to test the hypothesis in part (ii)?
(iv) Estimate a model that directly gives the $t$ statistic for testing the hypothesis in part (ii). What do you conclude? (Use a two-sided alternative.)

Krish Desai

Krish Desai

Numerade Educator

Problem 2

Use the data in LAWSCH85 for this exercise.
(i) Using the same model as in Problem 4 in Chapter $3,$ state and test the null hypothesis that the rank of law schools has no ceteris paribus effect on median starting salary.
(ii) Are features of the incoming class of students- namely, $L S A T$ and $G P A-$ individually or jointly significant for explaining salary? (Be sure to account for missing data on $L S A T$ and $G P A . )$
(iii) Test whether the size of the entering class (clsize) or the size of the faculty (faculty) needs to be added to this equation; carry out a single test. (Be careful to account for missing data on clsize and faculty.)
(iv) What factors might influence the rank of the law school that are not included in the salary regression?

Lyle Anderson

Numerade Educator

Problem 3

Refer to Computer Exercise $\mathrm{C} 2$ in Chapter $3 .$ Now, use the log of the housing price as the dependent variable:
$$\log (price)=\beta_{0}+\beta_{1} \operatorname{sqr} f t+\beta_{2} b d r m s+u$$
(i) You are interested in estimating and obtaining a confidence interval for the percentage change in price when a 150 -square-foot bedroom is added to a house. In decimal form, this is $\theta_{1}=150 \beta_{1}+\beta_{2} .$ Use the data in HPRICE 1 to estimate $\theta_{1}$ .
(ii) Write $\beta_{2}$ in terms of $\theta_{1}$ and $\beta_{1}$ and plug this into the log(price) equation.
(iii) Use part (ii) to obtain a standard error for $\hat{\theta}_{2}$ and use this standard error to construct a 95$\%$ confidence interval.

Lyle Anderson

Numerade Educator

Problem 4

In Example $4.9,$ the restricted version of the model can be estimated using all $1,388$ observations in the sample. Compute the $R$ -squared from the regression of bwght on cigs, parity, and faminc using all observations. Compare this to the $R$ -squared reported for the restricted model in Example $4.9 .$

Lyle Anderson

Numerade Educator

Problem 5

Use the data in MLB1 for this exercise.
(i) Use the model estimated in equation $(4.31)$ and drop the variable rbisyr. What happens to the statistical significance of hrunsyr? What about the size of the coefficient on hrunsyr?
(ii) Add the variables runsyr(runs per year), fldper (fielding percentage), and sbasesyr (stolen bases per year) to the model from part (i). Which of these factors are individually significant?
(iii) In the model from part (ii), test the joint significance of bavg, fldperc, and sbasesyr.

Lyle Anderson

Numerade Educator

Problem 6

Use the data in WAGE2 for this exercise.
(i) Consider the standard wage equation
$$\log (wage)=\beta_{0}+\beta_{1} e d u c+\beta_{2}$exper$+\beta_{3}$tenure$+u$$
State the null hypothesis that another year of general workforce experience has the same effect on log(wage) as another year of tenure with the current employer.
(ii) Test the null hypothesis in part (i) against a two-sided alternative, at the 5$\%$ significance level, by constructing a 95$\%$ confidence interval. What do you conclude?

Lyle Anderson

Numerade Educator

Problem 7

Refer to the example used in Section $4-4 .$ You will use the data set TWOYEAR.
(i) The variable $p h s r a n k$ is the person's high school percentile. (A higher number is better. For example, 90 means you are ranked better than 90 percent of your graduating class.) Find the smallest, largest, and average phsrank in the sample.
(ii) Add phsrank to equation $(4.26)$ and report the OLS estimates in the usual form. Is phsrank statistically significant? How much is 10 percentage points of high school rank worth in terms of wage?
(iii) Does adding phsrank to $(4.26)$ substantively change the conclusions on the returns to two- and four-year colleges? Explain.
(iv) The data set contains a variable called $i d .$ Explain why if you add $i d$ to equation $(4.17)$ or $(4.26)$ you expect it to be statistically insignificant. What is the two-sided p-value?

Lyle Anderson

Numerade Educator

Problem 8

The data set 401 $\mathrm{KSUBS}$ contains information on net financial wealth (nettfa), age of the survey respondent $(a g e),$ annual family income (inc), family size (fsize), and participation in certain pension plans for people in the United States. The wealth and income variables are both recorded in thousands of dollars. For this question, use only the data for single-person households (so fsize$=1 )$ .
(i) How many single-person households are there in the data set?
(ii) Use OLS to estimate the model
nettfa$=\beta_{0}+\beta_{1} i n c+\beta_{2} a g e+u$
and report the results using the usual format. Be sure to use only the single-person households in the sample. Interpret the slope coefficients. Are there any surprises in the slope estimates?
(iii) Does the intercept from the regression in part (ii) have an interesting meaning? Explain.
(iv) Find the $p$ -value for the test $\mathrm{H}_{0} : \beta_{2}=1$ against $\mathrm{H}_{1} : \beta_{2}<1 .$ Do you reject $\mathrm{H}_{0}$ at the 1$\%$ significance level?
(v) If you do a simple regression of netfa on inc, is the estimated coefficient on $i n c$ much different from the estimate in part (ii)? Why or why not?

Lyle Anderson

Numerade Educator

Problem 9

Use the data in DISCRIM to answer this question. (See also Computer Exercise $\mathrm{C} 8$ in Chapter $3 . )$
(i) Use OLS to estimate the model
$\log (p s o d a)=\beta_{0}+\beta_{\mathrm{i}} p r p b l c k+\beta_{2} \log ($income$)+\beta_{3}$ prppov $+u$
and report the results in the usual form. Is $\hat{\beta}_{1}$ statistically different from zero at the 5$\%$ level against a two-sided alternative? What about at the 1$\%$ level?
(ii) What is the correlation between log(income) and prppov? Is each variable statistically significant in any case? Report the two-sided $p$ -values.
(iii) To the regression in part (i), add the variable log(hseval). Interpret its coefficient and report the two-sided $p$ -value for $\mathrm{H}_{0} : \beta_{\text { loghseral }}=0$
(iv) In the regression in part (ii), what happens to the individual statistical significance of $\quad$ log(income) and prppov? Are these variables jointly significant? (Compute a p-value.) What do you make of your answers?
(v) Given the results of the previous regressions, which one would you report as most reliable in determining whether the racial makeup of a zip code influences local fast-food prices?

Lyle Anderson

Numerade Educator

Problem 10

Use the data in ELEM94 95 to answer this question. The findings can be compared with those in Table $4.1 .$ The dependent variable lavgsal is the log of average teacher salary and $b s$ is the ratio of average benefits to average salary (by school).
(i) Run the simple regression of lavgsal on bs. Is the estimated slope statistically different from zero? Is it statistically different from $-1 ?$
(ii) Add the variables lenrol and lstaff to the regression from part (i). What happens to the coefficient on $b s ?$ How does the situation compare with that in Table 4.1$?$
(iii) How come the standard error on the $b s$ coefficient is smaller in part (ii) than in part (i)? (Hint: What happens to the error variance versus multicollinearity when lenrol and Istaff are added?)
(iv) How come the coefficient on lstaff is negative? Is it large in magnitude?
(v) Now add the variable lunch to the regression. Holding other factors fixed, are teachers being compensated for teaching students from disadvantaged backgrounds? Explain.
(vi) Overall, is the pattern of results that you find with ELEM94 95 consistent with the pattern in Table 4.1$?$

Lyle Anderson

Numerade Educator

Problem 11

Use the data in $\mathrm{HTV}$ to answer this question. See also Computer Exercise $\mathrm{C} 10$ in Chapter $3 .$
(i) Estimate the regression model
$e d u c=\beta_{0}+\beta_{1}$ motheduc $+\beta_{2} f a t h e d u c+\beta_{3} a b i l+\beta_{4} a b i l^{2}+u$
by OLS and report the results in the usual form. Test the null hypothesis that educ is linearly related to abil against the alternative that the relationship is quadratic.
(ii) Using the equation in part (i), test $\mathrm{H}_{0} : \beta_{1}=\beta_{2}$ against a two-sided alternative. What is the $p$ -value of the test?
(iii) Add the two college tuition variables to the regression from part (i) and determine whether they are jointly statistically significant.
(iv) What is the correlation between tuit17 and ttuit18? Explain why using the average of the tuition over the two years might be preferred to adding each separately. What happens when you do use the average?
(v) Do the findings for the average tuition variable in part (iv) make sense when interpreted causally? What might be going on?

Lyle Anderson

Numerade Educator

Problem 12

Use the data in ECONMATH to answer the following questions.
(i) Estimate a model explaining colgpa to hsgpa, actmth, and acteng. Report the results in the usual form. Are all explanatory variables statistically significant?
(ii) Consider an increase in $h s g p a$ of one standard deviation, about $343 .$ By how much does $\widehat{c o l g p a}$ increase, holding actmth and acteng fixed. About how many standard deviations would the actmth have to increase to change colgpa by the same amount as a one standard deviation in
hsgpa? Comment.
(iii) Test the null hypothesis that actmth and acteng have the same effect (in the population) against a two-sided alternative. Report the $p$ -value and describe your conclusions.
(iv) Suppose the college admissions officer wants you to use the data on the variables in part (i) to create an equation that explains at least 50 percent of the variation in colgpa. What would you tell the officer?

Lyle Anderson

Numerade Educator