Problems: Multivariate Regression
2
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6. Below is a passage from the book Fringe Banking:
CheckCashing Outlets, Pawnshops, and the Poor by John P
Caskey (New York: Russell Sage Foundation, 1994). The
numbers in parenthesis under the Estimated Coefficients are
tvalues. (A dummy variable takes values of 0 or 1—it
means that a variable can be coded as on or off.) Click on
the image to get a larger version.
 a. The author is trying to explain why some states
have more pawnshops per capita than other states have.
How successful is he in explaining this?
 b. He lists four hypotheses about the signs he
expects on the coefficients. One of them is contradicted.
Which one? Explain.
 c. Which of his hypotheses is most strongly supported
by his results? Explain.
7. Use this information to answer the questions below.
(Click on the image for a larger version.)
 a) Would this equation be useful for prediction
purposes? Why or why not? (Us the R^{2} in your
explanation.)
 b) Why is the tvalue under EF (education of father)
not significant at the .05 level?
 c) What would the standard error of the regression
coefficient for S (hours of study) be? (You have to
compute it from the information given.)
 d) What does the negative sign in front of the
regression coefficient of M (number of moves during
childhood) indicate?
 e) The author has six a priori hypotheses. Which are
confirmed? Explain for each hypothesis.
8. The authors of a study in the Canadian Journal of
Economics in the Spring of 1975 wanted to examine the
factors responsible for learning among
principlesofeconomics students. The variable they were
trying to explain, the amount a student had learned, was
measured by a score on the test at the end of the semester.
Some of the independent variables that the authors used are
selfexplanatory and were presumably obtained from student
records. The authors also gave a test at the beginning of
the year, called the pretest, which was meant to measure a
student's initial knowledge of economics. Other information
was obtained from a questionnaire given the students that
asked their evaluation of the usefulness of the course,
their rating of the teacher (was he interesting?
organized?), the number of classes they attended, and the
time they spent on the course. These were all rated on a
onetofive scale.
Below is the regression equation that the authors
estimated (tvalues are in parentheses):
Test Score =

2.91 +

.350 H.S.grades +

0.0184 V&M_SAT_Score

+ .375 pretest

(0.41)

(3.77)

(4.08)

(7.08)

 4.86 girl 

1.87 yr in school

 .32 Q stimulating +

1.02 Q organized

(3.65)

(1.78)

(0.50)

(1.32)

+ 1.88 Q useful

+ 0.21 attendance

 1.54 time spent


(2.79)

(0.29)

(2.55)


R^{2} = .44

9. What factors seem to be important in determining the
amount a typical student learned? Which may not be
important? Does the small size of the regression coefficient
on the verbalmath SAT score mean that this was not
important? What else do you conclude?
Below are the regression results that a student obtained
when he sought to explain the number of runs baseball
players scored in a year by considering their hits,
homeruns, and stolen bases.
Dependent variable is runs scored

Independent
Variable

Regression
Coefficient

Standard Error
of Regression
Coefficient

tValue

Constant

7.4232

12.2636

0.605

Hits

0.0464

0.0972

4.772

Homeruns

0.5359

0.2496

2.147

Stolen Bases

1.4040

0.5327

2.786

Standard error of estimate: 10.705; Number of
observations = 25
Coefficient of Determination (R^{2}):
.817
Corrected R^{2}: .791
Multiple Correlation: .904

 a) How well do the independent variables explain the
dependent variable? Explain what number(s) you use to
decide this.
 b) If a player had 60 hits, 5 home runs, and 10
stolen bases, how many runs could we expect him to
score?
 c) Construct a 95% confidence interval for the true
regression coefficient of stolen bases.
 d) What does the 4.772 mean in the tvalue
column?
 e) There are a number of things that are questionable
with the way the student has set up this study. For
example, we are estimating a regression coefficient for
home runs, but we actually know what it should be. What
is it? All of these regression coefficients may be
capturing the influences of variables that are not
included in the study, especially the number of at bats a
player had. How might you improve this study if you were
to start over?
10. A large company divides its market into a number of
districts. It is interested in knowing how sales in these
districts depend on the population and on the amount spent
on advertising. Using least squares regression, it finds
Y = 10.7 +

0.4X_{1} +

4.2X_{2 }

 n = 18 districts


(0.15)

( 2.0)

 R^{2} = .76

where Y is sales in millions of dollars
X_{1} is population in hundreds of thousands
X_{2} is the amount spend on advertising in
millions of dollars
numbers in parentheses are standard errors of the
regression coefficients
 a) Explain what the regression coefficients
mean.
 b) Compute the tvalues for the regression
coefficients and explain what they tell us.
 c) What does the R^{2} tell us?
 d) If you wanted to test the hypothesis that the true
value of the R^{2} is zero, that is, that this
regression explains nothing at all but whatever we are
seeing is just chance, what value would you want to
have?
11. A regression uses 100 observations to determine the
relationship between the GPA of freshmen and their SAT test
scores. It gives the following equation:
GPA = 1.43 +

.0004(Math SAT) +

.0018(Verbal SAT)


(.00015)

(0004)

The numbers in parentheses are the standard errors of the
regression coefficients. Test the hypothesis that the
population regression coefficient for Math SAT is equal to
zero (which is the hypothesis that there is no effect of
Math SAT scores on GPA).
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