4630 Problem Set 8 SOLUTIONS
1.
You are interested in knowing whether there is a relationship between
performance in Dr. McPherson’s statistics class (ECON 4630) and performance in
Dr. Tieslau’s introductory econometrics class (ECON 4870). The following
information represents a random sample of students who have taken both classes:
Grade in
ECON Grade in
Student 4870 (Y) 4630 (X)
1
69.75
71
2
76.25
80
3
90.5
94
4
80.25
86
5
74.25
82
6
93.25
98
7
94
97
8
58
17
9
100
97
10
100
96
11
75
87
12
89
96
13
88
94
14
97
100
15
65
88
Using Excel, do an OLS regression.
a.
Interpret the estimated coefficients
If a person were to receive a zero in 4630, we would expect him or her
to earn a grade of 41.4 in 4870.
Each additional point that a student earns in 4630 will cause his or
her grade in 4870 to rise by 0.49 points, ceteris paribus.
b.
Are the estimated coefficients significantly different from zero?
We have two hypotheses to test here:
1.
H0: β1 = 0 vesus HA: β1 ≠ 0
2.
H0: β2 = 0 vesus HA: β2 ≠ 0
The test statistic for the first test is 4.14, and the test statistic for the
second test is 4.30.
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Here we have n-2 = 13 degrees of freedom. The critical value for a
two-tailed test at the 95% confidence level is ±2.16. Since in each case
the test statistic is larger (in absolute value) than the critical value, we
reject the null in both tests: both β1 and β2 are significantly different
from zero.
c.
What is the R2, and what is its interpretation?
R2 is 0.587 in this case. This means that 58.7% of the variation in the
grade received in 4870 is explained by the variation in the grade
received in 4630.
2.
Let’s look at the issue above in a different light. Consider the following data from
Dr. Tieslau’s ECON 4870 class.
GPA
Grade in going in
Student 4870 (Y) (X1)
1
75.5
3.532
2
89.5
3.404
3
94.75
3.63
4
91
3.096
5
82.5
2.965
6
84.25
3.266
7
95.25
3.912
8
80.5
3.238
9
69.75
2.425
10
90.75
3.5
11
76.25
2.759
12
90.5
3.968
13
80.25
2.287
14
57
2.636
15
80.5
3.614
16
94.75
3.6
17
55
2.397
18
74.25
3.285
19
74
3.391
20
82.25
3.824
21
79.25
3.8
22
94
4
23
93.25
4
24
90.75
3.651
25
76.75
3.056
26
94
4
27
58
2.076
did student
take 4630?
(D1)
0
0
0
0
0
0
0
0
1
0
1
1
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
Gender (D2)
1
0
0
1
0
0
0
1
1
1
0
0
1
1
0
0
1
1
1
1
0
0
0
1
1
0
1
217
Using Excel, do an OLS regression.
a. Interpret the estimated coefficients
If a person were to have a GPA of 0.00, had taken 4630, and was male,
we would expect him or her to earn a grade of 38.33 in 4870.
If a student’s GPA were to rise by 1 point on a 4-point scale, we would
expect his or her grade in 4870 to rise by 13.78 points, ceteris paribus.
That is, we would expect a female student who had not taken 4630
with a 3.0 GPA to earn a grade of 79.67. A female student who had
not taken 4630 with a 4.0 GPA should earn a grade of 93.45.
The grades earned in 4870 by students who have taken 4630 are 0.93
points higher than students who have not taken 4630, ceteris paribus.
The grades earned in 4870 by male students are 4.93 points lower
than female students, ceteris paribus.
b. Are the estimated coefficients significantly different from zero?
We have four hypotheses to test here:
1. H0: β1 = 0 vesus HA: β1 ≠ 0
2. H0: β2 = 0 vesus HA: β2 ≠ 0
3. H0: β3 = 0 vesus HA: β3 ≠ 0
4. H0: β4 = 0 vesus HA: β4 ≠ 0
The test statistic for the first test is 3.14, the test statistic for the
second test is 4.26, the test statistic for the third is 0.28, and the test
statistic for the fourth is –1.40.
Here we have n-4 = 23 degrees of freedom. The critical value for a
two-tailed test at the 95% confidence level is ±2.069. For the first two
pairs of hypotheses the test statistic is larger (in absolute value) than
the critical value, so we reject the null in those tests: both β1 and β2 are
significantly different from zero. For the second two pairs of
hypotheses the test statistic is smaller (in absolute value) than the
critical value, so we fail to reject the null in those tests: neither β3 nor
β4 is significantly different from zero. This means that the GPA of a
student going in to 4870 is a significant determinant of grade in 4870,
but whether or not a person has taken 4630 is not a significant
determinant of grade in 4870, and the student’s gender also doesn’t
affect grade in 4870.
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c. What is the R2, and what is its interpretation?
R2 is 0.626 in this case. This means that 62.6% of the variation in the
grade received in 4870 is explained by the variation in all three
explanatory variables.
219