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Problem Set 5
Introduction to Econometrics  Fall 2006
Due Date: Thursday, November 30th, BEFORE CLASS.
Each student is responsible for handing one handwritten solution. If working in groups, indicate the
members of the group (to facilitate grading).
PLEASE SHOW YOUR WORK
.
1. The following multiple regression function was estimated from a random sample of 2000 observations:
b
Y
i
=10
.
9
(2
.
3)
+2
.
35
(0
.
99)
X
1
i
+1
.
07
(0
.
61)
X
2
1
i
+1
.
32
(0
.
44)
X
2
i
R
2
=0
.
26
This function is quadratic in
X
1
, and linear in
X
2
. Assume that all the least squares assumptions hold.
a. What is the predicted change in
Y
i
when
X
1
i
increases by
1
unit and
X
2
i
is constant? Does it depend
on the value of
X
1
i
?
Does it depend on the value of
X
2
i
?
b. Is the coef
f
cient on the squared term (
X
2
1
i
) statistically signi
f
cant at the 5% signi
f
cance level? Can
we say that this nonlinear model improves upon a linear model? Explain.
c. From the same random sample, you estimate the following multiple regression function:
b
Y
i
=11
.
5
(1
.
9)
+2
.
21
(0
.
89)
X
1
i
+1
.
17
(0
.
41)
X
2
i
+1
.
23
(0
.
47)
X
1
i
X
2
i
R
2
=0
.
32
which includes an interaction term between the regressors. What is the predicted change in
Y
i
when
X
1
i
increases by
1
unit and
X
2
i
is constant in this case? Does it depend on the value of
X
1
i
?
Does it
depend on the value of
X
2
i
?
d. Is the coef
f
cient on the interaction term (
X
1
i
X
2
i
) statistically signi
f
cant at the 5% signi
f
cance level?
Can we say that the nonlinear model improves upon a linear model in this case? Explain.
2. From a random sample of 5000 observations you estimated the following quadratic regression function:
b
Y
i
=2
.
52
(0
.
96)
+1
.
47
(0
.
54)
X
i
+1
.
01
(0
.
38)
X
2
i
R
2
=0
.
51
Assume that the least squares assumptions hold.
a. Use hypothesis testing to check whether the quadratic model improves upon the linear model. Use a
1% signi
f
cance level. Is the quadratic speci
f
cation appropriate? (Recall that the critical value for a
1% signi
f
cance level test is
2
.
58
)
b. From the same random sample used to estimate the quadratic regression function, you estimate the
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 Fall '08
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 Econometrics

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