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STA 6208 – Spring 2003 – Exam 1
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SSN:
All questions are based on the following two regression models, where SIMPLE REGRESSION
refers to the case where
p
=1
, and X is of full column rank (no linear dependencies among the
predictor variables)
Model 1:
Y
i
=
β
0
+
β
1
X
i
1
+
···
+
β
p
X
ip
+
ε
i
i
,...,n ε
i
∼
NID
(0
,σ
2
)
Model 2: Y
=
X
β
+
ε
X
≡
n
×
p
±
β
≡
p
±
×
1
ε
∼
N
(
0
2
I
)
1) For
Model 1
, derive the normal equations.
2) A researcher is interested in the relationship between the education level and salaries in rural counties in
the U.S. He obtains the percentage of adults over 25 with a college education in each county (
X
) and the per
capita income of the county (
Y
, in $1000s). He obtains the following summary statistics, based on a sample of
n
= 30 counties:
(12 points)
±
(
X

X
)
2
= 2207
.
45
±
(
X

X
)(
Y

Y
)=658
.
37
±
(
Y

Y
)
2
= 654
.
86
X
=41
.
92
Y
=35
.
83
a) Give least squares estimates of the parameters of
Model 1
:
β
0
,
β
1
.
b) It can be shown that:
±
(
Y

ˆ
Y
)
2
=
±
(
Y

Y
)
2

(
∑
(
X

X
)(
Y

Y
))
2
∑
(
X

X
)
2
Give an unbiased estimate of
σ
2
.
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 Fall '08
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 Regression Analysis

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