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Problem Set 4
ECON 837
Prof. Simon Woodcock, Spring 2006
Due: Friday Feb 17
1. [A warmup question from the 2004 Midterm] Suppose the data generating process is
y
=
X
+
"
;
where
E
[
"
] =
0
,
E
[
""
0
] =
2
I
n
;
and
X
includes an intercept term. You
do not observe the data set
Z
= [
y X
]
:
Instead you observe
Z
0
Z
=
2
4
100 10 25
10
20
0
25
0
75
3
5
:
Compute the least squares estimators
^
; s
2
;
and
R
2
. Is there anything to be gained
by observing the full data set?
2. [A followup question] Find a (matrixvalued) su¢ cient statistic for
and
2
in the
multivariate linear regression under normality. Comment on your solution to Question
1.
3. [2004 Midterm] Suppose the data generating process is
y
i
=
x
0
i
+
"
i
where the errors
are spherical and have mean zero. The data fall into one of two groups of equal size. In
n=
2
observations,
x
0
i
= [1 1]
:
In the second group (group
2),
x
0
i
= [1
1]
:
Despite your knowledge of least squares, you devise a new estimator
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 Spring '09
 Thewalt
 Physics

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