PROF HONG
FALL 2006
ECONOMICS 619
FINAL EXAM
Notes: (1) This is a closed book/notes exam. There are 7 questions, with a total of 100
points; (2) you have
150
minutes; (3) suggestion: have a look at all problems and °rst
solve the problems you feel easiest; (4) good luck!
1. [15 pts]
Suppose
Y
=
°
+
±X
+
j
X
j
";
where
E
(
X
) = 0
;
var
(
X
) =
²
2
X
; E
(
"
) = 0
;
var
(
"
) =
²
2
"
;
and
"
and
X
are independent.
Both
°
and
±
are constants.
(a) [5 pts]
Find
E
(
Y
j
X
)
:
(b) [5 pts]
Find var
(
Y
j
X
)
:
(c) [5 pts]
Show cov
(
Y; X
) = 0
if and only if
±
= 0
:
ANS:
(a)
E
(
Y
j
X
) =
E
[
°
+
±X
+
j
X
j
"
j
X
] =
°
+
±X
+
j
X
j
E
(
"
j
X
) =
°
+
±X:
(b) var
(
Y
j
X
) =
var
(
°
+
±X
+
j
X
j
"
j
X
) =
var
(
j
X
j
"
j
X
) =
X
2
²
2
"
:
(c)cov
(
Y; X
) =
cov
(
°
+
±X
+
j
X
j
"; X
) =
±cov
(
X; X
) +
cov
(
j
X
j
"; X
)
=
±cov
(
X; X
) +
E
[(
j
X
j
"
°
E
(
j
X
j
"
))
X
] =
±cov
(
X; X
) +
E
(
X
j
X
j
"
)
=
±²
2
X
= 0
if and only if
±
= 0
:
Grading policy: 5 points each. Correct formula gets 1 point.
2. [10 pts]
Suppose
f
X
i
g
n
i
=1
is an i.i.d.
N
(
³; ²
2
)
random sample, where both
³
and
²
2
are unknown parameters. Contruct an unbiased estimator for
´
=
³
2
, and justify it is
unbiased.
ANS: A possible answer:
Let
b
´
=
X
2
i
°
S
2
n
;
where
S
2
n
=
P
n
i
=1
(
X
i
°
X
n
)
2
n
°
1
and
X
n
=
P
n
i
=1
X
i
n
. We could check the unbiasedness by
E
(
b
´
) =
E
(
X
2
i
)
°
E
(
S
2
n
) =
³
2
+
²
2
°
²
2
=
³
2
1
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(Remark:
There are many solutions for this construction, say, we could also let
b
´
=
P
n
i
=1
X
2
i
n
°
S
2
n
).
Grading policy: Knowing the concept of unbiasedness 2 points. Correct construction
8 points.
3.
[12 pts] (a) [5 pts]
Suppose
X
n
=
f
X
i
g
n
i
=1
is an i.i.d.
random sample with
a population probability density
f
(
x; µ
)
:
Is
X
n
a su¢ cient statistic for
µ
?
Give your
reasoning.
(b) [7 pts]
Suppose
X
n
=
f
X
i
g
n
i
=1
is an i.i.d.
random sample from a
N
(
µ; µ
)
population, where
µ
is unknown. Find a
ONE DIMENSIONAL
su¢ cient statistic
for
µ
. Give your reasoning.
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 Fall '07
 HONG
 Economics, Econometrics, Trigraph, Maximum likelihood, Likelihood function, Mean squared error

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