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
:
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.
3. [15 pts] (a)
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)
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 su¢ cient statistic for
µ
. Give your reasoning.
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 Fall '07
 HONG
 Economics, Econometrics, Probability theory, pts, 5 pts, 4 pts

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