Economics 241B
Exercise 6
1.
Asymptotic Behavior of Hypothesis Tests
Your °eldwork requires that you test
H
0
:
°
k
=
°
°
k
vs.
H
1
:
°
k
6
=
°
°
k
.
a.
You know that under
H
0
,
p
n
°
B
k
°
°
°
k
±
d
!
N
(0
; Avar
(
B
k
))
(1)
\
Avar
(
B
k
)
p
!
Avar
(
B
k
)
;
where
\
Avar
(
B
k
) =
S
°
1
xx
^
SS
°
1
xx
.
To perform your test, you wish to use a
t
statistic
and you want to make sure that the limit distribution is standard normal.
Show
that (1) implies that under
H
0
,
B
k
°
°
°
k
SE
±
(
B
k
)
d
!
N
(0
;
1)
;
(2)
where
SE
±
(
B
k
) =
q
1
n
\
Avar
(
B
k
)
.
b.
When performing the test, under what circumstances would you select the
critical values from a normal distribution versus a
t
distribution?
c.
Show what
SE
±
(
B
k
)
converges in probability to.
d.
With your answer from part
c
, intuitively explain the convergence in (2).
2.
Consider the model
Y
t
=
°
01
X
t
1
+
°
02
X
t
2
+
U
t
t
= 1
; : : : ; n
(3)
where
U
t
iid
±
N
(0
; ±
2
)
while
X
t
1
and
X
t
2
are scalars such that
E
(
X
t
1
X
t
2
)
6
= 0
but
E
(
X
t
1
U
t
) =
E
(
X
t
2
U
t
) = 0
.
Suppose you fail to observe
X
2
, estimating instead
the model
Y
t
=
°
01
X
t
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 Fall '08
 Staff
 Economics, Normal Distribution, Avar, true model, Bk ) !

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