Economics 241B
Hypothesis Testing: Large Sample Inference
Statistical inference in largesample theory is based on test statistics whose dis
tributions are known under the truth of the null hypothesis.
Derivation of these
distributions is easier than in °nitesample theory because we are only concerned
with the largesample approximation to the exact distribution.
In what follows we assume that a consistent estimator of
S
exists, which we term
^
S
.
Recall that
S
=
E
(
g
t
g
0
t
)
, where
g
t
=
X
t
U
t
.
Testing Linear Hypotheses
Consider testing a hypothesis regarding the
k
th coe¢ cient
°
k
.
Proposition 2.1,
which established the asymptotic distribution of the OLS estimator, implies that
under
H
0
:
°
k
=
°
°
k
,
p
n
°
B
k
°
°
°
k
±
!
d
N
(0
; Avar
(
B
k
))
and
\
Avar
(
B
k
)
!
p
Avar
(
B
k
)
:
Here
B
k
is the
k
th element of the OLS estimator
B
and
Avar
(
B
k
)
is the
(
k; k
)
element of the
K
±
K
matrix
Avar
(
B
)
.
The key issue here is that we have not
assumed conditional homoskedasticity, hence
\
Avar
(
B
k
) =
S
°
1
XX
^
S S
°
1
XX
;
which is the (heteroskedasticityconsistent) robust asymptotic variance.
Under
the Slutsky result (Lemma 2.4c), the resultant robust
t
ratio
t
k
²
p
n
°
B
k
°
°
°
k
±
q
\
Avar
(
B
k
)
=
°
B
k
°
°
°
k
±
SE
±
(
B
k
)
!
d
N
(0
;
1)
;
where the robust standard error is
SE
±
=
q
1
n
\
Avar
(
B
k
)
.
Note this robust
t

ratio is distinct from the
t
ratio introduced under the °nitesample assumptions
in earlier lectures.
To test
H
0
:
°
k
=
°
°
k
, simply follow these steps:
Step 1:
Calculate the robust
t
ratio
Step 2:
Obtain the critical value from the
N
(0
;
1)
distribution
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Step 3:
Reject the null hypothesis if
j
t
k
j
exceeds the critical value
There are several di/erences from the °nitesample test that relies on conditional
homoskedasticity.
³
The standard error is calculated in a di/erent way, to accommodate condi
tional heteroskedasticity.
³
The normal distribution is used to obtain critical values, rather than the
t
distribution.
³
The actual (or empirical) size of the test is not necessarily equal to the
nominal size.
The di/erence between the actual size and the nominal size
is the size distortion.
Because the asymptotic distribution of the robust
t
ratio is standard normal, the size distortion shrinks to zero as the sample
size goes to in°nity.
To summarize these results, together with the behavior of the Wald statistic let
us brie±y recall the assumptions required for Proposition 2.1:
Assumption 2.1 (linearity):
Y
t
=
X
0
t
°
+
U
t
(
t
= 1
; : : : ; n;
)
where
X
t
is a
K
dimensional vector of regressors,
°
is a
K
dimensional vector of
coe¢ cients and
U
t
is the latent error.
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 Fall '08
 Staff
 Economics, Normal Distribution, Null hypothesis, Avar

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