Economics 241B
Hypothesis Testing Under Normality
Often, the economic theory that gives rise to a model also yields the values
of the coe¢ cients.
As the probability that the estimate equals the hypothesized
value is zero, we cannot conclude the theory is false simply because the estimated
coe¢ cient is not equal to the value speci°ed by theory. To determine the adequacy
of the theory, we study the sampling error
B
°
°
= (
X
0
X
)
°
1
X
0
U:
To determine if the sampling error is so large that we reject the theory, we need
to construct (from the sampling error) a test statistic whose distribution is known
given the truth of the hypothesis.
It seems that to do so we need to know the
joint distribution of
(
X; U
)
as the sampling error is a function of both random
variables.
Surprisingly, if the distribution of
U
conditional on
X
is Gaussian,
then we can derive the distribution of the test statistic without knowledge of the
distribution of
X
.
The restriction to be tested, such as
H
0
:
°
2
= 0
is termed the null hypothesis.
It is a restriction on the set of maintained hypothe
ses, a set of assumptions, which together with the null, produce the test statistic
with known distribution.
For testing hypotheses about regression coe¢ cients, we
need add only that the error, conditional on
X
, is Gaussian
Assumption 5 (Normality of the Error):
The distribution of U, condi
tional on X, is jointly normal.
Together, Assumptions 15 form the maintained hypotheses.
The model is cor
rectly speci°ed if the maintained hypotheses are true.
Too large a value of the test statistic is taken as evidence that the null is false.
Such a conclusion is only correct if the model is correctly speci°ed.
If one of the
maintained hypotheses is false, then it is possible that the test statistic does not
have the supposed distribution under the null.
A large value of the test statistic
can arise from the failure of the other maintained hypothesis, rather than the null.
Implications of Normality Assumption
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
±
The distribution depends only on the mean and variance.
The mean and
variance can be functions of
X
.
If , however, the mean and variance do
not depend on
X
, then the marginal distribution of
U
is identical to the
conditional distribution.
±
If two random variables are jointly Gaussian, then lack of correlation im
plies independence.
This carries over to the conditional distribution, if two
random variables are jointly Gaussian and uncorrelated conditional on
X
,
then they are independent conditional on
X
.
±
Linear combinations preserve normality.
If
A
is a function of
X
, then
AU
is Gaussian conditional on
X
.
With the speci°ed mean and variance of
U
from preceding assumptions, we
have
U
j
X
²
N
°
0
; ±
2
I
n
±
:
Because the conditional distribution does not depend on
X
,
U
and
X
are inde
pendent, which implies
U
²
N
°
0
; ±
2
I
n
±
:
It also follows that the sampling error is Gaussian, conditional on
X
(
B
°
°
)
j
X
²
N
²
0
; ±
2
(
X
0
X
)
°
1
³
:
(0.1)
Tests for Individual Regression Coe¢ cient Hypotheses
We consider
H
0
:
°
2
=
c;
where
c
is a known value speci°ed by the null hypothesis.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 Staff
 Economics, Normal Distribution, Null hypothesis, Statistical hypothesis testing, Statistical tests, Abraham Wald

Click to edit the document details