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Introduction to Financial Econometrics
Hypothesis Testing in the Market Model
Eric Zivot
Department of Economics
University of Washington
February 29, 2000
1
Hypothesis Testing in the Market Model
In this chapter, we illustrate how to carry out some simple hypothesis tests concerning
the parameters of the excess returns market model regression.
1.1
A Review of Hypothesis Testing Concepts
To be completed.
1.2
Testing the Restriction
α
=0
.
Using the market model regression,
R
t
=
α
+
β
R
Mt
+
ε
t
,t
=1
,...,T
ε
t
∼
iid N
(0
,
σ
2
ε
)
,
ε
t
is independent of
R
Mt
(1)
consider testing the null or maintained hypothesis
α
= 0 against the alternative that
α
6
=0
H
0
:
α
=0
vs. H
1
:
α
6
=0
.
If
H
0
is true then the market model regression becomes
R
t
=
β
R
Mt
+
ε
t
and
E
[
R
t

R
Mt
=
r
Mt
]=
β
r
Mt
.
We will reject the null hypothesis,
H
0
:
α
=0
,i
f
the estimated value of
α
is either much larger than zero or much smaller than zero.
Assuming
H
0
:
α
= 0 is true, ˆ
α
∼
N
(0
,SE
(ˆ
α
)
2
) and so is fairly unlikely that ˆ
α
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View Full Documentbe more than 2 values of
SE
(ˆ
α
) from zero. To determine how big the estimated value
of
α
needs to be in order to reject the null hypothesis we use the tstatistic
t
α
=0
=
b
α
−
0
d
SE
(
b
α
)
,
where
b
α
is the least squares estimate of
α
and
d
SE
(
b
α
) is its estimated standard error.
The value of the tstatistic,
t
α
=0
, gives the number of estimated standard errors that
b
α
is from zero. If the absolute value of
t
α
=0
is much larger than 2 then the data cast
considerable doubt on the null hypothesis
α
=0whereasi
fitislessthan2thedata
are in support of the null hypothesis
1
. To determine how big

t
α
=0

needs to be to
reject the null, we use the fact that under the statistical assumptions of the market
model and
assuming the null hypothesis is true
t
α
=0
∼
Student
−
t
with
T
−
2 degrees of freedom
If we set the signi
f
cance level (the probability that we reject the null given that the
null is true) of our test at, say, 5% then our decision rule is
Reject
H
0
:
α
=0atthe5%leve
lif

t
α
=0

>t
T
−
2
(0
.
025)
where
t
T
−
2
is the 2
1
2
% critical value from a Studentt distribution with
T
−
2degrees
of freedom.
Example 1
Market Model Regression for IBM
Consider the estimated MM regression equation for IBM using monthly data from
January 1978 through December 1982:
b
R
IBM,t
=
−
0
.
0002
(0
.
0068)
+0
.
3390
(0
.
0888)
·
R
Mt
,R
2
=0
.
20
,
b
σ
ε
=0
.
0524
where the estimated standard errors are in parentheses. Here
b
α
=
−
0
.
0002, which is
very close to zero, and the estimated standard error,
d
SE
(ˆ
α
) = 0.0068, is much larger
than
b
α
. The tstatistic for testing
H
0
:
α
=0vs
.
H
1
:
α
6
=0is
t
α
=0
=
−
0
.
0002
−
0
0
.
0068
=
−
0
.
0363
so that
b
α
is only 0.0363 estimated standard errors from zero. Using a 5% signi
f
cance
level,
t
58
(0
.
025)
≈
2and

t
α
=0

=0
.
0363
<
2
so we do not reject
H
0
:
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