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Introduction to Econometrics
71
Technical Notes
Introduction to Econometrics
Lecture 7: Multiple Regression  Hypotheses about Coefficients and Sets of Coefficients
An Outline of the Specification Problem
With multiple regression we need to decide which explanatory variables to include in an equation. This is
the problem of choosing a specification. Although economic theory can suggest variables, it is not
informative enough to solve the problem, so we need statistical techniques. In a simple case these
techniques can be regarded as a form of hypothesis testing.
We need first to distinguish between nested
and nonnested
hypotheses. Consider three linear models
(alternative regressions) explaining Y
1
Y =
α
+
β
X +
θ
Z + U
1
2
Y =
α
+
β
X + U
2
3
Y =
α
+
θ
Z + U
3
Model 2 is nested
in Model 1: it is the special case in which
θ
= 0. Similarly Model 3 is nested
in Model 1.
But Model 2 and Model 3 are not nested: neither is a special case of the other.
Choosing between Model 1 and Model 2 can be based on a test of whether the null hypothesis H
0
:
θ
= 0 is
rejected by the data at an appropriate significance level when Model 1 is estimated. Similarly the choice
between Model 1 and Model 3 can be based on a test of H
0
:
β
= 0. Choosing between nonnested models
(for example, choosing between Models 2 and 3) is much more difficult, and is not covered in this course.
Significance Testing for Individual Coefficients
If the specification problem can be reduced to a significance test on an individual coefficient then we can
use a ttest.
To test
H
0
:
β
= 0
H
1
:
β
≠
0
use the tratio
β
e
/se[
β
e
], which has the tdistribution with NK degrees of freedom (K is the number of
regressors including
the intercept). To test
H
0
:
β
= 1
H
1
:
β
≠
1
the relevant statistic is
t
*
= (
β
e
 1)/se[
β
e
]
which also has a tdistribution if H
0
is true. A version of this test is applicable whenever the null hypothesis
is that the single coefficient
β
e
takes a known
constant value.
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View Full DocumentIntroduction to Econometrics
72
Technical Notes
Significance Testing for Groups of Coefficients
Consider the slightly more complex model
1
Y =
α
+
β
X +
θ
Z +
φ
W + U
and the joint
null hypothesis about the pair of coefficients
β
and
θ
.
H
0
:
β
= 0
and
θ
= 0
H
1:
not
(
β
= 0
and
θ
= 0)
which is equivalent to
β
≠
0
or
θ
≠
0 (or both)
This implies that the model which corresponds to the null is
2
Y =
α
+
φ
W + U
From the
AN
alysis
O
f
VA
riance Tables for the two regressions we know that
SSE
Model 1
/
σ
2
=
{
β
2
Var(X) +
θ
2
Var(Z) +
φ
2
Var(W) + 2
βθ
Cov(X, Z) + 2
βφ
Cov(X,W) +
2
θφ
Cov(Z,W)}
which is distributed as
χ
2
(3)
SSE
Model 2
/
σ
2
=
φ
2
Var(W)/
σ
2
which is distributed as
χ
2
(1)
Hence under the null H
0
, which implies
β
=
θ
= 0, the difference between the explained sums of squares,
{SSE
Model 1
 SSE
Model 2
}/
σ
2
, has zero mean and is distributed as
χ
2
(2). Unfortunately we can’t use this result
directly, because we don’t know
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 Spring '10
 Cowell
 Economics, Econometrics

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