Introduction to Econometrics
7-
1
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 non-nested
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 non-nested 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 t-test.
To test
H
0
:
β
= 0
H
1
:
β
≠
0
use the t-ratio
β
e
/se[
β
e
], which has the t-distribution with N-K 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 t-distribution 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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