Chapter 6
Specification of regression variables: a
preliminary skirmish
Overview
This chapter treats a variety of topics relating to the specification of the variables in a regression model.
First
there are the consequences for the regression coefficients, their standard errors, and
R
2
of failing to include a
relevant variable, and of including an irrelevant one.
This leads to a discussion of the use of proxy variables to
alleviate a problem of omitted variable bias.
Next come
F
and
t
tests of the validity of a restriction, the use of
which was advocated in Chapter 3 as a means of improving efficiency and perhaps mitigating a problem of
multicollinearity.
The chapter concludes by outlining the potential benefit to be derived from examining
observations with large residuals after fitting a regression model.
Further material
This section contains some new material on the following topics:
1. The reparameterization of a regression model
2. The application of reparameterization to
t
tests of linear restrictions
3. The testing of multiple restrictions
4. Tests of zero restrictions
The reparameterization of a regression model
Suppose that you have fitted the regression model
(1)
u
X
Y
k
j
j
j
+
+
=
∑
=
2
1
β
β
and that the regression model assumptions are valid.
Let the fitted model be
(2)
∑
=
+
=
k
j
j
j
X
b
b
Y
2
1
ˆ
as usual.
Suppose that, as well as the individual parameter estimates, you are interested in some linear
combination:
(3)
∑
=
=
k
j
j
j
1
β
λ
θ
To obtain a point estimate of
θ
, it is natural to construct the statistic
, and indeed, given that the
regression model assumptions are valid, it can easily be shown that this is unbiased and the most efficient
estimator of
θ
.
However you do not have information on its standard error and hence you are not able to
construct confidence intervals or to perform
t
tests.
There are three ways that you might use to obtain such
information:
∑
=
=
k
j
j
j
b
1
ˆ
λ
θ
(1)
Some regression applications have a special command that produces it.
For example, Stata has the lincom
command.
(2)
Given the appropriate command, most regression applications will produce the variance-covariance matrix
for the estimates of the parameters.
This is the complete list of the estimates of their variances and
October 2007

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