The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

The general rule is this:
df
=
(
n
−
number of parameters estimated).
guj75772_ch03.qxd 23/08/2008 02:34 PM Page 70

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Chapter 3
Two-Variable Regression Model: The Problem of Estimation
71
Note the following features of the variances (and therefore the standard errors) of 
ˆ
β
1
and
ˆ
β
2
.
1. The variance of
ˆ
β
2
is directly proportional to
σ
2
but inversely proportional to
x
2
i
.
That is, given
σ
2
, the larger the variation in the
X
values, the smaller the variance of
ˆ
β
2
and
hence the greater the precision with which
β
2
can be estimated. In short, given
σ
2
, if there is
substantial variation in the
X
values,
β
2
can be measured more accurately than when the
X
i
do
not vary substantially. Also, given
x
2
i
, the larger the variance of
σ
2
, the larger the variance
of
β
2
. Note that as the sample size
n
increases, the number of terms in the sum,
x
2
i
, will in-
crease. As
n
increases, the precision with which
β
2
can be estimated also increases. (Why?)
2. The variance of 
ˆ
β
1
is directly proportional to 
σ
2
and 
X
2
i
but inversely proportional
to 
x
2
i
and the sample size 
n
.
3. Since 
ˆ
β
1
and 
ˆ
β
2
are estimators, they will not only vary from sample to sample but in
a given sample they are likely to be dependent on each other, this dependence being mea-
sured by the covariance between them. It is shown in Appendix 3A, Section 3A.4 that
(3.3.9)
Since var (
ˆ
β
2
) is always positive, as is the variance of any variable, the nature of the
covariance between 
ˆ
β
1
and 
ˆ
β
2
depends on the sign of 
¯
X
.
If 
¯
X
is positive, then as the
formula shows, the covariance will be negative. Thus, if the slope coefficient 
β
2
is 
overes-
timated
(i.e., the slope is too steep), the intercept coefficient 
β
1
will be 
underestimated
(i.e.,
the intercept will be too small). Later on (especially in the chapter on multicollinearity,
Chapter 10), we will see the utility of studying the covariances between the estimated
regression coefficients.
How do the variances and standard errors of the estimated regression coefficients
enable one to judge the reliability of these estimates? This is a problem in statistical
inference, and it will be pursued in Chapters 4 and 5.