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

2
X
3
1
2
4
2
0
2
3
4
0
4
6
12
5
8
16
14
These standard errors are obtained from the formula
se (
ˆ
β
2
+ ˆ
β
3
)
=
var (
ˆ
β
2
)
+
var (
ˆ
β
3
)
+
2 cov (
ˆ
β
2
,
ˆ
β
3
)
Note that increasing collinearity increases the variances of 
ˆ
β
2
and 
ˆ
β
3
, but these variances may be
offset if there is high negative covariance between the two, as our results clearly point out.
15
Goldberger, op. cit., pp. 248–250.
EXAMPLE 10.1
Consumption
Expenditure 
in Relation to
Income and
Wealth
To illustrate the various points made thus far, let us consider the consumption–income ex-
ample from the introduction. Table 10.5 contains hypothetical data on consumption,
income, and wealth. If we assume that consumption expenditure is linearly related to
income and wealth, then, from Table 10.5 we obtain the following regression:
ˆ
Y
i
=
24.7747
+
0.9415
X
2
i
−
0.0424
X
3
i
(6.7525)
(0.8229)
(0.0807)
t
=
(3.6690)
(1.1442)
(
−
0.5261)
(10.6.1)
R
2
=
0.9635
¯
R
2
=
0.9531
df
=
7
guj75772_ch10.qxd 12/08/2008 02:45 PM Page 332

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Chapter 10
Multicollinearity: What Happens If the Regressors Are Correlated?
333
Regression (10.6.1) shows that income and wealth together explain about 96 percent
of the variation in consumption expenditure, and yet neither of the slope coefficients is
individually statistically significant. Moreover, not only is the wealth variable statistically
insignificant but also it has the wrong sign. A priori, one would expect a positive relation-
ship between consumption and wealth. Although 
ˆ
β
2
and 
ˆ
β
3
are individually statistically
insignificant, if we test the hypothesis that 
β
2
=
β
3
=
0 simultaneously, this hypothesis can
be rejected, as Table 10.6 shows. Under the usual assumption we obtain
F
=
4282
.
7770
46
.
3494
=
92
.
4019