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

Part One
Single-Equation Regression Models
Now
ˆ
u
1
i
=
(CM
i
−
263
.
8635
+
2
.
3905 FLR
i
)
(7.3.3)
represents that part of CM left after removing from it the (linear) influence of FLR. Likewise,
ˆ
u
2
i
=
(PGNP
i
+
39
.
3033
−
28
.
1427 FLR
i
)
(7.3.4)
represents that part of PGNP left after removing from it the (linear) influence of FLR.
Therefore, if we now regress
ˆ
u
1
i
on
ˆ
u
2
i
, which are “purified” of the (linear) influence of
FLR, wouldn’t we obtain the net effect of PGNP on CM? That is indeed the case (see
Appendix 7A, Section 7A.2). The regression results are as follows:
ˆˆ
u
1
i
= −
0.0056
ˆ
u
2
i
(7.3.5)
se
=
(0.0019)
r
2
=
0.1152
Note:
This regression has no intercept term because the mean value of the OLS residuals
ˆ
u
1
i
and
ˆ
u
2
i
is zero. (Why?)
The slope coefficient of 
−
0.0056 now gives the “true” or net effect of a unit change in
PGNP on CM or the true slope of CM with respect to PGNP. That is, it gives the partial
regression coefficient of CM with respect to PGNP, 
β
2
.
Readers who want to get the partial regression coefficient of CM with respect to FLR
can replicate the above procedure by first regressing CM on PGNP and getting the residu-
als from this regression (
ˆ
u
1
i
), then regressing FLR on PGNP and obtaining the residuals
from this regression (
ˆ
u
2
i
), and then regressing
ˆ
u
1
i
on
ˆ
u
2
i
. I am sure readers get the idea.
Do we have to go through this multistep procedure every time we want to find out the
true partial regression coefficient? Fortunately, we do not have to do that, for the same job
can be accomplished fairly quickly and routinely by the OLS procedure discussed in the
next section. The multistep procedure just outlined is merely for pedagogic purposes to
drive home the meaning of “partial” regression coefficient.