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

Part One
Single-Equation Regression Models
These partial correlations can be easily obtained from the simple or zero-order, correlation
coefficients as follows (for proofs, see the exercises):
19
r
1 2
.
3
=
r
1 2
−
r
1 3
r
2 3
1
−
r
2
1 3
1
−
r
2
2 3
(7.11.1)
r
1 3
.
2
=
r
1 3
−
r
1 2
r
2 3
1
−
r
2
1 2
1
−
r
2
2 3
(7.11.2)
r
2 3
.
1
=
r
2 3
−
r
1 2
r
1 3
1
−
r
2
1 2
1
−
r
2
1 3
(7.11.3)
The partial correlations given in Eqs. (7.11.1) to (7.11.3) are called 
first-order correlation
coefficients.
By 
order
we mean the number of secondary subscripts. Thus 
r
1 2.3 4
would be
the correlation coefficient of order two, 
r
1 2.3 4 5
would be the correlation coefficient of order
three, and so on. As noted previously, 
r
1 2
, 
r
1 3
, and so on are called 
simple
or 
zero
-
order
correlations.
The interpretation of, say, 
r
1 2.3 4
is that it gives the coefficient of correlation
between 
Y
and 
X
2
, holding 
X
3
and 
X
4
constant.
Interpretation of Simple and Partial
Correlation Coefficients 
In the two-variable case, the simple 
r
had a straightforward meaning: It measured the
degree of (linear) association (and not causation) between the dependent variable 
Y
and the
single explanatory variable 
X
. But once we go beyond the two-variable case, we need to
pay careful attention to the interpretation of the simple correlation coefficient. From
Eq. (7.11.1), for example, we observe the following:
1. Even if 
r
1 2
=
0, 
r
1 2.3
will not be zero unless 
r
1 3
or 
r
2 3
or both are zero.
2. If 
r
1 2
=
0 and 
r
1 3
and 
r
2 3
are nonzero and are of the same sign, 
r
1 2.3
will be negative,
whereas if they are of the opposite signs, it will be positive. An example will make this
point clear. Let 
Y
=
crop yield, 
X
2
=
rainfall, and 
X
3
=
temperature. Assume 
r
1 2
=
0, that
is, no association between crop yield and rainfall. Assume further that 
r
1 3
is positive and
r
2 3
is negative. Then, as Eq. (7.11.1) shows, 
r
1 2.3
will be positive; that is, holding tempera-
ture constant, there is a positive association between yield and rainfall. This seemingly
paradoxical result, however, is not surprising. Since temperature 
X
3
affects both yield 
Y
and
rainfall 
X
2
, in order to find out the net relationship between crop yield and rainfall, we need
to remove the influence of the “nuisance” variable temperature. This example shows how
one might be misled by the simple coefficient of correlation.
3. The terms 
r
1 2.3
and 
r
1 2
(and similar comparisons) need not have the same sign.
4. In the two-variable case we have seen that 
r
2
lies between 0 and 1. The same property
holds true of the squared partial correlation coefficients. Using this fact, the reader should
verify that one can obtain the following expression from Eq. (7.11.1):
0
≤
r
2
1 2
+
r
2
1 3
+
r
2
2 3
−
2
r
1 2
r
1 3
r
2 3
≤
1
(7.11.4)
19
Most computer programs for multiple regression analysis routinely compute the simple correlation
coefficients; hence the partial correlation coefficients can be readily computed.
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Chapter 7
Multiple Regression Analysis: The Problem of Estimation
215
which gives the interrelationships among the three zero-order correlation coefficients. Sim-
ilar expressions can be derived from Eqs. (7.11.2) and (7.11.3).
5. Suppose that 
r
1 3
=
r
2 3
=
0. Does this mean that 
r
1 2
is also zero? The answer is
obvious from Eq. (7.11.4). The fact that 
Y
and 
X
3
and 
X
2
and 
X
3
are uncorrelated does not
mean that 
Y
and 
X
2
are uncorrelated.
In passing, note that the expression 
r
2
1 2
.
3
may be called the