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beta coefficients.
7
Incidentally, notice that (6.3.5) is a regression through the origin.
How do we interpret the beta coefficients? The interpretation is that if the (standardized)
regressor increases by one standard deviation, on average, the (standardized) regressand
increases by
β
∗
2
standard deviation units. Thus, unlike the traditional model in Eq. (6.3.3), we
measure the effect not in terms of the original units in which
Y
and
X
are expressed, but in
standard deviation units.
To show the difference between Eqs. (6.3.3) and (6.3.5), let us return to the GPDI and
GDP example discussed in the preceding section. The results of (6.2.21) discussed previ-
ously are reproduced here for convenience.
GPDI
t
= −
926.090
+
0.2535 GDP
t
(6.3.6)
se
=
(116.358)
(0.0129)
r
2
=
0.9648
where GPDI and GDP are measured in billions of dollars.
The results corresponding to Eq. (6.3.5) are as follows, where the starred variables are
standardized variables:
GPDI
∗
t
=
0
.
9822 GDP
∗
t
(6.3.7)
se
=
(0.0485)
We know how to interpret Eq. (6.3.6): If GDP goes up by a dollar, on average GPDI goes
up by about 25 cents. How about Eq. (6.3.7)? Here the interpretation is that if the (stan-
dardized) GDP increases by one standard deviation, on average, the (standardized) GPDI
increases by about 0.98 standard deviations.
What is the advantage of the standardized regression model over the traditional model?
The advantage becomes more apparent if there is more than one regressor, a topic we
will take up in Chapter 7. By standardizing all regressors, we put them on an equal basis
and therefore can compare them directly. If the coefficient of a standardized regressor is
larger than that of another standardized regressor appearing in that model, then the latter
contributes more relatively to the explanation of the regressand than the former. In other
words, we can use the beta coefficients as a measure of relative strength of the various
regressors. But more on this in the next two chapters.
Before we leave this topic, two points may be noted. First, for the standardized regres-
sion in Eq. (6.3.7) we have not given the 
r
2
value because this is a regression through
the origin for which the usual 
r
2
is not applicable, as pointed out in Section 6.1. Second,
there is an interesting relationship between the 
β
coefficients of the conventional model
and the beta coefficients. For the bivariate case, the relationship is as follows:
ˆ
β
∗
2
= ˆ
β
2
S
x
S
y
(6.3.8)
where 
S
x
=
the sample standard deviation of the 
X
regressor and 
S
y
=
the sample standard
deviation of the regressand. Therefore, we can crisscross between the 
β
and beta coefficients
6
Recall from Eq. (3.1.7) that Intercept
=
Mean value of the dependent variable
−
Slope 
×
Mean
value of the regressor. But for the standardized variables the mean values of the dependent variable
and the regressor are zero. Hence the intercept value is zero.
7
Do not confuse these beta coefficients with the beta coefficients of finance theory.
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Chapter 6
Extensions of the Two-Variable Linear Regression Model
159
if we know the (sample) standard deviation of the regressor and regressand. We will see in the
next chapter that this relationship holds true in the multiple regression also. It is left as an
exercise for the reader to verify Eq. (6.3.8) for our illustrative example.