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

(6.6.11)
For descriptive purposes we call such a model a 
lin–log model.
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Chapter 6
Extensions of the Two-Variable Linear Regression Model
165
Let us interpret the slope coefficient 
β
2
.
17
As usual,
β
2
=
Change in
Y
Change in ln
X
=
Change in
Y
relative change in
X
The second step follows from the fact that 
a change in the log of a number is a relative
change.
Symbolically, we have
β
2
=

Y

X
/
X
(6.6.12)
where, as usual, 
 
denotes a small change. Equation 6.6.12 can be written, equivalently, as

Y
=
β
2
(

X
/
X
)
(6.6.13)
This equation states that the absolute change in 
Y
(
=

Y
) is equal to slope times the rela-
tive change in 
X.
If the latter is multiplied by 100, then Eq. (6.6.13) gives the absolute
change in 
Y
for a percentage change in 
X
. Thus, if (

X
/
X
) changes by 0.01 unit (or 1 per-
cent), the absolute change in 
Y 
is 0.01(
β
2
)
;
if in an application one finds that 
β
2
=
500, the
absolute change in 
Y
is (0.01)(500)
=
5.0. Therefore, when regression (6.6.11) is estimated
by OLS, do not forget to multiply the value of the estimated slope coefficient by 0.01, or,
what amounts to the same thing, divide it by 100.
If you do not keep this in mind, your in-
terpretation in an application will be highly misleading.
The practical question is: When is a lin–log model like Eq. (6.6.11) useful? An interest-
ing application has been found in the so-called