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Dougherty: Introduction to Econometrics 3e
Study Guide
Chapter 4
Transformations of variables
Overview
This chapter shows how least squares regression analysis can be extended to fit nonlinear models.
Sometimes an
apparently nonlinear model can be linearized by taking logarithms.
and
are examples.
Because they can be fitted using linear regression analysis, they have proved very popular in the literature, there
usually being little to be gained from using more sophisticated specifications.
If you plot earnings on schooling,
using the
EAEF
data set, or expenditure on a given category of expenditure on total household expenditure, using
the CES data set, you will see that there is so much randomness in the data that one nonlinear specification is
likely to be just as good as another, and indeed a linear specification may not be obviously inferior.
Often the
real reason for preferring a nonlinear specification to a linear one is that it makes more sense theoretically.
The
chapter shows how the least squares principle can be applied when the model cannot be linearized, and it
concludes with a transformation for comparing the fits of linear and logarithmic specifications.
2
1
β
X
Y
=
X
e
Y
2
1
=
Further material
This section has three unrelated topics: (1) the use of interactive explanatory variables, and the interpretation of
their coefficients, (2) Ramsey’s RESET test for functional form, and (3) Box–Cox tests of functional
specification.
Interactive explanatory variables
Consider the model
u
X
X
X
X
Y
+
+
+
+
=
3
2
4
3
3
2
2
1
This is linear in parameters and it may be fitted using straightforward OLS, provided that the regression model
assumptions are satisfied.
However, the fact that it is nonlinear in variables has implications for the
interpretation of the parameters.
When multiple regression was introduced at the beginning of the previous
chapter, it was stated that the slope coefficients represented the separate, individual marginal effects of the
variables on
Y
, holding the other variables constant.
In this model, such an interpretation is not possible.
In
particular, it is not possible to interpret
2
as the effect of
X
2
on
Y
, holding
X
3
and
X
2
X
3
constant, because it is not
possible to hold both
X
3
and
X
2
X
3
constant if
X
2
changes.
To give a proper interpretation to the coefficients, we can rewrite the model as
( )
u
X
X
X
Y
+
+
+
+
=
3
3
2
3
4
2
1
The coefficient of
X
2
, (
2
+
4
X
3
), can now be interpreted as the marginal effect of
X
2
on
Y
, holding
X
3
constant.
This representation makes explicit the fact that the marginal effect of
X
2
depends on the value of
X
3
.
The
interpretation of
2
now becomes the marginal effect of
X
2
on
Y
,
when
X
3
is equal to zero
.
One may equally well rewrite the model as
( )
u
X
X
X
Y
+
+
+
+
=
3
2
4
3
2
2
1
From this it may be seen that the marginal effect of
X
3
on
Y
, holding
X
2
constant, is (
3
+
4
X
2
) and that
3
may
be interpreted as the marginal effect of
X
3
on
Y
, when
X
2
is equal to zero.
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 Spring '10
 öcal
 Econometrics

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