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CHAPTER
9
FUNCTIONAL FORMS OF REGRESSION MODELS
QUESTIONS
9.1.
(
a
)
In a log-log model the dependent and all explanatory variables are in the
logarithmic form.
(
b
)
In the log-lin model the dependent variable is in the logarithmic form but
the explanatory variables are in the linear form.
(
c
)
In the lin-log model the dependent variable is in the linear form, whereas
the explanatory variables are in the logarithmic form.
(
d
)
It is the percentage change in the value of one variable for a (small)
percentage change in the value of another variable.
For the log-log model,
the slope coefficient of an explanatory variable gives a direct estimate of the
elasticity coefficient of the dependent variable with respect to the given
explanatory variable.
(
e
)
For the lin-lin model, elasticity = slope
Y
X
.
Therefore the elasticity
will depend on the values of
X
and
Y
.
But if we choose
X
and
Y
, the mean
values of
X
and
Y
, at which to measure the elasticity, the elasticity at mean
values will be: slope
Y
X
.
9.2.
The slope coefficient gives the rate of change in (mean)
Y
with respect to
X
,
whereas the elasticity coefficient is the percentage change in (mean)
Y
for a
(small) percentage change in
X
. The
relationship between two is: Elasticity
= slope
Y
X
.
For the log-linear, or log-log, model only, the elasticity and
slope coefficients are identical.
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Model 1
:
i
i
X
B
B
Y
ln
ln
2
1
+
=
:
If the scattergram of ln
Y
on ln
X
shows a
linear relationship, then this model is appropriate.
In practice, such models
are used to estimate the elasticities, for the
slope coefficient gives a direct
estimate of the elasticity coefficient.
Model 2
:
i
i
X
B
B
Y
ln
2
1
+
=
:
Such a model is generally used if the objective
of the study is to measure the rate of growth of
Y
with respect to
X
.
Often,
the
X
variable represents time in such models.
Model 3
:
i
i
X
B
B
Y
ln
2
1
+
=
:
If the objective is to find out the absolute
change in
Y
for a relative or percentage change in
X
, this model is often
chosen.
Model 4
:
)
(1/
2
1
i
i
X
B
B
Y
+
=
:
If the relationship between
Y
and
X
is
curvilinear, as in the case of the Phillips curve, this model generally gives a
good fit.
9.4.
(
a
)
Elasticity.
(
b
)
The absolute change in the mean value of the dependent variable for a
proportional change in the explanatory variable.
(
c
)
The growth rate.
(
d
)
Y
X
dX
dY
(
e
)
The percentage change in the quantity demanded for a (small) percentage
change in the price.
(
f
)
Greater than 1;
less than 1.
9.5.
(
a
)
True
.
X
d
Y
d
ln
ln
=
Y
X
dX
dY
, which, by definition, is elasticity.
(
b

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