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

Part One
Single-Equation Regression Models
It may be noted that sometimes logarithmic transformation is used to reduce
heteroscedasticity as well as skewness. (See Chapter 11.) A common feature of many
economic variables, is that they are positively skewed (e.g., size distribution of firms or
distribution of income or wealth) and they are heteroscedastic. A logarithmic transforma-
tion of such variables reduces both skewness and heteroscedasticity. That is why labor
economists often use the logarithms of wages in the regression of wages on, say, schooling,
as measured by years of education.
6.7
Reciprocal Models
Models of the following type are known as 
reciprocal
models.
(6.7.1)
Although this model is nonlinear in the variable 
X
because it enters inversely or recipro-
cally, the model is linear in 
β
1
and 
β
2
and is therefore a linear regression model.
19
This model has these features: As 
X
increases indefinitely, the term 
β
2
(l
/
X
) appro-
aches zero (
note:
β
2
is a constant) and 
Y
approaches the limiting or 
asymptotic
value 
β
1
.
Y
i
=
β
1
+
β
2
1
X
i
+
u
i
Interpreted in the manner described earlier, the slope coefficient of about 257 means
that an increase in the total food expenditure of 1 percent, on average, leads to about
2.57 rupees increase in the expenditure on food of the 55 families included in the sample.
(
Note:
We have divided the estimated slope coefficient by 100.)
Before proceeding further, note that if you want to compute the elasticity coefficient
for the log–lin or lin–log models, you can do so from the definition of the elasticity coeffi-
cient given before, namely,
Elasticity
=
dY
dX
X
Y
As a matter of fact, once the functional form of a model is known, one can compute elas-
ticities by applying the preceding definition. (Table 6.6, given later, summarizes the elas-
ticity coefficients for the various models.)
300
100
400 500 600
Total expenditure (Rs.)
700 800 900
200
300
Expenditure on food (Rs.)
500
700
600
400
FIGURE 6.5
19
If we let 
X
∗
i
=
(1
/
X
i
), then Eq. (6.7.1) is linear in the parameters as well as the variables 
Y
i
and
X
∗
i
.
EXAMPLE 6.5
(
Continued
)
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Chapter 6
Extensions of the Two-Variable Linear Regression Model