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CHAPTER 2
FUNCTIONAL FORMS
OF REGRESSION MODELS
Damodar Gujarati
Econometrics by Example
LOG-LINEAR, DOUBLE LOG, OR
CONSTANT ELASTICITY MODELS
The Cobb-Douglas Production Function:
B3
B2
Q i B1 L i K i
can be transformed into a linear model by taking natural
logs of both sides:
ln Q i ln B1 B 2 ln Li B 3 ln K i
The slope coefficients can be interpreted as elasticities.
If (B2 + B3) = 1, we have constant returns to scale.
If (B2 + B3) > 1, we have increasing returns to scale.
If (B2 + B3) < 1, we have decreasing returns to scale.
Damodar Gujarati
Econometrics by Example
LOG-LIN OR GROWTH MODELS
The rate of growth of real GDP:
R G D Pt R G D P1960 (1 r )
t
can be transformed into a linear model by taking natural
logs of both sides:
ln RG D Pt ln RG D P1960 t ln(1 r )
Letting B1 = ln RGDP1960 and B2 = ln (l+r), this can be
rewritten as:
ln RGDPt = B1 +B2 t
B2 is considered a semi-elasticity or an instantaneous growth rate.
The compound growth rate (r) is equal to (eB2 – 1).
Damodar Gujarati
Econometrics by Example
LIN-LOG MODELS
Lin-log models follow this general form:
Yi B1 B 2 ln X i u i
Note that B2 is the absolute change in Y responding to a
percentage (or relative) change in X
If X increases by 100%, predicted Y increases by B2 units
Used in Engel expenditure functions: “The total expenditure
that is devoted to food tends to increase in arithmetic
progression as total expenditure increases in geometric
proportion.”
Damodar Gujarati
Econometrics by Example
RECIPROCAL MODELS
Lin-log models follow this general form:
Yi B1 B 2 (
1
Xi
) ui
Note that:
1
As X increases indefinitely, the term B 2 ( ) approaches zero and Y approaches
Xi
the limiting or asymptotic value B1.
The slope is:
dY
dX
B2 (
1
X
2
)
Therefore, if B2 is positive, the slope is negative throughout, and if B2 is negative,
the slope is positive throughout.
Damodar Gujarati
Econometrics by Example
POLYNOMIAL REGRESSION MODELS
The following regression predicting GDP is an example of
a quadratic function, or more generally, a second-degree
polynomial in the variable time:
R G D Pt A1 A2 tim e A3 tim e u t
2
The slope is nonlinear and equal to:
dR G D P
tim e
Damodar Gujarati
Econometrics by Example
A2 2 A3 tim e
SUMMARY OF FUNCTIONAL FORMS
MODEL
FORM
SLOPE
(
dY
)
dX
B2
B2 (
Y =B1 + B2 X
Log-linear
lnY =B1 + ln X
B2 (
Log-lin
lnY =B1 + B2 X
B 2 (Y )
Lin-log
Y B1 B 2 ln X
Y B1 B 2 (
Damodar Gujarati
Econometrics by Example
1
X
)
dY
dX
Linear
Reciprocal
ELASTICITY
B2 (
B2 (
Y
)
X
1
)
X
1
X
)
2
X
.
Y
X
)
Y
B2
B2 ( X )
B2 (
B2 (
1
)
Y
1
XY
)
COMPARING ON BASIS OF R2
We cannot directly compare two models that have different
dependent variables.
We can transform the models as follows and compare RSS:
Step 1: Compute the geometric mean (GM) of the dependent
variable, call it Y*.
Step 2: Divide Yi by Y* to obtain: Yi Y~
Y
*
i
Step 3: Estimate the equation with lnYi as the dependent variable
using Y~i in lieu of Yi as the dependent variable (i.e., use ln Y~ as
i
the dependent variable).
Step 4: Estimate the equation with Yi as the dependent variable
using Y~i as the dependent variable instead of Yi.
Damodar Gujarati
Econometrics by Example
STANDARDIZED VARIABLES
We can avoid the problem of having variables
measured in different units by expressing them in
standardized form:
Yi
*
Yi Y
SY
_
; X
*
i
Xi X
SX
where_ SY and SX are the sample standard deviations
_
and Y and X are the sample means of Y and X,
respectively
The mean value of a standardized variable is always
zero and its standard deviation value is always 1.
Damodar Gujarati
Econometrics by Example
MEASURES OF GOODNESS OF FIT
R2: Measures the proportion of the variation in the regressand explained
by the regressors.
2
Adjusted R2: Denoted as R , it takes degrees of freedom into account:
_
R 1 (1 R )
2
2
n 1
nk
Akaike’s Information Criterion (AIC): Adds harsher penalty for
adding more variables to the model, defined as:
ln A IC
2k
n
ln(
R SS
)
n
The model with the lowest AIC is usually chosen.
Schwarz’s Information Criterion (SIC): Alternative to the AIC criterion,
expressed as:
k
R SS
ln SIC
ln n ln(
n
)
n
The penalty factor here is harsher than that of AIC.
Damodar Gujarati
Econometrics by Example