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.
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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).
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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.”
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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.
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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
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 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 (
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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.
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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.
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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
nk
 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