Chapter 8
Functional Forms of
Regression Model
The models we discussed are models that are linear in
parameters; variables Y and Xs do not necessarily have to be linear.
The price elasticity of demand～ the log-linear models
The rate of growth～ semilog model
? Functional forms of regression models which are linear in
parameters,but not necessarily linear in variables:
1,Log-linear of constant elasticity models,(Section 8.1)
2,Semilog models (Sections 8.4 and 8.5)
3,Reciprocal models (Section 8.6)
4,Polynomial regression models (Section 8.7)
For a regression model linear in explanatory variable(s),for a
unit change in the explanatory variable,the rate of change(i.e.,the
slope) of the dependent variable remains constant;
For regression models nonlinear in explanatory variable(s),the
slope does not remain constant.
8.1 How to Measure Elasticity:
The Log-Linear Model
?1,Model and its transformations:
Nonlinear model:
(8.1)
(8.2)
B1=lnA (8.3)
lnYi=B1+B2lnXi (8.4)
Double-log or log-linear model----linear model:
lnYi= B1+B2lnXi+ui (8.5)
letting and
then
(8.6)
2Bii AXY ?
i2i ln XBln Aln Y ??
i*i lnYY ? i*i lnYX ?
i*i21*i uXBBY ???
2,Estimation
（ 1） Under the CLRM assumptions,we can get OLS
estimators of the log-linear model,and they are BLUE.
（ 2） Slope coefficient B2 measures the elasticity of Y with
respect to X,that is,the percentage change in Y for a given
(small) percentage change in X.
How to compute the elasticity coefficient,E
let △ Y stand for a small change in Y and △ X for a
small change in X
= = = slope·
(8.7)
（ 3） Hypothesis Testing in log-Linear Models — the same as
linear models
Xin ch an g e %
Yin ch an g e % E ?
100XX
100YY
??
??
Y
X
X
Y ?
?
? ??????YX
8.2 Comparing Linear and Log-Linear Regression Models
——Question,which model is better?）
lnYi= B1+B2lnXi+ui (8.5)
Yi= B1+B2Xi+ui
1.Use the scattergram
plot the data,Y～ X,if they are nonlinear,then
plot the log of Y against the log of X.
－－ to find which model is the best estimate of the PRF.
Problem,this principle works only in the two-variable
regression models
2,Compare the two models on the basis of r2
Problem:
（ 1） To compare the r2 values of two models,the dependent
variable must be in the same form.
（ 2） High r2 value criterion,An r2 (=R2) can always be increased
by adding more explanatory variables to the model.
?3,Compare the two slope coefficients,B2
In the linear model
In the log-linear model
How to compare the elasticity of the two models?
In log-linear model,the elasticity（ which is the slope coefficient）
remains the same,that is,B2,no matter at what price the
elasticity is measured,So it is called constant elasticity model.
In linear model:
= = = slope·
The elasticity changes from point to point on the linear
curve because the ratio X/Y changes from point to point,In
practice,the elasticity coefficient for the linear model is often
computed by average elasticity.
Average elasticity = (8.9)
Xin ch an g e %
Yin ch an g e % E ? 100XX 100YY ?? ??
Y
X
X
Y ?
?
? ??????YX
Y
X?
?
?
X
Y
8.3 Multiple Log-Linear Regression Models
A three-variable log-linear model:
lnYi=B1+B2lnX2i+B3lnX3i+ui (8.10)
B2,B3,Partial elasticity coefficients.
B2,measures the elasticity of Y with respect to X2 holding
the influence of X3 constant; that is,it measures the
percentage change in Y for a percentage change in X2,
holding the influence of X3 constant.
B3,measures the (Partial) elasticity of Y with respect to X3,
holding the influence of X2 constant.
2,Instantaneous versus Compound Rate of Growth
b2=the estimate of B2=ln(1+r)
antilog (b2) = (1+r)
r = antilog(b 2) –1 (8.21)
r－－ the compound （ over a period of time） rate of growth
b2 in semilog model,the instantaneous（ at a point in time）
growth rate.
In practice,we generally use the instantaneous growth rate,
3,The Linear Trend Model
Yt=B1+B2t +ut (8.23)
time variable t,the trend variable.
If B2＞ 0,there is an upward in Y
If B2＜ 0,there is a downward in Y.
（ 1） Compare to linear trend model,the growth model is more
useful.
（ 2） We cannot compare r2 values of the two models because
the dependent variables in the two models are not the same
8.4 How to Measure the Growth Rate,The Semilog Model
1,The Semilog Model
● Nonlinear model:
Yt=Y0(1+r)t (8.13)
lnYt= lnY0+tln (1+r) (8.14)
Then lnYt=B1+B2t (8.17)
Semilog Model,growth model－ Linear model:
lnYt= B1+B2t + ut (8.18)
● Estimation
(1) Under CLRM,the semi-log model can be estimated by using
OLS method,and the OLS estimators are BLUE.
(2) In a semilog model,the slope coefficient measures the
proportional or relative change in Y for a given absolute
change in the explanatory variable
(3) Semilog models are called growth models because they are
routinely used to measure the growth rate of many variables.
8.5The Lin-log Model,When the Explanatory Variable is
Logarithmic
Yt=B1+B2lnX2t+ut (8.25)
the slope coefficient B2 measures:
B2= = （ 8.27）
where Δ Y and Δ X represent (small) changes in Y and X.
equivalently,
△ Y =B2 (8.28)
Xin ch a n g e r e la t iv e
in ch a n g e ab s o lu t e ?
XX
Y
?
?
X
X?
8.6 Reciprocal Models:
(8.29)
（ 1） average fixed cost (AFC) of production
（ 2） Engel expenditure curve
Y, expenditure on a commodity
X,the total income
（ 3） Phillips curve
Y,the percent rate of change of money wages
X,the unemployment rate in percent.
The slope coefficient in reciprocal and linear model is
different,a positive slope in the reciprocal model is analogous
to the negative slope in the linear model.
The dependent variable in the two models is the same,
so we can compare the two r2 values
i
i
i uXBBY ????
?
???
??? 1
21
Summary:
Model Meaning of B2
? 1.Linear model,Yi= B1+B2Xi+ui X→ Y
? 2.Double-logmodel → lnYi= B1+B2lnXi+ui %X→ %Y
Log-Linear model
Constant elasticity model,lnYi= B1+B2lnXi+ui
Yi= B1+B2Xi+ui
? 3.Semilog model Yt=Y0(1+r)t → lnYt= B1+B2t + ut t
→ %Y
Log-lin model
Growth model
r = antilog(b 2)–1,the compound （ over a period of time） rate of growth
b2 in semilog model,the instantaneous（ at a point in time） growth rate.
The Linear Trend Model
Yt=B1+B2t +ut t → %Y
lnYt= B1+B2t + ut
2Bii AXY ?
? 4.The Lin-log Model,Yt=B1+B2lnX2t+ut %X→ Y
? 5.Reciprocal Models,X→∞,
Y→ B1
? 6.Polynomial Models:
i
i
i uXBBY ????
???
?
??? 1
21
342321 iiii XBXBXBBY ????
Functional Forms of
Regression Model
The models we discussed are models that are linear in
parameters; variables Y and Xs do not necessarily have to be linear.
The price elasticity of demand～ the log-linear models
The rate of growth～ semilog model
? Functional forms of regression models which are linear in
parameters,but not necessarily linear in variables:
1,Log-linear of constant elasticity models,(Section 8.1)
2,Semilog models (Sections 8.4 and 8.5)
3,Reciprocal models (Section 8.6)
4,Polynomial regression models (Section 8.7)
For a regression model linear in explanatory variable(s),for a
unit change in the explanatory variable,the rate of change(i.e.,the
slope) of the dependent variable remains constant;
For regression models nonlinear in explanatory variable(s),the
slope does not remain constant.
8.1 How to Measure Elasticity:
The Log-Linear Model
?1,Model and its transformations:
Nonlinear model:
(8.1)
(8.2)
B1=lnA (8.3)
lnYi=B1+B2lnXi (8.4)
Double-log or log-linear model----linear model:
lnYi= B1+B2lnXi+ui (8.5)
letting and
then
(8.6)
2Bii AXY ?
i2i ln XBln Aln Y ??
i*i lnYY ? i*i lnYX ?
i*i21*i uXBBY ???
2,Estimation
（ 1） Under the CLRM assumptions,we can get OLS
estimators of the log-linear model,and they are BLUE.
（ 2） Slope coefficient B2 measures the elasticity of Y with
respect to X,that is,the percentage change in Y for a given
(small) percentage change in X.
How to compute the elasticity coefficient,E
let △ Y stand for a small change in Y and △ X for a
small change in X
= = = slope·
(8.7)
（ 3） Hypothesis Testing in log-Linear Models — the same as
linear models
Xin ch an g e %
Yin ch an g e % E ?
100XX
100YY
??
??
Y
X
X
Y ?
?
? ??????YX
8.2 Comparing Linear and Log-Linear Regression Models
——Question,which model is better?）
lnYi= B1+B2lnXi+ui (8.5)
Yi= B1+B2Xi+ui
1.Use the scattergram
plot the data,Y～ X,if they are nonlinear,then
plot the log of Y against the log of X.
－－ to find which model is the best estimate of the PRF.
Problem,this principle works only in the two-variable
regression models
2,Compare the two models on the basis of r2
Problem:
（ 1） To compare the r2 values of two models,the dependent
variable must be in the same form.
（ 2） High r2 value criterion,An r2 (=R2) can always be increased
by adding more explanatory variables to the model.
?3,Compare the two slope coefficients,B2
In the linear model
In the log-linear model
How to compare the elasticity of the two models?
In log-linear model,the elasticity（ which is the slope coefficient）
remains the same,that is,B2,no matter at what price the
elasticity is measured,So it is called constant elasticity model.
In linear model:
= = = slope·
The elasticity changes from point to point on the linear
curve because the ratio X/Y changes from point to point,In
practice,the elasticity coefficient for the linear model is often
computed by average elasticity.
Average elasticity = (8.9)
Xin ch an g e %
Yin ch an g e % E ? 100XX 100YY ?? ??
Y
X
X
Y ?
?
? ??????YX
Y
X?
?
?
X
Y
8.3 Multiple Log-Linear Regression Models
A three-variable log-linear model:
lnYi=B1+B2lnX2i+B3lnX3i+ui (8.10)
B2,B3,Partial elasticity coefficients.
B2,measures the elasticity of Y with respect to X2 holding
the influence of X3 constant; that is,it measures the
percentage change in Y for a percentage change in X2,
holding the influence of X3 constant.
B3,measures the (Partial) elasticity of Y with respect to X3,
holding the influence of X2 constant.
2,Instantaneous versus Compound Rate of Growth
b2=the estimate of B2=ln(1+r)
antilog (b2) = (1+r)
r = antilog(b 2) –1 (8.21)
r－－ the compound （ over a period of time） rate of growth
b2 in semilog model,the instantaneous（ at a point in time）
growth rate.
In practice,we generally use the instantaneous growth rate,
3,The Linear Trend Model
Yt=B1+B2t +ut (8.23)
time variable t,the trend variable.
If B2＞ 0,there is an upward in Y
If B2＜ 0,there is a downward in Y.
（ 1） Compare to linear trend model,the growth model is more
useful.
（ 2） We cannot compare r2 values of the two models because
the dependent variables in the two models are not the same
8.4 How to Measure the Growth Rate,The Semilog Model
1,The Semilog Model
● Nonlinear model:
Yt=Y0(1+r)t (8.13)
lnYt= lnY0+tln (1+r) (8.14)
Then lnYt=B1+B2t (8.17)
Semilog Model,growth model－ Linear model:
lnYt= B1+B2t + ut (8.18)
● Estimation
(1) Under CLRM,the semi-log model can be estimated by using
OLS method,and the OLS estimators are BLUE.
(2) In a semilog model,the slope coefficient measures the
proportional or relative change in Y for a given absolute
change in the explanatory variable
(3) Semilog models are called growth models because they are
routinely used to measure the growth rate of many variables.
8.5The Lin-log Model,When the Explanatory Variable is
Logarithmic
Yt=B1+B2lnX2t+ut (8.25)
the slope coefficient B2 measures:
B2= = （ 8.27）
where Δ Y and Δ X represent (small) changes in Y and X.
equivalently,
△ Y =B2 (8.28)
Xin ch a n g e r e la t iv e
in ch a n g e ab s o lu t e ?
XX
Y
?
?
X
X?
8.6 Reciprocal Models:
(8.29)
（ 1） average fixed cost (AFC) of production
（ 2） Engel expenditure curve
Y, expenditure on a commodity
X,the total income
（ 3） Phillips curve
Y,the percent rate of change of money wages
X,the unemployment rate in percent.
The slope coefficient in reciprocal and linear model is
different,a positive slope in the reciprocal model is analogous
to the negative slope in the linear model.
The dependent variable in the two models is the same,
so we can compare the two r2 values
i
i
i uXBBY ????
?
???
??? 1
21
Summary:
Model Meaning of B2
? 1.Linear model,Yi= B1+B2Xi+ui X→ Y
? 2.Double-logmodel → lnYi= B1+B2lnXi+ui %X→ %Y
Log-Linear model
Constant elasticity model,lnYi= B1+B2lnXi+ui
Yi= B1+B2Xi+ui
? 3.Semilog model Yt=Y0(1+r)t → lnYt= B1+B2t + ut t
→ %Y
Log-lin model
Growth model
r = antilog(b 2)–1,the compound （ over a period of time） rate of growth
b2 in semilog model,the instantaneous（ at a point in time） growth rate.
The Linear Trend Model
Yt=B1+B2t +ut t → %Y
lnYt= B1+B2t + ut
2Bii AXY ?
? 4.The Lin-log Model,Yt=B1+B2lnX2t+ut %X→ Y
? 5.Reciprocal Models,X→∞,
Y→ B1
? 6.Polynomial Models:
i
i
i uXBBY ????
???
?
??? 1
21
342321 iiii XBXBXBBY ????
