LECTURE 7
Introduction to Econometrics
Nonlinear specifications and dummy
variables
October 27, 2017
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ON THE PREVIOUS LECTURE
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ON THE PREVIOUS LECTURE
We showed how restrictions are incorporated in regression
models
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ON THE PREVIOUS LECTURE
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
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ON THE PREVIOUS LECTURE
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
We defined the notion of the overall significance of a
regression
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ON THE PREVIOUS LECTURE
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
We defined the notion of the overall significance of a
regression
We introduced the measure or the goodness of fit - R2
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ON THE PREVIOUS LECTURE
We showed how restrictions are incorporated in regression
models
We explained the idea of the F-test
We defined the notion of the overall significance of a
regression
We introduced the measure or the goodness of fit - R2
We showed how the F-test and the R2 are related
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ON TODAY’S LECTURE
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ON TODAY’S LECTURE
We will discuss different specifications nonlinear in
dependent and independent variables and their
interpretation
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ON TODAY’S LECTURE
We will discuss different specifications nonlinear in
dependent and independent variables and their
interpretation
We will define the notion of a dummy variable and we will
show its different uses in linear regression models
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NONLINEAR SPECIFICATION
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NONLINEAR SPECIFICATION
There is not always a linear relationship between
dependent variable and explanatory variables
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NONLINEAR SPECIFICATION
There is not always a linear relationship between
dependent variable and explanatory variables
The use of OLS requires that the equation be linear in
coefficients
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NONLINEAR SPECIFICATION
There is not always a linear relationship between
dependent variable and explanatory variables
The use of OLS requires that the equation be linear in
coefficients
However, there is a wide variety of functional forms that
are linear in coefficients while being nonlinear in variables!
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NONLINEAR SPECIFICATION
There is not always a linear relationship between
dependent variable and explanatory variables
The use of OLS requires that the equation be linear in
coefficients
However, there is a wide variety of functional forms that
are linear in coefficients while being nonlinear in variables!
We have to choose carefully the functional form of the
relationship between the dependent variable and each
explanatory variable
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NONLINEAR SPECIFICATION
There is not always a linear relationship between
dependent variable and explanatory variables
The use of OLS requires that the equation be linear in
coefficients
However, there is a wide variety of functional forms that
are linear in coefficients while being nonlinear in variables!
We have to choose carefully the functional form of the
relationship between the dependent variable and each
explanatory variable
The choice of a functional form should be based on the
underlying economic theory and/or intuition
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NONLINEAR SPECIFICATION
There is not always a linear relationship between
dependent variable and explanatory variables
The use of OLS requires that the equation be linear in
coefficients
However, there is a wide variety of functional forms that
are linear in coefficients while being nonlinear in variables!
We have to choose carefully the functional form of the
relationship between the dependent variable and each
explanatory variable
The choice of a functional form should be based on the
underlying economic theory and/or intuition
Do we expect a curve instead of a straight line? Does the
effect of a variable peak at some point and then start to
decline?
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LINEAR FORM
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LINEAR FORM
y = β0 + β1x1 + β2x2 + ε
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LINEAR FORM
y = β0 + β1x1 + β2x2 + ε
Assumes that the effect of the explanatory variable on the
dependent variable is constant:
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LINEAR FORM
y = β0 + β1x1 + β2x2 + ε
Assumes that the effect of the explanatory variable on the
dependent variable is constant:
∂y
∂xk
= βk k = 1, 2
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LINEAR FORM
y = β0 + β1x1 + β2x2 + ε
Assumes that the effect of the explanatory variable on the
dependent variable is constant:
∂y
∂xk
= βk k = 1, 2
Interpretation: if xk increases by 1 unit (in which xk is
measured), then y will change by βk units (in which y is
measured)
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LINEAR FORM
y = β0 + β1x1 + β2x2 + ε
Assumes that the effect of the explanatory variable on the
dependent variable is constant:
∂y
∂xk
= βk k = 1, 2
Interpretation: if xk increases by 1 unit (in which xk is
measured), then y will change by βk units (in which y is
measured)
Linear form is used as default functional form until strong
evidence that it is inappropriate is found
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DOUBLE-LOG FORM
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DOUBLE-LOG FORM
ln y = β0 + β1 ln x1 + β2 ln x2 + ε
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DOUBLE-LOG FORM
ln y = β0 + β1 ln x1 + β2 ln x2 + ε
Assumes that the elasticity of the dependent variable with
respect to the explanatory variable is constant:
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DOUBLE-LOG FORM
ln y = β0 + β1 ln x1 + β2 ln x2 + ε
Assumes that the elasticity of the dependent variable with
respect to the explanatory variable is constant:
∂ ln y
∂ ln xk
=
∂y/y
∂xk/xk
= βk k = 1, 2
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DOUBLE-LOG FORM
ln y = β0 + β1 ln x1 + β2 ln x2 + ε
Assumes that the elasticity of the dependent variable with
respect to the explanatory variable is constant:
∂ ln y
∂ ln xk
=
∂y/y
∂xk/xk
= βk k = 1, 2
Interpretation: if xk increases by 1 percent, then y will
change by βk percents
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DOUBLE-LOG FORM
ln y = β0 + β1 ln x1 + β2 ln x2 + ε
Assumes that the elasticity of the dependent variable with
respect to the explanatory variable is constant:
∂ ln y
∂ ln xk
=
∂y/y
∂xk/xk
= βk k = 1, 2
Interpretation: if xk increases by 1 percent, then y will
change by βk percents
Before using a double-log model, make sure that there are
no negative or zero observations in the data set
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EXAMPLE
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EXAMPLE
Estimating the production function of Indian sugar
industry:
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EXAMPLE
Estimating the production function of Indian sugar
industry:
ln Q = 2.70 + 0.59
0.14)
ln L + 0.33
0.17)
ln K
Q . . . output
L . . . labor
K . . . capital employed
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EXAMPLE
Estimating the production function of Indian sugar
industry:
ln Q = 2.70 + 0.59
0.14)
ln L + 0.33
0.17)
ln K
Q . . . output
L . . . labor
K . . . capital employed
Interpretation: if we increase the amount of labor by 1%, the
production of sugar will increase by 0.59%, ceteris paribus.
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EXAMPLE
Estimating the production function of Indian sugar
industry:
ln Q = 2.70 + 0.59
0.14)
ln L + 0.33
0.17)
ln K
Q . . . output
L . . . labor
K . . . capital employed
Interpretation: if we increase the amount of labor by 1%, the
production of sugar will increase by 0.59%, ceteris paribus.
Ceteris paribus is a Latin phrase meaning ’other things
being equal’.
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SEMILOG FORMS
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SEMILOG FORMS
Linear-log form:
y = β0 + β1 ln x1 + β2 ln x2 + ε
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SEMILOG FORMS
Linear-log form:
y = β0 + β1 ln x1 + β2 ln x2 + ε
Interpretation: if xk increases by 1 percent, then y will
change by (βk/100) units (k = 1, 2)
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SEMILOG FORMS
Linear-log form:
y = β0 + β1 ln x1 + β2 ln x2 + ε
Interpretation: if xk increases by 1 percent, then y will
change by (βk/100) units (k = 1, 2)
Log-linear form:
ln y = β0 + β1x1 + β2x2 + ε
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SEMILOG FORMS
Linear-log form:
y = β0 + β1 ln x1 + β2 ln x2 + ε
Interpretation: if xk increases by 1 percent, then y will
change by (βk/100) units (k = 1, 2)
Log-linear form:
ln y = β0 + β1x1 + β2x2 + ε
Interpretation: if xk increases by 1 unit, then y will change
by (βk ∗ 100) percent (k = 1, 2)
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EXAMPLES OF SEMILOG FORMS
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EXAMPLES OF SEMILOG FORMS
Estimating demand for chicken meat:
Y = −6.94 − 0.57
0.19)
PC + 0.25
0.11)
PB + 12.2
2.81)
ln YD
Y . . . annual chicken consumption (kg.)
PC . . . price of chicken
PB . . . price of beef
YD . . . annual disposable income
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EXAMPLES OF SEMILOG FORMS
Estimating demand for chicken meat:
Y = −6.94 − 0.57
0.19)
PC + 0.25
0.11)
PB + 12.2
2.81)
ln YD
Y . . . annual chicken consumption (kg.)
PC . . . price of chicken
PB . . . price of beef
YD . . . annual disposable income
Interpretation: An increase in the annual disposable income by
1% increases chicken consumption by 0.12 kg per year, ceteris
paribus.
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EXAMPLES OF SEMILOG FORMS
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EXAMPLES OF SEMILOG FORMS
Estimating the influence of education and experience on
wages:
ln wage = 0.217 + 0.098
0.008)
educ + 0.010
0.002)
exper
wage . . . annual wage (USD)
educ . . . years of education
exper . . . years of experience
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EXAMPLES OF SEMILOG FORMS
Estimating the influence of education and experience on
wages:
ln wage = 0.217 + 0.098
0.008)
educ + 0.010
0.002)
exper
wage . . . annual wage (USD)
educ . . . years of education
exper . . . years of experience
Interpretation: An increase in education by one year increases
annual wage by 9.8%, ceteris paribus. An increase in experience
by one year increases annual wage by 1%, ceteris paribus.
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POLYNOMIAL FORM
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POLYNOMIAL FORM
y = β0 + β1x1 + β2x2
1 + ε
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POLYNOMIAL FORM
y = β0 + β1x1 + β2x2
1 + ε
To determine the effect of x1 on y, we need to calculate the
derivative:
∂y
∂x1
= β1 + 2 · β2 · x1
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POLYNOMIAL FORM
y = β0 + β1x1 + β2x2
1 + ε
To determine the effect of x1 on y, we need to calculate the
derivative:
∂y
∂x1
= β1 + 2 · β2 · x1
Clearly, the effect of x1 on y is not constant, but changes
with the level of x1
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POLYNOMIAL FORM
y = β0 + β1x1 + β2x2
1 + ε
To determine the effect of x1 on y, we need to calculate the
derivative:
∂y
∂x1
= β1 + 2 · β2 · x1
Clearly, the effect of x1 on y is not constant, but changes
with the level of x1
We might also have higher order polynomials, e.g.:
y = β0 + β1x1 + β2x2
1 + β3x3
1 + β4x4
1 + ε
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EXAMPLE OF POLYNOMIAL FORM
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EXAMPLE OF POLYNOMIAL FORM
The impact of the number of hours of studying on the
grade from Introductory Econometrics:
grade = 30 + 1.4 · hours − 0.009 · hours2
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EXAMPLE OF POLYNOMIAL FORM
The impact of the number of hours of studying on the
grade from Introductory Econometrics:
grade = 30 + 1.4 · hours − 0.009 · hours2
To determine the effect of hours on grade, calculate the
derivative:
∂y
∂x
=
∂grade
∂hours
= 1.4 − 2 · 0.009 · hours = 1.4 − 0.018 · hours
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EXAMPLE OF POLYNOMIAL FORM
The impact of the number of hours of studying on the
grade from Introductory Econometrics:
grade = 30 + 1.4 · hours − 0.009 · hours2
To determine the effect of hours on grade, calculate the
derivative:
∂y
∂x
=
∂grade
∂hours
= 1.4 − 2 · 0.009 · hours = 1.4 − 0.018 · hours
Decreasing returns to hours of studying: more hours
implies higher grade, but the positive effect of additional
hour of studying decreases with more hours
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CHOICE OF CORRECT FUNCTIONAL FORM
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CHOICE OF CORRECT FUNCTIONAL FORM
The functional form has to be correctly specified in order
to avoid biased and inconsistent estimates
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CHOICE OF CORRECT FUNCTIONAL FORM
The functional form has to be correctly specified in order
to avoid biased and inconsistent estimates
Remember that one of the OLS assumptions is that the
model is correctly specified
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CHOICE OF CORRECT FUNCTIONAL FORM
The functional form has to be correctly specified in order
to avoid biased and inconsistent estimates
Remember that one of the OLS assumptions is that the
model is correctly specified
Ideally: the specification is given by underlying theory of
the equation
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CHOICE OF CORRECT FUNCTIONAL FORM
The functional form has to be correctly specified in order
to avoid biased and inconsistent estimates
Remember that one of the OLS assumptions is that the
model is correctly specified
Ideally: the specification is given by underlying theory of
the equation
In reality: underlying theory does not give precise
functional form
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CHOICE OF CORRECT FUNCTIONAL FORM
The functional form has to be correctly specified in order
to avoid biased and inconsistent estimates
Remember that one of the OLS assumptions is that the
model is correctly specified
Ideally: the specification is given by underlying theory of
the equation
In reality: underlying theory does not give precise
functional form
In most cases, either linear form is adequate, or common
sense will point out an easy choice from among the
alternatives
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
often approximated by polynomial form
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
often approximated by polynomial form
missing higher powers of a variable can be detected as
omitted variables (see next lecture)
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
often approximated by polynomial form
missing higher powers of a variable can be detected as
omitted variables (see next lecture)
Nonlinearity of dependent variable
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
often approximated by polynomial form
missing higher powers of a variable can be detected as
omitted variables (see next lecture)
Nonlinearity of dependent variable
harder to detect based on statistical fit of the regression
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
often approximated by polynomial form
missing higher powers of a variable can be detected as
omitted variables (see next lecture)
Nonlinearity of dependent variable
harder to detect based on statistical fit of the regression
R2
is incomparable across models where the y is
transformed
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CHOICE OF CORRECT FUNCTIONAL FORM
Nonlinearity of explanatory variables
often approximated by polynomial form
missing higher powers of a variable can be detected as
omitted variables (see next lecture)
Nonlinearity of dependent variable
harder to detect based on statistical fit of the regression
R2
is incomparable across models where the y is
transformed
dependent variables are often transformed to log-form in
order to make their distribution closer to the normal
distribution
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DUMMY VARIABLES
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DUMMY VARIABLES
Dummy variable - takes on the values of 0 or 1, depending
on a qualitative attribute
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DUMMY VARIABLES
Dummy variable - takes on the values of 0 or 1, depending
on a qualitative attribute
Examples of dummy variables:
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DUMMY VARIABLES
Dummy variable - takes on the values of 0 or 1, depending
on a qualitative attribute
Examples of dummy variables:
Male =
1 if the person is male
0 if the person is female
Weekend =
1 if the day is on weekend
0 if the day is a work day
NewStadium =
1 if the team plays on new stadium
0 if the team plays on old stadium
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INTERCEPT DUMMY
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INTERCEPT DUMMY
Dummy variable included in a regression alone (not
interacted with other variables) is an intercept dummy
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INTERCEPT DUMMY
Dummy variable included in a regression alone (not
interacted with other variables) is an intercept dummy
It changes the intercept for the subset of data defined by a
dummy variable condition:
yi = β0 + β1Di + β2xi + εi
where
Di =
1 if the i-th observation meets a particular condition
0 otherwise
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INTERCEPT DUMMY
Dummy variable included in a regression alone (not
interacted with other variables) is an intercept dummy
It changes the intercept for the subset of data defined by a
dummy variable condition:
yi = β0 + β1Di + β2xi + εi
where
Di =
1 if the i-th observation meets a particular condition
0 otherwise
We have
yi = (β0 + β1) + β2xi + εi if Di = 1
yi = β0 + β2xi + εi if Di = 0
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INTERCEPT DUMMY
X
Y
β0+β1
β0
Di=0
Di=1
Slope = β2
Slope = β2
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EXAMPLE
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EXAMPLE
Estimating the determinants of wages:
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EXAMPLE
Estimating the determinants of wages:
wagei = −3.890 + 2.156
0.270)
Mi + 0.603
0.051)
educi + 0.010
0.064)
experi
where Mi =
1 if the i-th person is male
0 if the i-th person is female
wage . . . average hourly wage in USD
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EXAMPLE
Estimating the determinants of wages:
wagei = −3.890 + 2.156
0.270)
Mi + 0.603
0.051)
educi + 0.010
0.064)
experi
where Mi =
1 if the i-th person is male
0 if the i-th person is female
wage . . . average hourly wage in USD
Interpretation of the dummy variable M: men earn on
average $2.156 per hour more than women, ceteris paribus
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SLOPE DUMMY
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SLOPE DUMMY
If a dummy variable is interacted with another variable (x),
it is a slope dummy.
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SLOPE DUMMY
If a dummy variable is interacted with another variable (x),
it is a slope dummy.
It changes the relationship between x and y for a subset of
data defined by a dummy variable condition:
yi = β0 + β1xi + β2(xi · Di) + εi
where
Di =
1 if the i-th observation meets a particular condition
0 otherwise
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SLOPE DUMMY
If a dummy variable is interacted with another variable (x),
it is a slope dummy.
It changes the relationship between x and y for a subset of
data defined by a dummy variable condition:
yi = β0 + β1xi + β2(xi · Di) + εi
where
Di =
1 if the i-th observation meets a particular condition
0 otherwise
We have
yi = β0 + (β1 + β2)xi + εi if Di = 1
yi = β0 + β1xi + εi if Di = 0
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SLOPE DUMMY
X
Y
Slope = β1+β2
β0
Di=0
Di=1
Slope = β1
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EXAMPLE
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EXAMPLE
Estimating the determinants of wages:
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EXAMPLE
Estimating the determinants of wages:
wagei = −2.620+ 0.450
0.054)
educi+ 0.170
0.021)
Mi·educi+ 0.010
0.065)
experi
where Mi =
1 if the i-th person is male
0 if the i-th person is female
wage . . . average hourly wage in USD
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EXAMPLE
Estimating the determinants of wages:
wagei = −2.620+ 0.450
0.054)
educi+ 0.170
0.021)
Mi·educi+ 0.010
0.065)
experi
where Mi =
1 if the i-th person is male
0 if the i-th person is female
wage . . . average hourly wage in USD
Interpretation: men gain on average 17 cents per hour
more than women for each additional year of education,
ceteris paribus
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SLOPE AND INTERCEPT DUMMIES
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SLOPE AND INTERCEPT DUMMIES
Allow both for different slope and intercept for two
subsets of data distinguished by a qualitative condition:
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SLOPE AND INTERCEPT DUMMIES
Allow both for different slope and intercept for two
subsets of data distinguished by a qualitative condition:
yi = β0 + β1Di + β2xi + β3(xi · Di) + εi
where
Di =
1 if the i-th observation meets a particular condition
0 otherwise
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SLOPE AND INTERCEPT DUMMIES
Allow both for different slope and intercept for two
subsets of data distinguished by a qualitative condition:
yi = β0 + β1Di + β2xi + β3(xi · Di) + εi
where
Di =
1 if the i-th observation meets a particular condition
0 otherwise
We have
yi = (β0 + β1) + (β2 + β3)xi + εi if Di = 1
yi = β0 + β2xi + εi if Di = 0
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SLOPE AND INTERCEPT DUMMIES
X
Y
Slope = β2+β3
β0
Di=0
Di=1
Slope = β2
β0+β1
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DUMMY VARIABLES - MULTIPLE CATEGORIES
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DUMMY VARIABLES - MULTIPLE CATEGORIES
What if a variable defines three or more qualitative
attributes?
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DUMMY VARIABLES - MULTIPLE CATEGORIES
What if a variable defines three or more qualitative
attributes?
Example: level of education - elementary school, high
school, and college
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DUMMY VARIABLES - MULTIPLE CATEGORIES
What if a variable defines three or more qualitative
attributes?
Example: level of education - elementary school, high
school, and college
Define and use a set of dummy variables:
H =
1 if high school
0 otherwise
and C =
1 if college
0 otherwise
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DUMMY VARIABLES - MULTIPLE CATEGORIES
What if a variable defines three or more qualitative
attributes?
Example: level of education - elementary school, high
school, and college
Define and use a set of dummy variables:
H =
1 if high school
0 otherwise
and C =
1 if college
0 otherwise
Should we include also a third dummy in the regression,
which is equal to 1 for people with elementary education?
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DUMMY VARIABLES - MULTIPLE CATEGORIES
What if a variable defines three or more qualitative
attributes?
Example: level of education - elementary school, high
school, and college
Define and use a set of dummy variables:
H =
1 if high school
0 otherwise
and C =
1 if college
0 otherwise
Should we include also a third dummy in the regression,
which is equal to 1 for people with elementary education?
No, unless we exclude the intercept!
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DUMMY VARIABLES - MULTIPLE CATEGORIES
What if a variable defines three or more qualitative
attributes?
Example: level of education - elementary school, high
school, and college
Define and use a set of dummy variables:
H =
1 if high school
0 otherwise
and C =
1 if college
0 otherwise
Should we include also a third dummy in the regression,
which is equal to 1 for people with elementary education?
No, unless we exclude the intercept!
Using full set of dummies leads to perfect multicollinearity
(dummy variable trap, see next lectures)
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SUMMARY
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SUMMARY
We discussed different nonlinear specifications of a
regression equation and their interpretation
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SUMMARY
We discussed different nonlinear specifications of a
regression equation and their interpretation
We defined the concept of a dummy variable and we
showed its use
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SUMMARY
We discussed different nonlinear specifications of a
regression equation and their interpretation
We defined the concept of a dummy variable and we
showed its use
Further readings:
Studenmund, Chapter 7
Wooldridge, Chapters 6 & 7
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