LECTURE 6
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Introduction to Econometrics
Nonlinear specifications and dummy  variables
Hieu Nguyen
Fall semester, 2024

NONLINEAR SPECIFICATION
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e We will discuss different specifications nonlinear in  dependent and independent variables and
their  interpretation
e We will define the notion of a dummy variable and we will  show its different uses in linear
regression models

NONLINEAR SPECIFICATION
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e  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!
e  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?

LINEAR FORM
y = β0 + β1x1 + β2x2 + ε
e Assumes that the effect of the explanatory variable on the  dependent variable is constant:
∂y
∂xk  = βk k = 1, 2
e Interpretation: if xk increases by 1 unit (in which xk is  measured), then y will change by βk
units (in which y is  measured)
e Linear form is used as default functional form until strong  evidence that it is inappropriate is
found
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LOG-LOG FORM
ln y = β0 + β1 ln x1 + β2 ln x2 + ε
e Assumes that the elasticity of the dependent variable with  respect to the explanatory variable
is constant:
∂ ln y ∂y/y
∂ ln xk = ∂xk/xk = βk
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k = 1, 2
e Interpretation: if xk increases by 1 percent, then y will  change by βk percents
e Before using a double-log model, make sure that there are  no negative or zero observations in
the data set

EXAMPLE
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e Estimating the production function of Indian sugar  industry:
ˆ
ln Q = 2.70 + 0
.
(0.14)       (0.17)
.59 ln L + 0.33 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’.

LOG-LINEAR FORMS
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e 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)
e 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)

EXAMPLES OF LOG LINEAR FORMS
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e Estimating demand for chicken meat:
Y . . . annual chicken consumption (kg.)
PC . . . price of chicken
PB . . . price of beef
YD . . . annual disposable income
e Interpretation: An increase in the annual disposable income by  1% increases chicken consumption
by 0.12 kg per year, ceteris  paribus.

EXAMPLES OF LOG LINEAR FORMS
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e Estimating the influence of education and experience on  wages:
wage  educ  exper
. . . annual wage (USD)
. . . years of education
. . . years of experience
e 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.

POLYNOMIAL FORM
1
y = β0 + β1x1 + β2x2 + ε
e To determine the effect of x1 on y, we need to calculate the  derivative:
∂y
∂x1 = β1 + 2 · β2 · x1
e Clearly, the effect of x1 on y is not constant, but changes  with the level of x1
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e We might also have higher order polynomials, e.g.:
y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ε
1 1 1

EXAMPLE OF POLYNOMIAL FORM
e The impact of the number of hours of studying on the  grade from Introductory Econometrics:
e To determine the effect of hours on grade, calculate the  derivative:
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Decreasing returns to hours of studying: more hours  implies higher grade, but the positive effect
of additional  hour of studying decreases with more hours

CHOICE OF CORRECT FUNCTIONAL FORM
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e 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
e Ideally: the specification is given by underlying theory of  the equation
e In reality: underlying theory does not give precise  functional form
e In most cases, either linear form is adequate, or common  sense will point out an easy choice
from among the  alternatives

CHOICE OF CORRECT FUNCTIONAL FORM
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e Nonlinearity of independent  variables
often approximated by polynomial form
missing higher powers of a variable can be detected as  omitted variables (see next lecture)
e 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

DUMMY VARIABLES
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e Dummy variable - takes on the values of 0 or 1, depending  on a qualitative attribute
e Examples of dummy variables:

INTERCEPT DUMMY
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e Dummy variable included in a regression alone (not  interacted with other variables) is an
intercept dummy
e It changes the intercept for the subset of data defined by a  dummy variable condition:
yi = β0 + β1Di + β2xi + εi
where
e We  have
yi =  (β0 + β1) + β2xi + εi    if Di = 1
yi = β0 + β2xi + εi    if Di = 0

INTERCEPT DUMMY
X
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β0+β1
β0
Di=1
Slope = β2
Di=0
Slope = β2

EXAMPLE
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Estimating the determinants of wages:
Interpretation of the dummy variable M: men earn on  average $2.156 per hour more than women,
ceteris paribus

SLOPE DUMMY
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e If a dummy variable is interacted with another variable (x),  it is a slope dummy.
e It changes the relationship between x and y for a subset of  data defined by a dummy variable
condition:
e We      have
yi =  β0 + (β1 + β2)xi + εi    if Di = 1
yi =  β0 + β1xi + εi   if Di = 0

SLOPE DUMMY
X
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β0
Di=0
Slope = β1+β2
Di=1
Slope = β1

EXAMPLE
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e Estimating the determinants of wages:
e Interpretation: men gain on average 17 cents per hour  more than women for each additional year
of education,  ceteris paribus

SLOPE AND INTERCEPT DUMMIES
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e 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
i
D =
.
1 if the i-th observation meets a particular condition
0 otherwise
e We have
yi =  (β0 + β1) + (β2 + β3)xi + εi    if Di = 1
yi = β0 + β2xi + εi   if Di = 0

SLOPE AND INTERCEPT DUMMIES
X
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Di=0
Slope = β2+β3
Di=1
Slope = β2
β0+β1
β0

DUMMY VARIABLES - MULTIPLE CATEGORIES
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e What if a variable defines three or more qualitative  attributes?
e Example: level of education - elementary school, high  school, and college
e Define and use a set of dummy variables:
e 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)

SUMMARY
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e We discussed different nonlinear specifications of a  regression equation and their
interpretation
e We defined the concept of a dummy variable and we  showed its use
e Further readings:
Studenmund, Chapter 7
Wooldridge, Chapters 6 & 7