Qualitative and Limited
Dependent Variable
Models
Ketevani Kapanadze
Brno, 2020
A Single Dummy Independent Variable
• Qualitative Information
• Examples: gender, race, industry, region, rating grade, …
• A way to incorporate qualitative information is to use dummy variables
• They may appear as the dependent or as independent variables
Dummy Variables
• Dummy variable - takes on the values of 0 or 1, depending on a
qualitative attribute;
• Examples of dummy variables are:
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 + ui
• We have: (on the board)
• Graphical Illustration
Intercept Dummy
Example
• Estimating the determinant of wages:
wagei = -3.89 + 2.156 Mi + 0.603 educi + 0.010 experi
(0.270) (0.051) (0.064)
• Interpretation of the dummy variable M: men earn on average $2.156
per hour more than women, ceteris paribus
• Estimated wage equation with intercept shift
• Does that mean that women are discriminated against?
• Not necessarily. Being female may be correlated with other productivity
characteristics that have not been controlled for.
Holding education, experience, and tenure
fixed, women earn 1.81$ less per hour than
men
A Single Dummy Independent Variable
• Comparing means of subpopulations described by dummies
• Discussion
• It can easily be tested whether difference in means is significant
• The wage difference between men and women is larger if no other things are controlled for;
i.e. part of the difference is due to differences in education, experience and tenure between
men and women
Not holding other factors constant, women earn 2.51$ per hour
less than men, i.e. the difference between the mean wage of
men and that of women is 2.51$.
A Single Dummy Independent Variable
• 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) + ui
• We have: (on the board)
Slope Dummy
Example
• Estimating the determinant of wages:
wagei = -2.620 + 0.450 educi + 0.17 Mi * educi + 0.010 experi
(0.054) (0.021) (0.065)
• Interpretation: men gain on average 17 cents per hour more than
women for each additional year of education, ceteris paribus
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:
• 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)
• Dummy variable trap This model cannot be estimated (perfect collinearity)
When using dummy variables, one category always has to be omitted:
Alternatively, one could omit the intercept:
The base category are men
The base category are women
Disadvantages:
1) More difficult to test for differences
between the parameters
2) R-squared formula only valid
if regression contains intercept
A Single Dummy Independent Variable
• Allowing for different slopes
• Interesting hypotheses
= intercept men
= intercept women
= slope men
= slope women
The return to education is the same
for men and women
The whole wage equation is the same
for men and women
Interaction term
Interactions Involving Dummy Variables
A Binary Dependent Variable:
The Linear Probability Model
• Linear regression when the dependent variable is binary
Linear probability model
(LPM)
If the dependent variable only takes on
the values 1 and 0
In the linear probability model, the coefficients describe
the effect of the explanatory variables on the probability
that y=1 (=the probability of „success“)
A Binary Dependent Variable: The Linear
Probability Model
• Example: Labor force participation of married women
=1 if in labor force, =0 otherwise Non-wife income (in thousand dollars per year)
If the number of kids under six years
increases by one, the proprobability
that the woman works falls by 26.2%
Not significant
A Binary Dependent Variable: The Linear
Probability Model
• Example: Female labor participation of married women (cont.)
Graph for nwifeinc=50, exper=5, age=30,
kindslt6=1, kidsge6=0
Negative predicted probability but no
problem because no woman in the sample
has educ < 5.
The maximum level of education in the
sample is educ=17. For the given case, this
leads to a predicted probability to be in the
labor force of about 50%.
A Binary Dependent Variable: The Linear
Probability Model
• Disadvantages of the linear probability model
• Predicted probabilities may be larger than one or smaller than zero
• Marginal probability effects sometimes logically impossible
• The linear probability model is necessarily heteroskedastic
• Heterosceasticity consistent standard errors need to be computed
• Advantanges of the linear probability model
• Easy estimation and interpretation
• Estimated effects and predictions often reasonably good in practice
Variance of Bernoulli
variable
A Binary Dependent Variable: The Linear
Probability Model
• Disadvantages of the LPM for binary dependent variables
• Predictions sometimes outside the unit interval
• Partial effects of explanatory variables are constant
• Nonlinear models for binary response
• Response probability is a nonlinear function of explanat. variables
Probability of a
„success“ given
explanatory
variables
A cumulative distribution function
. . The response
probability is thus a function of the
explanatory variables x.
Shorthand vector notation: the
vector of explanatory variables x
also contains the constant of
the model.
Logit and Probit Models for Binary Response
• Choices for the link function
Logit: (logistic function)
Probit: (standard normal distribution)
Logit and Probit Models for Binary Response
• Interpretation of coefficients in Logit and Probit models
• Marginal effects are nonlinear and depend on the level of X !
where
How does the probability for y=1 change if
explanatory variable xj changes by one unit?
Logit and Probit Models for Binary Response
Interpretation of marginal effects
• An increase in x increases (decreases) the probability that y=1 by the
marginal effect expressed as a percent.
• For dummy independent variables, the marginal effect is expressed in
comparison to the base category (x=0).
• For continuous independent variables, the marginal effect is
expressed for a one-unit change in x.
• We interpret both the sign and the magnitude of the marginal
effects.
• The probit and logit models produce almost identical marginal
effects.
• Goodness-of-fit measures for Logit and Probit models
• Percent correctly predicted
• Pseudo R-squared
Individual i‘s outcome is predicted as one if the
probability for this event is larger than .5, then
percentage of correctly predicted y=1 and y=0
is counted
Compare maximized log-likelihood of the
model with that of a model that only
contains a constant (and no explanatory
variables)
Logit and Probit Models for Binary Response
Discussion about binary outcome models
Choice between the logit and probit model
• The choice depends on the data generating process, which is unknown.
• The models produce almost identical results (different coefficients but
similar marginal effects).
• The choice is up to you.
Coding of the dependent variable
If we reverse the categories 0 and 1, the signs of the coefficients are
reversed (positive become negative and vice versa) but the magnitudes
are the same.
• Example: Married women‘s labor force participation
The coefficients are not comparable across models
Often, Logit estimated coefficients are 1.6 times Probit
estimated because .
The biggest difference between the LPM and Logit/Probit is
that partial effects are nonconstant in Logit/Probit:
(Larger decrease in probability for the first child)
Logit and Probit Models for Binary Response
Next Class – 10.04
In the Zoom at 1pm
Regression Analysis with Time Series Data