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Econometrics
Qualitative and
Limited Dependent
Variable Models
Anna Donina
Lecture 8
Introduction
So far, the dependent variable (Y) was continuous:
• Average wage
• Number of children
• Money growth rate
But what if it is a binary variable?
Y = 1, if person has college degree, 0 otherwise;
Y = 1, if person smokes, 0 otherwise;
The linear probability model (LPM)
Non-linear probability model
• Probit
• Logit
• Limited dependent variables (LDV)
• LDV are variables whose range is substantively
restricted
• Binary variables, e.g. employed/not employed
• Nonnegative variables, e.g. wages, prices,
interest rates
• Nonnegative variables with excess zeros, e.g.
labor supply
• Count variables, e.g. the number of arrests in a
year
• Censored variables, e.g. unemployment
durations
Limited Dependent Variable Models
• 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
The Linear Probability Model:
Heteroskedasticity
Yi = β0 + β1X1i + · · · + βk Xki + ui
The variance of a Bernoulli random variable:
Var (Y ) = Pr (Y = 1) × (1 − Pr (Y = 1))
We can use this to find the conditional variance of the error term
Solution: always use heteroskedasticity robust standard errors when
estimating a LPM
• 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
Logit and Probit Models for Binary Response
Pr (Y = 1) = Pr (Z ≤ −0.8) = Φ(−0.8) = 0.2119
Logit and Probit Models for Binary Response
Pr 𝑌 = 1 = Pr 𝑍 ≤ −0.8 =
1
1 + 𝑒0.8
= 0.31
Logit and Probit Models for Binary Response
• Interpretation of coefficients in Logit and Probit models
• Partial 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?
Discrete explanatory variables:
For example, explanatory variable xk increases by one unit.
Logit and Probit Models for Binary Response
• So far, we used OLS to estimate models
• Logit and Probit models are nonlinear in parameters:
• Hence, in this case the OLS cannot be used
• The method used to estimate Logit and Probit models is
Maximum Likelihood Estimation (MLE)
• The MLE are the values of parameters that best describe the full
distribution of the data
• The likelihood function is the joint probability distribution of
the data, treated as a function of the unknown coefficients
• The MLE are the values of the coefficients that maximize the
likelihood function
• MLE’s are the parameter values “most likely” to have
produced the data
Logit and Probit Models: Estimation
• Goodness-of-fit measures for Logit and Probit models
• Percent correctly predicted
• Pseudo R-squared
• Correlation based measures
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)
Look at correlation (or squared
correlation) between predictions or
predicted prob. and true values
Logit and Probit Models for Binary Response
• Reporting partial effects of explanatory variables
• The difficulty is that partial effects are not constant but
depend on x
• Partial effects at the average:
• Average partial effects:
The partial effect of explanatory variable xj is
considered for an „average individual“ (this is
problematic in the case of explanatory variables
such as gender)
The partial effect of explanatory variable xj is
computed for each individual in the sample and
then averaged across all sample members
(makes more sense)
Logit and Probit Models for Binary Response
• 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