1. Binary Response
Lecture 22
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2. Today’s plan Three models: Linear probability model Probit model
Logit model
L22.xls provides an example of a linear probability model and a logit model
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3. Discrete choice variable
Defining variables:
Yi = 1 if individual : Yi = 0 if individual:
The discrete choice variable Yi is a function of individual characteristics: Yi = a + bXi + ei
Takes BART
Buys a car
Joins a union
Does not take BART
Does not buy a car
Does not join a union
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4. Graphical representation
X = years of labor market experience
Y = 1 [if person joins union]
= 0 [if person doesn’t join union] X Y 1
Observed data with OLS
regression line
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5. Linear probability model
The OLS regression line in the previous slide is called the linear probability model
predicting the probability that an individual will join a union given their years of labor market experience
Using the linear probability model, we estimate the equation:
using we can predict the probability
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6. Linear probability model (2)
Problems with the linear probability model
1) Predicted probabilities don’t necessarily lie within the 0 to 1 range
2) We get a very specific form of heteroskedasticity
errors for this model are
note: values are along the continuous OLS line, but Yi values jump between 0 and 1 - this creates large variation in errors
3) Errors are non-normal
We can use the linear probability model as a first guess
can be used for start values in a maximum likelihood problem
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7. McFadden’s Contribution
Suggestion: curve that runs strictly between 0 and 1 and tails off at the boundaries like so: Y 1
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8. McFadden’s Contribution
Recall the probability distribution function and cumulative distribution function for a standard normal: 1
PDF
CDF
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9. Probit model
For the standard normal, we have the probit model using the PDF
The density function for the normal is:
where Z = a + bX
For the probit model, we want to find
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10. Probit model (2)
The probit model imposes the distributional form of the CDF in order to estimate a and b
The values have to be estimated as part of the maximum likelihood procedure
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11. Logit model The logit model uses the logistic distribution Density:
Cumulative: 1
Standard normal F(Z)
Logistic G(Z)
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12. Maximum likelihood
Alternative estimation that assumes you know the form of the population
Using maximum likelihood, we will be specifying the model as part of the distribution
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13. Maximum likelihood (2)
For example: Bernoulli distribution where: (with a parameter )
We have an outcome
The probability expression is:
We pick a sample of Y1….Yn
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14. Maximum likelihood (3)
Probability of getting observed Yi is based on the form we’ve assumed:
If we multiply across the observed sample:
Given we think that an outcome of one occurs r times:
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15. Maximum likelihood (3) If we take logs, we get
This is the log-likelihood
We can differentiate this and obtain a solution for
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16. L(a, b) = Si [Yi log(Gi) + (1 - Yi) log(1 - Gi)]
Maximum likelihood (4)
In a more complex example, the logit model gives
Instead of looking for estimates of we are looking for estimates of a and b
Think of G(Zi) as :
we get a log-likelihood
L(a, b) = Si [Yi log(Gi) + (1 - Yi) log(1 - Gi)]
solve for a and b
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17. Example
Data on union membership and years of labor market experience (L22.xls)
To build the maximum likelihood form, we can think of:
intercept: a
coefficient on experience : b
There are three columns
Predicted value Z
Estimated probability
Estimated likelihood as given by the model
The Solver from the Tools menu calculates estimates of a and b
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18. Example (2) How the solver works: Defining a and b using start values
Choose start values of a and b equal to zero
Define our model: Z = a + bX
Define the predictive possibilities:
Define the log-likelihood and sum it
Can use Solver to change the values on a and b
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19. Comparing parameters How do we compare parameters across these models?
The linear probability form is: Y = a + bX
where
Recall the graphs associated with each model
Consequently
This is the same for the probit and logit forms
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20. L22.xls example Predicting the linear probability model:
If we wanted to predict the probability given 20 years of experience, we’d have:
For the logit form:
use logit distribution:
logit estimated equation is:
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21. L22.xls example (2) At 20 years of experience:
Thus the slope at 20 years of experience is:
0.234 x 0.06 = 0.014
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