1. Ulf H. Olsson Professor of Statistics
GRA 6020 Multivariate Statistics; The Linear Probability model and The Logit Model (Probit)
Ulf H. Olsson
Professor of Statistics

2. Binary Response Models
The Goal is to estimate the parameters
Ulf H. Olsson

3. The Linear Probability Model
Ulf H. Olsson

4. The Linear Probability Model
Number of problems
The predicted value can be outside the interval (0,1)
The error term is not normally distributed
=> Heteroscedasticity =>Non-efficient estimates
T-test is not reliable
Ulf H. Olsson

5. The Logit Model
The Logistic Function
Ulf H. Olsson

6. The Probit Model
Ulf H. Olsson

7. The Logistic Curve G (The Cumulative Normal Distribution)
Ulf H. Olsson

8. The Logit Model
Ulf H. Olsson

9. Logit Model for Pi
Ulf H. Olsson

10. The Logit Model Non-linear => Non-linear Estimation =>ML
Comparing estimates of the linear probability model and the logit model ?
Amemiya (1981) proposes:
Multiply the logit estimates with 0.25 and further adding 0.5 to the constant term.
Model can be tested, but R-sq. does not work. Some pseudo R.sq. have been proposed.
Ulf H. Olsson

11. The Logit Model (example)
Dependent variable: emp=1 if a person has a job, emp=0 if a person is unemployed
Independent variables: (x1) edu = yrs. at a university; (x2) score= score on a dancing contest.
Estimate a model to predict the probability that a person has a job, given yrs. at a university and score at the dancing contest. (data see SPSS-file:Biomgra1.sav)
Ulf H. Olsson

12. The Latent Variable Model
Ulf H. Olsson

13. The Latent Variable Model
Ulf H. Olsson

14. Binary Response Models
The magnitude of each effect is not especially useful since y* rarely has a well-defined unit of measurement.
But, it is possible to find the partial effects on the probabilities by partial derivatives.
We are interested in significance and directions (positive or negative)
To find the partial effects of roughly continuous variables on the response probability:
Ulf H. Olsson

15. Binary Response Models
The partial effecs will always have the same sign as
Ulf H. Olsson