1. GRA 6020 Multivariate Statistics; The Logit Model Introduction to Multilevel Models Ulf H. Olsson Professor of Statistics

2. Ulf H. Olsson The Logit Model The Logistic Function e ~ 2.7 1828 1828

3. Ulf H. Olsson The Logit Model

4. Ulf H. Olsson Logit Model for P i

5. Ulf H. Olsson 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:

6. Ulf H. Olsson The Marginal effect

7. Ulf H. Olsson Simple interaction Model; X- continuous and D dichotomous

8. Ulf H. Olsson The Logit Model Non-linear => Non-linear Estimation =>ML Model can be tested, but R-sq. does not work. Some pseudo R.sq. have been proposed. Estimate a model to predict the probability

9. Ulf H. Olsson Introduction to the ML-estimator

10. Ulf H. Olsson Introduction to the ML-estimator The value of the parameters that maximizes this function are the maximum likelihood estimates Since the logarithm is a monotonic function, the values that maximizes L are the same as those that minimizes ln L

11. Ulf H. Olsson Goodness of Fit The lower the better (0 – perfect fit) Some Pseudo R-sq. The Wald test for the individual parameters

12. Multilevel Models More general than panel models

13. Ulf H. Olsson

14. Classifying Structures Simple hierarchy Cross classifications Multiple membership School Pupil NeighbourhoodSchool Pupil

15. Ulf H. Olsson

17. Simple panel model Studying nine individuals and five time periodes; i= 1,2,..,9. t= 1,..,5; Y: Wages; X:Education level