GRA 6020 Multivariate Statistics The Linear Probability model

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

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

The Linear Probability Model Ulf H. Olsson

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

The Logit Model • The Logistic Function Ulf H. Olsson

The Probit Model Ulf H. Olsson

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

The Logit Model Ulf H. Olsson

Logit Model for Pi Ulf H. Olsson

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

The Logit Model (example) • Dependent variable: emp=1 if a person has a job, emp=0 if a person is unemployed • Independent variables: (x 1) edu = yrs. at a university; (x 2) 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: Binomgra 1. sav) Ulf H. Olsson

The Logit Model (example) Ulf H. Olsson

The Latent Variable Model Ulf H. Olsson

The Latent Variable Model 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: Ulf H. Olsson

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