Ulf H. Olsson Professor of Statistics

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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

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: (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

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