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

Ulf H. Olsson Statistical Models Statistical models are mathematical representations of population behavior; they describe salient features of the hypothesized process of interest among individuals in the target population. When you use a particular statistical model to analyze a particular set of data, you implicitly declare that this population model gave rise to these sample data.

Ulf H. Olsson Regression Analysis

Ulf H. Olsson Regression analysis OLS Regression parameter St.error T-value P-value Confidence interval R-sq R-sq.adj F-value The error term

Ulf H. Olsson Regression Analysis The error term has constant variance The error term follows a normal distribution with expectation equal to zero The x-variables are independent of the error term The x-variables are linearly independent The dependent variable is normally distributed

Ulf H. Olsson OLS example (affairs)

Ulf H. Olsson OLS example (affairs)

Ulf H. Olsson Kleins (OLS) CT = *PT *PT_ *WT, (1.303) (0.0912) (0.0906) (0.0399) R² = 0.981

Ulf H. Olsson 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 P i

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:Binomgra1.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