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GRA 6020 Multivariate Statistics; The Linear Probability model and The Logit Model (Probit)

This resource delves into statistical models to analyze population behavior, including regression analysis, OLS examples, binary response models like linear probability and logit models. It covers parameter estimation and model comparisons.

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GRA 6020 Multivariate Statistics; The Linear Probability model and The Logit Model (Probit)

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

  2. 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

  3. Regression Analysis Ulf H. Olsson

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

  5. 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 Regression Analysis Ulf H. Olsson

  6. OLS example (affairs) Ulf H. Olsson

  7. OLS example (affairs) Ulf H. Olsson

  8. Kleins (OLS) • CT = 16.237+0.193*PT+0.0899*PT_1+0.796*WT, • (1.303) (0.0912) (0.0906) (0.0399) • 12.464 2.115 0.992 19.933 • R² = 0.981 Ulf H. Olsson

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

  10. The Linear Probability Model Ulf H. Olsson

  11. 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

  12. The Logit Model • The Logistic Function Ulf H. Olsson

  13. The Probit Model Ulf H. Olsson

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

  15. The Logit Model Ulf H. Olsson

  16. Logit Model for Pi Ulf H. Olsson

  17. 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

  18. 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

  19. The Logit Model (example) Ulf H. Olsson

  20. The Latent Variable Model Ulf H. Olsson

  21. The Latent Variable Model Ulf H. Olsson

  22. 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

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

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