Logit Models

Logit Models
An Introduction

Prof. David Bernstein
James Madison University

Computer Science Department

bernstdh@jmu.edu

Motivation

Review:
- Regression analysis can be used to determine the dependence of one variable (called the endogenous or dependent variable) on other variables (called the exogenous or independent or explanatory variables)
- Linear regression models are linear in parameters, assume the dependent variable is continuous, and the errors are identically and normally distributed
Our Interest:
- Situations in which the dependent variable is not continuous (e.g., 0-1)

A 0-1 Example

The Data

A 0-1 Example (cont.)

Using Linear Regression

Implications of the 0-1 Example for Linear Regression

Improper Probabilities:
- The predicted probabilities can be outside of \([0, 1]\)
Heteroscedasticity:
- The variance of the dependent variable is different for different values of the independent variable
Non-Normality:
- The error terms are not normally distributed

Toward an Alternative Approach

Model the Probabilities:
- \(\frac{p_i}{1-p_i} = e^{\alpha + \beta X_i + \epsilon_i}\)
Interpretation of the Results:
- \(e^{\beta}\) is the effect of the independent variable on the odds ratio (i.e., the probability of the event divided by the probability of the non-event)

The 0-1 Example Revisited

Using Logistic Regression (i.e., A Binary Logit Model)