CHAPTER 15: QUALITATIVE RESPONSE REGRESSION MODELS

ü  The Nature of Qualitative Response Models

Three approaches to developing a probability model for a binary response variable:

1.      The linear probability model

2.      The logit model

3.      The probit model

ü  The Linear Probability Model

Y_i = β₁ + β₂X_i + u_i

Model looks like a typical linear regression model but because the regressand is binary or dichotomous, it is called a linear probability model (LPM).

ü  The Logit Model

P_i        = 1 + e²ⁱ = e²ⁱ

I – P_i      1 + e^-2i

Take the natural log,

L_i = ln       P_i          = 2_i

               1 – P_i

     = β₁ + β₂X_i

Features of the Logit Model:

1.      As P goes from 0 to 1, the logit 1 goes from -      to +         .

2.      Although L is linear in X, the probabilities themselves are not.

3.      Although we have included only a single X variable, or regressor, in the preceding model, one can add as many regressors as may be dictated by the underlying theory.

4.      If L, the logit, is positive, it means that when the value of the regressor(s) increases, the odds that the regressand equals 1 increases.

5.      The slope measures the change in L for a unit change in X.

6.      Whereas the LPM assumes that P_i is linearly related to X_i the logit model assumes that the log of the odds ratio is linearly related to X_i.

ü  Estimation of the Logit Model

L_i = ln       P_i          = β₁ + β₂X_i + ui

               1 – P_i

2 Types of Data

1.      Data of the individual, or micro, level

2.      Grouped or replicated data

ü  Probit Model/ Normit Model

The estimated model that emerges from the normal CDF.

ü  The Tobit Model

An extension of the probit model originally developed by James Tobin, a Nobel laureate economist.

ü  Further Topics in Qualitative Response Regression Models

·         Original Logit and Probit Models

·         Multinomial Logit and Probit Models

·         Duration Models

Popularly known as survival analysis or time-to-event data analysis.