Extreme Value Distribution and Logit Regression

Logit regression has been used a lot in modeling selection problems. To review its relationship with the extreme value distribution, it is better to derive the analytical form of Logit regression from its error assumption of type I extreme value distribution. Actually, this is a homework exercise from my Ph.D. econometrics course. It is a great chance to refresh my mind by doing this exercise again.

Generalized Extreme Value Distribution

Reference Link: Generalized extreme value distribution - Wikipedia

Notions:

• Location parameter and scale parameter

• Standardized variable

• Cumulative distribution function

• Probability density function

Logit Function

Indirect Utility

Suppose the indirect utility of choosing alternative is specified as

where is a vector of features of alternative and follows the GEV.

Derivation

Alternative is selected if for any . Hence, the probability of choosing is

Note that

where is a constant.

The probability of choosing is then written as

where , , and .

Logit Regression

Reference Link: Logistic regression - Wikipedia

Let one of the alternatives be an opt-out and normalize the opt-out be zero. The logit form is given by

where is a set of mutually exclusive alternatives included in the analysis.