# On the representation of the nested logit model

## TL;DR

This paper offers a simple proof of a longstanding conjecture on the nested logit model's representation, and derives a closed-form formula for correlation in Gumbel copulas, enhancing theoretical understanding.

## Contribution

It provides a concise proof of the nested logit model's representation and introduces a closed-form correlation formula for Gumbel copulas, simplifying existing approaches.

## Key findings

- A two-line proof of the nested logit model's representation.
- A closed-form formula for correlation between Fréchet variables with Gumbel copula.
- Enhanced theoretical clarity and simplicity in nested logit modeling.

## Abstract

We give a two-line proof of a long-standing conjecture of Ben-Akiva and Lerman (1985) regarding the random utility representation of the nested logit model, thus providing a renewed and straightforward textbook treatment of that model. As an application, we provide a closed-form formula for the correlation between two Fr\'echet random variables coupled by a Gumbel copula.