Another Look at the Linear Probability Model and Nonlinear Index Models

TL;DR

This paper critically examines the use of linear probability models and nonlinear index models for binary response data, providing insights into their approximation accuracy, estimation properties, and practical implications.

Contribution

It clarifies when linear models approximate average partial effects and introduces a consistent nonlinear least squares estimation method for ramp models.

Findings

Linear projection parameters can match APEs in specific cases.

OLS may or may not approximate APEs depending on data.

Nonlinear least squares for ramp models is consistent and asymptotically normal.

Abstract

We reassess the use of linear models to approximate response probabilities of binary outcomes, focusing on average partial effects (APE). We confirm that linear projection parameters coincide with APEs in certain scenarios. Through simulations, we identify other cases where OLS does or does not approximate APEs and find that having large fraction of fitted values in [0, 1] is neither necessary nor sufficient. We also show nonlinear least squares estimation of the ramp model is consistent and asymptotically normal and is equivalent to using OLS on an iteratively trimmed sample to reduce bias. Our findings offer practical guidance for empirical research.