Nonlinear Binscatter Methods

TL;DR

This paper introduces advanced nonlinear binscatter methods for flexible, interpretable visualization and analysis of regression functions, including treatment effects and hypothesis testing, with theoretical guarantees and software implementations.

Contribution

It develops novel nonlinear binscatter techniques based on M-estimation, covering generalized linear, robust, and quantile regression, with optimal tuning, confidence bands, and formal tests.

Findings

New methods handle nonlinear, nonsmooth models.

Theoretical results include strong approximations for partitioning estimators.

Software packages are provided in Python, R, and Stata.

Abstract

Binned scatter plots are a powerful statistical tool for empirical work in the social, behavioral, and biomedical sciences. Available methods rely on a quantile-based partitioning estimator of the conditional mean regression function to primarily construct flexible yet interpretable visualization methods, but they can also be used to estimate treatment effects, assess uncertainty, and test substantive domain-specific hypotheses. This paper introduces novel binscatter methods based on nonlinear, possibly nonsmooth M-estimation methods, covering generalized linear, robust, and quantile regression models. We provide a host of theoretical results and practical tools for local constant estimation along with piecewise polynomial and spline approximations, including (i) optimal tuning parameter (number of bins) selection, (ii) confidence bands, and (iii) formal statistical tests regarding functional form or shape restrictions. Our main results rely on novel strong approximations for general partitioning-based estimators covering random, data-driven partitions, which may be of independent interest. We demonstrate our methods with an empirical application studying the relation between the percentage of individuals without health insurance and per capita income at the zip-code level. We provide general-purpose software packages implementing our methods in Python, R, and Stata.