The Sorted Effects Method: Discovering Heterogeneous Effects Beyond Their Averages
Victor Chernozhukov, Iván Fernández-Val, Ye Luo
Econometrica · 2018 · Volume 86 · Issue 6
Abstract
The partial (ceteris paribus) effects of interest in nonlinear and interactive linear models are heterogeneous as they can vary dramatically with the underlying observed or unobserved covariates. Despite the apparent importance of heterogeneity, a common practice in modern empirical work is to largelyignore itby reporting average partial effects (or, at best, average effects for some groups). While average effects provide very convenient scalar summaries of typical effects, by definition they fail to reflect the entire variety of the heterogeneous effects. In order to discover these effects much more fully, we propose to estimate and reportsorted effects—a collection of estimated partial effects sorted in increasing order and indexed by percentiles. By construction, the sorted effect curvescompletely representand help visualize the range of the heterogeneous effects in one plot. They are as convenient and easy to report in practice as the conventional average partial effects. They also serve as a basis forclassification analysis, where we divide the observational units into most or least affected groups and summarize their characteristics. We provide a quantification of uncertainty (standard errors and confidence bands) for the estimated sorted effects and related classification analysis, and provide confidence sets for the most and least affected groups. The derived statistical results rely on establishing key, new mathematical results on Hadamard differentiability of a multivariate sorting operator and a related classification operator, which are of independent interest. We apply thesorted effects methodand classification analysis to demonstrate several striking patterns in the gender wage gap. We find that this gap is particularly strong for married women, ranging fromtobetween theandpercentiles, as a function of observed and unobserved characteristics; while the gap for never married women ranges fromto. The most adversely affected women tend to be married, do not have college degrees, work in sales, and have high levels of potential experience.
Keywords
Sorting, partial effect, marginal effect, sorted effect, classification analysis, nonlinear model, functional analysis, differential geometry, gender wage gap
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