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Register a new userThe Sorted Effects Method: Discovering Heterogeneous Effects Beyond Their Averages
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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 largely ignore it by 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 report sorted effects -- a collection of estimated partial effects sorted in increasing order and indexed by percentiles. By construction the sorted effect curves completely represent and 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 for classification 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 the sorted effects method and classification analysis to demonstrate several striking patterns in the gender wage gap.
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We develop new semiparametric methods for estimating treatment effects. We focus on a setting where the outcome distributions may be thick tailed, where treatment effects are small, where sample sizes are large and where assignment is completely random. This setting is of particular interest in recent experimentation in tech companies. We propose using parametric models for the treatment effects, as opposed to parametric models for the full outcome distributions. This leads to semiparametric models for the outcome distributions. We derive the semiparametric efficiency bound for this setting, and propose efficient estimators. In the case with a constant treatment effect one of the proposed estimators has an interesting interpretation as a weighted average of quantile treatment effects, with the weights proportional to (minus) the second derivative of the log of the density of the potential outcomes. Our analysis also results in an extension of Hubers model and trimmed mean to include asymmetry and a simplified condition on linear combinations of order statistics, which may be of independent interest.
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