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, see e.g. Angrist and Pischke (2008)). While average eﬀects provide very convenient scalar summaries of typical eﬀects, by deﬁnition they fail to reﬂect the entire variety of the heterogenous 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 all of the heterogeneous eﬀects in one plot. They are as convenient and easy to report in practice as the conventional average partial eﬀects. We also provide a quantiﬁcation of uncertainty (standard errors and conﬁdence bands) for the estimated sorted eﬀects. We apply the sorted eﬀects method to demonstrate several striking patterns of gender-based discrimination in wages, and of race-based discrimination in mortgage lending.
Using diﬀerential geometry and functional delta methods, we establish that the estimated sorted eﬀects are consistent for the true sorted eﬀects, and derive asymptotic normality and bootstrap approximation results, enabling construction of pointwise conﬁdence bands (point-wise with respect to percentile indices). We also derive functional central limit theorems and bootstrap approximation results, enabling construction of simultaneous conﬁdence bands (simultaneous with respect to percentile indices). The derived statistical results in turn rely on establishing Hadamard diﬀerentiability of the multivariate sorting operator, a result of independent mathematical interest.