Distributional Effects of a Continuous Treatment with an Application on Intergenerational Mobility

Abstract

This paper considers the effect of a continuous treatment on the entire distribution of outcomes after adjusting for differences in the distribution of covariates across different levels of the treatment. Our methodology encompasses dose‐response functions, counterfactual distributions, and ‘distributional policy effects’ depending on the assumptions invoked by the researcher. We propose a three‐step estimator that consists of (i) estimating the distribution of the outcome conditional on the treatment and other covariates using quantile regression; (ii) for each value of the treatment, averaging over a counterfactual distribution of the covariates holding the treatment fixed; (iii) converting the resulting counterfactual distribution into parameters of interest that are easy to interpret. We show that our estimators converge uniformly to Gaussian processes and that the empirical bootstrap can be used to conduct uniformly valid inference across a range of values of the treatment. We use our method to study intergenerational income mobility where we consider effects of parents’ income on features of their child's income distribution such as (i) the fraction of children with income below the poverty line; (ii) the variance of child's income; and (iii) the inter‐quantile range of child's income–all as a function of parents’ income.

Difference-in-Differences with a Continuous Treatment

“…as reasonable candidates for interpreting as average causal response parameters in the case where the treatment is continuous or multi-valued discrete, respectively. 8 The following proposition relates these expressions to ACRT and ACR. , (a-Cont)…”

“…where Equation (a-Cont) holds when the dose is continuous and Equation (a-MV) holds when the dose is discrete. 8 Notice that E[∆Yt|D = 0] is common across doses and cancels when taking differences across doses.…”

“…First, the interpretation issues that we point out regarding comparisons of AT T (d|d) at different values of d are related to existing points made on comparing "local" treatment effect parameters to each other (e.g., (Oreopoulos, 2006; Angrist and Fernandez-Val, 2013; Mogstad, Santos, and Torgovitsky, 2018) in the context of local average treatment effects or (Frölich, 2004;Fricke, 2017) in the context of difference-in-differences with multiple treatments). Second, our paper is broadly related to the literature on continuous treatments in cross-sectional setups, see, e.g., Hirano and Imbens (2004), Flores (2007), Flores, Flores-Lagunes, Gonzalez, and Neumann (2012), Galvao and Wang (2015), Kennedy, Ma, McHugh, and Small (2017), Su, Ura, and Zhang (2019), and Callaway and Huang (2020) in the context of an unconfounded treatment; and Florens, Heckman, Meghir, and Vytlacil (2008) in the context of an endogenous treatment. Our paper is also related to D'Haultfoeuille, Hoderlein, and Sasaki (2021), as they considered setups with repeated cross-sectional data.…”

Intergenerational education mobility: A machine learning perspective

“…The data here is sourced from the Panel Study of Income Dynamics (PSID) frequently studied by the literature on intergenerational mobility through a variety of channels such as income, education, and other factors. Following the processing procedure described in Callaway and Huang (2020) the base sample consists of 3,168 child-parent pairs after dropping observations with an extraordinary and unreasonably high level of educational attainment.…”

Dissecting distributions: Applications of quantile regression in empirical asset pricing and causal inference

“…See, for example,Kon (1984),Nelson (1991),Peiro (1994),Baixauli and Alvarez (2004),Jiang (2014), andHohberg et al (2020), to cite just a few.6 Melly (2005),Chernozhukov et al (2013), andCallaway andHuang (2020) among others apply quantile regression to causal inference in labor economics, whileFortin et al (2011) provide a review of distributional methods in economics Machado and Mata (2005). applies linear quantile regression to predict conditional distributions using small panel data, whileZhao (2013) applies linear quantile regression to estimate conditional distributions using time series data.…”