Distributional Effects of a Continuous Treatment with an Application on Intergenerational Mobility
| Published date | 01 August 2020 |
| Date | 01 August 2020 |
| Author | Brantly Callaway,Weige Huang |
| DOI | http://doi.org/10.1111/obes.12355 |
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©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 4 (2020) 0305–9049
doi: 10.1111/obes.12355
Distributional Effects of a Continuous Treatment with
an Application on Intergenerational Mobility*
Brantly Callaway† and Weige Huang‡
†Assistant Professor, Department of Economics, University of Mississippi, University,
Mississippi, USA (e-mail: bmcallaw@olemiss.edu)
‡Assistant Professor, Wenlan School of Business, Zhongnan University of Economics and
Law, Wuhan, Hubei, China (e-mail: weige huang@zuel.edu.cn)
Abstract
This paper considers the effect of a continuous treatment on the entire distribution of out-
comes after adjusting for differences in the distribution of covariatesacross 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 dis-
tribution 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 esti-
mators 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.
I. Introduction
Researchers in economics often consider the effect of one variable (a treatment) on some
outcome. Most often, researchers consider the effect of the treatment on the mean of the
outcome, but there are also many cases in economics where understanding the effect of
the treatment on the entire distribution is important. For example, in labour economics,
JEL Classification numbers: C21, J62.
Corresponding Author: Weige Huang.
*Wethank Afrouz Azadikhah-Jahromi, David Kaplan, Martin Lopez-Daneri, Catherine Maclean, Irina Murtaza-
shvili, Pedro Sant’Anna, Tymon Sloczynski, Emmanuel Tsyawo and DougWebber, participants in TempleUniver-
sity’sEconometrics Reading Group, and seminar participants at the 2017 Midwest Econometrics Group for valuable
comments. This paper replaces ‘Intergenerational Income Mobility: Counterfactual Distributions with a Continu-
ous Treatment’. Code for our method is available in the R package ccfa available on CRAN and Github (https://
github.com/WeigeHuangEcon/ccfaor devtools::install github (“WeigeHuangEcon/ccfa”)).
Distributions and continuous treatments 809
understanding inequality or povertyis inherently related to distributions of outcomes (rather
than just the mean of outcomes). There have been many recent advances on developing
approaches to understand the effect of a binary treatment on the distribution of an outcome;1
however, understanding the effect of a continuous treatment on the entire distribution of
outcomes has received considerably less attention. In addition, considering continuous
treatments introduces some new identification and estimation challenges that do not occur
with a binary treatment.
Methods for dealing with a continuous treatment are likely to be of interest to empirical
researchers across a variety of areas. To give some examples of applications with a contin-
uous treatment, Imbens, Rubin and Sacerdote (2001) study the effect of unearned income
on labour market outcomes; Galvao and Wang (2015) consider an application on the effect
of mother’s weight gain during pregnancy on child’s birth weight; Jasova, Mendicino and
Supera (2018) study the effect of the term structure of debt on banks’ lending behaviour;
and local labour markets approaches common in labour economics often involvea continu-
ous treatment (e.g. Autor, Dorn and Hanson, 2013;Acemoglu and Restrepo, 2017; Collins
and Niemesh, 2019).
This paper proposes simple, but flexible, semi-parametric estimators of distributional
effects2of a continuous treatment while adjusting for differences in the distribution of co-
variates across differentlevels of treatment. Our procedure requires three steps. First, we es-
timate the distribution of the outcome conditional on the treatment and other
observed characteristics by inverting quantile regression (e.g. Koenker and Bassett Jr
(1978); Koenker (2005); Chernozhukov, Fernandez-Val, and Melly (2013)) estimates of
the conditional quantiles. The second step is to estimate the ‘counterfactual distribution’
which involves integrating over the first-step estimates while changing the distribution
of observable characteristics. This step obtains the entire distribution of the outcome as
a function of the continuous treatment after adjusting for differences in the distribution
of covariates across different levels of the treatment. Finally, parameters of interest such
as measures of the spread of the outcome and the probability of having a very low out-
come (e.g. child’s income being below the poverty line) are obtained as functions of the
counterfactual distribution.
The literature on continuous treatment effects includes Hirano and Imbens (2004),
Flores (2007); Florens et al. (2008); Flores et al. (2012); Galvao andWang (2015); Kennedy
et al. (2017). Within this literature, the only paper that we are aware of that looks at
1To give some examples, Firpo (2007); Donald and Hsu (2014) propose weighting estimators of distributional
effects under the assumption of unconfoundedness; Abadie,Angrist and Imbens (2002); Cher nozhukov and Hansen
(2005); Carneiro and Lee (2009); Frolich and Melly (2013) consider the case with an instrumental variable;and Athey
and Imbens (2006); Bonhomme and Sauder (2011); Chernozhukov et al. (2013); Callaway and Li (2019) identify
distributional effects when panel data are available.
2We use the term distributional effects broadly here. It encompasses counterfactual distributions, dose-response
functions, treatment effects, and distributional policy effects with a continuous treatment depending on the assump-
tions invoked by the researcher. Dose-response functions and treatment effects are defined in terms of potential
outcomes, and we consider these under the assumption of unconfoundedness. Counterfactual distributions are sim-
ilar, but do not rely on potential outcomes notation or the assumption of unconfoundedness. The counterfactual
distributions that we consider fix the return to characteristics (for a particular value of the treatment) but change the
distribution of the characteristics. Distributional policy effects compare (functionals of) the counterfactual distribu-
tion to (functionals of) the observed distribution of outcomes conditional on the treatment. They are closely related
to composition effects in the literature on decompositions.
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
810 Bulletin
distributional parameters is Galvao andWang (2015), which proposes a weightingestimator
of quantile dose–response functions with a continuous treatment under the assumption of
unconfoundedness. Our approach is different in that our estimators are based on first-step
quantile regression and do not require estimating conditional densities in the first step.3
There are trade-offs to using quantile regression relative to weighting estimators based
on conditional densities. Quantile regression imposes stronger parametric assumptions
than non-parametrically estimating conditional densities, though it is much simpler to
implement in practice; on the other hand, quantile regression is much more flexible (though
perhaps somewhat harder to implement) than assuming a fully parametric model for a
conditional density.4
We obtain the limiting processes for each of our parameters of interest and develop
inference procedures using results from the empirical processes literature (see, for exam-
ple, van der Vaart and Wellner, 1996; Kosorok, 2007) and results on first-step quantile
regression estimators (Chernozhukov, Fernandez-Val and Melly, 2013). We show that the
limiting processes can be approximated using the empirical bootstrap. In the context of in-
tergenerational mobility, these results allow us to test functional hypotheses about income
mobility such as: (i) whether adjusting for covariates has any effect on any particular pa-
rameter (e.g. the percentage of children with income below the poverty line as a function
of parents’ income); or (ii) whether parameters of interest are the same at all values of
parents’ income (e.g. the variance of child’s income).
We use our method to study the effect of parents’ income on child’s income. It is well
known that children from families with high income tend to have higher incomes than
children from families with low income (see Solon, 1992; Solon, 1999, among many oth-
ers). However, much less is known about the distribution of child’s income across parents’
income levels. And learning about this distribution provides much more information to
researchers and policy makers about the effect of parents’ income on child’s income.
To give an example, our baseline estimates suggest that a child whose parents’ income
is at the poverty line (we set this to be $22,100 and discuss why below) has an income
of $33,800 on average. If this is all that a researcher knows about outcomes for children
from families right at the poverty line, it could be the case that: (i) the variance of these
individuals’income is low implying that many of them haveincomes ver y close to $33,800;
or (ii) the variance of these individuals’ income is high implying that some of them have
much higher incomes than $33,800 and others have much lower incomes. In the first case,
most children from low-income families would be moving out of poverty and into the
lower middle class; whilein the second case, many children would remain in poverty while
others might have substantially higher incomes. These two scenarios have quite different
3Another primary difference between our approach and that of Galvao and Wang (2015) is that quantile dose–
response functions are not our primary object of interest. For studying intergenerational mobility, we found that
several other parameters (discussed in detail in section II) that are functions of the counterfactual distribution are
more useful. However, it seems that it would be possible to extend the results in Galvao and Wang (2015) to cover
these parameters as well.
4Our paper is also related to the literature on decompositions with a continuous treatment. ˜
Nopo (2008); Ulrick
(2012) provide decompositions for the mean with a continuous treatment. Ao, Calonico and Lee (2019) consider
decompositions with a multi-valued discrete treatment. Bowles and Gintis (2002); Groves (2005); Blanden, Gregg
and Macmillan (2007); Richey and Rosburg (2018) have decomposed intergenerational mobility parameters into
parts that are explained by various background characteristics.
©2019 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
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