Probability of Causation with Sample Selection A Reanalysis of the Impacts of J ovenes en Acci on on Formality

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Probability of Causation with Sample

Selection: A Reanalysis of the Impacts of

J´ovenes en Acci´on on Formality

Vitor Possebom∗

Sao Paulo School of Economics - FGV

and

Flavio Riva†

Instituto Mobilidade e Desenvolvimento Social - Imds

July 8, 2024

Abstract

This paper identiﬁes the probability of causation when there is sample selection.

We show that the probability of causation is partially identiﬁed for individuals who

are always observed regardless of treatment status and derive sharp bounds under

three increasingly restrictive sets of assumptions. The ﬁrst set imposes an exogenous

treatment and a monotone sample selection mechanism. To tighten these bounds,

the second set also imposes the monotone treatment response assumption, while

the third set additionally imposes a stochastic dominance assumption. Finally, we

use experimental data from the Colombian job training program J´ovenes en Acci´on

to empirically illustrate our approach’s usefulness. We ﬁnd that, among always-

employed women, at least 10.2% and at most 13.4% transitioned to the formal labor

market because of the program. However, our 90%-conﬁdence region does not reject

the null hypothesis that the lower bound is equal to zero.

Keywords: Probability of Causation, Sample Selection, Partial Identiﬁcation, Job Training

Programs.

∗vitor.possebom@fgv.br

†ﬂaviorussoriva@gmail.com

arXiv:2210.01938v6 [econ.EM] 3 Jul 2024

1 Introduction

Many policy evaluation questions involve two simultaneous identiﬁcation challenges: the

causal parameter of interest depends on the joint distribution of potential outcomes (Heck-

man et al., 1997; Pearl, 1999; Tian and Pearl, 2000; Jun and Lee, 2022; Cinelli and Pearl,

2021), and sample selection is present (Lee, 2009; Chen and Flores, 2015; Bartalotti et al.,

2023). For example, when evaluating the eﬀects of job training programs (Heckman et al.,

1999; Attanasio et al., 2011, 2017; Blanco and Flores-Lagunes, 2018), the researcher may

be interested in learning to what extent the transition from informal to formal employ-

ment can be attributed to the policy. Still, she only observes formality status among those

who are employed. This double identiﬁcation challenge also arises when researchers ana-

lyze the eﬀects of a political campaign on agents’ opinions (DellaVigna and Kaplan, 2007;

DellaVigna and Gentzkow, 2010) if agents may not reply to the researchers’ survey.

In this paper, we derive novel sharp bounds around the probability of causation param-

eter (Pearl, 1999; Tian and Pearl, 2000; Jun and Lee, 2022; Cinelli and Pearl, 2021) for

individuals who self-select into the sample regardless of their treatment assignment. The

probability of causation parameter summarizes one crucial aspect of the eﬀects of treat-

ments on binary outcomes: the proportion of individuals who beneﬁt from being treated

within the subgroup who would, counterfactually, experience a negative untreated outcome.

Thus, our target parameter helps researchers gauge to what extent the transition from one

state to another can be attributed to the treatment in a relevant latent sub-population.

Our partial identiﬁcation strategies are based on three increasingly restrictive sets of

assumptions. They extend the identiﬁcation of probabilities of causation to scenarios with

endogenous sample selection. In our model, treatment eﬀects can be related to the sample

selection mechanism even though treatment take-up is exogenous. We also discuss when our

assumptions have identiﬁcation power and how to test them through necessary observable

conditions.

Our ﬁrst identiﬁcation result relies on a monotone sample selection mechanism. This

condition imposes that treatment has a non-negative eﬀect on the sample selection indicator

for all individuals. In the job training example, this restriction implies that the treatment

can move workers into employment but never out of employment.

Our second result further assumes a monotone treatment response to tighten the iden-

tiﬁed bounds. This condition imposes that treatment has a non-negative eﬀect on the

potential outcomes for all individuals. In the job training example, this restriction implies

that the treatment can move workers into formal jobs but never into informal jobs.

Our ﬁnal result additionally relies on a stochastic dominance assumption to further

reduce the identiﬁed set. This condition imposes that the sub-population that self-selects

into the sample regardless of the treatment status has higher treated potential outcomes

than the sub-population that self-selects into the sample only when treated. In the job

training example, this restriction implies that the agents who are always employed are

more likely to have a formal job if treated than those who are employed only when treated.

Additionally, we propose parametric estimators for all these bounds. We also combine

the precision-corrected bounds proposed by Chernozhukov et al. (2013) with a Bonferroni-

style correction to derive conﬁdence regions that contain the identiﬁed region with a pre-

speciﬁed conﬁdence level.

To empirically illustrate the usefulness of our approach, we provide bounds for the

probability of causation of an intensive training program: J´ovenes en Acci´on. This program

aimed to improve the labor market prospects and, in particular, the quality of jobs held by

disadvantaged youths in seven large cities in Colombia. It oﬀered in-classroom intensive

training in occupational skills to qualify unemployed individuals for locally demanded jobs.

Additionally, it focused on socioemotional development and oﬀered on-the-job internships

with formal employers.

Previous research (Attanasio et al., 2011, 2017) ﬁnds that this program positively af-

fects employment and unconditional formality. However, less is known about whether the

program achieves its goal of improving job quality conditioning on having a job. We study

its eﬀects on the job quality margin by considering the share of women that transitioned to

the formal labor market because they participated in the training program. We ﬁnd that

incorporating selection and bounding the probability of causation leads to a pessimistic

view of the program’s impacts. More precisely, we ﬁnd that at most 13.4% of the always-

employed women switched their formality status because they were assigned to the J´ovenes

en Acci´on training program. Moreover, our 90%-conﬁdence region includes the zero, im-

plying that we cannot reject the null hypothesis that our target parameter’s lower bound

is equal to zero.

Concerning its theoretical contribution, our work is inserted in two research areas: iden-

tiﬁcation of probabilities of causation and identiﬁcation in the presence of sample selection.

Heckman et al. (1997) motivate the focus on a parameter closely connected to the

probability of causation based on the political economy of policy evaluation. They argue

that a program would only be adopted in a democracy if it beneﬁted most people in the

population. They either make strong probabilistic assumptions or impose model restrictions

on treatment take-up decisions to point-identify this parameter, while we focus entirely on

partial identiﬁcation strategies based on a menu of easily interpretable assumptions.

Pearl (1999) and Tian and Pearl (2000) discuss how to interpret and partially identify

probabilities of causation in a single population where agents are always observed. Cinelli

and Pearl (2021) extend their work by combining experimental results from multiple trials

to extrapolate probabilities of causation from one population to a diﬀerent population.

Moreover, Jun and Lee (2022) extend their work by considering endogenous selection into

treatment.

We extend the work by Pearl (1999) and Tian and Pearl (2000) in a diﬀerent direction.

We identify probabilities of causation when the agents’ realized outcomes may not be

observed due to endogenous sample selection. To do so, we combine the tools developed in

the literature about probabilities of causation with the trimming bounds developed in the

sample selection literature (Horowitz and Manski, 1995; Lee, 2009; Chen and Flores, 2015;

Bartalotti et al., 2023).

Concerning its empirical contribution, our work is inserted in the literature about job

training programs. Attanasio et al. (2011) and Attanasio et al. (2017) analyze the average

treatment eﬀect (ATE) of J´ovenes en Acci´on on short and long-term outcomes associated

with labor force attachment. We extend their work by analyzing a treatment eﬀect pa-

rameter that focuses on job quality instead of labor force attachment. Importantly, Blanco

and Flores-Lagunes (2018) also analyze the impact of a job training program on job qual-

ity using partial identiﬁcation strategies. However, we focus on diﬀerent contexts (Job

Corps v. J´ovenes en Acci´on) and diﬀerent target parameters (Quantile Treatment Eﬀects

v. Probabilities of Causation).

This paper is organized as follows. Section 2 presents our structural model, sample

selection mechanism, and identifying assumptions. It also discusses the testable restrictions

imposed by our model. Section 3 describes our main identiﬁcation results, while Section 4

proposes a parametric estimator for our bounds and discusses an inferential method for the

identiﬁed region. Moreover, Section 5 discusses the results of our empirical application. In

the end, Section 6 concludes.

Moreover, we also have an online appendix with additional details and results. Appendix

A presents the proofs of all our identiﬁcation results, while Appendix B intuitively explains

them using a numerical example. Moreover, Appendix C brings a detailed discussion about

the testable restrictions of our identifying assumptions, while Appendix D compares our

target parameter against other causal parameters. Furthermore, Appendix E detailedly

explains our estimator and inferential method. Finally, Appendix F presents additional

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ProbabilityofCausationwithSampleSelection:AReanalysisoftheImpactsofJ´ovenesenAcci´ononFormalityVitorPossebom∗SaoPauloSchoolofEconomics-FGVandFlavioRiva†InstitutoMobilidadeeDesenvolvimentoSocial-ImdsJuly8,2024AbstractThispaperidentifiestheprobabilityofcausationwhenthereissampleselection.Weshowthatthe...

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