Partial Identification of Marginal Treatment Effects with Discrete Instruments and Misreported Treatment*
| Published date | 01 February 2024 |
| Author | Santiago Acerenza |
| Date | 01 February 2024 |
| DOI | http://doi.org/10.1111/obes.12581 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 86, 1 (2024) 0305-9049
doi: 10.1111/obes.12581
Partial Identification of Marginal Treatment
Effects with Discrete Instruments and
Misreported Treatment*
SANTIAGO ACERENZA
Departamento de Econom´
ıa, Universidad ORT Uruguay, Blvr. Espa˜
na 2633, 11300 Montevideo,
Uruguay (e-mail: acerenza@ort.edu.uy)
Abstract
This paper provides partial identification results for the marginal treatment effect (MTE)
when the binary treatment variable is potentially misreported and the instrumental variable
is discrete. Identification results are derived under smoothness assumptions. Bounds for
both the case of misreported treatment and the case of no misreported treatment are
derived. The identification results are illustrated by identifying the marginal treatment
effects of food stamps on health.
I. Introduction
This paper provides partial identification results for the marginal treatment effect (MTE)
in the presence of measurement error in the treatment variable when only a discrete
instrument is available. The discrete instrument case is relevant as many applications in
the literature rely on these types of instruments (Angrist, 1990; Angrist and Krueger, 1991;
Angrist and Evans, 1998; Krueger, 1999). The discrete nature of the instrument requires
identification strategies for recovering the MTE that differ from those explored in the
previous literature with continuous instruments (Heckman and Vytlacil, 1999; Heckman,
Urzua, and Vytlacil, 2006).
The results of this paper are relevant since instrumental variables with discrete variation
are common (e.g. assignment to treatment via eligibility requirements), and misreporting
is a common problem in survey data, which are one of the main sources of empirical
research (Gundersen and Kreider, 2008;Kreideret al.,2012).
Our novel results can serve as a sensitivity analysis tool for recovering the MTE in the
presence of a discrete instrument and the potential existence of measurement error.
The combination of measurement error and discrete instruments has not been
explored in the literature, although it is a fairly common situation. Researchers work
JEL Classification numbers: C21, C26.
*We would like to thank D´
esir´
eK
´
edagni and Otavio Bartalotti for their guidance; Nestor Gandelman, Joydeep
Bhattacharya, Kyunghoon Ban, Vitor Possebom, Hannah Wich, three anonymous referees, and participants at the
events of Sociedad de Economistas del Uruguay (SEU) and seminars at Iowa State University for their useful
comments; and Brent Kreider for sharing the data of the empirical application.
74
©2023 Oxford University and John Wiley & Sons Ltd.
Partial identification of marginal treatment effects 75
mostly with self-reported data from surveys; such data systematically have reporting
problems that lead to measurement error of the treatment status and, consequently,
to bias in the treatment effect of interest. The results in this paper are useful
for identifying the MTE (which can be used to recover average effects or policy-
relevant effects) in the presence of the two previously mentioned problems for
identification.
In most cases, researchers observe a discrete (often binary) instrument, such as
assignment to treatment. In this case, point identification of the MTE (even without
measurement error) is not possible when only relying on the standard assumptions of
instrument exogeneity and relevance (Brinch, Mogstad, and Wiswall, 2017). Under a set
of restrictions on the severity of measurement error and shape restrictions, we provide
partial identification results for the MTE in the presence of measurement error when a
discrete instrument is available.
The MTE can help reveal the heterogeneity in the treatment effect, which has value
in itself since it can help reveal the way latent characteristics of individuals affect the
benefit of a particular treatment or policy. This could then be used to design more specific
and efficient policies. Besides having value in itself, the MTE is relevant in recovering
policy-relevant treatment effect parameters (PRTEs), the average treatment effect (ATE),
the average treatment on the treated (ATT), the average treatment on the untreated (ATU),
the local average treatment effect (LATE), etc.1
To achieve partial identification, we introduce smoothness conditions on the
marginal treatment responses. To deal with the misreporting of the binary treatment,
we assume that the unconditional probability of misreporting is fixed.2This can be
interpreted either as the researcher having prior knowledge of the possible value of the
misclassification rates or as a sensitivity analysis in which the researcher allows for
the possibility of misclassification up to a certain level. Relevance and independence
of the instrument are required for all the identification results of the paper. Although
partial identification of the MTE does not imply sharp bounds on the ATE, it is
still a useful tool for generating bounds on an aggregate-relevant effect from local
effects.
Empirical research usually combines a measurement error problem with endogeneity
and heterogeneity. For example, Ura (2018) documents that there is a substantial
measurement error in educational attainments in the 1990 U.S. Census. Educational
attainments are endogenous as treatment variables in return to schooling analyses
because, among other possibilities, unobserved individual ability affects both schooling
decisions and wages. A similar issue exists in labour supply to welfare programme
participation. Further, Hernandez and Pudney (2007) highlight that self-reported
programme participation in survey data is subject to misreporting. The psychological cost
of welfare programme participation affects job search behaviour and welfare programme
participation simultaneously.
1See Heckman and Vytlacil (2005) and Heckman et al. (2006), who show the link between the MTE and those
parameters via properly weighting the MTE.
2Alternatively, one could take the results of this paper and assume a known upper bound of this probability and take
the union of the bounds derived here.
©2023 Oxford University and John Wiley & Sons Ltd.
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