Identification Based on Difference‐in‐Differences Approaches with Multiple Treatments
| Published date | 01 June 2017 |
| Date | 01 June 2017 |
| DOI | http://doi.org/10.1111/obes.12178 |
| Author | Hans Fricke |
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©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 79, 3 (2017) 0305–9049
doi: 10.1111/obes.12178
Identification Based on Difference-in-Differences
Approaches with MultipleTreatments*
Hans Fricke†
†Center for Education Policy Analysis, Stanford University, 520 Galvez Mall, Stanford, CA
94305-3001, USA (e-mail: hfricke@stanford.edu)
Abstract
This paper discusses identification based on difference-in-differences (DiD) approaches
with multiple treatments. It shows that an appropriate adaptation of the common trend
assumption underlying the DiD strategy for the comparison of two treatments restricts the
possibility of effect heterogeneity for at least one of the treatments. The required assump-
tion of effect homogeneity is likely to be violated because of non-random assignment to
treatment based on both observables and unobservables. However, this paper shows that,
under certain conditions, the DiD estimate comparing two treatments identifies a lower
bound in absolute values on the average treatment effect on the treated compared to the
unobserved non-treatment state, even if effect homogeneity is violated. This is possible if
the treatments have ordered treatment effects, that is, in expectation, the effects of both
treatments compared to no treatment have the same sign, and one treatment has a stronger
effect than the other treatment on the respective recipients. Such assumptions are plausible
if treatments are ordered or vary in intensity.
I. Introduction
The main challenge in policy evaluation is to identify what would have happened to the
treated group in absence of the policy. Difference-in-differences (DiD) strategies identify
this hypothetical situation by using the development in outcomes of a control group that
was unaffected bythe policy. Identification requires that both groups would havedeveloped
equally over time in absence of the policy. In other words, the effect of time (other time
varying factors) on the outcome of interest is equal for both groups. This assumption is
known as the common trend (CT) assumption.
In case of multiple treatments, the effect of different treatments in comparison to no
treatment can be identified precisely under this standard CT assumption (Fr¨olich, 2004).The
comparison of different treatments with DiD has not received much theoretical attention.
JEL Classification numbers: C21, C23
*I thank Martin Huber, Michael Lechner,Giovanni Mellace and Conny Wunsch for helpful comments and sugges-
tions. Furthermore, I am grateful for valuable remarks by JoshuaAng rist, Eric Bettinger,Charlotte Cabane, Christina
Felfe,Angela Sun Johnson, Shailee Pradhan, Andreas Steinmayr, and two anonymousreferees. Hans Fricke received
funding from the Swiss National Science Foundation through grant P1SGP1 158810. The usual disclaimer applies.
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