Testing for homogeneous treatment effects in linear and nonparametric instrumental variable models

Abstract.

The hypothesis of homogeneous treatment effects is central to the instrumental variables literature. This assumption signifies that treatment effects are constant across all subjects. It allows to interpret instrumental variable estimates as average treatment effects over the whole population of the study. When this assumption does not hold, the bias of instrumental variable estimators can be greater than that of naive estimators ignoring endogeneity. This article develops two tests for the assumption of homogeneous treatment effects when the treatment is endogenous and an instrumental variable is available. The tests leverage a covariable that is (jointly with the error terms) independent of a coordinate of the instrument. This covariate does not need to be exogenous. The first test assumes that the potential outcomes are linear in the regressors and is computationally simple. The second test is nonparametric and relies on Tikhonov regularization. The treatment can be either discrete or continuous. We show that the tests have asymptotically correct level and asymptotic power equal to one against a range of alternatives. Simulations demonstrate that the proposed tests attain excellent finite sample performances. The methodology is also applied to the evaluation of returns to schooling and demand estimation in a fish market.

Keywords:

• Homogeneous treatment effects
• hypothesis testing
• nonparametric instrumental variables

Notes

1. By {U(z)}_z, we mean the process consisting of all the random variables U(z) for all z∈𝒵.

2. For more details on the role of recentering in bootstrap tests, see, e.g., Bickel and Ren (2001).

Additional information

Funding

Financial support from the European Research Council (2016-2021, Horizon 2020 / ERC grant agreement No. 694409) is gratefully acknowledged. Jean-Pierre Florens acknowledges funding from the French National Research Agency (ANR) under the Investments for the Future program (Investissements d’Avenir, grant ANR-17-EURE-0010). Elia Lapenta acknowledges funding from the French National Research Agency (ANR) under grant ANR-23-CE26-0008-01.