  This paper proposes new nonparametric diagnostic tools to assess the
asymptotic validity of different treatment effects estimators that rely on the
correct specification of the propensity score. We derive a particular
restriction relating the propensity score distribution of treated and control
groups, and develop specification tests based upon it. The resulting tests do
not suffer from the "curse of dimensionality" when the vector of covariates is
high-dimensional, are fully data-driven, do not require tuning parameters such
as bandwidths, and are able to detect a broad class of local alternatives
converging to the null at the parametric rate $n^{-1/2}$, with $n$ the sample
size. We show that the use of an orthogonal projection on the tangent space of
nuisance parameters facilitates the simulation of critical values by means of a
multiplier bootstrap procedure, and can lead to power gains. The finite sample
performance of the tests is examined by means of a Monte Carlo experiment and
an empirical application. Open-source software is available for implementing
the proposed tests.
