Specification tests for the propensity score

Abstract

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.JEL: C12, C31, C35, C52.

Kernel-weighted specification testing under general distributions

“…It is shown in [10] (see also [17,18,8] for related applications) that when the distribution is absolutely continuous and certain smoothness regularity conditions hold on the density, Condition (9) reduces to the usual restriction on the bandwidth: h n ↓ 0 and nh q n ↑ ∞. Our starting point in generalizing beyond absolutely continuous measures is to derive the distributional restrictions that are implicitly imposed through Condition (9). This requires us to express and bound the moments that appear in (7,8) in terms of interpretable functionals of F X .…”

Specification tests for generalized propensity scores using double projections

“…We label P n,t 1 β ⊤ X ≤ u as a double-projection because, as it is evident from (9), it involves first using the projection proposed by Escanciano (2006), 1 β ⊤ X ≤ u , and then projecting 1 β ⊤ X ≤ u onto the the tangent space of the nuisance parameters θ n Goh, 2014, Sant'Anna andSong, 2019). To the best of our knowledge, this paper is the first to incorporate this double-projection argument, which, in practice, translates to test statistics that are robust against the "curse of dimensionality" and whose limiting null distributions are asymptotically invariant to θ n since, for each t ∈ T ,…”

“…In order to facilitate its practical implementation, we obtain closed-form expressions for our test statistics, and show that critical values can be computed with the assistance of an easy-to-use multiplier-type bootstrap. To the best of our knowledge, no other (specification) test available in the literature enjoys all these attractive properties (e.g., Escanciano, 2006, Mora and Moro-Egido, 2008, Shaikh et al, 2009, Escanciano and Goh, 2014, García-Portugués et al, 2014, Sant'Anna and Song, 2019, and Kim et al, 2020. The results of Monte Carlo simulations indicate that these attractive properties translate to tests with excellent finite sample properties, even the dimension of covariates is relatively high.…”

“…More specifically, our specification tests build on the dimension-reduction projection introduced by Escanciano (2006), allowing us to effectively handle situations with moderate number of covariates. Our tests also builds on Neyman (1959), Bickel et al (2006), Goh (2014), andSant'Anna andSong (2019) in the sense that we explicitly acknowledge we do not know the "true" correct specification of the generalize propensity score by making use of an orthogonal projection onto the tangent space of nuisance parameters.…”

Covariate distribution balance via propensity scores

“…Related literature: Our proposal builds on different branches of the econometrics literature. For instance, this paper is related to Shaikh et al (2009) and Sant'Anna and Song (2019), who exploit the covariate balancing of the PS to propose specification tests for a given PS model. Here, instead of checking if a given PS estimator balances the covariate distribution among different treatment groups, we propose to estimate the PS unknown parameters by maximizing the covariate balancing.…”