A Smooth Specification Test for the Propensity Score

This paper proposes a simple smooth specification test for the propensity score in the spirit of Neyman (1937). We derive a particular restriction on the propensity score and construct the specification test based on it. Using the density-weighting method, we transform the original specification testing problem into a test of the joint significance of the generalized Fourier coefficients. We propose a novel projection method to eliminate the parameter estimation effect arising from estimating unknown nuisance parameters. The proposed test is shown to follow a standard distribution under the null hypothesis and to possess nontrivial asymptotic power against local alternatives that converge to the null at the parametric rate , where is the sample size. To enhance the practical applicability of the procedure, we further introduce a data-driven selection method for determining the testing order, enabling automatic selection based on the observed data. Compared to existing methods, our test does not suffer from the "curse of dimensionality" faced by Shaikh et al. (2009); at the same time, our test is both asymptotically pivotal and more computationally efficient than that of Sant'Anna and Song (2019). Extensive simulations demonstrate that the proposed test yields satisfactory empirical size and power. Compared with previous tests, our test exhibits comparable overall performance and delivers clear improvements in some cases, especially for high-frequency alternatives. Two empirical applications further illustrate the practical applicability of our proposal, examining propensity score specifications in the Soviet 156 Program and the US Job Corps Program.