A Unified Framework for Specification Tests of Continuous Treatment Effect Models

Abstract

We propose a general framework for the specification testing of continuous treatment effect models. We assume a general residual function, which includes the average and quantile treatment effect models as special cases. The null models are identified under the unconfoundedness condition and contain a nonparametric weighting function. We propose a test statistic for the null model in which the weighting function is estimated by solving an expanding set of moment equations. We establish the asymptotic distributions of our test statistic under the null hypothesis and under fixed and local alternatives. The proposed test statistic is shown to be more efficient than that constructed from the true weighting function and can detect local alternatives deviated from the null models at the rate of $O(N^{-1/2})$. A simulation method is provided to approximate the null distribution of the test statistic. Monte-Carlo simulations show that our test exhibits a satisfactory finite-sample performance, and an application shows its practical value.

Keywords:

• Bootstrap
• Consistent tests
• Continuous treatment effect
• Series estimation

Supplementary Material

Supplementary materials are only for online publication. The supplementary file contains the simulation results of the KS-type statistic, the details of the estimation and the test statistics for the Tobit model used in Section 6.3, the proofs of Theorems 1, 2, 5, and the asymptotic properties of ${\widehat{J}}_N(t;{\widehat{\mathit{\theta }}}_{\text{opt}})$ and ${\widehat{\text{CM}}}_N({\widehat{\mathit{\theta }}}_{\text{opt}})$.

Acknowledgments

We thank two anonymous referees, an associate editor, and the editor Christian Hansen for constructive comments which lead to substantial improvement of the article.

Additional information

Funding

The first author, Wei Huang’s research was supported by the Professor Maurice H. Belz Fund of the University of Melbourne. The second author, Oliver Linton, acknowledges Cambridge INET for financial support. The final author, Zheng Zhang, acknowledges financial support from the National Natural Science Foundation of China through project 12001535, and the fund for building world-class universities (disciplines) of the Renmin University of China.