Testing Conditional Mean Independence Under Symmetry

Abstract

Conditional mean independence (CMI) is one of the most widely used assumptions in the treatment effect literature to achieve model identification. We propose a Kolmogorov–Smirnov-type statistic to test CMI under a specific symmetry condition. We also propose a bootstrap procedure to obtain the p-values and critical values that are required to carry out the test. Results from a simulation study suggest that our test can work very well even in small to moderately sized samples. As an empirical illustration, we apply our test to a dataset that has been used in the literature to estimate the return on college education in China, to check whether the assumption of CMI is supported by the dataset and show the plausibility of the extra symmetry condition that is necessary for this new test.

KEY WORDS:

• Conditional mean independence
• Kolmogorov–Smirnov-type statistic
• Symmetry
• Treatment effect

ACKNOWLEDGMENTS

We would like to thank the co-editor, associate editor, and two referees for their insightful comments. Financial supports for this research are generously provided through the National Natural Science Foundation of China (NSFC) Grant 71171127 and 71471108 and the Open Project Program in the Key Laboratory of Mathematical Economics (SUFE) (201309KF02), Ministry of Education of the People's Republic of China. We are also supported by the Program for Changjiang Scholars and Innovative Research Team in SUFE (PCSIRT, Grant No. IRT13077) and the Innovative Research Team of Econometrics in Shanghai Academy of Social Sciences.

Notes

1 The only other relevant nonparametric $\widehat{g}\left(x\right)$ in modeling sample selection literature that we are aware of was proposed by Das, Newey, and Vella (2003), where no shape restriction on the errors was imposed. However, as Das, Newey, and Vella (2003) only identify g(x) up to a scale, their estimator is not a good candidate for testing CMI.

2 For a d-dimensional vector, $t={(t_1,\dots t_d)}^{\text{'}},$ and a corresponding vector of integers, j = (j ₁, …, j_d ), t^j denotes ⋅⋅⋅, and |j| = j ₁ + ⋅⋅⋅ + j_d , j! = j ₁!⋅⋅⋅j_d !

3 CHNS is conducted jointly by Carolina Population Center at the University of North Carolina at Chapel Hill and the National Institute of Nutrition and Food Safety at the Chinese Center for Disease Control and Prevention, which usually investigated by researchers, for example, Wang (2011), Wang (2012), and Chamon, Liu, and Prasad (2013).