Testing independence between exogenous variables and unobserved errors

Abstract

Although the exogeneity condition is usually used in many econometric models to identify parameters, the stronger restriction that the error term is independent of a vector of exogenous variables might lead to theoretical benefits. In this paper, we develop a unified methodology for testing the independence assumption. Our methodology can deal with a wide class of parametric models and allows for endogeneity and instrumental variables. In the first-step development, we construct tests that are continuous functionals of the estimated difference of the joint distribution and the product marginal distributions. Next, to remedy the dimensionality issue that arises when the dimension of the exogenous random vector is large, we propose a multiple testing approach which combines marginal p-values obtained by employing the original tests to test independence between the error term and each exogenous variable, while taking full account of the multiplicity nature of the testing problem. We obtain null limiting distributions of our tests, establish the testing consistency, and justify the sensitivity to $n^{-1/2}$-local alternatives, with n the sample size. The multiplier bootstrap is employed to estimate the critical values. Our methodology is illustrated in the linear regression, the instrumental variables regression, and the nonlinear quantile regression. Our tests are found to perform well in simulations and are demonstrated via an empirical example.

Keywords:

• Instrumental variables estimation
• multiple testing
• multiplier bootstrap
• quantile regression

JEL CLASSIFICATION:

• C12
• C21
• C26

Notes

1 To illustrate this point, consider a simple situation where $\mathbf{Z}={(Z_1,Z_2)}^{\prime }$ with Z₁ independent of Z₂ and ε is independent of Z₂. Thus independence between ε and Z is in fact equivalent to independence between ε and Z₁, thereby making Z₂ irrelevant in the testing problem.

2 The idea of conducting multiple testing by combining marginal p-values has been adopted by several studies; see Hansen and Timmermann (2012) and Bergamelli, Bianchi, Khalaf, and Urga (2019) for similar treatments in the contexts of forecast evaluation and multiple structural break detection, respectively.

3 The data are available at https://davidcard.berkeley.edu/data_sets/proximity.zip.

4 When X contains mixed categorical and continuous data, the uniform convergence of the conditional density estimator still holds following the proof of Theorem 2.1 of Li and Ouyang (2005). Therefore, our results can be readily extended to the case where X contains mixed categorical and continuous data.

Additional information

Funding

Li’s research was supported by the National Natural Science Foundation of China (Grants 11801399). Tu would like to thank support from National Natural Science Foundation of China (Grant nos. 72073002, 12026607, 92046021), the Center for Statistical Science at Peking University, and Key Laboratory of Mathematical Economics and Quantitative Finance (Peking University), Ministry of Education. We thank the editor Prof. Esfandiar Maasoumi and three anonymous referees for constructive comments which lead to great improvement of the paper.