Testing independence between exogenous variables and unobserved errors

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.

Automated summary

In this paper, a unified methodology is developed to test the independence assumption between the error term and exogenous variables. The methodology can be applied to a wide range of parametric models and can handle endogeneity and instrumental variables. Tests are constructed using continuous functionals and a multiple testing approach is proposed to accommodate high-dimensional exogenous random vectors. The tests are shown to be consistent and sensitive to locally alternative hypotheses.