Conditional Independence Specication Testing for Dependent Processes with Local Polynomial Quantile Regression

We provide straightforward new nonparametric methods for testing conditional independence using local polynomial quantile regression, allowing weakly dependent data. Inspired by Hausman’s (1978) speci…cation testing ideas, our methods essentially compare two collections of estimators that converge to the same limits under correct speci…cation (conditional independence) and that diverge under the alternative. To establish the properties of our estimators, we generalize the existing nonparametric quantile literature not only by allowing for dependent heterogeneous data but also by establishing a weak consistency rate for the local Bahadur representation that is uniform in both the conditioning variables and the quantile index. We also show that, despite our nonparametric approach, our tests can detect local alternatives to conditional independence that decay to zero at the parametric rate. Our approach gives the …rst nonparametric tests for time-series conditional independence that can detect local alternatives at the parametric rate. Monte Carlo simulations suggest that our tests perform well in …nite samples. Our tests have a variety of uses in applications, such as testing conditional exogeneity or Granger non-causality. Key Words: Conditional independence, Empirical process, Granger causality, Local polynomial, Quantile regression, Speci…cation test, Uniform local Bahadur representation. 1 Introduction Hausman’s (1978) seminal paper on speci…cation testing opened the way to a broad array of methods for assessing the validity of econometric models and their resulting insights. The fundamental idea of comparing two estimators, both consistent under correct speci…cation, but divergent under misspeci…cation applies not only to detecting incorrect parametric functional form for conditional means, variances, or other aspects of the conditional distribution of a variable of interest, but also to detecting failures of exogeneity –the stochastic orthogonality condition between observable and unobservable drivers of Address correspondence to: Halbert White, Department of Economics, UCSD, La Jolla, CA 92093-0508, USA. Phone: +1 858 534-3502; e-mail: [email protected]. The …rst author gratefully acknowledges the …nancial support from a research grant (Grant number: C244/MSS8E004) from Singapore Management University.