Nonparametric inference for quantile cointegrations with stationary covariates

This paper considers the inference problems in nonlinear quantile regressions with both stationary and nonstationary covariates. The nonparametric local constant quantile estimator is proposed to estimate the unknown quantile regression function, whose asymptotic properties are established under quite general conditions. Specification testing of the quantile regression function is further considered through a statistic constructed based on the integrated squared distance between the parametric and the nonparametric estimators for the regression function. The test statistic is shown to converge to a random variable related to the local time of an Ornstein–Uhlenbeck process under the parametric null. The power of the test against local alternatives is also investigated. Additional asymptotic results on the null parametric quantile estimators and a bootstrap test are developed as well. Numerical results demonstrate that the proposed nonparametric estimator and the specification test enjoy attractive finite sample performance.