Regression quantiles for unstable autoregressive models

This paper investigates regression quantiles (RQ) for unstable autoregressive models. The uniform Bahadur representation of the RQ process is obtained. The joint asymptotic distribution of the RQ process is derived in a unified manner for all types of characteristic roots on or outside the unit circle. Unlike the results already available for the regression and stationary autoregression quantiles, the joint asymptotic distribution involves stochastic integrals in terms of a series of independent and identically distributed multivariate Brownian motions with correlated components. The related L-estimator is also discussed. As an auxiliary theorem, a weak convergence of a randomly weighted residual empirical process to the stochastic integral of a Kiefer process is established. The results obtained in this paper provide an asymptotic theory for nonstationary time series processes, which can be used to construct robust unit root tests.