  We establish the asymptotic theory in quantile autoregression when the model
parameter is specified with respect to moderate deviations from the unit
boundary of the form (1 + c / k) with a convergence sequence that diverges at a
rate slower than the sample size n. Then, extending the framework proposed by
Phillips and Magdalinos (2007), we consider the limit theory for the
near-stationary and the near-explosive cases when the model is estimated with a
conditional quantile specification function and model parameters are
quantile-dependent. Additionally, a Bahadur-type representation and limiting
distributions based on the M-estimators of the model parameters are derived.
Specifically, we show that the serial correlation coefficient converges in
distribution to a ratio of two independent random variables. Monte Carlo
simulations illustrate the finite-sample performance of the estimation
procedure under investigation.
