  Quantile regression continues to increase in usage, providing a useful
alternative to customary mean regression. Primary implementation takes the form
of so-called multiple quantile regression, creating a separate regression for
each quantile of interest. However, recently, advances have been made in joint
quantile regression, supplying a quantile function which avoids crossing of the
regression across quantiles. Here, we turn to quantile autoregression (QAR),
offering a fully Bayesian version. We extend the initial quantile regression
work of Koenker and Xiao (2006) in the spirit of Tokdar and Kadane (2012). We
offer a directly interpretable parametric model specification for QAR. Further,
we offer a p-th order QAR(p) version, a multivariate QAR(1) version, and a
spatial QAR(1) version. We illustrate with simulation as well as a temperature
dataset collected in Arag\'on, Spain.
