  In this paper, we propose two important measures, quantile correlation (QCOR)
and quantile partial correlation (QPCOR). We then apply them to quantile
autoregressive (QAR) models, and introduce two valuable quantities, the
quantile autocorrelation function (QACF) and the quantile partial
autocorrelation function (QPACF). This allows us to extend the classical
Box-Jenkins approach to quantile autoregressive models. Specifically, the QPACF
of an observed time series can be employed to identify the autoregressive
order, while the QACF of residuals obtained from the fitted model can be used
to assess the model adequacy. We not only demonstrate the asymptotic properties
of QCOR, QPCOR, QACF, and PQACF, but also show the large sample results of the
QAR estimates and the quantile version of the Ljung-Box test. Simulation
studies indicate that the proposed methods perform well in finite samples, and
an empirical example is presented to illustrate usefulness.
