  Many financial time series have varying structures at different quantile
levels, and also exhibit the phenomenon of conditional heteroscedasticity at
the same time. In the meanwhile, it is still lack of a time series model to
accommodate both of the above features simultaneously. This paper fills the gap
by proposing a novel conditional heteroscedastic model, which is called the
quantile double autoregression. The strict stationarity of the new model is
derived, and a self-weighted conditional quantile estimation is suggested. Two
promising properties of the original double autoregressive model are shown to
be preserved. Based on the quantile autocorrelation function and self-weighting
concept, two portmanteau tests are constructed, and they can be used in
conjunction to check the adequacy of fitted conditional quantiles. The
finite-sample performance of the proposed inference tools is examined by
simulation studies, and the necessity of the new model is further demonstrated
by analyzing the S&P500 Index.
