  This paper proposes a novel conditional heteroscedastic time series model by
applying the framework of quantile regression processes to the ARCH(\infty)
form of the GARCH model. This model can provide varying structures for
conditional quantiles of the time series across different quantile levels,
while including the commonly used GARCH model as a special case. The strict
stationarity of the model is discussed. For robustness against heavy-tailed
distributions, a self-weighted quantile regression (QR) estimator is proposed.
While QR performs satisfactorily at intermediate quantile levels, its accuracy
deteriorates at high quantile levels due to data scarcity. As a remedy, a
self-weighted composite quantile regression (CQR) estimator is further
introduced and, based on an approximate GARCH model with a flexible
Tukey-lambda distribution for the innovations, we can extrapolate the high
quantile levels by borrowing information from intermediate ones. Asymptotic
properties for the proposed estimators are established. Simulation experiments
are carried out to access the finite sample performance of the proposed
methods, and an empirical example is presented to illustrate the usefulness of
the new model.
