  Motivated by the need for effectively summarising, modelling, and forecasting
the distributional characteristics of intra-daily returns, as well as the
recent work on forecasting histogram-valued time-series in the area of symbolic
data analysis, we develop a time-series model for forecasting
quantile-function-valued (QF-valued) daily summaries for intra-daily returns.
We call this model the dynamic quantile function (DQF) model. Instead of a
histogram, we propose to use a $g$-and-$h$ quantile function to summarise the
distribution of intra-daily returns. We work with a Bayesian formulation of the
DQF model in order to make statistical inference while accounting for parameter
uncertainty; an efficient MCMC algorithm is developed for sampling-based
posterior inference. Using ten international market indices and approximately
2,000 days of out-of-sample data from each market, the performance of the DQF
model compares favourably, in terms of forecasting VaR of intra-daily returns,
against the interval-valued and histogram-valued time-series models.
Additionally, we demonstrate that the QF-valued forecasts can be used to
forecast VaR measures at the daily timescale via a simple quantile regression
model on daily returns (QR-DQF). In certain markets, the resulting QR-DQF model
is able to provide competitive VaR forecasts for daily returns.
