Before revision:  In order to make statistical inferences and account for parameter uncertainty , we propose a method whereby a likelihood function can be constructed for QF-valued data, and develop an adaptive MCMC sampling algorithm for simulating from the posterior distribution. Compared to modelling realised measures,modelling the entire quantile functions of intra-daily returns allows one to gain more insight into the dynamic structure of price movements.
After revision:  In order to make statistical inferences and account 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 allows one to gain more insight into the dynamic structure of price movements.