In recent years, autoregressive conditional duration models (ACD models) introduced by Engle and Russell in 1998 have become very popular in modelling of the durations between selected events of the transaction process (trade durations or price durations) and modelling of financial market microstructure effects. The aim of the paper is to develop Bayesian inference for the ACD models. Different specifications of ACD models will be considered and compared with particular emphasis on the linear ACD model, Box-Cox ACD model, augmented Box-Cox ACD model and augmented (Hentschel) ACD model. The analysis will consider models with the Burr distribution and the generalized Gamma distribution for the innovation term. Bayesian inference will be presented and practically used in estimation of and prediction within ACD models describing trade durations. The MCMC methods including Metropolis-Hastings algorithm are suitably adopted to obtain samples from the posterior densities of interest. The empirical part of the work includes modelling of trade durations of selected equities from the Polish stock market.

In empirical research on financial market microstructure and in testing some predictions from the market microstructure literature, the behavior of some characteristics of trading process can be very important and useful. Among all characteristics associated with tick-by-tick data, the trading time and the price seem the most important. The very first joint model for prices and durations, the so-called UHF-GARCH, has been introduced by Engle (2000). The main aim of this paper is to propose a simple, novel extension of Engle’s specification based on trade-to-trade data and to develop and apply the Bayesian approach to estimation of this model. The intraday dynamics of the return volatility is modelled by an EGARCH-type specification adapted to irregularly time-spaced data. In the analysis of price durations, the Box-Cox ACD model with the generalized gamma distribution for the error term is considered. To the best of our knowledge, the UHF-GARCH model with such a combination of the EGARCH and the Box-Cox ACD structures has not been studied in the literature so far. To estimate the model, the Bayesian approach is adopted. Finally, the methodology developed in the paper is employed to analyze transaction data from the Polish Stock Market.