In this paper we present a copula-based model for a binary and a continuous variable in a time series setup. Within this modeling framework both marginals can be equipped with their own dynamics whereas the contemporaneous dependence between both processes can be flexibly captured via a copula function. We propose a method for testing the goodness-of-fit of such a time series model using probability integral transforms (PIT). This verification procedure allows not only a verification of the goodness-of-fit of the estimated marginal distribution for a continuous variable but also the conditional distribution of a continuous variable given the outcome of its binary counterpart (i.e. the adequacy of the copula choice). We test the model on an empirical example: investigating the relationship between trading volume and the indicators of arbitrarily ’large’ price movements on the interbank EUR/PLN spot market.

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.