| Presented here is a multivariate micro-movement model for asset prices. The model, as it is developed, reflects many of the stylized features of ultra-high frequency transaction (UHF), or trade-by-trade, data. In our model, we describe the trade-by trade data as counting point process and then explicitly incorporate three types of 'noise' present in UHF data via a random transformation. We first study the likelihood, the marginal or integrated likelihood, the likelihood ratios, and the posterior and Bayes factors of the model and characterize them by evolution equations, such as filtering equations. We then develop the Bayesian inference including estimation and model selection via filtering for the model. Since those likelihoods, posteriors and Bayes factor are continuous time and computationally unfeasible, we construct recursive algorithms to approximate them, and show the consistency of these algorithms. A simple model for two correlated stocks is studied in detail with both simulated demonstration and empirical results. We also prove the consistency of the Bayes estimates for this simple two stock model. Additionally, we construct a series of simple models for the single-stock case for Bayesian inference via filtering with simulated demonstration and empirical results. |
