In this paper we develop an analysis of multivariate time series that exhibit reduced rank cointegration, implying that a lower dimensional linear projection of the process can be obtained in which the projected process becomes stationary. Detection of the rank and basis upon which to project the process for stationarity to hold is a critical problem when working with such settings in practice. There is a range of practice when performing estimation of in such multivariate time series settings. In this paper we provide a review of a few selected different models and estimation techniques for these multivariate time series. Having presented an overview of important new directions with regard to estimation of cointegration relationships we then turn our attention to the performance of a range of estimation procedures. In particular we design a range of numerical studies in order to assess some of these approaches in terms of efficiency and accuracy. In particular, we study the question related to examining the robustness of these classes of estimation procedure in Bayesian and non-parametric estimation approaches to the influence of the model noise in the estimation of the cointegration space. In this context we develop a novel Bayesian inference procedure not previously studied in cointegration models to estimate the cointegration space. This is based on a Markov Chain Monte Carlo sampling method, that consists of a novel extension of Hamiltonian and Geodesic Monte Carlo for the present problem. We will illustrate the performance of this method numerically and show that it produces results on par with an efficient Gibbs Sampler.
- Hamiltonian Monte Carlo