Many economic time series exhibit random walk or trend dynamics and other persistent non-stationary behaviour (e.g. stock prices, exchange rates, unemployment rate and net trading). If a time series is not stationary, then any shock can be permanent and there is no tendency for its level to return to a constant mean over time; moreover, in the long run, the volatility of the process is expected to grow without bound, and the time series cannot be predicted based on historical observations. Cointegration allows the identification of economic integrated time series that exhibit similar dynamics in the long run and the estimation of their relationships, by exploiting the stationary linear combinations of these time series.
This thesis proposes three Bayesian estimation methods of the well-known Vector Error Correction Model (VECM) about difference stationary time series in order to extract the long-run equilibrium relationships. Each method used in this thesis is implemented using Markov Chain Monte Carlo (MCMC) and illustrated on synthetic data, and then on real economic data sets. The first method consists of a static model, where we compare comovements between Eurozone economic time series comprising net trading, long-term interest rates and the harmonised unemployment rate. Primiceri (2005) established a time-varying model for the vector autoregressive model. Following Primiceri and the idea of the static model seen in the first method, we are constructing a time-varying model for our VECM, from which we extract information about the time-varying cointegration matrix, and more interestingly about its time-varying rank (i.e. the cointegration rank) and independent cointegration relationships. These two first methods are based on the singular value decomposition of the cointegration matrix from the error correction model and the so-called irrelevance criterion, a flexible thresholding approach to determine its rank. In these two methods, the joint estimation of the cointegration rank and the cointegration relationships is deducted from synthetic data sets before applying them to real data sets (European economies and major stock market exchange indices). The last main chapter of this thesis covers the use of a prior singular distribution on the long-run relationship matrix of the VECM given the cointegration rank. Based on the definition of the singular matrix normal distribution, we also learn about the space definition and the density of such a distribution. We also remind the singular Inverse-Wishart distribution and in our discussion, we eventually open the issues arising in implementing a dynamic model, by developing the idea of a singular Inverse-Wishart distribution on the variance covariance matrix of the transition equation (see Chapter 6).
