Mixed Causal-Noncausal AR Processes and the Modelling of Explosive Bubbles

Noncausal autoregressive models with heavy-tailed errors generate locally explosive processes and therefore provide a natural framework for modelling bubbles in economic and financial time series. We investigate the probability properties of mixed causal-noncausal autoregressive processes, assuming the errors follow a stable non-Gaussian distribution. We show that the tails of the conditional distribution are lighter than those of the errors, and we emphasize the presence of ARCH effects and unit roots in a causal representation of the process. Under the assumption that the errors belong to the domain of attraction of a stable distribution, we show that a weak AR causal representation of the process can be consistently estimated by classical least-squares. We derive a Monte Carlo Portmanteau test to check the validity of the weak AR representation and propose a method based on extreme residuals clustering to determine whether the AR generating process is causal, noncausal or mixed. An empirical study on simulated and real data illustrates the potential usefulness of the results.