Mixed Causal-Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing

This paper stresses the bimodality of the widely used Student's t likelihood function applied in modelling Mixed causal-noncausal AutoRegressions (MAR). It first shows that a local maximum is very often to be found in addition to the global Maximum Likelihood Estimator (MLE), and that standard estimation algorithms could end up in this local maximum. It then shows that the issue becomes more salient as the causal root of the process approaches unity from below. The consequences are important as the local maximum estimated roots are typically interchanged, attributing the noncausal one to the causal component and vice-versa, which severely changes the interpretation of the results. The properties of unit root tests based on this Student's t MLE of the backward root are obviously affected as well. To circumvent this issues, this paper proposes an estimation strategy which i) increases noticeably the probability to end up in the global MLE and ii) retains the maximum relevant for the unit root test against a MAR stationary alternative. An application to Brent crude oil price illustrates the relevance of the proposed approach.