Ubiquitous multimodality in mixed causal-noncausal processes

According to the literature, the bimodality of estimates in mixed causal–non-causal autoregressive processes is due to unlucky starting values and happens only ocassionally. This paper shows that a unique and convergent solution is not always the case for models of this class. Instead, the likelihood function is not convex leading to the multimodality of estimated parameters. It can be attributed to the magnitude and sign of the autoregressive coefficients. Simultaneously, the number of local modes grows with the number of autoregressive parameters in the model. This multimodality depends on the parameters of the process and the chosen error distribution. We have to apply grid search methods to extract candidate solutions. The independence of residuals is a necessary hypothesis for the proper identification of the processes. A simple AIC criterion helps to select an independent model. Finally, I sketch a roadmap on estimating mixed causal-noncausal autoregressive models and illustrate the approach with Brent spot oil price returns.