Spectral estimation for mixed causal-noncausal autoregressive models

Mixed causal-noncausal autoregressive (MAR) processes driven by non Gaussian noise can replicate the non linear dynamics induced by local explosive episodes observed in financial bubbles. MAR models cannot be identified using second-order moments because they share spectral density with a set of different representations. In this study, we propose an identification and estimation method based on the third-order spectral density cumulant that can recover the complete probability structure of the errors without assuming any prior knowledge of the probability distribution function. Monte Carlo experiments demonstrated the estimation and identification performances. Furthermore, we illustrated the adequacy of our method through an empirical application to eight monthly commodity prices. The results show that MAR models can effectively capture the explosiveness and bubble phenomena generated in the commodities market.