Misspecification of Causal and Noncausal Orders in Autoregressive Processes

This paper examines the consequences of estimating a past-dependent (causal) AR model from data generated by a stationary noncausal process with a future-dependent component. We show that the outcomes of that estimation depend on the noncausal persistence. When the noncausal persistence is strong, the (pseudo)-ML estimator of the misspecified causal model is consistent, and reveals the presence of a noncausal component as long as the sample is sufficiently non-Normal. When the noncausal persistence is weak, the (pseudo)-ML estimator is inconsistent and leads to a misleading conclusion that the fit of the past-dependent model is correct. The results are derived theoretically from examining the binding function, and illustrated by simulations of noncausal AR processes with errors that follow mixture distributions with varying proportions of Normal and Cauchy variables.