Mixed causal-noncausal autoregressions with exogenous regressors

Abstract

The mixed causal-noncausal autoregressive (MAR) model has been proposed to estimate time series processes involving explosive roots in the autoregressive part, as it allows for stationary forward and backward solutions. Possible exogenous variables are substituted into the error term to ensure the univariate MAR structure of the variable of interest. To study the impact of fundamental exogenous variables directly, we instead consider a MARX representation which allows for the inclusion of exogenous regressors. We argue that, contrary to MAR models, MARX models might be identified using second-order properties. The asymptotic distribution of the MARX parameters is derived assuming a class of nonGaussian densities. We assume a Student’s t-likelihood to derive closed form solutions of the corresponding standard errors. By means of Monte Carlo simulations, we evaluate the accuracy of MARX model selection based on information criteria. We examine the influence of the U.S. exchange rate and industrial production index on several commodity prices.