Laurent Series Expansion for MA(∞) Representation of Mixed Causal–Noncausal Autoregressive Processes

This note develops a rigorous analytical framework for computing exact MA() coefficients of mixed causal–noncausal autoregressive MAR processes. While analytical solutions exist only for the MAR specification in the existing literature (Gouriéroux and Jasiak, 2016), general MAR processes are typically handled through recursive approximation algorithms that suffer from numerical approximations. Using complex contour integration and the residue theorem, we derive explicit closed‐form expressions valid for arbitrary orders . The derived expressions enable direct simulation algorithms, eliminating the recursive approximation bias while retaining only the standard truncation error inherent to any finite‐order approximation, and facilitate implementation of forecasting methodologies. Numerical comparisons with existing recursive methods demonstrate improvements in accuracy, and an experiment with ‐stable innovations illustrate the empirical relevance of our results.