Journal of Applied Mathematics and Physics, 2022, 10, 589-609 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 DOI: 10.4236/jamp.2022.102043 Feb. 28, 2022 589 Journal of Applied Mathematics and Physics On the Paradox of the Duality of Autoregressive and Moving Average Processes Elechi Onyemachi, Iheanyi Sylvester Iwueze, Eleazar Chukwunenye Nwogu Department of Statistics, Federal University of Technology, Owerri, Nigeria Abstract A widely held view in time series analysis is the concept of duality that a finite order stationary autoregressive process of order p (AR(p)) is equivalent to an infinite order moving average (MA) process and a finite order invertible moving average of order q (MA(q)) is equivalent to an infinite order autore- gressive (AR) process. The purpose of this paper is to demonstrate that the concept is not universally true. Thus, a finite order stationary autoregressive process of order p (AR(p)) can be written as an finite order moving average process and a finite order moving average process of order q (MA(q)) can be written as a finite order stationary autoregressive process. The regions of breakdown of concept of duality were determined for 1, 2 p q = = using me- thod of moments. The method involves equating non-zero autocovariances of the stationary AR(p) to the equivalent non-zero autocovariances of the in- vertible MA(p) to determine the region of non-duality. In such region of breakdown in duality, 1) both the Autocorrelation function and the Partial Autocorrelation function of the AR process and MA process cuts off after equal lags 2) a finite AR model can be adequately represented by a finite MA model of equal order and conversely with the same error variance and 3) negative values of the parameters of the AR process are equal in magnitude but opposite in direction to the parameters of the equivalent MA process and conversely. Empirical examples (simulation and real life examples) were used to illustrate these. Therefore, it has been recommended that caution should be exercised in using the concept of duality in time series analysis until future research proves otherwise. Keywords Duality, Non-Duality, Method of Moments, Quadratic Inequality, Stationarity Region, Invertibility Region How to cite this paper: Onyemachi, E., Iwueze, I.S. and Nwogu, E.C. (2022) On the Paradox of the Duality of Autoregressive and Moving Average Processes. Journal of Applied Mathematics and Physics, 10, 589- 609. https://doi.org/10.4236/jamp.2022.102043 Received: March 18, 2021 Accepted: February 25, 2022 Published: February 28, 2022 Copyright © 2022 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access