Duality of Autoregressive and Moving Average Models with Application to Economic Time Series

The need to compare the efficiency between the duality of autoregressive and moving average models when modelling economic time series is the motivation behind the research. The research focuses on the investigation and derivation of different orders of autoregressive and moving average models with their dual forms. Error variances for both the original and the dual AR(p =1,2), MA(q=1,2), and Autoregressive Moving Average (ARMA(p,q)) models were also derived as the basis for model performance comparison. Box and Jenkins methodology was adopted for the Empirical analysis. The Autocorrelation function (ACF) and Partial autocorrelation function (PACF) of the stationary time series were the basis for the suggested Autoregressive (AR), Moving Average (MA) and Autoregressive Moving Average (ARMA) models. Empirically, Crude Oil Price, Consumers Price Index, External Reserves and Exchange Rate data extracted from the Central Bank of Nigeria website were used for the analysis. Findings from the study revealed that, the dual models outperformed the original models. The work has further established different regions of duality for autoregressive and moving average models. For AR model, defines the duality interval ( is the parameter estimate of AR model), while for MA model, establishes the region of duality ( is the parameter estimate of MA model). Beyond the intervals are the non-duality regions for autoregressive and moving average models. Further investigation to confirm the duality and non-duality regions for the moving average models is evident in the negative error variances. That means, parameter estimate of MA (θ) that falls outside the interval of duality has a negative error variance in their respective dual models, signifying that such models are non-dual models. In the event of non-duality, the original models with the least error variance are considered the best. This work underscores the need to present the dual forms of the suggested AR, MA and ARMA for model comparison using error variances of both the original and dual models so as to identify, if there exists, region of non-duality.