Assessing Forecast Accuracy Through Invertibility in Univariate ARIMA Models: The Case of MA(1) and MA(2) Without Linear Trend

Abstract

This study presents a generalized framework for the invertibility of univariate Moving Average models of order one and two, MA(1) and MA(2), through their infinite autoregressive representations. Table 1 summarizes the derived AR(∞) coefficients obtained by inverting the MA(1) and MA(2) models, where the autoregressive parameters follow systematic patterns governed by the Pascal triangle principle for MA(2). The study further determined the order q required for the inverted models to ensure convergence, emphasizing that invertibility holds when the MA parameters satisfy the constraints [-1≤θ≤1] for MA(1) and appropriate bounds on (and) for MA(2). Microsoft Excel was employed to compute the coefficients, and Minitab was used to assess model adequacy through AIC and BIC criteria derived from ACF and PACF plots. The results address the study objectives by establishing parameter adequacy for invertibility and summarizing the parameter estimates for MA(1) and MA(2) to AR(∞) model. This study recommended the summarized table 23 to statisticians, econometrists and time series researchers to explore when the inverting MA(q) model to AR(∞) model. It will help identify those MA(q) models which cannot be inverted, because the parameters are not interpretable, this is essential for accurate forecasting.