ARMA Process and Box–Jenkins Model

This chapter extends the foundational AR and MA concepts by presenting the autoregressive moving average (ARMA) model, which combines both components into a unified framework. The ARMA(p,q) model is developed mathematically using the lag operator formulation, demonstrating how the AR and MA polynomials interact to describe stationary stochastic processes. The chapter traces the historical development of these models from Peter Whittle’s 1951 thesis through their popularization by Box and Jenkins in 1970. A detailed exposition of the Box–Jenkins methodology is provided, covering the iterative three-stage approach: model identification, parameter estimation, and diagnostic checking. The chapter discusses methods for selecting appropriate model orders p and q, including the use of ACF/PACF plots and information criteria (AIC, BIC). The spectral density of ARMA processes is derived, and extensions such as ARMAX models (incorporating exogenous inputs) are introduced. Implementation guidance using statistical software packages including R, MATLAB, Python, Stata, and SAS is provided. The chapter concludes by highlighting the Box–Jenkins model as a bridge between ARMA and the more general ARIMA framework discussed in the following chapter.