Integer Differencing and ARIMA Process with White Noise

This chapter introduces the concept of integer differencing and develops the autoregressive integrated moving average (ARIMA) model for handling nonstationary time series. The chapter begins by establishing the importance of stationarity in time series modeling and presents various methods for detecting nonstationarity, including visual inspection, summary statistics comparison, autocorrelation plot analysis, and the augmented Dickey–Fuller (ADF) test for unit roots. The mathematical foundation of integer differencing is developed, showing how first-order, second-order, and seasonal differencing transformations can achieve stationarity. The ARIMA(p,d,q) model is formally defined, where d represents the integer degree of differencing required to transform a nonstationary series to stationarity. Special cases including the random walk, random walk with drift, and exponential smoothing models are identified as ARIMA specifications. The chapter discusses order selection using AIC, AICc, and BIC criteria, and presents forecasting methodology using the cascade representation. An important extension, the seasonal ARIMA (SARIMA) model, is introduced with its characteristic polynomial factorization for handling series with seasonal patterns. The chapter establishes the mathematical groundwork for the fractional differencing concepts that distinguish long memory processes in subsequent chapters.