Description

- **Core Topic:** The chapter focuses on extending ARIMA models to handle seasonality, introducing Seasonal ARIMA (SARIMA) and REGARMA models for nonstationary data.
- **Box-Jenkins Method:** It begins by outlining the Box-Jenkins approach for seasonal data, emphasizing seasonal differencing to remove periodic patterns.
- **SARIMA Structure:** A SARIMA model is defined by both non-seasonal (p,d,q) and seasonal (P,D,Q)s components, capturing regular and seasonal autocorrelation.
- **Model Building:** The process involves achieving stationarity through differencing, identifying orders from ACF/PACF plots, fitting the model, and performing residual diagnostics (Ljung-Box test).
- **Example 5.1 (Female Employment):** A SARIMA(2,1,2)(0,1,1)12 is initially fitted but rejected after diagnostics, leading to a more parsimonious SARIMA(2,1,0)(0,1,1)12.
- **Simulation Examples:** Several simulated SARMA processes (e.g., SMA(1)(1)4, SARMA(1,1)(1,1)4) are used to demonstrate model identification, fitting, and validation.
- **Model Comparison (GDP Data):** SARIMA(2,1,0)(0,1,0)4 and SARIMA(1,1,1)(0,1,0)4 are fitted to log-transformed Chinese quarterly GDP and compared using AIC/BIC.
- **Parsimony Principle:** Overly complex models are shown to be inferior; dropping insignificant seasonal terms (e.g., in SARIMA(0,1,2)(0,1,3)4) improves fit and information criteria.
- **REGARMA Models:** The chapter introduces REGARMA models, which combine harmonic seasonal regression with ARMA errors to handle both deterministic and stochastic seasonality.
- **Temperature Volatility Case Study:** A Southern Hemisphere temperature series is modeled by first fitting a harmonic seasonal regression, then applying ARMA(2,[4]) to the residuals, culminating in a validated REGARMA model.