Description

- **Core Objective:** To detail the process of building ARMA and ARIMA models for time series analysis and forecasting.
- **Model Building Steps:** The chapter outlines a structured approach: model identification, parameter estimation, diagnostic checking, and forecasting.
- **Estimation Methods:** Several techniques are explained for estimating model parameters, including the Method of Moments (Yule-Walker), Conditional Least Squares (LS), and Maximum Likelihood (ML).
- **Order Determination:** Information criteria like AIC (Akaike Information Criterion) and BIC (Bayesian Information Criterion) are used to select the optimal ARMA(p,q) orders by balancing model fit with complexity.
- **Model Diagnosis:** The Ljung-Box test is the primary tool for checking model adequacy. It tests the null hypothesis that the model's residuals are independent (white noise).
- **Worked Example (NAO):** The North Atlantic Oscillation (NAO) index is used as a case study to demonstrate fitting an ARMA(3,2) model and validating it using ACF/PACF plots and the Ljung-Box test.
- **Worked Example (US Bills):** Monthly U.S. government bill rates are analyzed to compare different ARIMA models, specifically ARIMA(6,1,7) versus ARIMA(0,1,1), using residual analysis and Ljung-Box tests.
- **Mathematical Foundation:** The document provides the theoretical background for key statistics, such as the confidence intervals for ACF under a white noise null hypothesis.
- **Likelihood in Detail:** A detailed comparison is made between the exact and conditional likelihood functions, illustrated with the estimation challenges of an MA(1) process.
- **Practical Implementation:** References to Python functions (e.g., from the `statsmodels` library) are provided throughout for tasks like order selection, model fitting, and diagnostic plotting.