1. The problem is to understand the methodology of the ARIMA (AutoRegressive Integrated Moving Average) model used in time series forecasting. 2. ARIMA models are used to analyze and forecast time series data by combining three components: autoregression (AR), differencing (I for integration), and moving average (MA). 3. The general ARIMA model is denoted as ARIMA(p,d,q), where: - $p$ is the order of the autoregressive part, - $d$ is the degree of differencing needed to make the series stationary, - $q$ is the order of the moving average part. 4. The methodology involves these steps: 1. **Identification:** Determine if the time series is stationary. If not, apply differencing $d$ times to achieve stationarity. 2. **Estimation:** Choose appropriate values for $p$ and $q$ by examining autocorrelation function (ACF) and partial autocorrelation function (PACF) plots. 3. **Model fitting:** Fit the ARIMA model using the chosen parameters. 4. **Diagnostic checking:** Check residuals to ensure they behave like white noise (no patterns). 5. **Forecasting:** Use the fitted model to predict future values. 5. The AR part models the relationship between an observation and a number of lagged observations: $$ X_t = \phi_1 X_{t-1} + \phi_2 X_{t-2} + \cdots + \phi_p X_{t-p} + \epsilon_t $$ 6. The MA part models the relationship between an observation and a number of lagged forecast errors: $$ X_t = \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + \cdots + \theta_q \epsilon_{t-q} $$ 7. Differencing is applied to remove trends and make the series stationary: $$ Y_t = X_t - X_{t-1} $$ (repeated $d$ times if necessary). In summary, ARIMA models combine these components to capture different aspects of the time series data for effective forecasting.