The process of estimating and forecasting with ARIMA models encompasses several essential steps. After identifying and analyzing a time series, the next phase involves determining suitable values for the model parameters (p, d, q). This often entails scrutinizing autocorrelation and partial autocorrelation plots to guide the selection of autoregressive and moving average orders. To achieve stationarity, differencing is applied, and the order of differencing (d) is determined accordingly.
The estimation of ARIMA parameters typically employs maximum likelihood estimation (MLE) methods. Subsequently, the model is fitted to historical data, and the residuals (differences between observed and predicted values) undergo examination to ensure the absence of significant patterns, indicating a well-fitted model.
Once the ARIMA model is successfully estimated and validated, it becomes a valuable tool for forecasting future values of the time series. Forecasting involves advancing the model forward in time, generating predicted values based on the estimated autoregressive and moving average parameters. Additionally, confidence intervals can be computed to offer a measure of uncertainty around the point forecasts.
Despite the widespread utilization of ARIMA models, they have limitations, such as assuming linearity and stationarity. In practical applications, other advanced time series models like SARIMA (Seasonal ARIMA) or machine learning approaches may be employed to address these limitations and enhance forecasting accuracy. Nevertheless, ARIMA models retain their value as an accessible and valuable tool for time series analysis and forecasting.