5.5 Seasonal ARIMA (SARIMA) Models

SARIMA models take ARIMA to the next level by handling seasonal patterns in time series data. They add seasonal components to capture repeating patterns, like yearly cycles in monthly data. This extension makes SARIMA models super useful for forecasting seasonal trends.

SARIMA models use seasonal differencing to remove seasonal components and achieve stationarity. They can capture both short-term and long-term dependencies in data. The Box-Jenkins method helps build these models, involving identification, estimation, and diagnostic checking steps.

SARIMA Model Principles

Extending ARIMA Models for Seasonal Patterns

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• SARIMA models extend ARIMA models to handle seasonal patterns in time series data by incorporating seasonal autoregressive (SAR), seasonal moving average (SMA), and seasonal differencing terms
• The general form of a SARIMA model is denoted as SARIMA(p,d,q)(P,D,Q)m
  • p, d, q represent the non-seasonal components
  • P, D, Q represent the seasonal components
  • m is the seasonal period (e.g., m=12 for monthly data with a yearly seasonal cycle)

Components of SARIMA Models

• The seasonal autoregressive (SAR) term captures the relationship between an observation and its corresponding seasonal lag
• The seasonal moving average (SMA) term captures the relationship between an observation and the residual error from the corresponding seasonal lag
• Seasonal differencing is performed to remove the seasonal component and achieve stationarity in the time series
  • Involves subtracting the observation from the corresponding observation in the previous seasonal cycle (e.g., for monthly data with a yearly cycle, the seasonal difference is computed as $y_t - y_{t-12}$)

SARIMA Model Properties

Capturing Short-term and Long-term Dependencies

• SARIMA models are capable of capturing both short-term and long-term (seasonal) dependencies in time series data
• The seasonal period (m) represents the number of observations per seasonal cycle (e.g., m=12 for monthly data with a yearly seasonal cycle)
• The seasonal autoregressive order (P) determines the number of seasonal lags used in the model
• The seasonal moving average order (Q) determines the number of seasonal lags of the forecast errors

Achieving Stationarity through Seasonal Differencing

• The seasonal differencing order (D) specifies the number of seasonal differences required to remove the seasonal component and achieve stationarity
• Seasonal differencing is applied by subtracting the observation from the corresponding observation in the previous seasonal cycle
• If the time series exhibits both trend and seasonality, both regular differencing (d) and seasonal differencing (D) may be required

SARIMA Modeling for Forecasting

Box-Jenkins Methodology for Model Construction

• The Box-Jenkins methodology is commonly used to construct SARIMA models, which involves:
  1. Model identification: Examining the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots to determine the appropriate orders for the seasonal and non-seasonal components
  2. Parameter estimation: Using maximum likelihood estimation or other optimization techniques to estimate the coefficients of the SARIMA model
  3. Diagnostic checking: Assessing the adequacy of the fitted model by examining the residuals for normality, independence, and constant variance

Model Selection and Comparison

• Goodness-of-fit measures, such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), can be used to compare different SARIMA models and select the most appropriate one
  • Lower values of AIC and BIC indicate better model fit
• The chosen SARIMA model should be validated on a holdout sample or using cross-validation techniques to ensure its generalizability and robustness

Seasonal Differencing in SARIMA

Identifying Seasonal Differencing Order

• Seasonal differencing is performed to remove the seasonal component and achieve stationarity in the time series
• The seasonal differencing order (D) is determined by examining the ACF and PACF plots for significant spikes at seasonal lags
  • If the ACF shows a slow decay at seasonal lags, seasonal differencing may be necessary
• The number of seasonal differences required depends on the strength and persistence of the seasonal pattern

Applying Seasonal Differencing

• Seasonal differencing is applied by subtracting the observation from the corresponding observation in the previous seasonal cycle
  • For example, for monthly data with a yearly cycle, the seasonal difference is computed as $y_t - y_{t-12}$
• If the time series exhibits both trend and seasonality, both regular differencing (d) and seasonal differencing (D) may be required
  • Regular differencing removes the trend component, while seasonal differencing removes the seasonal component

SARIMA Model Evaluation

Interpreting Model Coefficients and Residuals

• The estimated coefficients of the SARIMA model provide insights into the significance and magnitude of the seasonal and non-seasonal components
• The residuals of the fitted SARIMA model should exhibit properties of white noise, indicating that the model has captured the relevant information in the time series
  • Residuals should be normally distributed, independent, and have constant variance

Assessing Forecast Accuracy

• Forecast accuracy metrics, such as mean squared error (MSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), can be used to assess the performance of the SARIMA model in generating out-of-sample forecasts
  • Lower values of these metrics indicate better forecast accuracy
• The chosen SARIMA model should be validated on a holdout sample or using cross-validation techniques to ensure its generalizability and robustness
  • This helps assess how well the model performs on unseen data and avoids overfitting