9.1 Seasonal differencing and SARIMA models

Seasonal differencing and SARIMA models are powerful tools for handling time series data with recurring patterns. They remove seasonal components, making data more stationary and easier to analyze. This technique is crucial for accurate forecasting and understanding underlying trends.

SARIMA models extend ARIMA by incorporating seasonal elements. They capture both short-term and long-term patterns in data, making them ideal for complex time series. Understanding these models helps in making better predictions and decisions based on seasonal data.

Seasonal Differencing and SARIMA Models

Seasonal differencing in time series

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• Technique to remove seasonal patterns from time series data by computing the difference between observations separated by a seasonal period (12 months, 4 quarters)
• Denoted as $\nabla_s^[D](https://www.fiveableKeyTerm:d) X_t = (1 - B^[s](https://www.fiveableKeyTerm:s))^D X_t$, where $s$ is the seasonal period and $D$ is the order of seasonal differencing
• Helps achieve stationarity, a key assumption for many time series models including SARIMA, by removing the seasonal component and leaving only trend and irregular components
• The order of seasonal differencing ($D$) depends on the strength and persistence of the seasonal pattern, with higher values necessary for strong and persistent patterns

Structure of SARIMA models

• Extend the ARIMA framework to incorporate seasonal components, denoted as SARIMA($[p](https://www.fiveableKeyTerm:p),d,[q](https://www.fiveableKeyTerm:q)$)($P,D,Q$)$_s$
  • $p$: Non-seasonal autoregressive order
  • $d$: Non-seasonal differencing order
  • $q$: Non-seasonal moving average order
  • $P$: Seasonal autoregressive order
  • $D$: Seasonal differencing order
  • $Q$: Seasonal moving average order
  • $s$: Seasonal period
• Seasonal autoregressive (SAR) component captures dependency between observations separated by seasonal periods, denoted as $\Phi_P(B^s) = 1 - \Phi_1 B^s - \Phi_2 B^{2s} - ... - \Phi_P B^{Ps}$
• Seasonal moving average (SMA) component captures dependency between error terms separated by seasonal periods, denoted as $\Theta_Q(B^s) = 1 - \Theta_1 B^s - \Theta_2 B^{2s} - ... - \Theta_Q B^{Qs}$

Order of seasonal differencing

• Determined by examining the seasonal pattern in the time series
  • $D=0$ (no seasonal differencing) if the seasonal pattern is constant over time
  • $D=1$ (first-order seasonal differencing) if the seasonal pattern varies proportionally to the level of the series
  • Higher-order seasonal differencing ($D>1$) may be necessary in rare cases
• Visual inspection of the time series plot helps identify the need for seasonal differencing
  • Constant seasonal patterns over time suggest $D=0$
  • Seasonal patterns that increase or decrease with the level of the series suggest $D=1$
• Autocorrelation function (ACF) and partial autocorrelation function (PACF) can also help determine the order of seasonal differencing, with significant spikes at seasonal lags suggesting the need for seasonal differencing

Interpretation of seasonal ARIMA terms

• Seasonal autoregressive (SAR) terms capture the relationship between observations separated by seasonal periods
  • The order of the SAR term ($P$) indicates the number of seasonal lags influencing the current observation
  • Coefficients of the SAR terms ($\Phi_1, \Phi_2, ..., \Phi_P$) represent the strength and direction of the seasonal autocorrelation
• Seasonal moving average (SMA) terms capture the relationship between error terms separated by seasonal periods
  • The order of the SMA term ($Q$) indicates the number of seasonal lags of error terms influencing the current observation
  • Coefficients of the SMA terms ($\Theta_1, \Theta_2, ..., \Theta_Q$) represent the strength and direction of the seasonal correlation between error terms
• Significance of SAR and SMA coefficients assessed using statistical tests (t-tests, confidence intervals)
  • Significant coefficients suggest presence of seasonal patterns in the time series
  • Non-significant coefficients may indicate the corresponding seasonal terms can be removed from the model