SARIMA Model

Describing a SARIMA Model

• A SARIMA model is an ARIMA model that accounts for seasonality
• Here, seasonality just refers to a repeating pattern within a year (i.e. can't be a two-year pattern)

• A SARIMA model is defined as the following:

  $$ARIMA(p,d,q)(P,D,Q)m$$
  • Where $p$ refers to the order of the autoregressive portion of the ARIMA model
  • Where $d$ refers to the order of the integrated portion of the ARIMA model (i.e. how many times we take a difference to get our model to be stationary)
  • Where $q$ refers to the order of the moving average portion of the ARIMA model
  • Where $m$ refers to the seasonal factor (i.e. the number of distinct periods within a year that make up a single seasonal pattern that is repeated over time)
  • Where $P$ refers to the order of the autoregressive portion of the ARIMA model, with an additional seasonal component
  • Where $D$ refers to the order of the integrated porition of the ARIMA model, with an additional seasonal component
  • Where $Q$ refers to the order of the order of the moving average porition of the ARIMA model, with an additional seasonal component
• A SARIMA model can also be written as an ARIMA model:

$$ARIMA(p,d,q)_{1}(P,D,Q)m$$

• This is because we're technically backshifting parameters $p, d,$ and $q$ by $1$ (i.e. $s_{t+1} - s_{t}$)

Example of SARIMA Model

• Let's say we define the following SARIMA model:

$$ARIMA(1,0,0)(0,1,1)_{4}$$

• The $p$ parameter with a value equal to $1$ can be translated to the following:

  $$(1-\phi_{1}B^{1})y_{t}$$
  • Where $B^{1}y_{t}$ is our lag operator representing $y_{t-1}$
  • Where $\phi_{1}$ is the slope of our variable $y_{t-1}$
  • Therefore, $(1-\phi_{1}B^{1})y_{t}$ is just shorthand for $y_{t}-\phi_{1}y_{t-1}$

• The $D$ parameter with a value equal to $1$ can be translated to the following:

  $$(1-B^{4})y_{t}$$
  • Where $B^{4}y_{t}$ is our lag operator representing $y_{t-4}$
  • Therefore, $(1-B^{4})y_{t}$ is just shorthand for $z = y_{t}-y_{t-4}$

• The $Q$ parameter with a value equal to $1$ can be translated to the following:

  $$(1+\theta_{1}B^{4})\epsilon_{t}$$

  • Where $B^{4}\epsilon_{t}$ is our lag operator representing $\epsilon_{t-4}$

    • Where $\epsilon_{t-4}$ is the error of the fourth previous predicted value and what we observed
    • Where $\theta_{1}$ represents the percentage of the error $\epsilon_{t-4}$ we should include in our model
    • Therefore, $(1-\theta_{1}B^{4})\epsilon_{t}$ is just shorthand for $\epsilon_{t}+\theta_{1}\epsilon_{t-4}$
• We can continue to simplify our SARIMA model to the following formulas:

$$(1-\phi_{1}B^{1})(1-B^{4})y_{t} = (1+\theta_{1}B^{4})\epsilon_{t}$$ $$(1-\phi_{1}B^{1}-B^{4}+\phi_{1}B^{5})y_{t} = \epsilon_{t}+\theta_{1}\epsilon_{t-4}$$ $$y_{t} - \phi_{1}y_{t-1} - y_{t-4} + \phi_{1}y_{t-5} = \epsilon_{t} + \theta_{1}\epsilon_{t-4}$$ $$y_{t} - y_{t-4} = \phi_{1}y_{t-1} - \phi_{1}y_{t-5} + \theta_{1}\epsilon_{t-4} + \epsilon_{t}$$ $$z_{t} = \phi_{1}z_{t-1} + \theta_{1}\epsilon_{t-4} + \epsilon_{t}$$

• Since $z = y_{t}-y_{t-4}$ is based on parameter $D$
• Now, we've transformed our ARIMA model to account for seasonality by taking differences $y_{t} - y_{t-4}$, which will lead to a more stationary model

References

• SARIMA Video
• Examples of SARIMA Models