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# SARIMAX Analysis

In principle, an SARIMAX^{i} model is a linear regression model that uses a SARIMA^{i}-type process (i.e. ) This model is useful in cases we suspect that residuals may exhibit a seasonal trend or pattern.

Where:

- is the lag
^{i}(aka back-shift) operator. - is the observed output at time t.
- is the k-th exogenous input variable at time t.
- is the coefficient value for the k-th exogenous (explanatory) input variable.
- is the number of exogenous input variables.
- is the auto-correlated regression residuals.
- is the order of the non-seasonal AR component.
- is the order of the seasonal AR component.
- is the order of the non-seasonal MA component.
- is the order of the seasonal MA component.
- is the seasonal length.
- is the seasonal integration order of the time series.
- is a constant in the SARIMA model
- is the innovation, shock or error term at time t.
- time series observations are independent and identically distributed (i.e. i.i.d) and follow a Gaussian distribution (i.e. )

Re-ordering the terms in the equation above and assuming the differenced (both seasonal and non-seasonal) results in a stationary time series () yields the following:

## notes

- The variance of the shocks is constant or time-invariant.
- The order of an AR component process is solely determined by the order of the last lagged auto-regressive variable with a non-zero coefficient (i.e. ).
- The order of an MA component process is solely determined by the order of the last moving average variable with a non-zero coefficient (i.e. ).
- In principle, you can have fewer parameters than the orders of the model.

## Files Examples

## References

- Hamilton, J .D.; Time Series Analysis , Princeton University Press (1994), ISBN 0-691-04289-6
- Tsay, Ruey S.; Analysis of Financial Time Series John Wiley & SONS. (2005), ISBN 0-471-690740