## Multiplicative ARIMA Model

Many time series collected periodically (e.g., quarterly or
monthly) exhibit a seasonal trend, meaning there is a relationship
between observations made during the same period in successive years.
In addition to this seasonal relationship, there can also be a relationship
between observations made during successive periods. The multiplicative
ARIMA model is an extension of the ARIMA model that addresses seasonality
and potential seasonal unit roots [1].

In lag operator polynomial notation, $${L}^{i}{y}_{t}={y}_{t-i}$$. For a series with periodicity *s*,
the multiplicative ARIMA(*p*,*D*,*q*)×(*p*_{s},*D*_{s},*q*_{s})_{s} is
given by

$$\varphi (L)\Phi (L){(1-L)}^{D}{(1-{L}^{s})}^{{D}_{s}}{y}_{t}=c+\theta (L)\Theta (L){\epsilon}_{t}.$$ | **(5-15)** |

Here, the stable, degree *p* AR operator polynomial $$\varphi (L)=(1-{\varphi}_{1}L-\dots -{\varphi}_{p}{L}^{p})$$, and $$\Phi (L)$$ is a stable, degree *p*_{s} AR
operator of the same form. Similarly, the invertible, degree *q* MA
operator polynomial $${\theta}_{q}(L)=(1+{\theta}_{1}L+\dots +{\theta}_{q}{L}^{q})$$, and $$\Theta (L)$$ is an invertible, degree *q*_{s} MA
operator of the same form.

When you specify a multiplicative ARIMA model using `arima`

,

Set the nonseasonal and seasonal AR coefficients with
the opposite signs from their respective AR operator polynomials.
That is, specify the coefficients as they would appear on the right
side of Equation 5-15.

Set the lags associated with the seasonal polynomials
in the periodicity of the observed data (e.g., 4, 8,... for quarterly
data, or 12, 24,... for monthly data), and not as multiples of the
seasonality (e.g., 1, 2,...). This convention does not conform to
standard Box and Jenkins notation, but is a more flexible approach
for incorporating multiplicative seasonality.

The nonseasonal differencing operator, $${(1-L)}^{D}$$ accounts for nonstationarity
in observations made in successive periods. The seasonal differencing
operator, $${(1-{L}^{s})}^{{D}_{s}}$$, accounts for nonstationarity
in observations made in the same period in successive years. Econometrics Toolbox™ supports
only the degrees of seasonal integration *D*_{s} = 0 or 1. When you specify *s* ≥ 0, Econometrics Toolbox sets *D*_{s} = 1. *D*_{s} = 0 otherwise.

## References

[1] Box, G. E. P., G. M. Jenkins, and G. C.
Reinsel. *Time Series Analysis: Forecasting and Control*.
3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

## See Also

`arima`

## Related Examples

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