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## 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)×(ps,Ds,qs)s is given by

 $\varphi \left(L\right)\Phi \left(L\right){\left(1-L\right)}^{D}{\left(1-{L}^{s}\right)}^{{D}_{s}}{y}_{t}=c+\theta \left(L\right)\Theta \left(L\right){\epsilon }_{t}.$ (6-15)

Here, the stable, degree p AR operator polynomial $\varphi \left(L\right)=\left(1-{\varphi }_{1}L-\dots -{\varphi }_{p}{L}^{p}\right)$, and $\Phi \left(L\right)$ is a stable, degree ps AR operator of the same form. Similarly, the invertible, degree q MA operator polynomial ${\theta }_{q}\left(L\right)=\left(1+{\theta }_{1}L+\dots +{\theta }_{q}{L}^{q}\right)$, and $\Theta \left(L\right)$ is an invertible, degree qs 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 6-15.

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