# why the non-seasonal and seasonal parts are multiplied in ARIMA models?

I would like to understand why the non-seasonal and seasonal parts are multiplied in Seasonal ARIMA models.

To be more specific: when we use the Seasonal ARIMA model we assume a multiplicative model and the ARIMA order is represented as: ARIMA (p,d,q)x(P,D,Q). (https://www.otexts.org/fpp/8/9)

so for e.g the order of SARIMA(1,0,0)x(1,0,0) can be represented mathematically as:

1. so why are they multiplied from the first place?
2. is there an additive model which can be applied here?
3. if there is, why is it not used?

all books of time series analysis that I've checked, just assume that the non-seasonal and seasonal parts should be multiplied, but non have given an explanation why.. I've asked around people who worked with ARIMA but non could help me with that, so I would really appreciate if someone could shed a light on that! a reference would be also very helpful.

why are they multiplied from the first place?

To produce a model where the seasonal component enters multiplicatively? It makes a kind of intuitive sense that it might work that way, and often seems to work okay in practice. Indeed, you so often see it (at least approximately) in the diagnostic plots (ACF and PACF) that trying this seems uncontroversial.

More prosaically, an ARIMA model is already conceived of as a product of terms:

$y_t = (1+\theta(B))(1-B)^{-\delta}(1-\phi(B))^{-1} \varepsilon_t$

So adding products in for seasonality keeps that nice structure - and with seasonal differencing, it stays in that same framework. In a sense this allows us to conceive of and model the seasonality and the "ordinary" ARIMA separately in a way we couldn't do so readily if the model were additive.

is there an additive model which can be applied here?

Sure, an AR model with lags at 1 and 60 is one such example

One with lags at 1, 60 and 61 is even more like the seasonal model and would include the multiplicative model as a special case.

if there is, why is it not used?

What makes you say it's not? Distributed lag models can be used on data with seasonality

• Given the OP's model an AR model with lags 1,60 and 61 would be a better example whereas his model implicitely constrains the coefficient for lag 61 to be the product of the other two coefficients ( 1 and 60). The multiplicative model has parsimony BUT may be too simple. – IrishStat Jan 27 '15 at 17:12
• @IrishStat I pondered for several minutes whether to discuss the simpler case or the case that was closer to the seasonal model (and indeed generalized it). I ended up choosing the simpler one but pointing out that it was one possibility (since that other one was at the back of my mind as a second one). I'll edit. – Glen_b Jan 27 '15 at 17:20