Does R's arima() fit / use multiplicative or additive seasonality?

Question

I have searched Cross Validated and read the documentation of the stats package in R, but I cannot figure out, whether the arima() implementation uses additive or multiplicative seasonality terms when fitting SARIMA models. It is also not clear how to choose one or the other through this function

Hope anyone knows and can help - thanks!

Regards

Answer 1

Yes, the man page could've included an example for additive models. Using p,d,q notation yields a multiplicative model:

 fitMult <- arima(x,order=c(0,0,1),seasonal=list(order=c(0,0,1),period=12))

An additive model would be :

fitAdd <- arima(x,order=c(0,0,13),fixed=c(NA,NA,0,0,0,0,0,0,0,0,0,0,NA,NA))

Thus the parameter at lag 13 is estimated freely instead of being the product of the parameters at lag 1 and lag 12. The fixed vector here has 14 terms to account for the mean, in position 1 which is estimated because there is no differencing and the option to include mean is left out. So an additive model is estimated like a non-seasonal model but there are lags at the seasonal periods.

Answer 2

Seasonal ARIMA models are just ARIMA models with AR & MA terms at particular lags & some constraints on the coefficients of those terms; so they're modelling additive seasonality—the variance of the error term doesn't increase with the level of the series. See e.g. here, where it's shown how to express $SARIMA(0,2,1)\times(0,0,1)_{12}$ as $$Y_t-2Y_{t-1}+Y_{t-2}=e_t+\Theta_1e_{t-12}+\theta_1e_{t-1}+\theta_1\Theta_1e_{t-13}$$

A log transformation of the observations is often used when variance does increase with the level: see When to log transform a time series before fitting an ARIMA model.