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

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

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 :

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 $$Y_t-2Y_{t-1}+Y_{t-2}=e_t+\Theta_1e_{t-12}+\theta_1e_{t-1}+\theta_1\Theta_1e_{t-13}$$