Additive models

We assumed earlier that the four time series components affect it additively.

Deseasonalised  =   Trend   +   Cyclical   +   Residual

This implies that:

The seasonal pattern has greater magnitude at the end of the following time series than at its start, so an additive model is not appropriate.

Multiplicative models

In many time series, the absolute differences in the values are of less interest and importance than the percentage changes. A multiplicative model assumes that seasonal and other effects act proportionally on the series:

Data   =   (Seasonal effect)   ×   Trend   ×   Cyclical   ×   Residual

After taking logarithms (either natural logarithms or to base 10), the four components of the time series again act additively.

log(Data)   =   log(Seasonal effect   ×   Trend   ×   Cyclical   ×   Residual)
=   log(Seasonal effect)  +  log(Trend)  +  log(Cyclical)  +  log(Residual)

To fit a multiplicative model, analyse the logarithms of the data with an additive model.