Difference between additive and multiplicative models

Multiplicative models for time series are very widely used. It is therefore worth ensuring that you can recognise the features that suggest a multiplicative model.

Additive and multiplicative models can only be easily distinguished when there is noticeable trend in the time series.

Additive model
The seasonal, cyclical and residual variation around the trend line are of approximately the same size whether the trend value is low or high.
Multiplicative model
The seasonal, cyclical and residual variation around the trend line are greater when the trend is at high values than when the trend is low.

Additive model for New Zealand visitors

The time series plot below describes visitor arrivals in New Zealand (in thousands).

Click the checkboxes Trend and then Seasonal to fit an additive model to these data. Observe that the seasonal fluctuations in the data are greater in recent years, but the 'best' additive model cannot match this.

As a result, the additive model over-estimates the seasonal fluctuations in the 1980's but under-estimates them in the 1990s. Click on values in the latest years to verify that the model under-estimates tourist numbers in the New Zealand summer (Q4 and Q1) and over-estimates tourist numbers in the winter (Q2 and Q3).

Quarterly forecasts based on the additive model will therefore similarly under-estimate tourist numbers in the New Zealand summer and over-estimate them in the winter.

Multiplicative model for New Zealand visitors

The time series plot below shows the visitor arrivals on a log scale.

The multiplicative model is equivalent to fitting an additive model to the log data. Click the checkboxes Trend and then Seasonal to display these components. On the log scale, the seasonal effect is the same throughout the time period.

Now use the slider to adjust the scale back to a plot of the original data. The multiplicative model implies lower seasonal variation when visitor arrivals are low, and higher seasonal variation when visitor numbers are high. This matches the pattern in the data well.

This multiplicative model is likely to give better forecasts than the additive model — there is no tendency to systematically over-estimate or under-estimate quarters in the most recent years.