Artificial data
Explain first that a multiplicative model is equivalent to an additive model fitted to the logarithms of the values.
log(Data) | = log(Seasonal effect × Trend × Cyclical × Residual) |
= log(Seasonal effect) + log(Trend) + log(Cyclical) + log(Residual) |
Illustrative example
In the first example, the values increase by 20% each year, except for two years — 1983 and 1998 — in each of which the values decrease by 20%. However it is easy to overlook the 20% decrease in 1983.
Drag the slider to Log scale and observe that the time series is linearised and the two years with decreases appear the same.
Personal disposable income
On a log scale, the trend is fairly linear until about 1985, implying a constant percentage increase each year. However there has been a flattening out of the time series in recent years — the annual percentage increase in personal disposable income has become smaller.
Visitor arrivals in New Zealand
Again it is percentage changes that are most meaningful, suggesting use of a multiplicative model (and therefore log scale). This makes the seasonal variation more similar in different years.
The first example is an artificial one.
The second example shows how US personal disposable income (in billions of dollars) changed between 1959 and 2001.
The third example is a seasonal time series. It describes the numbers of short-term visitor arrivals in New Zealand (in thousands) each quarter from 1980 to 2000.