# How to decide whether to model a time series additively or multiplicatively?

I've been reading a lot of theoretical content on additive and multiplicative decomposition, but when it comes to deciding what type of decomposition to use on my own data I find it kinda hard to do visually. For example, these are some of my data decomposed using STL, plotting the raw data (quarterly) vs. the trend cycle.

What kind of "pattern" should I pay attention to? Is there a more objective way to decide the type of decomposition, rather than visually?

For the visual decomposition, with quarterly data you can expect various scenarios when examining the time series plot of your data:

1) no trend and no seasonality;
2) trend but no seasonality;
3) no trend but seasonality;
4) trend and seasonality.


Trend can be deterministic (i.e., a straight line, whose slope does not change over time) or stochastic. The following link on this forum explains the difference between a deterministic and a stochastic trend: Explain what is meant by a deterministic and stochastic trend in relation to the following time series process?).

Similarly, seasonality can be deterministic or stochastic (as explained, for example, here: https://thesamuelsoncondition.com/2016/01/23/time-series-iii-deterministicstochastic-seasonality/).

For simplicity, let's say that scenarios 2) - 4) described above refer to deterministic trend and deterministic seasonality.

Let's also leave aside scenario 1), as there are no temporal components to extract for it, such as trend and seasonality. However, if you had to analyze a time series under scenario 1), you would have to examine the variability/spread of its values over time. If that variability is roughly the same across time, then you would not need to transform the series prior to analyzing it. Otherwise, you may need to transform the series first - perhaps using a Box-Cox transformation - and then analyze it.

Scenario 2

For scenario 2), you would examine the variability of the values of the time series over time about the trend line.

If that variability is roughly constant over time, use an additive decomposition on the time series.

If the variability roughly increases/decreases over time, apply a transformation to the time series first to stabilize its variability across time and then use an additive decomposition on the transformed time series to extract its trend plus the short-term fluctuations.

Scenario 3

For scenario 3, you would examine the variability of the values of the time series over time about the seasonal cycles and whether or not the cycles amplify/diminish over time.

If that variability is roughly constant over time and the cycles don't amplify/diminish over time, use an additive decomposition on the time series to extract its seasonal cycles plus the short-term fluctuations.

If the variability roughly increases/decreases over time but the cycles don't amplify/diminish over time, apply a transformation to the time series first to stabilize its variability across time and then use an additive decomposition on the transformed time series to extract its seasonal cycles plus the short-term fluctuations.

If the variability roughly increases/decreases over time and the cycles amplify/diminish over time, apply a transformation to the time series first to stabilize its variability across time and hopefully that transformation may also induce the seasonal cycles to not amplify/diminish over time. Then proceed as described earlier.

Scenario 4

Keep track of the variability of the values of the time series over time and also of whether or not the seasonal cycles amplify/diminish over time, then proceed accordingly.