# Why is there variation in the trend-cycle component despite using additive decomposition?

So, I've been reading Rob Hyndman's Forecasting book, and I'm now at the part of time series decomposition.

Hyndman states that we use additive decomposition if the trend-cycle component or the seasonal component does not vary with time, and to use multiplicative decomposition if the trend-cycle component or the seasonal component does vary with time.

First, what should we do if the trend-cycle component varies with time but the seasonal component does not vary with time? Could we use either additive decomposition or multiplicative decomposition?

Second, just for clarification, in Figure 6.2 Hyndman uses additive decomposition. Am I correct to say that the trend-cycle component varies with time but the magnitude of the seasonal component does not vary with time(the seasonal component seems to have a consistent pattern)?

The book actually says this (emphasis mine):

The additive decomposition is the most appropriate if the magnitude of the seasonal fluctuations, or the variation around the trend-cycle, does not vary with the level of the time series. When the variation in the seasonal pattern, or the variation around the trend-cycle, appears to be proportional to the level of the time series, then a multiplicative decomposition is more appropriate.

That is:

1. They are referring to the variation around the trend-cycle (the remainder), not the trend-cycle itself; it wouldn't make sense to consider when the "trend-cycle doesn't vary with time" because that wouldn't be a trend at all, it would be a constant.

2. The multiplicative decomposition is not appropriate whenever the remainder and seasonality change over time, but only when that change is proportional to the level of the series. If they change for a different reason (e.g. a sudden break in the seasonality), then that does not indicate that you should use the multiplicative decomposition.

For your first question, it's not inconceivable that a certain series could be better modeled like this:

$$y_t = (S_t + T_t) \times R_t$$

That is, the seasonality is additive but the remainder is multiplicative. You can still theoretically decompose these components but it does not fit into the "additive" vs "multiplicative" boxes. You can decompose much more general relationships between these and other components within the framework of general state space models, for example.

For the second question, look at the first panel, the raw data. Does the seasonal variation look bigger when the series is higher (e.g. higher in 2000/2007 than 1996/2009)? Not really, so yes, that's an indication that you might want to look at the additive decomposition. It's maybe harder to see that the remainder doesn't look like it becomes larger with the level of the series either; there are times when it is larger in magnitude (e.g. end of 2008) but it is not tied to the level of the series.

A series (datasets::AirPassengers) that requires multiplicative decomposition looks more like this:

Edited to add: You can tell because, in the raw data in the first panel, as the level increases, the seasonal variation increases (when the level is low, you can barely see the July-August peaks that become very well defined as the level grows). Once you've done the decomposition, the seasonal component in the second panel does not change with the level; it is the impact of the component on the data that increases with the level because of the multiplication, not the component itself.

It's harder to tell in the raw data that the remainder is also multiplicative, because its effect is small. However, if it had been additive and we assumed it was multiplicative, the estimated component would actually shrink with the level (it would become a smaller percentage of the variation since its size would be constant but the level would grow). The one we estimated is relatively stable (it does get smaller in the middle, but this is unrelated to the level since that is basically always increasing).

You can convince yourself of this by constructing a fictional alternate series out of these components which has the structure $$y_t = (S_t \times T_t) + R_t$$ and then doing the multiplicative decomposition like this (I also increased the size of the remainder so you can see it better):

library(forecast)

decomp <- decompose(AirPassengers, "multiplicative")
AirPassengers.alternate <- (decomp$$seasonal*decomp$$trend) + 1000*(decomp\$random-1)
autoplot(decompose(AirPassengers.alternate, "multiplicative"))


• Thanks, Chris! This is a well-written explanation. To clarify, is the last image an example of a multiplicative component because the remainder component varies with time? The remainder 1955-1958 seems to be smaller than the remainder pre-1955 and post-1958. Commented Sep 13, 2020 at 10:32
• @Andrew Not exactly, see my edit. Commented Sep 13, 2020 at 14:12
• Ok, thank you, Chris. Now I think I understand. Great explanation! Commented Sep 13, 2020 at 19:07

Hyndman states that we use additive decomposition if the trend-cycle component or the seasonal component does not vary with time, and to use multiplicative decomposition if the trend-cycle component or the seasonal component does vary with time.

This statement is incorrect. Additive decomposition is appropriate when both the seasonality component and the remainder component do not depend on the level of the trend. In other words, if the model is additive, the amplitude of both the seasonality component and the remainder component does not increase (decrease) as the trend increases (decreases). On the other hand, the multiplicative model is appropriate when both the seasonality component and the remainder component are affected by the level of the trend: their amplitude increase (decrease) as the trend increases (decreases).

First, what should we do if the trend-cycle component varies with time but the seasonal component does not vary with time? Could we use either additive decomposition or multiplicative decomposition?

If the amplitude of the seasonal component looks stable in spite of the trend component increasing or decreasing, an additive model is appropriate.

Second, just for clarification, in Figure 6.2 Hyndman uses additive decomposition. Am I correct to say that the trend-cycle component varies with time but the magnitude of the seasonal component does not vary with time(the seasonal component seems to have a consistent pattern)?

Yes, you are right.