## 7.1 Time series components

If we assume an additive decomposition, then we can write $y_{t} = S_{t} + T_{t} + R_t,$ where $$y_{t}$$ is the data, $$S_{t}$$ is the seasonal component, $$T_{t}$$ is the trend-cycle component, and $$R_t$$ is the remainder component, all at period $$t$$. Alternatively, a multiplicative decomposition would be written as $y_{t} = S_{t} \times T_{t} \times R_t.$

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. Multiplicative decompositions are common with economic time series.

An alternative to using a multiplicative decomposition is to first transform the data until the variation in the series appears to be stable over time, then use an additive decomposition. When a log transformation has been used, this is equivalent to using a multiplicative decomposition because $y_{t} = S_{t} \times T_{t} \times R_t \quad\text{is equivalent to}\quad \log y_{t} = \log S_{t} + \log T_{t} + \log R_t.$

### Employment in the US retail sector

We will look at several methods for obtaining the components $$S_{t}$$, $$T_{t}$$ and $$R_{t}$$ later in this chapter, but first, it is helpful to see an example. We will decompose the number of persons employed in retail as shown in Figure 7.1. The data shows the total monthly number of persons in thousands employed in the retail sector across the US. Figure 7.1: Total number of persons employed in US retail: the trend-cycle component (red) and the raw data (grey).

Figure 7.1 shows the trend-cycle component, $$T_t$$, in red and the original data, $$y_t$$, in grey. The trend-cycle shows the overall movement in the series, ignoring the seasonality and any small random fluctuations.

Figure 7.2 shows an additive decomposition of these data. The method used for estimating components in this example is STL, which is discussed in Section 7.6. Figure 7.2: The total number of persons employed in US retail (top) and its three additive components.

The three components are shown separately in the bottom three panels of Figure 7.2. These components can be added together to reconstruct the data shown in the top panel. Notice that the seasonal component changes over time, so that any two consecutive years have similar patterns, but years far apart may have different seasonal patterns. The remainder component shown in the bottom panel is what is left over when the seasonal and trend-cycle components have been subtracted from the data.

The grey bars to the left of each panel show the relative scales of the components. Each grey bar represents the same length but because the plots are on different scales, the bars vary in size. The large grey bar in the bottom panel shows that the variation in the remainder component is smallest compared to the variation in the data, which has a bar about one quarter the size. If we shrunk the bottom three panels until their bars became the same size as that in the data panel, then all the panels would be on the same scale.

If the seasonal component is removed from the original data, the resulting values are the “seasonally adjusted” data. For an additive decomposition, the seasonally adjusted data are given by $$y_{t}-S_{t}$$, and for multiplicative data, the seasonally adjusted values are obtained using $$y_{t}/S_{t}$$. 