# 6.1 Time series components

## Time series patterns

In this chapter, we will refer to three types of time series patterns.

Trend
A trend exists when there is a long-term increase or decrease in the data. It does not have to be linear. Sometimes we will refer to a trend “changing direction” when it might go from an increasing trend to a decreasing trend.
Seasonal
A seasonal pattern exists when a series is influenced by seasonal factors (e.g., the quarter of the year, the month, or day of the week). Seasonality is always of a fixed and known period.
Cyclic
A cyclic pattern exists when data exhibit rises and falls that are not of fixed period. The duration of these fluctuations is usually of at least 2 years.

Many people confuse cyclic behaviour with seasonal behaviour, but they are really quite different. If the fluctuations are not of fixed period then they are cyclic; if the period is unchanging and associated with some aspect of the calendar, then the pattern is seasonal. In general, the average length of cycles is longer than the length of a seasonal pattern, and the magnitude of cycles tends to be more variable than the magnitude of seasonal patterns.

The following four examples shows different combinations of the above components.

Figure 6.1: Four time series exhibiting different types of time series patterns.

R code
par(mfrow=c(2,2))
plot(hsales,xlab="Year",ylab="Monthly housing sales (millions)")
plot(ustreas,xlab="Day",ylab="US treasury bill contracts")
plot(elec,xlab="Year",ylab="Australian monthly electricity production")
plot(diff(dj),xlab="Day",ylab="Daily change in Dow Jones index")
1. The monthly housing sales (top left) show strong seasonality within each year, as well as some strong cyclic behaviour with period about 6–10 years. There is no apparent trend in the data over this period.
2. The US treasury bill contracts (top right) show results from the Chicago market for 100 consecutive trading days in 1981. Here there is no seasonality, but an obvious downward trend. Possibly, if we had a much longer series, we would see that this downward trend is actually part of a long cycle, but when viewed over only 100 days it appears to be a trend.
3. The Australian monthly electricity production (bottom left) shows a strong increasing trend, with strong seasonality. There is no evidence of any cyclic behaviour here.
4. The daily change in the Dow Jones index (bottom right) has no trend, seasonality or cyclic behaviour. There are random fluctuations which do not appear to be very predictable, and no strong patterns that would help with developing a forecasting model.

## Time series decomposition

We shall think of the time series $y_{t}$ as comprising three components: a seasonal component, a trend-cycle component (containing both trend and cycle), and a remainder component (containing anything else in the time series). For example, if we assume an additive model, then we can write $$y_{t} = S_{t} + T_{t} + E_{t},$$ where $y_{t}$ is the data at period $t$, $S_{t}$ is the seasonal component at period $t$, $T_{t}$ is the trend-cycle component at period $t$ and $E_{t}$ is the remainder (or irregular or error) component at period $t$. Alternatively, a multiplicative model would be written as $$y_{t} = S_{t} \times T_{t} \times E_{t}.$$

The additive model is 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 model is more appropriate. With economic time series, multiplicative models are common.

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

Sometimes, the trend-cycle component is simply called the “trend” component, even though it may contain cyclic behaviour as well.

## Example 6.1 Electrical equipment manufacturing

We will look at several methods for obtaining the components $S_{t}$, $T_{t}$ and $E_{t}$ later in this chapter. But first, it is helpful to see an example. We will decompose the new orders index for electrical equipment shown in Figure 6.2. These data show the number of new orders for electrical equipment (computer, electronic and optical products) in the Euro area (16 countries). The data have been adjusted by working days and normalized so a value of 100 corresponds to 2005.

Figure 6.2: Electrical equipment orders: the trend-cycle component (red) and raw data (grey).

R code
fit <- stl(elecequip, s.window=5)
plot(elecequip, col="gray",
main="Electrical equipment manufacturing",
ylab="New orders index", xlab="")
lines(fit$time.series[,2],col="red",ylab="Trend") Figure 6.2 shows the trend-cycle component,$T_{t}$, in red and the original data,$y_{t}$, in grey. The trend-cycle shows the overall movement in the series, ignoring the seasonality and any small random fluctuations. Figure 6.3 shows an additive decomposition of these data. The method used for extracting components in this example is STL which is discussed in Section 6/5. Figure 6.3: The electricial equipment orders (top) and its three additive components. R code plot(fit) All three components are shown in the bottom three panels of Figure 6.3. These three components can be added together to reconstruct the data shown in the top panel. Notice that the seasonal component changes very slowly over time, so any two consecutive years have very similar pattern, but years far apart may have different seasonal patterns. The remainder component shown in the bottom panel is what is left over when the seasonal and trend-cycle components have been subtracted from the data. The grey bars to the right of each panel show the relative scales of the components. Each grey bar represents the same length but because the plots are on different scales, the bars vary in size. The large grey bar in the bottom panel shows that the variation in the remainder component is small compared to the variation in the data which has a bar about one quarter the size. In other words, if we shrunk the bottom three panels until their bars became the same size as that in the data panel, then all the panels would be on the same scale. Figure 6.4: Seasonal sub-series plot of the seasonal component from the STL decomposition shown in Figure 6.2. R code monthplot(fit$time.series[,"seasonal"], main="", ylab="Seasonal")

It can be useful to use seasonal plots and seasonal sub-series plots of the seasonal component. These help us to visualize the variation in the seasonal component over time. Figure 6.4 shows a seasonal sub-series plot of the seasonal component from Figure 6.3. In this case, there are only very small changes over time.

If the seasonal component is removed from the original data, the resulting values are called the “seasonally adjusted” data. For an additive model, the seasonally adjusted data are given by $y_{t}-S_{t}$, and for multiplicative data, the seasonally adjusted values are obtained using $y_{t}/S_{t}$. The graph below shows the seasonally adjusted electrical equipment orders.

Figure 6.5: Seasonally adjusted electrical equipment orders (red) and the original data (grey).

R code
plot(elecequip, col="grey",
main="Electrical equipment manufacturing",
xlab="", ylab="New orders index")