Components of a Time Series

When we look at a series over a long stretch of time, its ups and downs are not random noise alone — they are the combined effect of four distinct forces called the components of a time series. Separating them is the first task of any analysis.

1. Secular trend (long-term trend), $T$. The smooth, regular, long-term movement of the series over a long period — the general direction in which the values move, whether steadily upward (population, national income), steadily downward (deaths from a disease as medicine improves) or roughly stable. It is the component we most often want to isolate and measure, and the whole of the rest of this chapter is about measuring it.

2. Seasonal variation, $S$. Regular, periodic fluctuations that complete themselves within one year and repeat year after year, caused by the seasons of the year or by social and religious customs. Examples: umbrella sales rising every monsoon, woollens selling every winter, sweet sales peaking at every festival. Seasonal variation can be seen only when the data are recorded for periods shorter than a year (monthly, quarterly, weekly).

3. Cyclical variation, $C$. Wave-like fluctuations that repeat over a period of more than one year, typically the business cycle of prosperity, recession, depression and recovery. Unlike seasonal variation, cyclical swings are not of fixed length — one cycle may last four years, the next seven.

4. Irregular (random) variation, $I$. Completely unpredictable, non-recurring movements caused by chance events — floods, earthquakes, strikes, wars, sudden policy changes. They follow no pattern and cannot be forecast.

How the components combine. Two models relate the observed value $Y$ to its components. In the additive model the components are added,

$Y = T + S + C + I,$ …