15.1 Time Series Components

A time series \(y_t\), is assumed to be decomposed in an additive way as:

\[ y_{t}=t_{t}+s_{t}+c_{t}+\varepsilon_{t} \]


  • \(t_t\) is the trend,
  • \(s_t\) is the seasonal component,
  • \(c_t\) is the cyclical component (or cycle); and
  • \(\varepsilon_{t}\) is the residual term.


Long-run increase or decrease over time (overall upward or downward movement). Captures the general direction of the time series.


It includes oscillations that occur within a period less than or equal to one year.

In other words, they are short-term oscillations that are repeated in successive years. The reasons for the seasonality of a series may be physical (climate, etc.), or or institutional (vacations, festivities, etc.).


The cycle is defined in different ways. From the macroeconomic point of view, they should be swings around the trend, which are due to the alternation between periods of crisis and prosperity. From the statistical point of view, the cycle includes any characteristic other than trend, seasonality and noise.


It captures transitory and irregular movements of the series. This component can be broken down into a clearly random and unexpected part and another part that is not always predictable, but which can be identified (such as a strike, a natural catastrophe, a political change, etc.).

If the series is analyzed using logarithms, the decomposition of the original variable will be of the multiplicative type, that is: \[ y_{t}=e^{t_{t}} \cdot e^{s_{t}} \cdot e^{c_{t}} \cdot e^{\varepsilon_{t}} \]