Census1

A procedure for determining a seasonal component. A source series is split into a seasonal component, a trend-cycle component and an irregular components.

General Model

Generally, a time series can be regarded as consisting of four different components:

Seasonal component (S(t), where t is a time moment).
Trend (Tt).
Cycle component (Ct).
Random irregular component or fluctuation (It).

The difference between a cycle component and a seasonal component is that the latter has regular (seasonal) frequency, whereas the effect of cyclic factors is usually more long-term and changes from cycle to cycle. The Census1 method commonly combines the trend and the cycle component to one trend-cycle component (TCt). Functional relations between these components can be various. Though two main types of interaction between them are additive and multiplicative.

Additive model:Xt = Tt + Ct + St + It

Multiplicative model: Xt = Tt * Ct * St * It / 100%

Output Series

Smoothed series. First, moving average of a time series is calculated, and window width must be equal to the seasonal period. If the seasonal period is an even number, the user can choose one of these two options: either use the moving average with equal weights or use the moving average with unequal weights, in which case the first observation and the last observation in the window have averaged weights.

Relation or difference. After the user has determined moving average values, all seasonal variation (that is, variation within a season) is excluded, and therefore, the difference (additive model) or the relation (multiplicative model) between the observed series and the smoothed series determine a seasonal component (plus an irregular component). To be more precise, a smoothed series is subtracted from an observed series (in additive model) or observed series values are divided into moving average values and multiplied by 100% (in multiplicative model).

Seasonal component. The next step calculates a seasonal component as the mean (for additive models) or the trimmed mean (for multiplicative models) of all series values for this point of a seasonal interval.

Seasonal adjustment. A source series can be adjusted by subtracting seasonal component values from it (additive model) or by dividing its values into seasonal component values in unit shares (multiplicative model). The output series is named seasonal series adjustment (a seasonal component is removed from the series).

Trend-cycle component. Remember that the difference between a cycle component and a seasonal component is that, as a rule, cycle period is longer than one seasonal period, and the duration of different cycles may differ. To calculate approximation for a combined trend-cycle component, the user can apply the 5-point (centered) weighed moving average with weights 1, 2, 3, 2, 1 to a series with applied seasonal adjustment.

Irregular component. The last step determines a random component or irregular component (error) by subtracting a trend-cycle component from a series with seasonal adjustment (additive model) or by dividing this series (multiplicative model) by a trend-cycle component.