Mean and seasonal effects of calendar variables

The calendar effects produced by the regression variables that fulfil the definition presented above include a mean effect (i.e. an effect that is independent of the period) and a seasonal effect (i.e. an effect that is dependent of the period and on average it is equal to 0). Such an outcome is inappropriate, as in the usual decomposition of a series the mean effect should be allocated to the trend component and the fixed seasonal effect should be affected to the corresponding component. Therefore, the actual calendar effect should only contain effects that don't belong to the other components.

In the context of JDemetra+ the mean effect and the seasonal effect are long term theoretical effects rather than the effects computed on the time span of the considered series (which should be continuously revised).

The mean effect of a calendar variable is the average number of days in its group. Taking into account that one year has on average 365.25 days, the monthly mean effects for a working days are, as shown in the table below, 21.7411 for week days and 8.696 for weekends.

Monthly mean effects for the Working day variable

The number of days by period is highly seasonal, as apart from February, the length of each month is the same every year. For this reason, any set of calendar variables will contain, at least in some variables, a significant seasonal effect, which is defined as the average number of days by period (Januaries..., first quarters...) outside the mean effect. Removing that fixed seasonal effects consists of removing for each period the long term average of days that belong to it. The calculation of a seasonal effect for the working days classification is presented in the table below.

The mean effect and the seasonal effect for the calendar periods

For a given time span, the actual calendar effect for week days can be easily calculated as the difference between the number of week days in a specific period and the sum of the mean effect and the seasonal effect assigned to this period, as it is shown in the table below for the period 01.2013 – 06.2013.

The calendar effect for the period 01.2013 - 06.2013

The distinction between the mean effect and the seasonal effect is usually unnecessary. Those effects can be considered together (simply called mean effects) and be computed by removing from each calendar variable its average number of days by period. These global means effect are considered in the next section.

Impact of the mean effects on the decomposition

When the ARIMA model contains a seasonal difference – something that should always happen with calendar variables – the mean effects contained in the calendar variables are automatically eliminated, so that they don't modify the estimation. The model is indeed estimated on the series/regression variables after differencing. However, they lead to a different linearised series ($y_{\text{lin}})$. The impact of other corrections (mean and/or fixed seasonal) on the decomposition is presented in the next paragraph. Such corrections could be obtained, for instance, by applying other solutions for the long term corrections or by computing them on the time span of the series.

Now the model with "correct" calendar effects (denoted as $C$), i.e. effects without mean and fixed seasonal effects, can be considered. To simplify the problem, the model has no other regression effects.

For such a model the following relations hold:

\(y_{\text{lin}} = \ y - C\)
\(T = \ F_{T}\left( y_{\text{lin}} \right)\)
\(S = \ F_{S}\left( y_{\text{lin}} \right) + C\)
\(I = \ F_{I}\left( y_{\text{lin}} \right)\)

where:

T - the trend;

S - the seasonal component;

I - the irregular component;

$F_{X}$ - the linear filter for the component X.

Consider next other calendar effects ($\widetilde{C}$) that contain some mean ($\text{cm}$, integrated to the final trend) and fixed seasonal effects ($\text{cs}$, integrated to the final seasonal). The modified equations are now:

\(\widetilde{C} = C + cm + cs\)
\({\widetilde{y}}_{\text{lin}} = \ y - \widetilde{C} = \ y_{\text{lin}} - cm - cs\)
\(\widetilde{T} = \ F_{T}\left( {\widetilde{y}}_{\text{lin}} \right) + cm\)
\(\widetilde{S} = \ F_{S}\left( {\widetilde{y}}_{\text{lin}} \right) + C + cs\)
\(\widetilde{I} = \ F_{I}\left( {\widetilde{y}}_{\text{lin}} \right)\)

Taking into account that $F_{X}$ is a linear transformation and that¹

\(F_{T}\left( \text{cm} \right) = cm\)
\(F_{T}\left( \text{cs} \right) = 0\)
\(F_{S}\left( \text{cm} \right) = 0\\)
\(F_{S}\left( \text{cs} \right) = cs\)
\(F_{I}\left( \text{cm} \right) = 0\)
\(F_{I}\left( \text{cs} \right) = 0\)

The following relationships hold:

\(\widetilde{T} = \ F_{T}\left( {\widetilde{y}}_{\text{lin}} \right) + cm = F_{T}\left( y_{\text{lin}} \right) - cm + cm = T\)
\(\widetilde{S} = \ F_{S}\left( {\widetilde{y}}_{\text{lin}} \right) + C + cs = F_{S}\left( y_{\text{lin}} \right) - cs + C + cs = S\)
\(\widetilde{I} = \ I\)

If we don’t take into account the effects and apply the same approach as in the “correct” calendar effects, we will get:

\(\breve{T} = \ F_{T}\left( {\widetilde{y}}_{\text{lin}} \right) = T - cm\)
\(\breve{S} = \ F_{S}\left( {\widetilde{y}}_{\text{lin}} \right) + \widetilde{C} = S + cm\)
\(\breve{I} = \ F_{I}\left( {\widetilde{y}}_{\text{lin}} \right) = I\)

The trend, seasonal and seasonally adjusted series will only differ by a (usually small) constant.

In summary, the decomposition does not depend on the mean and fixed seasonal effects used for the calendar effects, provided that those effects are integrated in the corresponding final components. If these corrections are not taken into account, the main series of the decomposition will only differ by a constant.