Linear transformations of the calendar variables

As far as the RegARIMA and the TRAMO models are considered, any non-degenerated linear transformation of the calendar variables can be used. It will produce the same results (likelihood, residuals, parameters, joint effect of the calendar variables, joint F-test on the coefficients of the calendar variables…). The linearised series that will be further decomposed is invariant to any linear transformation of the calendar variables.

However, it should be mentioned that choices of calendar corrections based on the tests on the individual t statistics are dependent on the transformation, which is rather arbitrary. This is the case in old versions of TRAMO-SEATS. That is why the joint F-test (as in the version of TRAMO-SEATS implemented in TSW+) should be preferred.

An example of a linear transformation is the calculation of the contrast variables. In the case of the usual trading day variables, they are defined by the following transformation: the 6 contrast variables ($\text{No.}\left( \text{Mondays} \right) - No.\left( \text{Sundays} \right),\ldots No.\left( \text{Saturdays} \right) - No.(Sundays)$) used with the length of period.

\[\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 1 & 0 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & - 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & - 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ \end{bmatrix}\begin{bmatrix} \text{Mon} \\ \text{Tue} \\ \text{Wed} \\ \text{Thu} \\ \text{Fri} \\ \text{Sat} \\ \text{Sun} \\ \end{bmatrix} = \begin{bmatrix} Mon - Sun \\ Tue - Sun \\ Wed - Sun \\ Thu - Sun \\ Fri - Sun \\ Sat - Sun \\ \text{Length of period} \\ \end{bmatrix}\]

For the usual working day variables, two variables are used: one contrast variable and the length of period

\[\begin{bmatrix} 1 & - \frac{5}{2} \\ 1 & 1 \\ \end{bmatrix}\begin{bmatrix} \text{Week} \\ \text{Weekend} \\ \end{bmatrix} = \begin{bmatrix} \text{Contrast week} \\ \text{Length of period} \\ \end{bmatrix}\]

The $\text{Length of period}$ variable is defined as a deviation from the length of the month (in days) and the average month length, which is equal to $30.4375.$ Instead, the leap-year variable can be used here (see Regression sections in RegARIMA or Tramo)1.

Such transformations have several advantages. They suppress from the contrast variables the mean and the seasonal effects, which are concentrated in the last variable. So, they lead to fewer correlated variables, which are more appropriate to be included in the regression model. The sum of the effects of each day of the week estimated with the trading (working) day contrast variables cancel out.

  1. GÓMEZ, V., and MARAVALL, A (2001b). ↩