Summary of the method used in JDemetra+ to compute trading day and working day effects
Summary of the method used in JDemetra+ to compute trading day and working day effects
The computation of trading day and working days effects is performed in four steps:
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Computation of the number of each weekday performed for all periods.
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Calculation of the usual contrast variables for trading day and working day.
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Correction of the contrast variables with specific holidays (for each holiday add +1 to the number of Sundays and subtract 1 from the number of days of the holiday). The correction is not performed if the holiday falls on a Sunday, taking into account the validity period of the holiday.
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Correction of the constant variables for long term mean effects,
taking into account the validity period of the holiday; see below for the different cases.
The corrections of the constant variables may receive a weight corresponding to the part of the holiday considered as a Sunday.
An example below illustrates the application of the above algorithm for the hypothetical country in which three holidays are celebrated:
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New Year (a fixed holiday, celebrated on 01 January);
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Shrove Tuesday (a moving holiday, which falls 47 days before Easter Sunday, celebrated until the end of 2012);
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Freedom day (a fixed holiday, celebrated on 25 April).
The consecutive steps in calculation of the calendar for 2012 and 2013 years are explained below.
First, the number of each day of the week in the given month is calculated as it is shown in table below.
Number of each weekday in different months
| Month | Mon | Tue | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|---|
| Jan-12 | 5 | 5 | 4 | 4 | 4 | 4 | 5 |
| Feb-12 | 4 | 4 | 5 | 4 | 4 | 4 | 4 |
| Mar-12 | 4 | 4 | 4 | 5 | 5 | 5 | 4 |
| Apr-12 | 5 | 4 | 4 | 4 | 4 | 4 | 5 |
| May-12 | 4 | 5 | 5 | 5 | 4 | 4 | 4 |
| Jun-12 | 4 | 4 | 4 | 4 | 5 | 5 | 4 |
| Jul-12 | 5 | 5 | 4 | 4 | 4 | 4 | 5 |
| Aug-12 | 4 | 4 | 5 | 5 | 5 | 4 | 4 |
| Sep-12 | 4 | 4 | 4 | 4 | 4 | 5 | 5 |
| Oct-12 | 5 | 5 | 5 | 4 | 4 | 4 | 4 |
| Nov-12 | 4 | 4 | 4 | 5 | 5 | 4 | 4 |
| Dec-12 | 5 | 4 | 4 | 4 | 4 | 5 | 5 |
| Jan-13 | 4 | 5 | 5 | 5 | 4 | 4 | 4 |
| Feb-13 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| Mar-13 | 4 | 4 | 4 | 4 | 5 | 5 | 5 |
| Apr-13 | 5 | 5 | 4 | 4 | 4 | 4 | 4 |
| May-13 | 4 | 4 | 5 | 5 | 5 | 4 | 4 |
| Jun-13 | 4 | 4 | 4 | 4 | 4 | 5 | 5 |
| Jul-13 | 5 | 5 | 5 | 4 | 4 | 4 | 4 |
| Aug-13 | 4 | 4 | 4 | 5 | 5 | 5 | 4 |
| Sep-13 | 5 | 4 | 4 | 4 | 4 | 4 | 5 |
| Oct-13 | 4 | 5 | 5 | 5 | 4 | 4 | 4 |
| Nov-13 | 4 | 4 | 4 | 4 | 5 | 5 | 4 |
| Dec-13 | 5 | 5 | 4 | 4 | 4 | 4 | 5 |
Next, the contrast variables are calculated (table below) as a result of the linear transformation applied to the variables presented in table below.
Contrast variables (series corrected for leap year effects)
| Month | Mon | Tue | Wed | Thu | Fri | Sat | Length |
|---|---|---|---|---|---|---|---|
| Jan-12 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |
| Feb-12 | 0 | 0 | 1 | 0 | 0 | 0 | 0.75 |
| Mar-12 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| Apr-12 | 0 | -1 | -1 | -1 | -1 | -1 | 0 |
| May-12 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Jun-12 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| Jul-12 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |
| Aug-12 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| Sep-12 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| Oct-12 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| Nov-12 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| Dec-12 | 0 | -1 | -1 | -1 | -1 | 0 | 0 |
| Jan-13 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Feb-13 | 0 | 0 | 0 | 0 | 0 | 0 | -0.25 |
| Mar-13 | -1 | -1 | -1 | -1 | 0 | 0 | 0 |
| Apr-13 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| May-13 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| Jun-13 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| Jul-13 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| Aug-13 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| Sep-13 | 0 | -1 | -1 | -1 | -1 | -1 | 0 |
| Oct-13 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Nov-13 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| Dec-13 | 5 | 5 | 4 | 4 | 4 | 4 | 0 |
In the next step the corrections for holidays is done in the following way:
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New Year: In 2012 it falls on a Sunday. Therefore no correction is applied. In 2013 it falls on a Tuesday. Consequently, the following corrections are applied to the number of each weekday in January: Tuesday -1, Sunday +1, so the following corrections are applied to the contrast variables: -2 for Tuesday and -1 for the other contrast variables.
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Shrove Tuesday: It is a fixed day of the week holiday that always falls on Tuesday. For this reason in 2012 the following corrections are applied to the number of each weekday in February: Tuesday -1, Sunday +1, so the following corrections are applied to the contrast variables: -2 for the contrast variable associated with Tuesday, and -1 for the other contrast variables. The holiday expires at the end of 2012. Therefore no corrections are made for 2013.
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Freedom Day: In 2012 it falls on a Wednesday. Consequently, the following corrections are applied to the number of each weekday in April: Wednesday -1, Sunday +1, so the following corrections are applied to the contrast variables: -2 for Wednesday and -1 for the other contrast variables. In 2013 it falls on Thursday. Therefore, the following corrections are applied to the number of each weekday in April: Thursday -1, Sunday +1, so the following corrections are applied to the contrast variables: -2 for Thursday, and -1 for the other contrast variables.
The result of these corrections is presented in table below.
Contrast variables corrected for holidays
| Month | Mon | Tue | Wed | Thu | Fri | Sat | Length |
|---|---|---|---|---|---|---|---|
| Jan-12 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |
| Feb-12 | -1 | -2 | 0 | -1 | -1 | -1 | 0.75 |
| Mar-12 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| Apr-12 | -1 | -2 | -3 | -2 | -2 | -2 | 0 |
| May-12 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Jun-12 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| Jul-12 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |
| Aug-12 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| Sep-12 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| Oct-12 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| Nov-12 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| Dec-12 | 0 | -1 | -1 | -1 | -1 | 0 | 0 |
| Jan-13 | -1 | -1 | 0 | 0 | -1 | -1 | 0 |
| Feb-13 | 0 | 0 | 0 | 0 | 0 | 0 | -0.25 |
| Mar-13 | -1 | -1 | -1 | -1 | 0 | 0 | 0 |
| Apr-13 | 0 | 0 | -1 | -2 | -1 | -1 | 0 |
| May-13 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| Jun-13 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| Jul-13 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| Aug-13 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| Sep-13 | 0 | -1 | -1 | -1 | -1 | -1 | 0 |
| Oct-13 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Nov-13 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| Dec-13 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |
Finally, the long term corrections are applied on each year of the validity period of the holiday.
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New Year: Correction on the contrasts: +1, to be applied to January of 2012 and 2013.
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Shrove Tuesday: It may fall either in February or in March. It will fall in March if Easter is on or after 17 April. Taking into account the theoretical distribution of Easter, it gives: prob(March) = +0.22147, prob(February) = +0.77853. The correction of the contrasts will be +1.55707 for Tuesday in February 2012 and +0.77853 for the other contrast variables. The correction of the contrasts will be +0.44293 for Tuesday in March 2012, +0.22147 for the other contrast variables.
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Freedom Day: Correction on the contrasts: +1, to be applied to April of 2012 and 2013.
The modifications due to the corrections described above are presented in table below.
Trading day variables corrected for the long term effects
| Month | Mon | Tue | Wed | Thu | Fri | Sat | Length |
|---|---|---|---|---|---|---|---|
| Jan-12 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| Feb-12 | -0.22115 | -0.44229 | 0.778853 | -0.22115 | -0.22115 | -0.22115 | 0.75 |
| Mar-12 | 0.221147 | 0.442293 | 0.221147 | 1.221147 | 1.221147 | 1.221147 | 0 |
| Apr-12 | 0 | -1 | -2 | -1 | -1 | -1 | 0 |
| May-12 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Jun-12 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| Jul-12 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |
| Aug-12 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| Sep-12 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| Oct-12 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| Nov-12 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| Dec-12 | 0 | -1 | -1 | -1 | -1 | 0 | 0 |
| Jan-13 | 0 | 0 | 1 | 1 | 0 | 0 | 0 |
| Feb-13 | 0 | 0 | 0 | 0 | 0 | 0 | -0.25 |
| Mar-13 | -1 | -1 | -1 | -1 | 0 | 0 | 0 |
| Apr-13 | 1 | 1 | 0 | -1 | 0 | 0 | 0 |
| May-13 | 0 | 0 | 1 | 1 | 1 | 0 | 0 |
| Jun-13 | -1 | -1 | -1 | -1 | -1 | 0 | 0 |
| Jul-13 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| Aug-13 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| Sep-13 | 0 | -1 | -1 | -1 | -1 | -1 | 0 |
| Oct-13 | 0 | 1 | 1 | 1 | 0 | 0 | 0 |
| Nov-13 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| Dec-13 | 0 | 0 | -1 | -1 | -1 | -1 | 0 |