Time-Series Modeling and Decomposition

Flow series may be affected by other variations associated with the composition of the calendar. The most important calendar variations are the trading-day variations, which are due to the fact that some days of the week are more important than others. Trading-day variations imply the existence of a daily pattern analogous to the seasonal pattern. However, these daily factors are usually referred to as daily coefficients.

Depending on the socio-economic variable considered, some days may be 60% more important than an average day and other days, 80% less important. If the more important days of the week appear five times in a month (instead of four), the month registers an excess of activity ceteris paribus. If the less important days appear five times, the month records a short-fall. As a result, the monthly tradingday component can cause increase of +8% or -8% (say) between neighbouring months and also between same-months of neighbouring years. The trading-day component is usually considered as negligible and very difficult to estimate in quarterly series.

For the multiplicative, the log-additive and the additive time series decomposition models, the monthly trading-day component is respectively obtained in the following manner

$D_{t}=\Sigma_{\tau \in t} d_{\tau} / n_{t} \equiv\left(2800+\Sigma_{\tau \in t 5 \text { times }} d_{\tau}\right) / n_{t}$                                    (21.a)

$D_{t}=\exp \left(\sum_{\tau \in t} d_{\tau} / n_{t}\right) \equiv \exp \left(\left(\sum_{\tau \in t 5 \text { times }} d_{\tau}\right) / n_{t}\right)$                        (21.b)

$D_{t}=\Sigma_{\tau \in t} d_{\tau} \equiv\left(\Sigma_{\tau \in t 5 \text { times }} d_{\tau}\right)$                                                              (21.c)

where $d_{\tau}$ are the daily coefficients in the month. The preferred option regarding $n_{t}$ is to set it equal to the number of days in month $t$, so that the length-of-month effect is captured by the multiplicative seasonal factors, except for Februaries. The other option is to set $n_{t}$ equal to 30.4375, so that the multiplicative trading-day component also accounts for the length-of-month effect. The number 2800 in Eq. (21.a) is the sum of the first 28 days of the months expressed in percentage.

Same-month year-ago comparisons are never valid in the presence of trading-day variations, not even as a rule of thumb. For a given set of daily coefficients, there are only 22 different monthly values for the trading-day component, for a given set of daily coefficients: seven values for 31-day months (depending on which day the month starts), seven for 30-day months, seven for 29-day months and one for 28-day months. In other words, there are at most 22 possible arrangements of days in monthly data.