Notes on centering holiday regressors

For simplicity, only regressors for monthly data are discussed. The quarterly and bimonthly cases are analogous.

Before centering, each regressor is a proportionality regressor whose value in any month is the proportion of the days of the specified interval that belong to the month. For example, if the interval is 10 days long and, in a certain year, the first two days of the interval fall in January, then the value of the uncentered regressor is 2/10 in January and 8/10 in February that year. Its value in March-December is 0.

Centering of the regressors is done to keep the yearly totals of the series obtained by removing the estimated holiday effects approximately equal to the yearly totals of the unadjusted data. If centering is not done, these two totals will differ by approximately the same amount each year, and user will conclude that combined seasonal and holiday adjustment is producing a biased estimate of the level of the observed series. (The bias is a crude estimate of what the data would be like if there were no holiday.) The type of holiday effect determines the type of centering.

Centering by removing the calendar-month means is appropriate for holidays like Easter or the Chinese New Year whose regressors are always zero in some calendar months because the holiday can occur only in a few calendar months. The calendar-month-centered regressors will be zero in exactly the same months as the regressor, and no adjustment will be done to data from these months. Also, holiday adjustment does remove fixed seasonal, with the result that all such effects are included in the seasonal factors produced by the seasonal adjustment procedure.

Centering by removing the overall mean is attractive for holiday periods like Ramadan that move through all of the calender months over time. With mean centering, the centered regressor's adjustment for every month outside the specified interval in a given year always has the same value and adjustment for the holiday effect has a usually negligible effect on the general level of the series.

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