Annex: Basic Features of the Multiplicative Version of X-11/X-12

(This annex is from ‘Compilation of Quarterly National Accounts,’ (draft) by A. Bloem, R. Dippelsman and N. Maehle, IMF, 2000.)
 
The three basic programs for seasonal adjustment applied by a majority of statistical agencies (X-11, X-11-Arima, and X-12-Arima) follow an iterative estimation procedure based on a series of moving averages¹. The estimation procedure comprises four main parts, in addition to various diagnostics and quality control statistics. First (part A), the series may optionally be “pre-adjusted” for calendar-related effects (moving holidays, working, and trading days), level shifts in the series, and the effect of known irregular events, using adjustment factors supplied by the user or estimated using the built-in estimation procedures. The pre-adjusted series then goes through three rounds of filtering and extreme value adjustments, the B, C, and D iterations in the X-11/X-12 jargon.

The main principles of the filtering procedure in the B, C, and D iteration are presented below. A summary of the the relationships and differences between these three iterations are also given, and last, the methods for pre-adjusting the series are briefly described.

The main steps of the multiplicative version of the filtering procedure in the B, C, and D iteration, assuming monthly data, are as follows:

Stage 1 - Initial Estimates
(i) Initial trend. The series is smoothed using a weighted 13-term (2x12)² centered moving average to produce a first estimate of the trend.
 

(ii) Initial SI ratios. The “original”³ series is divided by the smoothed series (T_t¹) to give an initial estimate of the seasonal and irregular component (S_t  I_t  ¹).

(iii) Initial preliminary seasonal factors, via a weighted 5-term (3x3) centered seasonal⁴ moving average⁵ of the initial SI ratios (S_tI_t¹).  
It is implicitly assumed that I_t is random and therefore eliminated by averaging.

(iv) Initial seasonal factors. A time series of initial seasonal factors is then derived by normalizing the initial preliminary seasonal factors.


This is done to ensure that the annual sum of seasonal factors is close to one.

(v) Initial seasonal adjustment. An initial estimate of the seasonally adjusted series is then derived as: 

Stage 2 - Revised Estimates
(i) Intermediate trend. A revised estimate of the trend (T_t²) is derived by applying a Henderson Moving Average⁶ to the initially seasonally adjusted series (A_t¹).

(ii) Revised SI ratios are derived by dividing the “original” series by the intermediate trend estimate (T_t²).

(iii) Revised preliminary seasonal factors are  derived by applying a 3x5 centered seasonal moving average⁷ to the revised SI ratios.

(iv) Revised seasonal factors. A time series of initial seasonal factors is then derived by normalizing the initial preliminary seasonal factors as in step 1.

(v) Revised seasonal adjustment. A revised estimate of the seasonally adjusted series is then derived as:


(vi) Tentative irregular factors are derived by de-trending the revised seasonally adjusted series


Stage 3 - Final Estimates

(D iteration only)

(i) Final trend. A final estimate of the trend-cycle component (T_t³) is derived by applying a Henderson Moving Average to the revised and final seasonally adjusted series (A_t²);

(ii) Final irregular. A final estimate of the irregular component is derived by de-trending the revised and final seasonally adjusted series



In addition, the filtering procedure described above is made more robust by a series of identification and adjustments for extreme values. First, for the B and D iteration, when estimating the seasonal factors in steps (ii) to (iv), based on analyses of implied irregular factors, extreme SI ratios are identified and temporally replaced. For the B iteration this is done both in stage 1 and in stage 2, while for the D iteration this is done only at stage 2. Second, after the B and C iterations, before the next round of filtering, based on analyses of the tentative irregular factors (I_t²) derived in step (vi) of stage 2, extreme values are identified and temporally removed from the original (or pre-adjusted) series (that is before the C and D iteration respectively).


As mentioned, the original series may be “pre-adjusted” for calendar-related effects (moving holidays trading days, length-of-month, number of Saturdays, and Sundays in the month, differences in the importance of certain working/trading days etc.), outlier and level shifts in the series, and the effect of known irregular events, before entering the filtering process described above. The pre-adjustment can be conducted in a multitude of ways. The user may adjust the data directly based on particular knowledge about the data before feeding them to the program⁸, or use the estimation procedures built into the program.


X-11 and X-11-Arima have built in models for estimation of trading day and Easter effects based on regression analysis (OLS) of the tentative irregular factors (I_t²). When requested, the program derives preliminary estimates and adjustments for trading days and Easter effects at the end of the B iteration, and final estimates and adjustments for trading days and Easter effects at the end of the C iteration.


X-12-Arima contains an extensive time series modeling block that allows the user to conduct regression analysis directly on the original series taking into account that the non-explained part of the series typically will be autocorrelated, non-stationary, and heteroscedastic. This is done by combining traditional linear regression techniques with ARIMA modeling, into what is labeled RegARIMA modeling. The RegARIMA part of X-12 allows the user to provide a set of user-defined regressor variables. In addition, the program contains a large set of pre-defined regressor variables to identify, e.g., trading days effects, Easter effects⁹, leap-year effects, length-of-month effects, level shifts, point outliers, and ramps in the series. As a simpler alternative to RegARIMA modeling, X-12-Arima has retained the traditional X-11 approach of regressing the tentative irregulars on explanatory variables, adding, regressors for point outliers and facilities for user-defined regressors to X-11's trading days and Easter effects.


X-12-Arima’s options for supplying user-defined regressors makes it possible for the users to construct custom-made moving holiday adjustment procedures, to take into account holidays particular to their country or region, or country specific effects of common holidays. Typical examples are regional moving holidays such as Chinese New Year and Ramadan, and the differences in timing and impact of Easter. Regarding the latter, while in some countries Easter is mainly a big shopping weekend creating a peak in retail trade during Easter, in other countries, most shops may be closed for more than a week creating a big drop in retail trade during Easter combined with a peak in retail trade before Easter.

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1} Also called "moving average filters" in the seasonal adjustment terminology.

² A 2x12 moving average is a 2-term moving average of a 12-term moving average:

 
where:


³ The series may be pre-adjusted, and, for the C and D iteration, extreme value adjusted.

⁴ A seasonal moving average is a moving average that is applied to each month separately, that is, as moving averages of neighboring
May’s, June’s and etc.

⁵ The 3x3 seasonal MA filter is the default. In addition, users can select a 3x5 or 3x9 MA filter (X-12-Arima also contain an optional 3x15 seasonal MA filter). The user-selected filter will then be used both in stage 1 and stage 2.

⁶ Henderson Moving Average is a particular type of weighted moving average where the weights are determined so as to produce the smoothest possible trend estimate. In X-11 and X-11-Arima Henderson filters of length 9, 13, and 23, could either be automatically chosen or be user determined. In X-12-Arima the users in addition can specify Henderson filters of length of any odd-number.

⁷ The 3x5 seasonal MA filter is the default. In the D iteration X-11-Arima, and X-12-Arima automatically select from among the four seasonal MA filters (3x3, 3x5, 3x9, and the average of all SI ratios for each calendar month (the stable seasonal average)), unless the user has specified that the program should use a particular MA filter.

⁸ Note that some countries have the tradition of publishing data as “original data” data that already have been adjusted for some calendar effects, particularly the number of working days, often in a rather primitive manner using fixed coefficients based on the ratio of the number of working days in the month to the number of working days in a standard month. This approach is not recommended. Data presented as original data should be original; showing what actually has happened, and not be partly adjusted for seasonal effects. Working days effects are part of the seasonal variation, and adjustment for these effects should be looked upon as an integral part of the seasonal adjustment process. Furthermore, X-11/X-12's trading days adjustment procedures are able to handle these effects in a much more sophisticated and realistic manner. It has been shown that the simple proportional method cited above overstates the effect of working days on the series.

⁹ The user can select from different Easter effect models.