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The X11 Procedure |
The following steps describe the analysis of a monthly time series using multiplicative adjustments. Additional steps used by the X-11-ARIMA method are also indicated. Equivalent descriptions apply for an additive model by replacing divide by subtract where applicable.
In the multiplicative adjustment, the original series O_{t} is assumed to be of the form
where C_{t} is the trend cycle component, S_{t} is the seasonal component, I_{t} is the irregular component, P_{t} is the prior monthly factors component and D_{t} is the trading-day component.
The trading-day component can be further factored as
where D_{tr,t} are the trading-day factors derived from the prior daily weights, and D_{r,t} are the residual trading-day factors estimated from the trading-day regression.
If prior factor or calendar effects are present, they must be eliminated from the series before the ARIMA estimation is done because these effects are not stochastic.
Prior factors, if present, are removed first. Calendar effects represented by prior daily weights are then removed. If there are no further calendar effects, the adjusted series is extended by the ARIMA model, and this extended series goes through the standard X-11 steps without repeating the removal of prior factors and calendar effects from prior daily weights.
If further calendar effects are present, a trading-day regression must be performed. In this case it is necessary to go through an initial pass of the X-11 steps to obtain a final trading-day adjustment. In this initial pass, the series, adjusted for prior factors and prior daily weights, goes through the standard X-11 steps. At the conclusion of these steps, a final series adjusted for prior factors and all calendar effects is available. This adjusted series is then extended by the ARIMA model, and this extended series goes through the standard X-11 steps again, without repeating the removal of prior factors and calendar effects from prior daily weights and trading day regression.
Seven daily weights can be specified to develop monthly factors to adjust the series for trading-day variation, D_{tr,t}; these factors are then divided into the original or prior adjusted series to obtain C_{t}S_{t}I_{t}D_{r,t}.
Sliding spans analysis attempts to quantify the stability of the seasonal adjustment process, and hence quantify the suitability of seasonal adjustment for a given series.
It is based on a very simple idea: for a stable series, deleting a small number of observations should not result in greatly different component estimates compared with the original, full series. Conversely, if deleting a small number of observations results in drastically different estimates, the series is unstable. For example, a drastic difference in the seasonal factors (Table D10) might result from a dominating irregular component, or sudden changes in the seasonally component. When the seasonal component estimates of a series is unstable in this manner, they have little meaning and the series is likely to be unsuitable for seasonal adjustment.
Sliding spans analysis, developed at the Statistical Research Division of the U.S. Census Bureau (see Findley, et al., 1990, and Findley and Monsell, 1986 ), performs a repeated seasonal adjustment on subsets or spans of the full series. In particular, an initial span of the data, typically eight years in length, is seasonally adjusted, and the tables C18, the trading day factors (if trading day regression performed), D10, the seasonal factors, and D11, the seasonally adjusted series are retained for further processing. Next, one year of data is deleted from the beginning of the initial span and one year of data is added. This new span is seasonally adjusted as before, with the same tables retained. This process continues until the end of the data is reached. The beginning and ending dates of the spans are such that the last observation in the original data is also the last observation in the last span. This is discussed in more detail below.
The following notation for the components or differences computed in the sliding spans analysis follows Findley et al., 1990. The meaning for the symbol X_{t}(k) is component X in month (or quarter) t, computed from data in the k-th span. These components are now defined.
Seasonal Factors (Table D10): S_{t}(k)
Trading Day Factor (Table C18): TD_{t}(k)
Seasonally Adjust Data (Table D11): SA_{t}(k)
Month-to-month changes in the Seasonally Adjust Data: MM_{t}(k)
Year-to-Year changes in the Seasonally Adjust Data: YY_{t}(k)
The key measure is the maximum percent difference across spans. For example, consider a series beginning in JAN72, ending in DEC84, and having four spans, each of length 8 years (see Figure 1. in Findley et al., 1990, page 346). Consider S_{t}(k) the seasonal factor (table D10) for month t for span k, and let N_{t} denote the number of spans containing month t, i.e.,
In the middle years of the series there is overlap of all four spans and N_{t} will be 4. The last year of the series will have but one span, while the beginning can have 1 or 0 spans depending on the original length.
Since we are interested in how much the seasonal factors vary for a given month across the spans, a natural quantity to consider is
In the case of the multiplicative model, it is useful to compute a percent difference; define the maximum percent difference (MPD) at time t as
The seasonal factor for month t is then unreliable if MPD_{t} is large. While no exact significance level can be computed for this statistic, empirical levels have been established by considering over 500 economic series (see Findley, et al. 1990 and Findley and Monsell, 1986). For these series it was found that for four spans, stable series typically had less than 15% of the MPD values exceeding 3.0%, while in marginally stable series, between 15% and 25% of the MPD values exceeded 3.0%. A series in which 25% or more of the MPD values exceeded 3.0% is almost always unstable.
While these empirical values cannot be considered an exact significance level, they provide a useful empirical basis for deciding if a series is suitable for seasonal adjustment. These percentage values are shifted down when less than four spans are used.
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