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The X11 Procedure |
The output from PROC X11, both printed tables and the series written to the OUT= data set, depends on whether the data is monthly or quarterly. For the printed tables, the output depends further on the value of the PRINTOUT= option and the TABLE statement, along with other options specified.
The printed output is organized into tables identified by a part letter and a sequence number within the part. The seven major parts of the X11 procedure are as follows.
Table | Description | Notes |
A1 | original series | M |
A2 | prior monthly adjustment factors | M |
A3 | original series adjusted for prior monthly factors | M |
A4 | prior trading-day adjustments | M |
A5 | prior adjusted or original series | M |
A13 | ARIMA forecasts | |
A14 | ARIMA backcasts | |
A15 | prior adjusted or original series extended by arima backcasts, forecasts | |
B1 | prior adjusted or original series | |
B2 | trend cycle | |
B3 | unmodified seasonal-irregular (S-I) ratios | |
B4 | replacement values for extreme S-I ratios | |
B5 | seasonal factors | |
B6 | seasonally adjusted series | |
B7 | trend cycle | |
B8 | unmodified S-I ratios | |
B9 | replacement values for extreme S-I ratios | |
B10 | seasonal factors | |
B11 | seasonally adjusted series | |
B13 | irregular series | |
B14 | extreme irregular values excluded from trading-day regression | M |
B15 | preliminary trading-day regression | M,P |
B16 | trading-day adjustment factors | M |
B17 | preliminary weights for irregular components | |
B18 | trading-day factors derived from combined daily weights | M |
B19 | original series adjusted for trading-day and prior variation | M |
C1 | original series modified by preliminary weights and adjusted for trading-day and prior variation | |
C2 | trend cycle | |
C4 | modified S-I ratios | |
C5 | seasonal factors | |
C6 | seasonally adjusted series | |
C7 | trend cycle | |
C9 | modified S-I ratios | |
C10 | seasonal factors | |
C11 | seasonally adjusted series | |
C13 | irregular series | |
C14 | extreme irregular values excluded from trading-day regression | M |
C15 | final trading-day regression | M,P |
C16 | final trading-day adjustment factors derived from regression coefficients | M |
C17 | final weight for irregular components | |
C18 | final trading-day factors derived from combined daily weights | M |
C19 | original series adjusted for trading-day and prior variation | M |
D1 | original series modified for final weights and adjusted for trading-day and prior variation | |
D2 | trend cycle | |
D4 | modified S-I ratios | |
D5 | seasonal factors | |
D6 | seasonally adjusted series | |
D7 | trend cycle | |
D8 | final unmodified S-I ratios | |
D9 | final replacement values for extreme S-I ratios | |
D10 | final seasonal factors | |
D11 | final seasonally adjusted series | |
D12 | final trend cycle | |
D13 | final irregular series | |
E1 | original series with outliers replaced | |
E2 | modified seasonally adjusted series | |
E3 | modified irregular series | |
E4 | ratios of annual totals | P |
E5 | percent changes in original series | |
E6 | percent changes in final seasonally adjusted series | |
F1 | MCD moving average | |
F2 | summary measures | P |
G1 | chart of final seasonally adjusted series and trend cycle | P |
G2 | chart of S-I ratios with extremes, S-I ratios without extremes, and final seasonal factors | P |
G3 | chart of S-I ratios with extremes, S-I ratios without extremes, and final seasonal factors in calendar order | P |
G4 | chart of final irregular and final modified irregular series | P |
The last numeric column of Table S 1.A is the average value over all spans for each calendar month, with the minimum and maximum row flagged as in the span columns.
Table S 1.B gives a summary of range measures for each span. The first column, Range Means, is calculated by computing the maximum and minimum over all months or quarters in a span, the taking the difference. The next column is the range ratio means, which is simply the ratio of the previously described maximum and minimum. The next two columns are the minimum and maximum seasonal factors over the entire span, while the range sf column is the difference of these. Finally, the last column is the ratio of the Max SF and Min SF columns.
Table S 2.A.1 begins the breakdown analysis for the various series considered in the sliding spans analysis. The key concept here is the MPD described in the Introduction and in "Computational Details" above. For a month or quarter that appears in two or more spans, the maximum percent difference is computed and tested against a cutoff level. If it exceeds this cutoff, it is counted as an instance of exceeding the level. It is of interest to see if such instances fall disproportionately in certain months and years. Tables S 2.A.1 - S 6.A.3 display this breakdown for all series considered.
Table S 2.A.1 gives the monthly (quarterly) breakdown for the seasonal factors (table D10). The first column identifies the month or quarter. The next column is the number of times the MPD for D10 exceeded 3.0%, followed by the total count. The last is the average maximum percentage difference for the corresponding month or quarter.
Table S 2.A.2 gives the same information as Table S 2.A.1, but on a yearly basis.
The description of Table S 2.A.3 requires the definition of "Sign Change" and "Turning Point".
First, some motivation. Recall that for a highly stable series, adding or deleting a small number of observations should not affect the estimation of the various components of a seasonal adjustment procedure.
Consider Table D10, the seasonal factors in a sliding spans analysis that uses 4 spans. For a given observation t, looking across the 4 spans, we can easily pick out large differences if they occur. More subtle differences can occur when estimates go from above to below (or vice versa) a base level. In the case of multiplicative model, the seasonal factors have a base level of 100.0. So it is useful to enumerate those instances where both a large change occurs (an MPD value exceeding 3.0%) and a change of sign (with respect to the base) occur.
Let B denote the base value (which in general depends on the component being considered and the model type, multiplicative or additive). If, for span 1, S_{t}(1) is below B (i.e., S_{t}(1)-B is negative) and for some subsequent span k, S_{t}(k) is above B (i.e., S_{t}(k)-B is positive), then an positive "Change in Sign" has occurred at observation t. Similarly, if, for span 1, S_{t}(1) is above B, and for some subsequent span k, S_{t}(k) is below B, then a negative "Change in Sign" has occurred. Both cases, positive or negative, constitute a "Change in Sign"; the actual direction indicated in tables S 7.A-S 7.E, which will be described below.
Another behavior of interest occurs when component estimates increase then decrease (or vice versa) across spans for a given observation. Using the example above, the seasonal factors at observation t could first increase, then decrease across the 4 spans. This behavior, combined with an MPD exceeding the level is of interest in questions of stability.
Again, consider Table D10, the seasonal factors in a sliding spans analysis that uses 4 spans. For a given observation t, (containing at least three spans), note the level of D10 for the first span. Continue across the spans until a difference of 1.0% or greater occurs (or no more spans are left), noting whether the difference is up or down. If the difference is up, continue until a difference of 1.0% or greater occurs downward (or no more spans are left). If such an up-down combination occurs, the observation is counted as an up-down turning point. A similar description occurs for a down-up turning point. Tables S 7.A-S 7.E, described below, show the occurrence of turning points, indicating whether up-down or down-up. Note that it requires at least three spans to test for a turning point. Hence Tables S 2.A.3 - S 6.A.3 show a reduced number in the "Turning Point" row for the "Total Tested" column, and in Tables S 7.A - S 7.E, the turning points symbols can only occur where three or more spans overlap.
With these descriptions of sign change and turning point, we now describe Table S 2.A.3. The first column gives the type or category, the second gives the total number of observations falling into the category, the third column gives the total number tested, and the last column gives the percentage for the number found in the category.
The first category (row) of the table is for flagged observations, i.e., those observations where the MPD exceeded the appropriate cutoff level (3.0% is default for the seasonal factors.) The second category is for level changes, while the third category is for turning points. The fourth category is for flagged sign changes, i.e., for those observations that are sign changes, how many are also flagged. Note the total tested column for this category equals the number found for sign change, reflecting the definition of the fourth category.
The fifth column is for flagged turning points, i.e., for those observations that are turning points, how many are also flagged.
The footnote to Table S 2.A.3 gives the Census Bureau recommendation for thresholds, as described in "Computational Details" earlier in this section.
Table S 2.B gives the histogram of flagged for seasonal factors (Table D10) using the appropriate cutoff value (default 3.0%). This table looks at the spread of the number of times the MPD exceeded the corresponding level. The range is divided up into four intervals: 3.0%-4.0%, 4.0%-5.0%, 5.0%-6.0% and greater than 6.0%. The first column shows the symbol used in table S 7.A; the second column gives the range in interval notation, and the last column gives the number found in the corresponding interval. Note that the sum of the last column should agree with the "Number Found" column of the "Flagged MPD" row in Table S 2.A.3.
Table S 2.C gives selected percentiles for the MPD for the seasonal factors (Table D10).
These table relate to the Trading Day Factors (Table C18), and follow the same format as Tables S 2.A.1-S 2.A.3. The only difference between these tables and S 2.A.1-S 2.A.3 is the default cutoff value of 2.0% instead of the 3.0% used for the Seasonal Factors.
These tables, applied to the Trading Day Factors (Table C18), are the same format as tables S 2.B - S 2.C. The default cutoff value is different, with corresponding differences in the intervals in S 3.B.
These table relate to the Seasonally Adjusted Series (Table D11), and follow the same format as Tables S 2.A.1-S 2.A.3. The same default cutoff value of 3.0% is used.
These tables, applied to the Seasonally Adjusted Series (Table D11) are the same format as tables S 2.B - S 2.C.
These table relate to the Month-to-Month (or Quarterly-to-Quarterly) differences in the Seasonally Adjusted Series, and follow the same format as Tables S 2.A.1-S 2.A.3. The same default cutoff value of 3.0% is used.
These tables, applied to the Month-to-Month (or Quarterly-to-Quarterly) differences in the Seasonally Adjusted Series, are the same format as tables S 2.B - S 2.C. The same default cutoff value of 3.0% is used.
These table relate to the Year-to-Year differences in the Seasonally Adjusted Series, and follow the same format as Tables S 2.A.1-S 2.A.3. The same default cutoff value of 3.0% is used.
These tables, applied to the Year-to-Year differences in the Seasonally Adjusted Series, are the same format as tables S 2.B - S 2.C. The same default cutoff value of 3.0% is used.
Table S 7.A gives the entire listing of the Seasonal Factors (Table D10) for each span. The first column gives the date for each observation included in the spans. Note that the dates do not cover the entire original data set. Only those observations included in one or more spans are listed.
The next N columns (where N is the number of spans) are the individual spans starting at the earliest span. The span columns are labeled by their beginning and ending dates.
Following the last span is the "Sign Change" column. As explained in the description of Table S 2.A.3, a sign change occurs at a given observation when the seasonal factor estimates go from above to below, or below to above, a base level. For the seasonal factors, 100.0 is the base level for the multiplicative model, 0.0 for the additive model. A blank value indicates no sign change, a "U" indicates a movement "upwards" from the base level and a "D" indicates a movement "downwards" from the base level.
The next column is the "Turning Point" column. As explained in the description of Table S 2.A.3, a turning point occurs when there is an upward then downward movement, or downward then upward movement of sufficient magnitude. A blank value indicates no turning point, a "U-D" indicates a movement "upwards then downwards" and a "D-U" indicates a movement "downwards then upwards".
The next column is the maximum percent difference (MPD). This quantity, described in "Computational Details" above, is the main computation for sliding spans analysis. A measure of how extreme the MPD value is given in the last column, the "Level of Excess" column. The symbols used and their meaning is described in Table S 2.A.3. If a given observation has exceeded the cutoff, the level of excess column is blank.
Table S 7.B gives the entire listing of the Trading Day Factors (Table C18) for each span. The format of this table is exactly like Table S 7.A.
Table S 7.C gives the entire listing of the Seasonally Adjusted Data (Table D11) for each span. The format of this table is exactly like Table S 7.A except for the "Sign Change" column, which is not printed. The Seasonally Adjusted Data has the same units as the original data; there is no natural base level as in the case of a percentage. Hence the sign change is not appropriate for D11.
Table S 7.D gives the entire listing of the Month-to-Month (or Quarter-to-Quarter) Changes in Seasonally Adjusted Data for each span. The format of this table is exactly like Table S 7.A.
Table S 7.E gives the entire listing of the Year-to-Year Changes in Seasonally Adjusted Data for each span. The format of this table is exactly like Table S 7.A.
If the PRINTALL option is specified, a summary of the Nonlinear Estimation Optimization and a table of Box-Ljung Statistics is also produced. If the automatic model selection is used, this information is printed for each of the five predefined models. Lastly, a Model Selection Summary is printed, showing the final model chosen.
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