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X11 Output: A 1. Original Series. This X-11 table will show the original series, prior to any initial user-defined or trading-day adjustment. Note that for quarterly series, no prior adjustment factors can be specified, and the original series will be shown as table B 1.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: A 2. Prior Monthly Adjustment Factors. For X-11 monthly series, the user may specify a second series that contains prior monthly adjustment factors, for example, in order to adjust for an unusual holiday etc. The factors specified here will be subtracted from the original series for additive models, or will be used to divide the original series if multiplicative seasonal adjustment was requested (thus, the values in this series must be unequal to zero in that case).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: A 3. Original Series Adjusted by Prior Monthly Adjustment Factors. In this X-11 monthly series, the factors specified in A 2 will be subtracted from the original series (additive adjustment) or they will be used to divide the values in the original series (multiplicative adjustment). The resulting adjusted series is shown in this table.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: A 4. Prior Trading-Day Adjustment Factors. This X-11 table is only available (applicable) when prior trading-day adjustment factors and a multiplicative model were specified. The user may specify a weight for each day (Monday through Friday); those weights are then proportionately adjusted so that they add to 7. The series (A 1 or A 3) is then divided by monthly calendar factors that are computed based on the number of the respective days in the respective month. Note that by default, the calendar factors are also adjusted for different lengths of different months; however, the length of month variability can also be included in the calendar factors (in which case a constant length of month of 30.4375 is used).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 1. Prior Adjusted Series or Original Series. This X-11 table shows the original series, or the initial adjusted series, depending on whether or not prior monthly adjustment factors and/or trading day adjustment factors where specified (for quarterly X-11, B 1 is always the original series).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 2. Trend-Cycle. The initial trend-cycle estimate is computed as a centered 12-term moving average of B 1.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 3. Unmodified S-I Differences or Ratios. An initial estimate of the combined irregular and seasonal component is obtained by subtracting B 2 from B 1 (additive model) or dividing B 1 by B 2 (multiplicative model).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 4. Replacement Values for Extreme S-I Differences (Ratios). First a preliminary estimate of the X-11 seasonal component is computed by applying a weighted 5-term moving average separately to the B 3 values for each month. Then a centered 12-term moving average of the preliminary factors for the entire series is computed, and the resulting values are adjusted to sum to zero (additive model) or 12.0 (multiplicative model) within each year. Next an initial estimate of the irregular component is obtained by subtracting from the S-I differences (additive model) or dividing the S-I ratios by the initial estimate for the seasonal component. For the resulting initial estimate of the irregular component, a 5-year sliding standard deviation (s, sigma) is computed, and extreme values in the central year that are beyond 2.5*s are removed. The 5-year sliding s is then recomputed and the process repeated; however, this time a zero weight is assigned to irregular values beyond 2.5*s, a full weight is assigned for values within 1.5*s, and linearly graduated weights between zero and one are assigned for values between 1.5 and 2.5 * s. Values receiving less than full weights are then recomputed as the average of the respective value times its weight and the nearest two full- weight values preceding and following the respective value in that month. Table B 4 shows the final replaced (re-computed) values, and the sliding 5-year s's.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 5. Seasonal Factors. The extreme values in the B 3 series are replaced by the values shown in B 4. From this X-11 series, preliminary seasonal factors are derived by applying a 5-term moving average to each month separately; then a 12-term moving average is computed for the entire series, and the resulting values adjusted to sum to zero (additive model) or 12.0 (multiplicative model) within each year.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 6. Seasonally Adjusted Series. The preliminary seasonally adjusted series is obtained by subtracting from B 1 (additive model) or dividing B 1 (multiplicative model) by the seasonal factors in B 5.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 7. Trend-Cycle. The X 11 seasonally adjusted series (B 6) is smoothed via a variable moving average procedure (see Shiskin, Young, & Musgrave, 1967, for details). Optionally, extremes can be removed from the smoothed series by a process analogous to that described under B 4. In general, the so- called Henderson curve moving average is applied, which is a weighted moving average with the magnitudes of the weights following a bell-shaped curve (see, for example, Makridakis and Wheelwright, 1978, or Shiskin, Young, and Musgrave, 1967). The choice of the appropriate length of the moving average is an important issue in the seasonal decomposition (i.e., the computation of the trend-cycle component). The general idea is to choose a longer moving average when there is a lot of random fluctuation in the data relative to the trend-cycle component, and to choose a shorter moving average when there is only relative little random fluctuation. By default the program will select a moving average transformation automatically. Specifically, first a preliminary 13-term Henderson (weighted) moving average of the seasonally adjusted series is computed (without extending to the ends of the series). A preliminary estimate of the irregular component is then computed by subtracting this series from (additive model) or dividing it into (multiplicative model) the seasonally adjusted series. Next, the average month- to-month difference (percent change) without regard to sign is computed for both the estimated irregular and trend-cycle components. The ratio of the average month-to-month differences (percent changes) in the two series reflects the relative importance of the irregular variations relative to the movements in the trend-cycle component. Depending on the value of this ratio, either a 9-term Henderson moving average is selected (if the ratio is between 0.0 and .99), a 13- term Henderson moving average is selected (if the ratio is between 1.0 and 3.49) or a 23-term Henderson moving average is selected (if the ratio is greater than 3.5).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 8. Unmodified S-I Differences (Ratios). This X-11 table is the same as B 3 except that it is based on the trend-cycle values computed in X-11 Census Method II Seasonal Adjustment.

X11 Output: B 9. Replacement Values for Extreme S-I Differences (Ratios). This X-11 table is the same as B 4 except that the differences (ratios) in B 8 are used to which a 7 term moving average is applied (to estimate the seasonal factors).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 10. Seasonal Factors. After replacing extreme values by the corresponding B 9 values, a 7-term weighted moving average is applied to the S-I differences (ratios) in B 8. The resulting estimate of the seasonal factors is then adjusted so that the sum for each year is equal to zero (additive model) or 12.0 (multiplicative model).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 11. Seasonally Adjusted Series. This X-11 table is the same as B 6, except that the seasonal factors in B 10 are used.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 13. Irregular Series. The trend-cycle estimates in B 7 are subtracted from the seasonally adjusted series in B 11 (additive model), or the B 7 values are used to divide the series in B 11 (multiplicative model). The resulting series is an improved estimate of the irregular series.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 14. Extreme Irregular Values Excluded from Trading-Day Regression. The months in the series are sorted into different groups, depending on the particular day when the month begins (30-day, 31-day months, and Februarys are treated separately). Then extreme values (beyond 2.5 * s; different s values can also be specified) are identified within each type of month in a two-step procedure. The final extreme values that will be excluded are shown in this X-11 table.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 15. Preliminary Trading-Day Regression. After removing the B 14 extreme values from B 13, least squares estimates for the seven daily weights are computed.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 16. Trading-Day Adjustment Factors Derived from Regression Coefficients. From the trading-day regression weights, monthly adjustment factors are computed based on the number of particular trading days (i.e., Mondays, Tuesdays, etc.) in the respective months. These factors are printed in this X-11 table, and are then used to adjust (i.e., subtracted from or divided into) the B 13 irregular series for trading-day variation.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 17. Preliminary Weights for Irregular Component. The estimates of the irregular component (in B 13 or adjusted by B 16, depending on whether or not a trading-day adjustment was performed) are further refined by computing graduated weights for extreme values, depending on their relative (in terms of a sliding 5-year s) distance from 0. Specifically, a process analogous to that described in B 4 above is used. This X-11 table (B17) contains the resulting adjustment factors.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 18. Trading-Day Factors Derived from Combined Daily Weights. This X-11 table contains the final trading day adjustment factors, computed from the least squares trading-day weights in B 15 and/or the prior trading-day weights in A 4.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: B 19. Original Series Adjusted for Trading-Day and Prior Variation. The values in B 18 are used to adjust the original (adjusted) series (in A 1, A 3, or B 1, depending on whether or not prior adjustment factors were specified). Specifically, the values in B 18 are subtracted from (additive model) or divided into (multiplicative model) the original series.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 1. Original Series Modified by Preliminary Weights and Adjusted for Trading-Day and Prior Variation. The series in B 19 (or B 1 if no trading-day adjustment was requested) is adjusted for extreme values by the weights computed in B 17 The resulting modified series in shown in this X-11 table (C 1).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 2. Trend-Cycle. An estimate of the combined trend-cycle component is computed from C 1 by applying a centered 12-term moving average.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 4. Modified S-I Differences (Ratios). To obtain the refined S-I differences (ratios), the values in C 2 are subtracted from (additive model) or divided into (multiplicative model) the modified series in C 1.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 5. Seasonal Factors. These values are the same as those in B 5, except that the C 4 differences (ratios) are used.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 6. Seasonally Adjusted Series. The preliminary seasonally adjusted series is computed by subtracting C 5 from (or dividing C 5 into) C 1.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 7. Trend-Cycle. The seasonally adjusted series (C 6) is smoothed via a variable moving average procedure (the same procedure used for B 7, see also Shiskin, Young, & Musgrave, 1967, for details) to derive the preliminary estimate of the trend-cycle component.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 9. Modified S-I Differences (Ratios). The modified S-I differences (ratios) are computed by subtracting C 7 from (additive models) or dividing C 7 into (multiplicative models) the C 1 series.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 10. Seasonal Factors. The seasonal factors are computed analogously to B 10, but based on the C 9 S-I differences (ratios).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 11. Seasonally Adjusted Series. The refined seasonally adjusted series is computed by subtracting from B 1 (additive model) or dividing B 1 by (multiplicative model) the values in C 10.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 13. Irregular Series. The refined estimate of the irregular (random) component is computed by subtracting from C 11 (additive model) or dividing C 11 by (multiplicative model) the values in C 7.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 14. Extreme Irregular Values Excluded from Trading-Day Regression. This table is analogous to table B 14, and it shows the extreme irregular values (usually beyond 2.5 * s) after re-applying the trading-day routine (based on the monthly trading-day factors shown in B 16).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 15. Final Trading-Day Regression. This X-11 table is the same as B 15, except that the computations are based on the values from table C 13.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 16. Final Trading-Day Adjustment Factors Derived from Regression X11 Output: Coefficients. This X-11 table is analogous to B 16, except that the factors are subtracted from (additive case) or divided into (multiplicative case) the values from table C 13.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 17. Final Weights for Irregular Component. This table is analogous to table B 17, except that it is computed based on the values in C 16 (or C 13 if no trading day adjustment is requested).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 18. Final Trading-Day Factors Derived from Combined Daily Weights. This X-11 table is analogous to B 18, except that the final weights shown in C 15 are used in the computations.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: C 19. Original Series Adjusted for Trading-Day and Prior Variation. The values in C 18 are used to adjust the original (adjusted) series (in A 3 or B 1). Specifically, the values in C 18 are subtracted from (additive model) or divided into (multiplicative model) the original series.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 1. Original Series Modified by Final Weights and Adjusted for Trading-Day and Prior Variation. This X-11 table is analogous to C 1, except that the C 17 weights and C 19 adjusted series are used in the computations.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 2. Trend-Cycle. A 12-term moving average of D 1 is computed to estimate the trend-cycle component.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 4. Modified S-I Differences (Ratios). The modified S-I differences (ratios) are computed by subtracting D 2 from (additive model) or dividing D 2 into (multiplicative model) the values in D 1.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 5. Seasonal Factors. This X-11 table is computed analogously to B 5, except that the computations are based on the values in D 4.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 6. Seasonally Adjusted Series. The values in this table are computed by subtracting D 5 from D 1 (additive model) or dividing D 1 by D5 (multiplicative model).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 7. Trend-Cycle. The values in this X-11 table are computed analogously to those in B 7, except that the computations are based on the values in D 6.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 8. Final Unmodified S-I Differences (Ratios). The values in the D 7 series are subtracted from (additive model) or divided into (multiplicative case) the values in C 19 (or B 1 if no adjustment for trading-day variation is applied). Then an analysis of variance by month (or quarter) is performed on this series, in order to test for the presence of stable significant seasonality.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 9. Final Replacement Values for Extreme S-I Differences (Ratios). The values in D 7 are subtracted from (additive model) or divided into (multiplicative model) D 1; values that are not identical to the corresponding entries in D 8 are then reported. Also, for each month, the year-to-year difference (additive model) or percent change (multiplicative mode) in the estimates of the irregular and the seasonal components and their ratio (called MSR, moving seasonality ratio) are computed. The MSR may be useful in order to determine the amount of moving seasonality present in each month.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 10. Final Seasonal Factors. This X-11 table is computed analogously to the values in B 10, except that it is computed based on the values reported in D 8 and D 9.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 11. Final Seasonally Adjusted Series. The final seasonally adjusted series is computed by subtracting D 10 from C 19 (additive model) or dividing C 19 by D 10 (multiplicative model).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 12. Final Trend-cycle. These values are computed by subtracting D 10 from D 1 (additive model), or by dividing D 1 by D 10 (multiplicative model).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: D 13. Final Irregular. These values are computed by subtracting D 12 from D 11 (additive model), or by dividing D 11 by D 12 (multiplicative model).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: E 1. Modified Original Series. The values in this X-11 table are computed by replacing in the original series extreme values (identified by a zero weight in C 17) by the values predicted from the final trend-cycle, seasonal, trading-day (if applicable), and prior adjustment (if applicable) components.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: E 2. Modified Seasonally Adjusted Series. These values are computed by replacing in the final seasonally adjusted series (D 11) extreme values (identified by a zero weight in C 17) with the D 12 final trend-cycle values.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: E 3. Modified Irregular Series. The values in this X-11 table are computed by replacing the values in D 13 with zero (additive model) or 1.0 (multiplicative model) if they were identified as extremes (i.e., assigned zero weight) in C 17.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: E 4. Differences (Ratios) of Annual Totals. These values are computed as the differences (additive model) or ratios (multiplicative model) of the annual totals of (a) the original series B 1 and the final seasonally adjusted series D 11, (b) the modified original series E 1 and the modified seasonally adjusted series E 2.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: E 5. Differences (Percent Changes) in Original Series. The values in this X-11 table are computed as the month-to-month (quarter-to-quarter) differences (additive model) or percent changes (multiplicative model) in B 1.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: E 6. Differences (Percent Changes) in Final Seasonally Adjusted Series. These values are the month-to-month (quarter-to-quarter) differences (additive model) or percent changes (multiplicative model) in D 11.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: F 1. MCD (QCD) Moving Average. The values in this series are computed by applying an unweighted moving average to the final seasonally adjusted series (D 11). The width of the smoothing window is determined by the month (quarter) for cyclical dominance, or MCD (QCD) for short. The MCD (QCD) is computed as the average span at which the changes in the random component are equal to the changes in the trend-cycle component

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: F 2. Summary Measures. Several final summary X-11 tables are computed:

1. The average differences (additive model) or percent changes (multiplicative model) are computed without regard to sign across spans 1, 2, 3 ..., 12 months (or four quarters) for the following series: Original series A 1 (B 1), final seasonally adjusted series (D 11), final irregular series (D 13), final trend-cycle (D 12), final seasonal factors (D 10), final prior monthly adjustment factors (A 2, monthly X-11 only), final trading-day adjustment factors (C 18, monthly X-11 only), modified original series (E 1), modified seasonally adjusted series (E 2), modified irregular series (E 3).
2. Next a table of relative contributions of the different components to the differences (additive model) or percent changes (multiplicative model) in the original series are computed.
3. The next table reports the average duration of run (the average number of consecutive monthly changes in the same direction; "no change" is counted as a change in the same direction) for the following series: Final seasonally adjusted series (D 11), final irregular series (D 13), final trend-cycle (D 12), and the MCD (QCD) moving average (F 1).
4. Finally, the means and standard deviations of differences (additive model) or percent changes (multiplicative model) are computed across different spans for each of the series mentioned above.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: G 1. Chart. This line graph will show the final seasonally adjusted series and final trend-cycle components (D 11 and D 12, respectively).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: G 2. Chart. This line graph will show the final S-I differences (additive model) or ratios (multiplicative model) with the extremes, the final S-I differences (ratios) without extremes, and the final seasonal factors (i.e., D 8, D 9, and D 10, respectively), categorized by month (X-11 monthly) or quarter (X-11 quarterly).

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: G 3. Chart. This plot shows the same values as G 2; however, this line plot shows those values in chronological order.

For more information, see X-11 Census Method II Seasonal Adjustment.

X11 Output: G 4. Chart. This is a line graph of the final irregular and final modified irregular series (D 13 and E 3, respectively).

For more information, see X-11 Census Method II Seasonal Adjustment.

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