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SELECTED TECHNIQUES OF SEASONAL ADJUSTMENT HA 33 F3cl https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Seminar on Seasonal Adjustment Federal Reserve System Washington, D C. June 5-6, 1962 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis n■ * 33 CONTENTS Computational Steps of Selected Methods of Seasonal Adjustment Bureau of the Census Seasonal Adjustment Technique (Method II) X-9 Version of Census Method II X-10 Version of Census Method II Seasonal Adjustment Method of the Bureau of Labor Statistics https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Prepared By: Research Department Federal Reserve Bank of Atlanta 18469 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis COMPUTATIONAL STEPS OF SELECTED METHODS OF SEASONAL ADJUSTMENT The tables appearing on the following four pages were compiled from published descriptions as follows: Census Method II, Seasonal Adjustment on Electronic Com puters, Organization for Economic Cooperation and Development, pp. 389-398 and Electronic Computers and Business Indicators, Occasional Paper 57, National Bureau of Economic Research, Inc, pp. 248-257. X-9 and X-10 Versions of Census II, mimeographed material from Office of Chief Economic Statistician, Bureau of the Census. BLS Method, ”BLS Seasonal Factor Method,” Abe Rothman, 1960 Proceedings of the Business and Economic Sta tistics .Section, American Statistical Association, pp. 2-12. Regression Method, ’’The Practice of Seasonal Adjustment with Regression Equations,” Deutsche Bundesbank, Frankfurt, October, 1960. BASIC CALCULATION STEPS USED ST SEIECTED METHODS CP SEASONAL ADJUSTMENT CENSUS METHOD II X-9 VERSION OP CENSUS METHOD II Computation of Preliminary Seasonally Adjusted Series Computation of Preliminary Seasonally Adjusted Series *1. Adjustment for trading days is optional. If used, daily averages became original data, *1, Steps 1-5 are the same as Census Method II. X-1O VERSION OP CENSUS METHOD H Computation of Preliminary Seasonally Adjusted Series *1, Steps 1-5 are the same as Census Method II. 2. Compute ratio of ori ginal to average of preceding and following months, BUREAU OF LABOR STATISTICS METHOD First Iteration *1, Develop a centered 12-month moving average (MA) of original. Six values at each end are computed by a series of steps. Preliminary estlmate of trend-cycle (TC), *2, Compute ratio of original to centered 12-month MA. First approximation of seasonal-irregular (SI), *3. Develop an uncentered 12-month moving average (MA) of original, *3, For each calendar month, compute a 5-term weighted moving average (WMA) of SI ratios in step 2. Un forced seasonals, first approximation, *4, Center 12-month MA. *5, Calculate ratio of ori ginal to centered 12-month MA. REGRESSION METHOD OP DEUTSCHE BUNDESBANK Basic Method *1, Develop an uncentered 12-month MA of original (a). This is used to represent trend (t). 2, Basic analysis based on following additive re lationship: an»tn + Pn + En Original values-trend values + seasonal com ponent + residual com ponent. 3, Other symbols: a’-seasonal values (re gression values) a"«residual values (a-a>) a*»seasonally adjusted values. *4, Force total to 1200, *6, Identify extreme values of step 5 and replace with more representative ones as follows: *6. Omit step 6 in Census II and substitute the following: *6. Omit step 6 in Census II and substitute the following: a. Compute 5-term MA for each month. To get MA for first two years, average the first two ratios available, MA for last two years are obtained similarly. a. Compute a 5-terni MA for each month of data in step 5* get MA for first two years, repeat MA of third year, MA for last two years are obtained similarly. b. For each month, compute 2-sigma control limits about 5-term MA. All ratios falling outside limits are extreme. b. For each month, compute 2-sigma control limits about 5-term MA. All ratios falling outside limits are extreme. Replace extremes as follows: (1) Ratio falling first of series, average of first three ratios; (2) Ratio falling in middle, average ex treme ratio and preceding and following ones; (3) Ratio falling at end, average extreme and two preceding ratios. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis c. Replace extremes as follows: (1) Ratio falling first of series average of second, third, and fourth ratios; (2) Falling second, average of first, third, and fourth ratios; (3) Falling middle, average two preceding and two following; (4) Falling next to last or last, similar to beginning. *5, Compute seasonally ad justed series, *4. Standard regression equa tion used for each month:. a»«tB + A, where B slope; A»Y intercept: , *6, Divide seasonally adjusted series by 12-month MA of original. First approxi ft = ~ mation of irregular (I) & with some TC. *5. With A and B values, com pute seasonal values (a’) Second Iteration for each original values: a’etB + A. *7, Smooth!ratios in step6by 7 -month WMA after extending *6. Graphical checking of com b. For each month, compute putations, Using (t) on 2-sigma control limits I for three months at each the X axis and (a) on the about 5-term MA. All end. Measure of residual TC. Multiply resulting ratios falling outside Y axis, plot the values values by TC of step 1. limits are extreme. and the regression line. Visual proof of the correct *8. Compute ratio of original c. Replace extremes as computation of A and B. follows: (1) Ratio to TC of previous step. falling first of Second approximation of *7. Compute residual values (a”) by comparing re series, average of SI. gression values (a*) with second, third and fourth ratios; (2) original values (a): *9. For each calendar month, a"=a-a*. If a" greater Falling second, aver compute 5-term WMA of SI age of first, third, ratios. Unforced seathan 0, there is superseasonal present; if a" and fourth ratios; sonals, second approxi (3) Falling middle, mation, is less than 0, there is average two preceding subseasonal; if 0, there is *10, Force total to 1200, and two following; is only purely seasonal. (4) Falling next to last or last, similar to beginning. a. Compute a 5-termMAfor each month of data in step 5» get MA for first two years, repeat MA of third year. MA for last two years are obtained similarly, 2 CENSUS METHOD II *6* (Continued) 1-9 VERSION OP CENSUS METHOD II *6. (Continued) X-1O VERSION OP CENSUS - METHOD II ♦6 d. Six missing ratios (due to step 4) at beginning are supplied by extending first available ratios for corresponding months back to Initial month of series. Six missing at end suppli ed similarly. d. For each month, com pute a 3-term MA. Missing values suppli ed for first year-aver age first three ratios} similar for end. e. Force total to 1200. e. Compute a centered 12-month MA. Missing values—repeat first available ratio six times. Similar for end. Divide Into step 6d. (Continued) d. For each month, com pute a 7-term HA of ratios In step fie. Missing ratios suppli ed In first 3 years by averaging first three years available. Simi lar for last years. MA values computed by using these estimates. e. For each month, com pute a 3-term MA. Missing values-use value in step 6e corresponding to the month missing. f. g. Six factors missing at end (due to step 4) are obtained by using the factor for the same month of the first or last available year. These are preliminary seasonal factors. g. f. Compute 3-term of 3f. term MA of ratios In fie for each month. Supply missing values at each end. The results are pre liminary seasonal factors. h. 1. (Step 7 follows on the next page) https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis j. BUREAU OF LABOR STATISTICS METHOD *11. Compute seasonally ad justed series. *12. Divide seasonally ad justed series by TC of step 7. Second approxi mation of I. Third Iteration REGRESSION METHOD OP DEUTSCHE BUNDESBANK *8. Compute seasonally ad justed values (a*) by adding residual values (a") to corresponding trend values: a*«t + a”. 9. To decompose time series: (seasonal) p-a’ - t; (residual) E~a"«a - a’. *13. Smooth I ratios In step Refinement of Trend Trans lation. 12 by 7-month WMA after extending I for 3 *10. Test whether trend Is For each month, compute months at each end. ’’true," i.e., whether the average, without Multiply smoothed I by TC in step 7. Result seasonal fluctuations regard to sign, of Is final TC unless ex year-to-year percent around trend are dis treme values are present. changes In MA of step torted. fid. *14. Compute ratio of ori *11. If refinement Is ginal to TC of previous necessary, improved trend values (tj) are step. Third approxi For each month, divide step 6d Into step fic* mation of SI. obtained by smoothing trend values (t): Estimate of I. ♦15. For each calendar month, compute 5-term WMA of So SI In step 14. Unforced Where Ao and Bo are ori seasonal, third approxi ginal regression equa mation. For each month, com tion coefficients. pute average, without *lfi. Force total to 1200. regard to sign, of *12. New Improved trend values (ti) are then used as These are final factors year-to-year percent changes In I. unless extreme values basis of a refined corre are present. lation between the trend For each month, com and original values. pute ratio of step fig *17. Compute seasonally ad Calculation techniques to step 6e. Designated justed series. for various values are as Moving Seasonality the same as in the Basic *18. Divide seasonally ad Ratios. Method. justed series by TC of step 13. Final I unless For each month, de Regression Method extreme values are pre Concluded pending upon siae of ratio in step fih, MA of sent. ratios yielded by step 6c Is computed using Fourth Iteration term indicated in the table at the end of the *19. Test for extreme values Instructions. Missing and, If found, replace ratios supplied. with substitute values. Tests involve developing, Compute a centered smoothing and analysing 12-month MA of fil. Irregular component to Missing values—repeat determine whether values first available ratio fall outside +2.8 sigma six times. Similar for limits. Replacements end. Divide Into step are calculated by multi 61. plying TC by S for a given month. 3 CENSUS METHOD II X-9 VERSION OP CENSUS METHOD II X-10 VERSION OF CENSUS METHOD II *6. (Continued) *7. Compute preliminary sea sonally adjusted series. Final Seasonally Adjusted Series *7. Same as Census II. Final Seasonally Adjusted Series Fifth Iteration k. For each month, com pute a 3-term MA. Missing values—use value in step 6j corre sponding to the month missing. Repeat basic steps 1-6 of First Iteration using re placements for extreme values. This Iteration develops first approximation to time series components. l. Six factors missing at end (due to step 4) are obtained by using the factor for the same month of the first or last available year. These are preliminary seasonal factors. Sixth Iteration •7. Same as Census II Final Seasonally Adjusted Series *8. Develop a 15-month WMA of preliminary seasonally adjusted series supplying missing values. *8. Same as Census II. *9. Compute ratio of original to 15-aonth WMA *9. Same as Census II *9. Same as Census II. *10. Compute ratio of pre liminary seasonally ad justed series to its 15-month WMA. *10. Same as Census II. *10. Omit from Census II. *11. Compute month-to-month percent changes of step 10 and average without regard to sign. Measures average ampli tude of I. *11. Same as Census II. *11. Omit from Census II. *8. Same as Census II. Steps 7-12 of Second Itera tion are repeated still using replacement values as original values. Seventh Iteration Steps 13-18 of Third Itera tion are repeated. After derivation of final measures, replacement values are re placed with original values and final seasonally adjusted series is derived. Bureau of Labor Statistics Method Concluded *12. Identify extreme values *12. Omit step 12 in Census in step 9 and replace II and apply steps 6a-6c above to the re in the same manner as explained in steps sults of step 10. 6a-6c above. Force total total to 1200. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis BUREAU OF LABOR STATISTICS METHOD *12. Omit steps 12 and 13 in Census II and substitute as explained above in steps 6a-6k using the results in step 9 above. These are final seasonal factors. - 4 - X-10 VERSION OP CENSUS METHOD II X-9 VERSION OF CENSUS METHOD II CENSUS METHOD II *13. Final seasonal factors are derived as follows: If irregular in step 11 averages under 2, use a 3-term MA of a 3-term MAj If I is 2 or more, use a 3-term MA of a 5-term MA. Missing values at each end are supplied. *13. Pinal seasonal factors are derived as follows: If I in *13. See step 12. step 11 averages under 2, use 3-term MA; if I is 2 or. more, compute a 5-terra MA. Missing values at each end are supplied. Then perform steps be and 6f. These are final seasonal factors. *14. Project seasonals in step 13 for year ahead on basis of the seasonal factors for the last two years. *14. The remaining steps are identical to the steps in Census II. *14. The remaining steps are Identical to the steps in Census II. 13. Compute seasonally adjusted series. 16. Compute ratio of final seasonally adjusted series to average of preceding and following month as test of residual seasonal. r TERM OP MOVING AVERAGE FOR DIFFERENT SEASONALITY RATIOS IN X-10. SEE STEP 6i. 17. Develop an uncentered 12-month MA of seasonally ad justed series. 18. Compute ratio of uncentered 12-month MA of final series te similar average of original series to provide test for bias. 19. Calculate ratio of each month to the preceding January in final series as test for residual seasonal of more than a month’s duration. Measures of Irregular (I), Cyclical (C), Seasonal (S) Moving Seasonality Ratio step 6h 0-1.49 1.50- 2.49 2.50- 4.49 4.50- 6.49 6.50- 8.49 8.50 and over Average of 6 c Values None (Leave 6c values unchanged) 3-term moving average 5-term moving average 9-term moving average 15-tena moving average Arithmetic average of all 60 values Number of beginning or ending 6 c values averaged to extend MA 2 2 3 3 20. Compute 15-month WMA of final series-yields (C). 21. Compute month-to-month percentage changes in original (0), seasonal factors (S), final seasonally adjusted series (CI), cyclical (C), and ratio of original to 12-month WMA. An asterisk (♦) identifies a step involved in derivation of seasonal factors. 22. Compute ratio of final series to 15-month WMA of final Series. Yields estimate of I. Calculate month-to-month percentage changes in I. 23. Derive mean of percentage changes in original fo), irre gular Cl). cyclical (C), seasonal C5), and seasonally adjusted (Cl). 24. Uslng_averages in step 23, calculate: "I/C, I/T, Vo; c/o; S/0". ~S/C~, 25. Compute ratio of VS”with percentage changes taken 2, 3, 4, and 5 months apart. The interval corresponding to the last ratio that is less than 1.00 is the "number of months for cyclical dominance." Calculate MA of final series, using this number as its period. 26. Derive average duration of run for CI, I, C, and CI, smoothed in step 25 • 27. Compute, without regard to sign, ratio of 12-month MA of month-to-month percent changes in I to 12-month MA of month-to-month percent changes in C. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis RESEARCH DEPARTMENT FEDERAL RESERVE BANK OF ATLANTA MAY, 1962 BUREAU OF THE CENSUS SEASONAL ADJUSTMENT TECHNIQUE* (METHOD II) I, Computation of Preliminary Seasonally Adjusted Series 1. Original observations. Where an adjustment for the number of working or trading days is made, these figures are shown after adjustment and all subsequent computations are based on these adjusted figures (Table I of sanple "print-out”). 2. Ratios of the original observations for each month to the average of the original observations for the preceding and following months are computed. Arithmetic means of these ratios for each month are given at the bottom of the table (Table 2). 3. A twelve-month moving average of the original series is computed. This curve provides a measure of the trend-cycle component of the series. It also provides annual averages of the original series (Table 3). 4. The twelve-month moving average is centered—that is, a two-month moving average of the twelve-month moving average is computed. This operation places the moving average values at mid-months. The first value of the centered moving average is placed at the seventh month of the original series. Thus six moving average values will be missing at the beginning and at the end of the series (Table 4). 5. Ratios of the original observations to the centered twelve-month moving average are computed. This computation results in a series which shows primarily the seasonal and irregular components of the original series (Table 5). 6. This step will provide a method for identifying extreme items among the ratios computed by step 5, substituting more representative ratios for these extreme ratios and fitting smooth curves to all ratios for each month. a. Fit a five-term moving average to the ratios for each month. This results in the loss of moving average values for the first two and the last two years for which ratios are avail able. To obtain moving averages for the first two years, use the average of the first two ratios as the estimated value of the ratio for each of the two years preceding the first year available. This is equivalent to weighting the first three years' ratios by 2/5, 2/5, and 1/5, respectively, *"A Description of the United States Bureau of the Census Method of Adjustment of Series of Monthly Data for Seasonal Variations," Seasonal Adjustment o Electronic Computers, pp. 391-398. This description is the same as cone tained in Electronic Computers and Business Indicators, Occasional Paper No. 57, National Bureau of Economic Research, 1957, pp. 248-257. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 2 (CM II) to obtain the first year’s moving average value, and to weighting the first four years’ ratios by 3/10, 2/10, 2/10, and 2/10, respectively, to obtain the second year’s moving average value. Moving average values for the last two years are obtained in a similar manner. b. For each month, compute two-sigma control limits about the five-term moving average line. All ratios falling outside these limits are designated as extreme. c. Replace extreme ratios as follows: for an extreme ratio falling at the first point in the series, substitute the average of the first three ratios of the series; for an extreme ratio falling in the middle of the series, sub stitute the average of the extreme ratio and the pre ceding and following ratios; for an extreme ratio falling at the end of the series, substitute the average of the extreme ratio and the two preceding ratios. d. The six missing ratios at supplied by extending the corresponding months back The six missing ratios at e. For each year, center the twelve ratios (i.e., adjust the twelve ratios so that their sum will be 1,200) by division of the twelve items by their arithmetic mean. If the ini tial year is incomplete, use as the ratio for any missing month the value of the average ratio for the same month in the next two years in centering the initial year's ratios. Treat the terminal year’s ratios in a similar manner. f. For each month, compute a three-term moving average of a three-term moving average of the centered ratios yielded by step 6e, above. This will result in the loss of two moving average values at the beginning and two at the end. To obtain the values missing at the beginning, use the average of the first two centered ratios as the estimated value of the centered ratio for each of the two years pre ceding the first year available. This is equivalent to weighting the first three years’ centered ratios by 9/18, 7/18, and 2/18, respectively, to obtain the first year’s moving average value, and to weighting the first four years’ centered ratios by 5/18, 7/18, 4/18, and 2/18, re spectively, to obtain the second year’s moving average value. The missing values at the end are obtained in a similar way. The values of these twelve curves constitute the preliminary seasonal adjustment factors (Table 6). https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis the beginning of the series are first available ratios for the to the initial month of the series. the end are supplied similarly. - 3 (CM II) 7. These seasonal factors are divided into the corresponding figures of the original series, month by month; i.e., the seasonal factor for January, 1947, is divided into the original observation for January, 1947; the factor for January, 1948, is divided into the original observation for January, 1948. Similarly, the factor for February, 1947, is divided in to the original observation for February, 1947; the factor for February, 1948, into the original observation for February, 1948; and so on. This yields the preliminary seasonally adjusted series (Table 7). II, Computation of Final Sesonally Adjusted Series 8. Compute a weighted fifteen-month moving average (Spencer’s fifteen-term formula) of the preliminary seasonally adjusted series. The weights are as follows: -3/320, -6/320, -5/320, 3/320, 21/320, 46/320, 67/320, 74/320, 67/320, 46/320, 21/320, 3/320, -5/320, -6/320, -3/320. This is equivalent to a weighted five-month moving average (weights are -3/4, 3/4, 1, 3/4, -3/4) of a five-month moving average, of a four month moving average, of a four-month moving average of the data. To obtain values for the beginning points of this curve, use the aver age of the first four values of the preliminary seasonally adjusted series as the estimated value of this series for each of the seven months pre ceding the first month available. The values for the end are supplied similarly. The preliminary seasonally adjusted series contains the cyclical, trend, and irregular components of the series with only a trace of the seasonal component. The weighted fifteen-month moving average can be used in place of a twelve-month moving average because there is no signi ficant seasonal factor to suppress. The weighted fifteen-month moving average is much more flexible than a twleve-month moving average and will, therefore, provide a better measure of the trend-cycle component; it is also much smoother than a simple five-month moving average (Table 8). 9. Ratios of the original observations to the weighted fifteen-month moving average are computed (Table 9). 10. Compute the ratios of the preliminary seasonally adjusted series (step 7) to its weighted fifteen-month moving average (step 8). Month-to-month changes in these ratios are computed and averaged without regard to sign. This yields a preliminary measure of the average amplitude of the irre gular component. 11. This step will provide a method for identifying extreme items among the ratios computed by step 9, substituting more representative ratios for these extreme ratios, and fitting smooth curves to all ratios for each month. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 4 (CM II) a. Fit a five-term moving average to the ratios for each month. This results in the loss of moving average values for the first two and the last two years. To obtain moving averages for the first two years, use the average of the first two ratios as the estimated value of the ratio for each of the two years preceding the first year available. This is equi valent to weighting the first three years’ ratios by 2/5, 2/5, and 1/5, respectively, to obtain the first year’s moving average value, and to weighting the first four years’ ratios by 3/10, 3/10, 2/10, and 2/10, respectively, to obtain the second year’s moving average value. The moving average values for the last two years are obtained in a similar manner. b. For each month, compute two-sigma control limits about the five-term moving average line. All ratios falling outside these limits are designated as "extreme." c. Replace extreme ratios as follows: for an extreme ratio falling at the first point in the series, substitute the average of the first three ratios of the series; for an ex treme ratio falling at the end of the series, substitute the average of the extreme ratio and the two preceding ratios (Table 10). d. For each year center the twelve ratios (i.e., adjust the twelve ratios so that their sum will be 1,200) by division of the twelve items by their arithmetic mean. If the ini tial year is incomplete, use as the ratio for any missing month the value of the average ratio for the same month in the next two years in centering the initial year’s ratios. Treat the terminal year's ratios in a similar manner (Table 11). e. If the average irregular amplitude, computed in step 10 above, is under 2, use step Ilf; if it is 2 or more, use step llg. f. For each month compute a three-term moving average of a three-term moving average of the centered ratios yielded by step lid, above. This will result in the loss of two moving average values at the beginning and two at the end. To ob tain the values missing at the beginning, use the average of the first two centered ratios as the estimated value of the centered ratio for each of the two years preceding the first year available. This is equivalent to weighting the first three year's centered ratios by 9/18, 7/18, and 2/18, re spectively, to obtain the first year’s moving average value, and to weighting the first four year’s centered ratios by 5/18, 7/18, 4/18, and 2/18, respectively, to obtain the second year’s moving average value. The missing values at https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 5 - (CM II) the end are obtained in a similar way. These smoothed ratios constitute the final seasonal adjustment factors. This series is identified later by the symbol S (Table 12). g. For each month compute a three-term moving average of a fiveterm moving average of the centered ratios yielded by step lid, above. This will result in the loss of three moving average values at the beginning and three at the end. To ob tain the values missing at the beginning, use the average of the first two centered ratios as the estimated value of the centered ratio for each of the three years preceding the first year available. This is equivalent to weighting the first four year’s centered ratios by 6/15, 6/15, 2/15, and 1/15, respectively, to obtain the first year’s moving average value; to weighting the first five years' centered ratios by 9/30, 9/30, 6/30, 4/30, and 2/30, respectively, to obtain the second year’s moving average value; and to weighting the first six years' centered ratios by 5/30, 7/30, 6/30j 6/30, 4/30, and 2/30, respectively, to obtain the third year's moving average value. The missing values at the end are obtained in a similar way. These smoothed ratios constitute the final seasonal adjustment factors. This series is later identified by symbol S (Table 12). h. Estimates of the seasonal factors one year ahead are given at the bottom of Table 12. These estimates are made by adding to the seasonal factor for the end year, one-half the trend between the factor for that year and the preceding year. If X=seasonal adjusment factor for year N, then X^ + is esti mated by the equation 3X - X 12. These seasonal factors are divided into the corresponding figures of the original series, month by month; i.e., the seasonal factor for January, 1947, is divided into the original observation for January, 1947; the factor for January, 1948, is divided into the original observation for January, 1948. Similarly, the factor for February, 1947, is divided into the original observation for February, 1947; the factor for February, 1948, into the original observation for February, 1948; and so on. This yields the final seasonally adjusted series. This series is later iden tified by the symbol CI (Table 13). 13. The ratios of the final seasonally adjusted series to the averages of the final seasonally adjusted series for the preceding and following months are computed. This is a rough test for residual seasonality, similar to that made on the original observations described in step 2 above. Arithmetic means of these ratios for each month are given at the bottom of the table (Table 14). https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 6 (CM II) 14. Compute an uncentered twelve-month moving average of the final season ally adjusted series. This step is required to carry out the test de scribed in step 15. It also provides annual averages of the seasonally adjusted series (Table 15). 15. Compute ratios of the uncentered twelve-month moving average of the standard seasonally adjusted series to the uncentered twelve-month moving average of the original series. This is a test of the effect of the seasonal adjustment on the level of the series, showing whether the adjustment has resulted in significant differences between the level of adjusted and the unadjusted series for any twelve-month per iod (Table 16). 16. Using the final seasonally adjusted series, compute the ratio of the value of each month, from February through the following January, to that of the preceding January. Such a table of ratios will disclose repetitive patterns in successive years of more than one month's dura tion (Table 17). III. Measures of the Irregular, Cyclical and Seasonal Components 17. Compute a weighted fifteen-month moving average (Spencer's fifteen-term formula) of the final seasonally adjusted series. The weights are as follows: -3/320, -6/320, -5/320, 3/320, 21/320, 46/320, 67/320, 74/320, 67/320, 46/320, 21/320, 3/320, -5/320, -6/320, -3/320. This is equivalent to a weighted five-month moving average (weights are -3/4, 3/4, 1, 3/4, -3/4), of a five-month moving average, of a four-month moving average, of a four-month moving average of the data. To obtain values for the beginning points of this curve, use the aver age of the first four values of the final seasonally adjusted series as the estimated value of this series for each of the seven months preceding the first month available. The values for the end are supplied similarly. The final seasonally adjusted series contains the cyclical, trend, and irregular components of the series. The weighted fifteen-month moving average can be used in place of a twelve-month moving average because there is no seasonal factor to suppress. The weighted fifteen-month moving average is much more flexible than a twelve-month moving average and will therefore provide a better measure of the trend-cycle component; it is also much smoother than a simple five-month moving average, and It fits the data about as closely as does the five-month moving average. This series is identified by the symbol C (Table 18). 18. Compute the month-to-month percentage changes in the original series (Table 19). 19.. Compute the month-to-month percentage changes in the final seasonal ad justment factors (Table 20). https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 7 (CM II) 20. Compute the month-to-month percentage changes in the final seasonally adjusted series (Table 21). 21. Compute the month-to-month percentage changes in the ratios (step 9) of the original observations to the weighted fifteen-month moving aver age (Table 22). 22. Compute the ratios of the final seasonally adjusted series (step 12) to its weighted fifteen-month moving average (step 17). This provides a measure of the irregular component of the series. This series is identified by the symbol I (Table 23). 23. Compute the month-to-month percentage changes in the irregular component (Table 24). 24. Compute the month-to-month percentage changes in the weighted fifteenmonth moving average of the final seasonally adjusted series (Table 25). 25. Compute the average, without regard to sign, of the percentage changes in steps 18, 19, 20, 23, and 24. This operation yields measures of the average monthly amplitude of the original series, the seasonal component, the seasonally adjusted series, the irregular component, and the cyclical component, respectively. The symbols used to represent these averages are original, 0; irregular, T; cyclical, (2; seasonal, ?»; and seasonally adjusted, CI (Table 27). 26. Compute the following ratios of the average monthly amplitudes of step 25: a. Irregular component to cyclical component (I/C); b. Irregular component to seasonal component (I/S); c. Seasonal component to cyclical component (S/C); d. Irregular component to original series (I/O); e. Cyclical con^onent to original series (C/0); f. Seasonal component to original series (S/0); See Table 27. 27. Compute the ratio of the average monthly amplitude of the irregular to the cyclical components when percentage changes are taken between entries two, three, four, and five months apart (Table 27). The interval corresponding to the last T/C ratio that is less than 1.00 is designated as "Number of Months for Cyclical Dominance,” and a moving average of the seasonally adjusted data is computed, using this interval as its period (Table 26). https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 8 (CM II) 28. The average duration of run, that is, the average number of months the series moves before changing direction, is computed for the following: a. Seasonally adjusted series; b. Irregular component; c. Cyclical component; d. Seasonally adjusted series smoothed by moving average with period as given by number of months for cyclical dominance; See Table 27. 29. Compute the ratios of a) the twelve-month moving average of the monthto-month percentage changes in the irregular component (step 23) to b) the twelve-month moving average of the month-to-month percentage changes in the cyclical component (step 24). In the computation of these moving averages, the signs of the percentage changes are dis regarded (Table 28). IV, 30. Notes** Where the average monthly amplitude of the irregular component is 4.0 or larger (on the basis of the preliminary seasonally adjusted series) and for special purposes, two additional tables are computed and inserted between Tables 10 and 11. In the first one, the stable adjustment factors are computed by averaging the modified ratios of step 11c for each month and then centering the average so that their sum will be 1,200. In the second table, these stable factors are divided into the corresponding values of the original data, yielding a seasonally adjusted series based on a constant seasonal pattern. These two additional tables do not affect the computations in any* other tables. ^Electronic Computers and Business Indicators, Occasional Paper No. 57, National Bureau of Economic Research, 1957, p. 252. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 9 (CM II) Listing of Tables Prepared by Census II Method Table Number 1 2 3 4 5 9 10 11 12 13 14 15 16 17 18: 19 20 21 22 23 24 25 26 27 28 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Title of Table Original series Ratios of original to preceding and following Averages of ratios Uncentered 12-month moving average of original Centered 12-month moving average of original Ratios of original to 12-month moving average Ratios of original to weighted 15-month moving average Modified ratios, original/WTD 15-month moving average Stable-seasonal adjustment factors Stable-seasonal adjusted series Centered ratios, original/WTD 15-month moving average Final seasonal adjusted factors, 3*5-month moving averages Estimated seasonal factors one year ahead Final seasonally adjusted series Ratios, final adjusted to preceding and following Averages Uncentered 12-month moving average, final adjustment Ratios, 12-month moving average, final adjustment to original Ratios, each month to preceding January, final adjustment Weighted 15-month moving average of final adjustment Percent change from preceding month, original Percent change from preceding month, seasonal Percent change from preceding month, final adjustment Percent change from preceding month, S-I ratios Irregular component Percent change from preceding month, irregular Percent change from preceding month, cyclical 2-month moving average, final adjusted series I, C, & S components, their relations, & average duration of run Ratios, 12-month moving averages of irregular and cycle amplitudes - 10 (CM II) of Tables Prepared under the 1401 Version of Census II Method (Philadelphia Program) e ; 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Title of Table Original series Ratios of original to preceding and following Averages Uncentered 12-month moving average of original Centered 12-month moving average of original Ratio of original to 12-month moving average Preliminary seasonal factors Preliminary adjusted series Weighted 15-month moving average of preliminary series Ratios of original to weighted 15-month moving average Percent change from preceding month, original Percent change from preceding month, S-I ratios Modified ratios of original/WTD, 15-month moving average Centered ratios of original/WTD, 15-month moving average Final seasonal adjusted factors, 3*3-month moving averages Estimated seasonal factors one year ahead Final seasonally adjusted series Percent change from preceding month, seasonal Percent change from preceding month, final adjustment Ratios, final adjustment to preceding and following Averages Uncentered 12-month moving average, final adjustment Ratios, 12-month moving averages, final adjustment to original Ratios, each month to preceding January, final adjustment Weighted 15-month moving average of final series Irregular component Percent change from preceding month, irregular Percent change from preceding month, cyclical Ratios, 12-month moving averages of irregular and cycle amplitudes 2-month moving average, final adjusted series I, C, & S components, their relations, and average duration of run BUREAU OF THE CENSUS SEASONAL ADJUSTMENT TECHNIQUE The X-9 Version of Census Method II* This procedure replaces steps 6 and 11 of Census Method II as described in the foregoing description, "A Description of the United States Bureau of the Census Method of Adjustment of series of Monthly Data for Seasonal Vari ations," Seasonal Adjustment on Electronic Computers. 6. This step will provide a method for identifying extreme items among the ratios computed by step 5, substituting more representative ratios for these extreme ratios and fitting smooth curves to all ratios for each month. a. Fit a five-term moving average to the ratios for each month. This results in the loss of moving average values for the first two and the last two years for which ratios are avail able. To obtain moving average values for the first two years, repeat the moving average value of the third year. This is equivalent to weighting the first five years’ ratios by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first and second years' moving average values. Moving average values for the last two years are obtained in a similar manner. b. For each month, compute two-sigma control limits about the five-term moving average line. All ratios falling outside these limits are designated as extreme. c. Replace extreme ratios for each month as follows: for an extreme ratio falling at the first point in the series, substitute the average of the second, third, and fourth ratios; for an extreme ratio falling at the second point of the series, substitute the average of the first, third, and fourth ratios; for an extreme ratio falling in the middle of the series, substitute the average of the two pre ceding and two following ratios; for an extreme ratio falling at the next to last or last point, follow a procedure similar to that for the beginning of the series (Table 5A, "Modified Ratios, Original/12-month Moving Average4’). ^’Specifications for the X-9 Version of Census II Method Seasonal Adjustment Program," Bureau of the Census, Office of Chief Economic Statistician, March 6, 1962. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 2 (X-9) d. For each month, compute a three-term moving average of the modified ratios yielded by step 6c. This results in the loss of moving average values for the first and last years for which ratios are available. To obtain the moving aver age value for the first year, use the average of the first three ratios as the estimated value for the ratio preceding the first year available. This is equivalent to weighting the first three years' ratios by 4/9, 4/9, and 1/9, re spectively, to obtain the first year’s moving average value. The missing value at the end is obtained in a similar way (Table 6B, "Preliminary Uncentered Seasonal Factors"), e. For the entire series, compute a centered twelve-month moving average (a two-term of a twelve-term moving average) of the preliminary uncentered seasonal factors yielded by step 6d (Table 60, "Preliminary Centering Factors"). For the six missing values at the beginning of the centered twelve-month moving average, repeat the first available value six times. The six missing values at the end are ob tained in a similar way. The values computed in step 6d are divided by these values (Table 6D, "Preliminary Centered Seasonal Factors"). f. For each month, compute a three-term moving average of the preliminary centered seasonal factors yielded by step 6e. This results in the loss of moving average values for the first and last years. To obtain the moving average value for the first year, use the first 6e value as an estimated value for the year preceding the first year for which a value is available. This is equivalent to weighting the first two years' values by 2/3 and 1/3, respectively, to obtain the first year's moving average value. The missing value at the end is obtained in a similar way. To obtain the six factors missing at the beginning of the series (due to the use of the twelve-term moving average in step 4), repeat the factor from the same month of the first available year. Fill in the six missing factors at the end of the series in a similar way (Table 6B, "Preliminary Seasonal Factors"). Continue with step 7 of "A Description of the United States Bureau of the Census Method of Adjustment of series of Monthly Data for Seasonal Variations," Seasonal Adjustment on Electronic Computers. 11. This step will provide a method for identifying extreme items among the ratios computed by step 9, substituting more representative ratios for these extreme ratios, and fitting smooth curves to all ratios for each month. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 3 (X-9) a. Fit a five-term moving average to the ratios for each month. This results in the loss of moving average values for the first two and the last two years for which ratios are avail able. To obtain moving average values for the first two years, repeat the moving average value of the third year. This is equivalent to weighting the first five years* ratios by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first and second years’ moving average values. Moving average values for the last two years are obtained in a similar manner. b. For each month, compute two-sigma control limits about the five-term moving average line. All ratios falling outside these limits are designated as extreme. c. Replace extreme ratios for each month as follows: for an extreme ratio falling at the first point in the series, substitute the average of the second, third, and fourth ratios; for an extreme ratio falling at the second point of the series, substitute the average of the first, third and fourth ratios; for an extreme ratio falling in the middle of the series, substitute the average of the two preceding and two following ratios; for an extreme ratio falling in the next to last or last point, follow a pro cedure similar to that for the beginning of the series (Table 10, "Modified Ratios, Original/Weighted 15-Month Moving Average"). d. If the average irregular amplitude, computed in step 10 above, is under 2, use step lie; if it is 2 or more, use step Ilf. e. For each month, compute a three-term moving average of the modified ratios yielded by step 11c. This results in the loss of moving average values for the first and last years for which ratios are available. To obtain the moving aver age value for the first year, use the average of the first three ratios as the estimated value for the ratio preceding the first year available. This is equivalent to weighting the first three years’ ratios by 4/9, 4/9, and 1/9, respectively, to obtain the first year’s moving average value. The missing value at the end is obtained in a simi lar way (Table 10D, "Final Uncentered Seasonal Factors"). f. For each month, compute a five-term moving average of the modified ratios yielded by step 11c. This results in the loss of moving average values for the first two and last two years for which ratios are available. To obtain moving average values for the first two years, use the average of the first four ratios as the estimated value for the ratios for each of the two years preceding the first year available. This is equivalent to weighting https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis _ 4 (X-9) the first four years’ ratios by 6/20, 6/20, 6/20, and 2/20, respectively, to obtain the first year’s moving average value and to weighting the first four years’ ratios by 5/20, 5/20, 5/20, and 5/20 to obtain the second year’s moving average value. The missing values at the end are obtained in a similar way (Table 10D, "Final Uricentered Seasonal Factors"). g# For the entire series, compute a centered twelve-month moving average (a two-term of a twelve-term moving average) of the final uncentered seasonal factors Yielded by step lie or Ilf (Table 10E, "Final Centering Factors"). For the six missing values at the beginning of the centered twelvemonth moving average, repeat the first available value six times. The six missing values at the end are obtained in a similar way. The values computed in step lie or Ilf are divided by these values (Table 11, "Final Centered Seasonal Factors"). h. For each month, compute a three-term moving average of the final centered seasonal factors yielded by step llg. This results in the loss of moving average values for the first and last years. To obtain the moving average value for the first year, use the first llg value as an estimated value for the year preceding the first year for which a value is available. This is equivalent to weighting the first two years’ values by 2/3 and 1/3, respectively, to obtain the first year’s moving average value. The missing value at the end is obtained in a similar way (Table 12, "Final Seasonal Factors"). i. Estimates of the seasonal factors one year ahead are given at the bottom of Table 12. These estimates are made by adding to the seasonal factor for the end year, one-half the trend between the factor for that year and the preceding year. If X=seasonal adjustment factor for year n, then Xjj + 1 is estimated by the equation Xj^ + 1 « 3Xh - Xn - 1. 2 Continue with step 12 of "A Description of the United States Bureau of the Census Method of Adjustment of series of Monthly Data for Seasonal Variations,” Seasonal Adjustment on Electronic Computers. NOTE: In these specifications, no description is given for Tables 6A, 10A, 10B, and 10C. In the Census Bureau's printout, Tables 10A and 10B are the "Stable-Seasonal Factors" and "Stable-Seasonal Adjusted Series" described in step 30 of Occasional Paper No. 57. They are printed out regardless of the size of the irregular component, not only when the average monthly ampli tude of the irregular component is 4.0 or larger as originally specified. Tables 6A and 10C are the Moving Seasonality Ratios described in the specificiations for X-10. In X-9, these ratios do not play a role in the https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 5 (X-9) selection of the seasonal factor curves; however, they are useful as a descriptive measure of the type of seasonality present in each month. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis BUREAU OF THE CENSUS SEASONAL ADJUSTMENT TECHNIQUE The X-10 Version of Census Method II* This procedure** replaces steps 6, 10, 1.1, of Census Method II as described in the foregoing description, "A Description of the United States Bureau of the Census Method of Adjustment of Series of Monthly Data for Seasonal Variations,11 Seasonal Adjustment on Electronic Computers. 6. This step will provide a method for identifying extreme items among the ratios computed by step 5, substituting more representative ratios for these extreme ratios and fitting smooth curves to all ratios for each month. a. Fit a five-term moving average to the ratios for each month. This results in the loss of moving average values for the first two and last two years for which ratios are available. To obtain moving average values for the first two years, re peat the moving average value of the third year. This is equivalent to weighting the first five years’ ratios by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first and second year’s moving average values. Moving average values for the last two years are obtained in a similar manner. b. For each month, compute two-sigma control limits about the five-term moving average line. All ratios falling outside these limits are designated as extreme. c. Replace extreme ratios for each month as follows: for an extreme ratio falling at the first point in the series, sub stitute the average of the second, third and fourth ratios; for an extreme ratio falling at the second point of the series, substitute the average of the first, third, and fourth ratios; for an extreme ratio falling in the middle of the series, substitute the average of the two preceding and two following ratios; for an extreme ratio falling at the next to last or last point, follow a procedure similar to that for the beginning of the series (Table 5A, "Modified Ratios, Original/12-Month Moving Average"). d. For each month, compute a seven-term moving average of the modified ratios yielded by step 6c. This results in the loss of moving average values for the first three and the last three years for which ratios are available. To obtain *”Specifications for the X-10 Version of the Census Method II Seasonal Adjust ment Program,” Bureau of the Census, Office of Chief Economic Statistician, March 6, 1962. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 2 (X-10) moving average values for the first three years, use the average of the first three ratios as the estimated value for the ratios for each of the three years preceding the first year available. Then the moving average values for the first three years are computed by including these estimated ratios in the moving average (see part (1) of Note at end of specifications). The missing values at the end are obtained in a similar way. e. For each month, compute the average, without regard to sign, of the year-to-year percentage changes in the moving average values of step 6d. This average is an estimate of the change in the seasonal component for a particular month and is referred to as 5y. f. For each month, divide the moving average values in step 6d into the modified ratios from step 6c. The resulting series is an estimate of the irregular component. g. For each month, compute the average, without regard to sign, of the year-to-year percentage changes in the irregular component yielded by step 6f. This average is an estimate of thexchange in the irregular component and is referred to as Iy. h. For each month, compute the ratio of the 6g value to the 6e value, Ty/Sy. These ratios are designated Moving Sea sonality Ratios (Table 6A, “Moving Seasonality Ratios"). i. For each month, depending upon the size of the moving seasonality ratio computed in step 6h, an average of the modified ratios yielded by step 6c is computed, as speci fied in the table below. When a moving average is selected and computed, there is a loss of moving average values at the beginning and end. The number of values lost depends upon the length of the moving average selected. To obtain the moving average values at the beginning, a specified number of beginning ratios are averaged to obtain estimated ratios for the years preceding the first available ratio. Then the moving average values for the first years are computed by including these estimated ratios in the average. The number of ratios to be averaged, in order to obtain the estimated ratios, is shown in the last column of the table (See part (1) of Note at end of specifications). The moving average values missing at the end are obtained in a similar way. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 3 (X-10) Moving seasonality ratio step 6h Average of 6c values No. of beginning or ending 6c values average to extend the moving average 0 to 1.49 None (Leave 6c values unchanged.) — 1.50 to 2.49 3-term moving average 2 2.50 to 4.49 5-term moving average 2 4.50 to 6.49 9-term moving average 3 6.50 to 8.49 15-term moving average 3 8.50 and over Arithmetic average of all 6c values — The values obtained in this step are printed out (Table 6B, ’’Pre liminary Uncentered Seasonal Factors"). j. For the entire series, compute a centered twelve-month moving average (a two-term of a twelve-term moving average) of the pre liminary uncentered seasonal factors yielded by step 6i (Table 6C, "Preliminary Centering Factors"). For the six missing values at the beginning of the centered twelve-month moving average, repeat the first available value six times. The six missing values at the end are obtained in a similar way. The values computed in step 6i are divided by these values (Table 6D, "Preliminary Centered Seasonal Factors"). k. For each month, compute a three-term moving average of the pre liminary centered seasonal factors yielded by step 6j. This results in the loss of moving average values for the first and last years. To obtain the moving average value for the first year, use the first 6j value as an estimated value for the year preceding the first year for which a value is available. This is equivalent to weighting the first two years’ values by 2/3 and 1/3, respectively, to obtain the first year’s moving average value. The missing value at the end is obtained in a similar way. To obtain the six factors missing at the beginning of the series (due to the use of the twelve-term moving average in step 4), re peat the factor from the same month of the first available year. Fill in the six missing factors at the end of the series in a similar way (Table 6E, "Preliminary Seasonal Factors"). Continue with step 7 of "A Description of the United States Bureau of the Census Method of Adjustment of Series of Monthly Data for Seasonal Variations," Seasonal Adjustment on Electronic Computers. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 4 (X-10) 10. Delete step 10. 11. This step will provide a method for identifying extreme items among the ratios computed by step 9, substituting more representative ratios for these extreme ratios, and fitting smooth curves to all ratios for each month. a. Fit a five-term moving average to the ratios for each month. This results in the loss of moving average values for the first two and the last two years for which ratios are avail able. To obtain moving average values for the first two years, repeat the moving average value of the third year. This is equivalent to weighting the first five years’ ratios by 1/5, 1/5, 1/5, 1/5, and 1/5 to obtain the first and second years' moving average values. Moving average values for the last two years are obtained in a similar manner. b. For each month, compute two-sigma control limits about the five-term moving average line. All ratios falling outside these limits are designated as extreme. c. Replace extreme ratios for each month as follows: For an extreme ratio falling at the first point in the series, substitute the average of the second, third, and fourth ratios; for an extreme ratio falling at the second point of the series, substitute the average of the first, third and fourth ratios; for an extreme ratio falling in the middle of the series, substitute the average of the two preceding and two following ratios; for an extreme ratio falling at the next to last or last point, follow a pro cedure similar to that for the beginning of the series (Table 10, ’’Modified Ratios, Original/Weighted 15-Month Moving Average"). d. For each month, compute a seven-term moving average of the modified ratios yielded by step 11c. This results in the loss of moving average values for the first three and the last three years for which ratios are available. To obtain moving average values for the first three years, use the average of the first three ratios as the estimated values for the ratios for each of the three years preceding the first year available. Then the moving average values for the first three years are computed by including these esti mated ratios in the moving average (See part 1 of Note at end of specifications). The missing values at the end are obtained in a similar way. e. For each month, compute the average, without regard to sign, of the year-to-year percentage changes in the moving aver age values of step lid. This average is an estimate of the https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 5 (X-10) change in the seasonal component for a particular month and is referred to as Ky. f. For each month, divide the moving average values in step lid into the modified ratios from step 11c. The resulting series is an estimate of the irregular component. g. For each month, compute the average, without regard to sign, of the year-to-year percentage changes in the irregular component series yielded by step Ilf. This average is an estimate of the change in the irregular component and is re ferred to as Iy. h. For each month, compute the ratio of the llg value to the lie values, Iy/Sy. These ratios are designated Moving Seasonality Ratios (Table 10G, ”Moving Seasonality Ratios”). i. For each month, depending upon the size of the moving sea sonality ratio computed in step llh, an average of the modified ratios yielded by step 11c is computed as specified in the table below. When a moving average is selected and computed, there is a loss of moving average values at the beginning and end. The number of values lost depends upon the length of the moving average selected. To obtain the moving average values at the beginning, a specified number of beginning ratios are averaged to obtain estimated ratios for the years preceding the first available ratio. Then the moving average values for the first years are computed by including these estimated ratios in the average. The number of ratios to be averaged, in order to obtain the esti mated ratios, is shown in the last column of the table (See part 1 of Note at end of specifications). The moving aver age values missing at the end are obtained in similar way. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Moving seasonality ratio step llh 0 to 1.49 Average of 11c values TIo. "of beginning or ending 11c values average to extend the roving average None (Leave 11c values unchanged.) 1.50 to 2.49 3-term moving average 2 2.50 to 4.49 5-term moving average 2 4.50 to 6.49 9-term moving average 3 6.50 to 8.49 15-term moving average 3 8.50 and over Arithmetic average of all 11c values - 6 (X-10) The values obtained in this step are printed out (Table 10D, "Final Uncentered Seasonal Factors"). j. For the entire series, compute a centered twelve-month moving average (a two-term of a twelve-term moving average) of the final uncentered seasonal factors yielded by step Hi (Table 10E, "Final Centering Factors"). For the six missing values at the beginning of the centered twelve-month moving average, repeat the first available value six times. The six missing values at the end are obtained in a similar way. The values computed in step lli are divided by these values (Table 11, "Final Centered Seasonal Factors"). k. For each month, compute a three-term moving average of the final centered seasonal factors yielded by step 11j. This results in the loss of moving average values for the first and last years. To obtain the moving average value for the first year, use the first 11j value as an estimated value for the year preceding the first year for which a value is available. This is equivalent to weighting the first two years’ values by 2/3 and 1/3, respectively, to obtain the first year’s moving average value. The missing value at the end is obtained in a similar way (Table 12, "Final Seasonal Factors"). l. Estimates of the seasonal factors one year ahead are given at the bottom of Table 12. These estimates are made by adding to the seasonal factor for the end year, one-half the trend between the factor for that year and the pre ceding year. If X^—seasonal adjustment factor for year n, then Xh + 1 is estimated by the equation Xn + 1 “ 3Xn - Xn - 1. Continue with step 12 of "A Description of the United States Bureau of the Census Method of Adjustment of series of Monthly Data for Seasonal Variations," Seasonal Adjustment on Electronic Computers. NOTE: 1. No implicit weights are given for steps 6d, 6i, lid, or lli, as are given for steps 6a, 6k, etc., because when the series is shorter than the moving average, the weights vary with the length of the series. The original Method II was programmed to accept series with a minimum of 72 months (six years) of data. For the 15-term moving average, different sets of weights are required for 14, 13,............ 6-year series; for the 9-term, sets for 8, 7, and 6-year series are required; and for the 7-term, sets for 6-year series are needed. The purpose in using a 15-term moving aver age with a series as short as six years is that it is a convenient way to fit a straight line within the framework of the method. 2. In these specifications, no description is given for Tables 10A and 10B. In the Census Bureau's printout, Tables 10A and 10B are the https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 7 £-10) ’’Stable-Seasonal Factors" and "Stable-Seasonal Adjusted Series" described in step 30 of Occasional Paper No. 57. They are printed out regardless of the size of the irregular component, not only when the average monthly amplitude of the irregular component is 4.0 or larger, as originally specified **The technique for selecting the seasonal factor curves on the basis of the moving seasonality ratios, which is incorporated in X-10, was developed by Stephen N. Marris, Head of Statistics Division of the Organization for Economic Cooperation and Development, Paris, France, and is described in Seasonal Adjustment on Electronic Computers, pages 257-309, OECD (Paris 1961). Hie Bureau of the Census and the Organization for Economic Cooperation and Development have cooperated in further theoretical and empirical development of this technique during the past two years. The X-10 program differs slightly from that described in the OECD paper. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis THE SEASONAL ADJUSTMENT METHOD OF THE BUREAU OF LABOR STATISTICS* Detailed Listing of Steps The following steps describe the method used by the Bureau of Labor Statistics in developing seasonal factors. The “Table No." reference pre ceding a description refers to the table in the print-out provided by the electronic computer** program. The BLS method may involve four or seven iterations, depending on extreme values detected in the original data. The table numbers have been assigned so that the first digit indicates the iteration; the third digit identifies the type of information contained in the table as follows: Table X01 always refers to trend-cycle X02 to seasonal-irregular ratios X03 to unforced seasonals X04 to forced seasonals X05 to irregular movements X07 to extreme values X08 to deseasonalized original values X09 to original data The computer program used with the BLS method permits selection of either a complete or partial record (print-out) of the values developed. The partial record includes the final trend-cycle, seasonal, and irregular components, the detected extreme original values and their substituted values, the deseasonalized series, and the centered 12-month moving average Tables included in the partial record (short print-out) are identified by an asterisk immediately preceding the table number. The complete record (long print-out) includes all the tables shown. *Table 101: 12-month moving average. This is a centered moving average of the original values (table 709), developed as a first approximation to the trend-cycle component. A centered moving average would being six months later than the original series. However, the difference has been reduced to three *“The BLS Factor Method," Abe Rothman, Proceedings of the 1960 American Statistical Association, pp. 8-11, **I.B.M. 650 basic installation. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 2 (BLS) months by the following series of steps. (Corresponding operations are applied at the end of the series. All operations in the entire procedure are symmetical with respect to the time scale.) Table 102: Table 103: https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis a. Seasonal-irregulars are computed as described for table 102. These seasonal-irregulars be gin with the following January, the first month for which the 12-month moving average is available. b. Unforced seasonals are computed as described for table 103. These begin with January. c. Forced seasonals are computed as described for table 104. These begin with January. d. A seasonally adjusted series is computed by dividing the original values (table 709) by the forced seasonal factors (step c). For the first six months of the original series, the seasonal factor for the same month of the following year is used. The adjusted series begins with July. e. The average of the first three seasonally ad justed values (those for July, August, and September) is multiplied by the seasonal factors (step c) for April, May, and June of the following year to provide synthetic ori ginal values for the three months preceding the beginning of the original series. The centered 12-month moving average of this ex tended original series is printed as table 101. Seasonal-irregular, first approximation. The original values (table 709) are divided by their 12-month moving average (table 101). Unforced seasonal, first approximation. For each calendar month, the seasonal-irregular ratios (table 102) are arranged by year and a weighted average is secured. The weights ,30, .30, .20, .10, .10 are applied to the first five seasonalirregulars. (the underline weight is applied to the term (year) whose seasonal is being computed.) For the second term, the weights .24, ,26, .20, .16, .14 are applied to the same first five values. For the third and all subsequent terms up to the last two, the weights .17, .20, .26, .20, .17 are applied to a centered group of five years. The - 3 (BLS) next-to-last term applies weights of .14, .16, .20, .26, .24 to the last five values. The last term applies weights of .10, .10, .20, .30, .30, to these same five end values. The weights for the central term are a compromise between a pattern with uniform weights (.20) and one with weights associated with a 3 x 3 moving average (.11, .22, .33, .22, .11). The actual pattern is very close to the average of these two patterns but is a little flatter in shape. Table 104j Forced seasonal, first approximation. Each unforced seasonal (table 103) is multiplied by an adjustment factor which is the ratio of 1200 to the sum of the unforced seasonals in the whole calendar year. This makes the average of the seasonal factors equal to 100. Table 105: Irregular, first approximation. A seasonally adjusted series is computed by dividing the original values (table 709) by the forced seasonal factors (table 104). This, in turn, is divided by the 12-month moving average (table 101) to produce an esti mate of the irregular component which also includes some resi dual trend-cycle. For the partial year at each end of the series, the seasonal factors of the adjacent year are used. Table 201: Moving average, modified.once. The irregulars (table 105) are extended three months at each end by tapering the first and last values to 100 percent. The extended series of irregulars, arranged in normal time sequence, is then smoothed by a weighted 7-month moving average to remove the irregular part and leave only the residual trend-cycle. The weighting pattern used, .090, .127, .183, .200, .183, .127, .090, is the average of a pattern with equal weights (.143) and a pattern associated with a 3-term of a 3-term of a 3-term (3x3x3) moving average (1, 3, 6, 7, 6, 3, 1 equal to .037, .111, .222, .259, .222, .111, .037). Table 202. Seasonal irregular, second approximation. The original values (table 709) are divided by the improved estimate of trend cycle (table 201). Table 203; Unforced seasonal, second approximation. This is a weighted 5-term moving average of the seasonal-irregulars (table 202) for each calendar month, using the same weights as for table 103. Table 204; Forced seasonal, second approximation. Each unforced seasonal (table 203) is multiplied by an adjustment factor which is the ratio of 1200 to the sum of the unforced seasonals in the whole calendar year. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 4 (BLS) Table 205: Irregular, second approximation. A seasonally adjusted series is computed by dividing the original values (table 709) by the forced seasonal factors (table 204). This is in turn divided by the trend-cycle (table 201) to estimate the irregular com ponent. For the partial year at each end of the series, the seasonal factors of the adjacent year are used. *Table 301: Moving average, modified twice (final trend if no extremes). The irregulars (table 205) are smoothed in the same way de scribed for table 201. The smoothed series of irregulars is multiplied by the previous estimate of trend-cycle (table 201) to produce table 301 as an improved estimate. This table gives the final trend-cycle component if there are no extreme values (revealed in next iteration). Table 302: Seasonal-irregular, third approximation. The original values (table 709) are divided by the latest estimate of trend-cycle (table 301). Table 303: Unforced seasonal, third approximation. This is a weighted 5-term moving average of the seasonal-irregulars (table 302) for each calendar month, using the same weights as for table 103. *Table 304: Forced seasonal, third approximation (final if no extremes). Each unforced seasonal (table 303) is multiplied by an adjust ment factor which is the ratio of 1200 to the sum of the un forced seasonals in the whole calendar year. This table gives the final seasonal component if there are no extreme values. *Table 305: Irregular, third approximation (final if no extremes). A seasonally adjusted series is computed by dividing the ori ginal values (table 709) by the forced seasonal factors (table 304). This, in turn, is divided by the trend-cycle (table 301) to yield the Irregular component. This table gives the final irregular component if there are no extreme values. For the partial year at each end of the series, the seasonal factors of the adjacent year are used. *Table 308: Seasonally adjusted series (final if no extremes). The original values (table 709) are divided by the forced seasonal factors (table 304). For the partial year at each end of the series, the seasonal factors are taken from the corresponding months of the adjacent year. *Table 407: Extreme values - tests and replacement values. This table con tains the results of the series of steps designed to determine whether the series contains any extreme values. If any are found, the procedure provides replacement values. If no extreme https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis -5 (BLS) values are found, tables 407 through 708 are omitted. test for extreme values includes the following steps: The a. The irregulars (table 305) are smoothed in the same way described for table 201, except that the central weight is zero instead of .200. The "mid-zero" weight pattern provides a trend-cycle which minimizes the effect of an extreme value on the test criterion. b. The smoothed series of irregulars (step a) is multi plied by the latest trend-cycle (table 301) to pro duce the test trend-cycle. These values are uniformly 20 percent too low, because the weights used in step a add to only .800. c. The original values (table 709) are divided by the test trend-cycle (step b) to yield test seasonalirregulars, which are uniformly 25 percent too high. d. The test seasonal-irregulars (step c) are smoothed by a weighted 5-term moving average for each calendar month to produce test seasonals, using the following "mid-zero" weights. For the first year, the weights are 0, .43, .29, .14,' .14. For the second year they are .32, 0, .27, .22, .19. For the third and subse quent years up to the last two, they are .23, .27, 0, .27, .23. For the next-to-last year, they are .19, .22, .27, 0, .32. For the last year they are .14, .14, .29, .43, 0. (These weights are proportional to those for table 103 except that the target year always receives zero weight.) The test seasonals, like the test seasonal-irregulars, are uniformly 25 percent too high. e. The test seasonal-irregulars (step c) are divided by the test seasonals (step d) to produce test irregulars. f. The mean and standard deviation are computed for the entire (all months of all complete calendar years) distribution of test irregulars (step e). Control limits are set at the mean +2.86* and are designed to provide a probability of about 50 percent that all "good" values will fall inside the limits. The 2.86* and the 50 percent probability are based on the assump tion that all values in the original series are "good" and belong to the series. However, since an original value not really belonging in the series may be en countered, a discriminating.test is needed which will detect the non-belonging observation without rejecting - 6 (BLS) too many acceptable values. The 2.8<f is the point in the distribution which will, in 50 percent of the cases, reject no values; in the other 50 percent, it will reject one or more (usually one) values. Initially, different sigma limits were calculated based on the length of the series. However, since our computer pro gram handled series of from 6-12 years, and the limits varied by .2 sigma, the single limit of 2.8 sigma was considered close enough for 6-12 year series. g. Particular months whose test irregulars (step e) fall outside the control limits (step f) are designated as extreme and are listed in table 407. The replacement value for each extreme value is obtained by multiplying the test trend-cycle (step b) by the test seasonal (step d). This provides a value whose extreme irregularity has been removed. *Table 501: 12-month moving average (extremes replaced). The set of ori ginal values (table 709) is modified by substituting for each extreme value the replacement value given in table 407. Table 501 is a centered moving average of these modified original values with extensions at the ends of series computed the same way as for table 101. Table 502: Seasonal-irregular, first approximation (extremes replaced). The modified original values are divided by the 12-month moving average (table 501). Table 503: Unforced seasonal, first approximation (extremes replaced). This is a weighted 5-term moving average of the seasonalirregulars (table 502) for each calendar month, using the same weights as for table 103. Table 504: Forced seasonal, first approximation (extremes replaced). Each un forced seasonal (table 503) is multiplied by an adjustment factor which is the ratio of 1200 to the sum of the unforced seasonals in the whole calendar year. Table 505: Irregular, first approximation (extremes replaced). A sea sonally adjusted series is computed by dividing the modified original values by the forced seasonal factors (table 504). This is in turn divided by the trend-cycle (table 501) to estimate the irregular component. For the partial year at each end of the series, the seasonal factors of the adjacent year are used. Table 601: Moving average, modified once (extremes replaced). The irre gulars (table 505) are smoothed in the same way described for table 201. The smoothed series of irregulars is multiplied by https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 7 (BLS) the previous estimate of trend-cycle (table 501) to produce table 601 as an improved estimate. Table 602: Seasonal-irregulars, second approximation (extremes replaced). The modified original values are divided by the latest estimate of trend-cycle (table 601). Table 603: Unforced seasonal, second approximation (extremes replaced). This is a weighted 5-term moving average o£ the seasonalirregulars (table 602) for each calendar month, using the same weights as for table 103. Table 604: Forced seasonal, second approximation (extremes replaced). Each unforced seasonal (table 603) is multiplied by an adjustment factor which is the ratio of 1200 to the sum of the un forced seasonals in the whole calendar year. Table 605: Irregular, second approximation (extremes replaced). A sea sonally adjusted series is computed by dividing the modified original values by the forced seasonal factors (table 604) This is in turn divided by the trend-cycle (table 601) to esti mate the irregular component. For the partial year at each end of the series, the seasonal factors of the adjacent year are used. ★Table 701: Final trend-cycle (extremes replaced). The irregulars (table 605) are smoothed in the same way described for table 201. The smoothed series of irregulars is multiplied by the previous estimate of trend-cycle (table 601) to produce this final esti mate. Table 702: Final seasonal-irregular (extremes replaced). The modified ori ginal values are divi.ded by the final trend-cycle (table 701). Table 703: Final unforced seasonal (extremes replaced). This is a weighted 5-term moving average of the final seasonal-irregulars (table 702) for each calendar month, using the same weights as for, table 103. ★Table 704: Final seasonal (extremes replaced). Each unforced seasonal (table 703) is multiplied by an adjustment factor which is the ratio of 1200 to the sum of the unforced seasonals in the whole calendar year. ★Table 705: Final irregular (extremes replaced). A seasonally adjusted series is computed by dividing the actual original values (table 709) by the final seasonal factors (table 704). This is, in turn divided by the final trend-cycle (table 701) to yield the final irregular comjponent. For the partial year at each end of the series, the seasonal factors of the adjacent year are used. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis - 8 (BLS) Seasonally adjusted series. The original values (table 709) are divided by the final seasonal factors (table 704). For the partial year at each end of the series, the seasonal factors are taken from the corresponding months of the adja cent year. Original series. values. This is the monthly series of original Listing of Tables Prepared Title of Table 101 102 103 104 105 12-month moving average Seasonal irregular, first approximation Unforced seasonal, first approximation Forced seasonal, first approximation Irregular, first approximation 201 202 203 204 205 Moving average modified once Seasonal irregular, second approximation Unforced seasonal, second approximation Forced seasonal, second approximation Irregular, second approximation 301 302 303 304 305 308 Moving average modified twice *final trend if no extremes* Seasonal irregular, third approximation Unforced seasonal, third approximation Forced seasonal, third approximation *final if no extremes* Irregular, third approximation *final if no extremes* Seasonally adjusted series *final if no extremes* 407 Extreme values - tests and replacement values 501 502 503 504 505 12-month moving average *extremes replaced* Seasonal irregular, first approximation *extremes replaced* Unforced seasonal, first approximation *extremes replaced* Forced seasonal, first approximationn *extremes replaced* Irregular, first approximation *extremes replaced* 601 602 603 604 605 Moving average modified once *extremes replaced* Seasonal irregular, second approximation *extremes replaced*' Unforced seasonal, second approximation *extremes replaced* Forced seasonal, second approximation *extremes replaced* Irregular, second approximation *extremes replaced* 701 702 703 704 705 708 Final trend cycle *extremes replaced* Final seasonal irregular *extremes replaced* Final unforced seasonal *extremes replaced* Final seasonal,*extremes replaced* Final irregular *extrernes replaced* Seasonally adjusted series https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Original series 18