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Row SELECTED TECHNIQUES OF SEASONAL ADJUSTMENT Research Department Federal Reserve Bank of Atlanta https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis May 1963 Atlanta, Georgia https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis SELECTED TECHNIQUES OF SEASONAL ADJUSTMENT A Revision of Selected Technique of Seasonal Adjustment Published June 1962 Compiled by W. M. Davis Senior Economist and Elizabeth Long Technical Assistant Research Department Federal Reserve Bank of Atlanta Atlanta, Georgia May 1963 F TABLE OF CONTENTS Page Preface Computational Steps of Selected Methods of Seasonal Adjustment ............................. 1 Bureau of the Census Seasonal Adjustment Technique (Method II) ........................... 8 Listing of Tables Prepared by Census Method II ......................................................16 New Tables in Census Method II ............................................................................................ 19 Bureau of the Census Seasonal Adjustment Technique (The X-9 Version of Census Method II) .................................................................................. .. ..................... 21 Listing of Tables Prepared by X-9 Version of Census Method II ................. 26 Bureau of the Census Seasonal Adjustment Technique (The X-10 Version of Census Method II) .......................................................................................................... 27 The Seasonal Adjustment Method of the Bureau of Labor Statistics ...................... 34 Listing of Tables Prepared by Bureau of Labor Statistics Method ............ 42 The Regression Method of Deutsche Bundesbank ...................................................................... 44 Bureau of the Census Seasonal Adjustment Technique (Method III) .........................45 Listing of Tables Prepared by Census Method III ................................................... 47 Bibliography ............................................................................................................................................................ 49 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis PREFACE This publication contains basic descriptions of various techniques of seasonal adjustment that are in common use. It was originally pre pared for the Federal Reserve System Seminar on Seasonal Adjustment held in Washington on June 5-6, 1962, and only a few minor changes have been made since that time. It is hoped that this compendium of techniques will prove useful to the technician and non-technician alike. In the last few years, there has been a renewed interest in seasonal adjustment of time series. Census Method II has been employed by users of seasonally adjusted data for quite sometime. This method adapted to the electronic computer the essence of the ratio-to-trend procedure, which the Federal Reserve System had used for many years. The availability of a computer program greatly facilitated seasonal analysis and led to a great expansion in the number and types of series for which seasonal factors were computed. As the number grew, inadequacies of existing methods came to light. These shortcomings of existing methods of seasonal adjustment have led to several developments in the last three or four years. The Bureau of Labor Statistics has designed a different version of the iterative, ratio-to-trend procedure. Many users have seriously studied the regression technique of seasonal adjustment, especially, as currently used by the Deutsche Bundesbank. The Bureau of the Census, moreover, has developed the X-9 and X-10 versions of Census Method II, and work is proceeding on Census Method III. In addition, basic research on seasonal methods is now being conducted by the Federal Reserve System, the Bureau of the Budget, and other institutions. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis COMPUTATIONAL STEPS OF SELECTED METHODS OF SEASONAL ADJUSTMENT CENSUS METHOD II X-9 VERSION OF CENSUS METHOD II X-1O VERSION OF CENSUS METHOD II BUREAU OF LABOR STATISTICS METHOD REGRESSION METHOD OF DEUTSCHE BUNDESBANK Preliminary Computation of Seasonally Adjusted Series Preliminary Computation of Seasonally Adjusted Series Preliminary Computation of Seasonally Adjusted Series First Iteration Basic Method *1 Adjustment for trading days is optional. If used, daily averages become original data. *1 Steps 1-5 are the same as Census Method II. *1 Steps 1-5 are the same as Census Method II. *1 Develop a centered 12-month moving average (MA) of original. Six values at each end are computed by a series of steps. Preliminary estimate of trend-cycle (TC). *1 Develop an uncentered 12-month moving average of original (a). This is used to represent trend (t). *2 Compute ratio of original to centered 12-month MA. First approximation of seasonal-irregular (SI). 2 Basic analysis based on following additive relationship: ansstn + Pn + En Original values—trend values + seasonal component + residual component. *3 For each calendar month, compute a 5-term weighted moving average(WMA) of SI ratios in step 2. Unforced seasonals, first approximation. 3 Other symbols: a'=seasonal values (regression values) a"«residual values (a-a1) a*=seasonally adjusted values 2 Compute ratio of original to average of preceding and following months. *3 Develop an uncentered 12-month moving average (MA) of original. *A step used in derivation of seasonal factors. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis r—4 CENSUS METHOD II____ X-9 VERSION OF CENSUS METHOD II X-1O VERSION OF CENSUS METHOD II BUREAU OF LABOR STATISTICS METHOD *4 Force total to 1200 *4 Center 12-month MA. REGRESSION METHOD OF DEUTSCHE BUNDESBANK *4 Standard regression equation used for each month: a'=tB + A, where B— slope; A—Y intercept: ’at - Et** Z a Etr-nEt1 -nA zt *5 Calculate ratio of original to centered 12-month MA. *6 Omit step 6 in Census Method II and substitute the following: *6 Omit step 6 in Census Method 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-term MA for each month of data in step 5. To get MA for first two years, repeat MA of third year. MA for last two years are obtained similarly. a Compute a 5-term MA for each month of data in step 5. To 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. b For each month, compute 2-sigma control limits about 5-term MA. All ratios falling outside limits are extreme. *6 Identify extreme values of step 5 and replace with more representative ones as follows: step used in derivation of seasonal factors. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis *5 Compute seasonally adjusted series. *5 With A and B values, compute seasonal values (a1) for each original value: a’~tB + A *6 Divide seasonally adjusted series by 12-month MA of original. First approximation of irregular (I) with s ome TC. *6 Graphical check of computations. Using (t) on the X axis and (a) on the Y axis, plot the values and the regression line. Visual proof of the correct computation of A and B. Second Iteration *7 Smooth I ratios in step 6 by 7-month WMA after extending I for three months at each end. Measure of residual TC. Multiply resuiting values by TC of step 1. *8 Compute ratio of original to TC of previous step. Second approximation of SI. *7 Compute residual values (a") by com paring regression values (a') with original values (a): a"=a-a’. If a" is greater than 0, there is superseasonal present; if a" is less than 0, there is subseasonal; if 0, there is only purely seasonal. ro CENSUS METHOD II X-9 VERSION OF CENSUS X-10 VERSION OF CENSUS METHOD II MFTUOn TT BUREAU OF LABOR STATISTICS METHOD c Replace extremes as follows: (1) Ratio falling first in 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, simi lar to beginning. c Replace extremes as follows: (1) Ratio falling first in 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, simi lar to beginning, d Six missing ratios (due to step 4) at beginning are sup plied by extending first available ratios for corre sponding months back to initial month of series. Six missing at end supplied similarly. d For each month, compute a 3-term MA. Missing values supplied for first year-average of first three ratios; similar for end. d For each month, compute a 7-term MA of ratios in step 6c. Missing ratios supplied in first 3 years by averaging first 3 years available. Similar for last years. MA values computed by using these estimates. 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. e For each month, compute the average, *14 Compute ratio of without regard to original to TC of sign, of year-toprevious step. year percent changes Third approxima in MA of step 6d. tion of SI. c Replace extremes as follows: (1) Ratio falling first in series, average of first three ratios; (2) Ratio falling in middle, average ex treme ratio and pre ceding and following ones; (3) Ratio falling at end, average extreme and two preceding ratios. *A step used in derivation https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis *9 For each calendar month, compute 5-term WMA of SI ratios. Unforced seasonals, second approximation. *10 Force total to 1200. *11 Compute seasonally adjusted series. *12 Divide seasonally adjusted series by TC of step 7. Second approxima tion of I. Third Iteration *13 Smooth I ratios in step 12 by 7-month WMA after extending I for 3 months at each end. Multiply smoothed I by TC in step 7. Result is final TC unless extreme values are present. REGRESSION METHOD OF DEUTSCHE 3UNDESBANK *8 Compute seasonally adjusted values (a*) by adding residual values (a'1) to corresponding trend values : a*?=t + a". 9 To decompose time series: (seasonal) p=a' - t; (residual) E-—a’~a - a'. Refinement of Trend Translation. *10 Test whether trend is "true," i.e., whether seasonal fluctuations around trend are distorted. *11 If refinement is necessary, improved trend values (t^) are obtained by smoothing trend values (t): Where Ao and Bo are original regression equation coeffi cients . of seasonal factors. UJ CENSUS METHOD II f Compute 3-term of 3-term MA of ratios in 6e for each month. Supply missing values at each end. The re sults are preliminary seasonal factors. X-9 VERSION OF CENSUS METHOD II X-10 VERSION OF CENSUS METHOD II f For each month, compute a 3-term MA. For missing values use value in step 6e corresponding to the month missing. f For each month, divide step 6d into step 6c. Estimate of I. *15 For each calendar month, compute 5-term WMA of SI in step 14. Un forced seasonal, third approxima tion . 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 prelimi nary seasonal factors. g For each month, compute average, without regard to sign, of year-toyear percent changes in I. *16 Force total to 1200. These are final factors un less extreme values are present. *A step used in derivation of seasonal factors. REGRESSION METHOD OF DEUTSCHE BUNDESBANK *12 New improved trend values (t|) are then used as basis of a refined corre lation between the trend and original values. Calculation techniques for various values are the same as in the Basic Method. *17 Compute seasonally adjusted series. h For each month, compute ratio of step 6g to step 6e. Designated as Moving Seasona lity Ratios. i For each month, depending upon the size of the ratio in step 6h, MA of ratios yielded by step 6c is com puted using the term indicated in the table at the end of this method (page 6). Missing ratios supplied. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis BUREAU OF LABOR STATISTICS METHOD *18 Divide seasonally adjusted series by TC of step 13. Final I unless extreme values are present. Fourth Iteration *19 Test for extreme values and, if found, re place with substitute values. Tests involve developing, smoothing, and analyzing irre gular component to determine whether values fall outside + 2.8 sigma limits. Replacement are calculated by multiplying TC by S for a given month. Fifth Iteration Repeat basic steps 1-6 of First Iteration using replacements for extreme values. This iteration develops first approximation to time series components. CENSUS METHOD II X-9 VERSION OF CENSUS METHOD II X-10 VERSION OF CENSUS METHOD II j Compute a centered 12-month MA of 6i. For missing values, repeat first avail able ratio six times. Similar for end. Divide into step bi. k For each month, compute a 3-term MA. For missing values, use value in step 6j corre sponding to the month missing. BUREAU OF LABOR STATISTICS METHOD Sixth Iteration Steps 7-12 of Second Iteration are repeated still using replacement values as original values. Seventh Iteration Steps 13-18 of Third Iteration are repeated. After derivation of final measures, original values are replaced and final seasonally adjusted series is derived. 1 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. *7 Compute preliminary seasonally adjusted series. *7 Same as Census II. *7 Same as Census II. Final Seasonally Adjusted Series Final Seasonally Ad justed Series Final Seasonally Ad justed Series *8 Develop a 15-month WMA of preliminary seasonally adjusted series supplying missing values. *8 Same as Census II. *8 Same as Census II. *9 Compute ratio of orig inal to 15-month WMA. *9 Same as Census II. *9 Same as Census II. *10 Compute ratio of pre liminary seasonally adjusted series to its 15-month WMA. *10 Same as Census II *10 Omit from Census II. *A step used in https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis derivation of seasonal factors. CENSUS METHOD II X-9 VERSION OF CENSUS METHOD II X-10 VERSION OF CENSUS METHOD II *11 Compute month-to-month percent changes of step 10 and average without re gard to sign. Measures average amplitude of I. *11 Same as Census II. *11 Omit from Census II. *12 Identify extreme values in step 9 and replace in the same manner as explained in steps 6a-6c above. Force total to 1200. *12 Omit step 12 in Census II and apply steps 6a-6c above to the re sults of step 10. *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. *13 Final seasonal factors are derived as follows: If irregular in step 11 average under 2, use a 3-term MA of a 3-term MA; If I is 2 or more, use a 3-term MA of a 5-term MA. Missing values at each end are supplied. *13 Final seasonal factors are derived as follows: If I in step 11 averages under 2, use a 3-terra MA; If I is 2 or more, compute a 5-term MA. Missing values at each end are supplied. Then per form steps 6e and 6f. These are final seasonal factors *13 See step 12 above. *14 Project seasonals in step 13 above 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. TERM OF MOVING AVERAGE FOR DIFFERENT SEASONALITY RATIOS IN X-10. SEE STEP 6i. 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 *A step used in derivation of https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis seasonal factors. Average of 6c Values None (Leave 6c values unchanged) 3-terra moving average 5-term moving average 9-term moving average 15-term moving average Arithmetic average of all 6c values Number of beginning or ending 6c values averaged to extend MA 2 2 3 3 O' CENSUS METHOD II 15 Compute seasonally adjusted series. 16 Compute ratio of final seasonally adjusted series to average of preceding and following month as test of residual seasonal. 17 Develop an uncentered 12-month MA of seasonally adjusted series. 18 Compute ratio of uncentered 12-month MA of final series to 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) 20 Compute 15-month WMA of final series--measure of cyclical component (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. 22 Compute ratio of final series to 15-month WMA of final series. percentage changes in I. Yields estimate of I. Calculate month-to-month 23 Derive mean of percentage changes in original (0), irregular (I), cyclical (C), seasonal (S), and seasonally adjusted (CI). 24 Using averages in step 23, calculate: I/C, I/S, S/C, I/O, C/0, S/0. 25 Compute ratio of I/C with percentage changes taken 2, 3, 4, and 5 months apart. The interval corresponding to the last I/C ratio that is less than 1.00 is the "number of months for cyclical dominance, (MCD).” 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 8 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 sample "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, i.e,, a two-month moving average of the twelve-month moving average is computed. This operation places the moving-average values at raid-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-terra 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 *"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, Paris, France, Organization for Economic Cooperation and Development, 1961, pp. 391-98, This description is the same as that pub lished in Electronic Computers and Business Indicators, Occasional Paper No. 57, New York, National Bureau of Economic Research, 1957, pp. 248-52 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 9 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, 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. Moving average values for the last two years are obtained in a similar manner. b. For each month, compute Lwo-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. 10 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 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 preliminary seasonally adjusted series (Table 7). II. Computation of Final Seasonally 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 signif icant seasonal factor to suppress. The weighted fifteen-month moving average is much more flexible then 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 (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 ir regular 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 11 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 extreme ratio falling in the middle of the series, sub stitute the average of the extreme ratio and the preceding 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 (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 years’ 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 years' 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 12 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 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 three years preceding the first year available. This is equivalent to weighting the first four years' centered ratios by 6/15, 6/15, 2/15, and 1/15, re spectively, 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, 6/30, 6/30, 4/30, and 2/30, respectively, to obtain the third year's moving average value. The missing values at the end are ob tained 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 of the trend between the factor for that year and the preceding year. If X?=seasonal adjustment factor for year n, then Xn 4- 1 is esti mated by the equation Xn + 1 - 3X„ . p 2 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 1943, 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 identified 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 13 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 the adjusted and the unadjusted series for any twelve-month period (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 14 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, C; seasonal, S’; 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 component 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 7/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 15 28. The average duration of run, i.e., 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, New York, National Bureau of Economic Research, 1957, p. 252. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 16 Listing of Tables Prepared by Census Method II* 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 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/weighted 15-month moving average Stable-seasonal adjustment factors Stable-seasonally adjusted series Centered ratios, original/weighted 15-month moving average Final seasonally adjusted factors, 3x5-month moving average Estimated seasonal factors, one year ahead Final seasonally adjusted series Ratios, final adjusted to preceding and following Averages Uncentered 12-month moving average, final adjusted series Ratios, 12-month moving averages, final adjusted series to original Ratios, each month to preceding January, final adjusted series Weighted 15-month moving average of final adjusted series Percent change from preceding month, original Percent change from preceding month, seasonal Percent change from preceding month, final adjusted series Percent change from preceding month, seasonal-irregular ratios Irregular component Percent change from preceding month, irregular Percent change from preceding month, cyclical Moving average with term of MCD, of final seasonally adjusted series Irregular, cyclical and seasonal components, their relationship and average duration of run Ratios, 12-month moving average of irregular and cyclical amplitudes *"Actual Sample Univac Print-out for Private Non-farm Dwelling Units Started, 1951-56,” Electronic Computers and Business Indicators, Occasional Paper No. 57, New York, National Bureau of Economic Research, 1957, pp. 253-57. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 17 Listing of Tables Prepared by Census Method II Long Program - Complete Shiskin Method* Table Number 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 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 'djustment factors Preliminary seasonally adjusted series Weighted 15-month moving average of preliminary seasonally adjusted series Ratios of original to weighted 15-month moving average Percent change from preceding month, original Percent change from preceding month, seasonal-irregular ratios Modified ratios of original/weighted 15-month moving average Centered ratios of original/weighted 15-month moving average Final seasonally adjusted factors, 3x3-month moving average Estimated seasonal factors, one year ahead Final seasonally adjusted series Percent change from preceding month, seasonal Percent change from preceding month, final seasonally adjusted series Ratios, final seasonally adjusted series to preceding and following Averages Uncentered 12-month moving average of final seasonally adjusted series Ratios of 12-month moving average--final seasonally adjusted series to original Ratios, each month to preceding January--final seasonally adjusted series Weighted 15-month moving average of final seasonally adjusted series Irregular component Percent change from preceding month, irregular Percent change from preceding month, cyclical Ratios, 12-month averages of irregular and cyclical amplitudes Moving average with term of MCD, of final seasonally adjusted series Irregular, cyclical, and seasonal components, their relationship and average duration of run <4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (Shiskin-Census II Method), Long program-Complete Shiskin Method, Research Department, Federal Reserve Bank of Philadelphia, December 1961. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 18 Listing of Tables Prepared by Census Method II* Short Program - Shiskin Method* Table Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Point Chart 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, seasonal-irregular ratios Modified ratios of original/weighted 15-month moving average Centered ratios of original/weighted 15-month moving average Final seasonally adjusted factors, 3x3-month moving average Estimated seasonal factors one year ahead Final seasonally adjusted series X indicates seasonally adjusted 0 indicates unadjusted *4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (Shiskin-Census II Method), Short Program with Point-Chart, Research Department, Federal Reserve Bank of Philadelphia, December 1961. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 19 New Tables In Census Method II * Since 1957, when the description of Census Method II was published in Electronic Computers and Business Indicators, "three new sets of tables have been added to provide additional information about the be havior of each series. The first set of tables shows by stages how the months for cyclical dominance, or MCD, curves are adjusted for amplitude. The series shown in the final Table (26C) of this set is adjusted for seasonality, irregularity and amplitude. Such series add another level to the adjustment process and facilitate comparisions of the cyclical timing and pattern of different series. Tables 26A, 26B and 26C are entitled Symmetrical Percentage Change from Preceding Month in Short-Term Moving Average, Standardized Symmetrical Percentage Change from Preceding Month in Short-Term Moving Average, and Standardized Short-Term Moving Average Index, respectively. Table 26A shows the month-to-month percentage changes in the short-term moving average of period equal to the MCD. These percentage changes are computed by the formula 200 (B-A)/(B+A), where A is the moving average value for the first month and B is the moving average value for the second month. Table 26B is obtained by dividing each value of Table 26A by the average (without re gard to sign) of all the values of Table 26A. This average value is shown after the title of Table 26B. Table 26C is obtained by setting the value for the first month for which the moving average is available at 100 and by obtaining values for subsequent months by application of the formula B'=A'(200+r)/(200-r), where A' is the Table 26C value for the month pre ceding the desired month and where r is the standardized, modified rate of change (shown in Table 26B) for the desired month. The second set of tables shows the original observations adjusted for large irregularities (Table 29) and then adjusted for seasonality (Table 30, see Note at end of this description). If the modified original observations are run through the program instead of the original observations, the whole series of computations will not be affected by large irregularities, and better measures of the seasonal, cyclical, and irregular components and a better seasonally adjusted series may be expected. Table 29 is obtained by multiplying the original observations (Table 1) by the ratio of the modified seasonal-irregular ratios (Table 10) to the un modified seasonal-irregular ratios (Table 9). Table 30 shows these modified original observations adjusted for seasonality. This table is computed by’ dividing the modified original observations (Table 29) by the final seasonal adjustment factors (Table 12). In addition a third set of tables will be computed upon request. They show the seasonal factors when converted to percentages of the yearly acti vity. Each of the monthly seasonal factors is divided by 12 to make up *"Test and Revisions of U.S. Census Methods," Seasonal Adjustment on Electronic Computers, Paris, France, Organization for Economic Cooperation and Development, 1961, pp. 138-39, https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 20 Table 12A (Monthly Seasonal Factors as Percent of Year), and appropriate values from Table 12A are added to obtain Table 12B, Quarterly Seasonal Factors as Percent of Year (e.g., first quarter values in Table 12B are sums of January, February and March values of Table 12A, etc). The sura of the percentages in each row (year) equals approximately 100. These tables indicate the percentage of the year’s activity that may be ’expected’ each month or each quarter as a result of the seasonal factor alone. These tables may be helpful in distributing an annual fore cast by month; however, in such a use an allowance should be made for the cycle and trend within the year. Other uses in which test tables may be helpful include distributing annual budgets seasonally and supervising monthly expenditures of annual appropriations." NOTE*: "The modified original observations are shown in a new table (print out Table 29) and the modified original observations adjusted for seasonality are shown in still another new table (print-out Table 30). Table 30 is computed by dividing the modified original observations (Table 29) by the final seasonal adjustment factors (print-out Table 12). A program which recognizes no extremes is also available through SAG (Seasonal Adjustment Generator program). If this program is applied to modified original ob servations, 'purer1 seasonal factors and trend-cycle curves, and a better adjusted series will be obtained." *"Test and Revisions of U.S. Census Methods," Seasonal Adjustment on Electronic Computers, Paris, France, Organization for Economic Cooperation and Development, 1961, p. 110. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 21 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-terra 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 average”). *”Specifications for the X-9 Version of the Census Method II Seasonal Adjustment Program,” Bureau of the Census, Office of Chief Economic Statistician, March 6, 1962. **This description is the same as that published in Electronic Computers and Business Indicators, Occasional Paper No. 57, New York, National Bureau of Economic Research, 1957, pp. 248-52. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 22 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 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 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 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. 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 23 - 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 2k 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 ODtained in a similar way (Table 10D, "Final Uncentered 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 Xn + 1 is estimated by the equation Xp + I - 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: 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 co.thly amplitude of the irregular component is 4.0 or larger as https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 25 originally specified. Tables 6A and IOC are the Moving Seasonality Ratios described in the specifications for X-10. In X-9, these ratios do not play a role in the 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 26 Listing of Tables Prepared by X-9 Version of Census Method II* Table Number 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 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 Preliminary seasonal adjustment factors Preliminary seasonally adjusted series Weighted 15-month moving average of preliminary seasonally adjusted series Ratios of original to weighted 15-month moving average Modified ratios of original to weighted 15-month moving average Stable-seasonal adjustment factors Stable-seasonally adjusted series Centered ratios of original to weighted 15-month moving average Final seasonally adjusted factors, 3x5-month moving average Estimated seasonal factors, one year ahead Final seasonally adjusted series Ratios of final seasonally adjusted series to preceding and following Averages of ratios Uncentered 12-month moving average of final seasonally adjusted series Ratios of 12-month moving averages--final seasonally adjusted series to original Ratios of each month to preceding January--final seasonally ad justed series Weighted 15-month moving average of final seasonally adjusted series Percent change from preceding month, original series Percent change from preceding month, final seasonally adjusted factors Percent change from preceding month, final seasonally adjusted series Percent change from preceding month, seasonal-irregular ratios Irregular component Percent change from preceding month, irregular component Percent change from preceding month, cyclical component Moving average with term of MCD, of final seasonally adjusted series Irregular, cyclical, and seasonal components, their relationship and average duration of run Ratios of 12-month moving average of irregular and cyclical amplitudes *1410 Program, Actual Print-out for Non-manufacturing Employment in Georgia, Washington, D. C., Board of Goversnors of the Federal Reserve System, 1962. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 27 BUREAU OF THE CENSUS SEASONAL ADJUSTMENT TECHNIQUE The X-10 Version of Census Method II* This procedure** replaces steps 6, 10, 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 Variations," 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 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, 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 *"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. **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). The 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. ***This description is the same as that published in Electronic Computers and Business Indicators, Occasional Paper No. 57, New York, National Bureau of Economic Research, 1957, pp. 248-52. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 28 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 the 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 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 Sy. 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_ the change 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, Iy/Sy. These ratios are designated Moving Seasonality Ratios (Table 6A, "Moving Seasonality Ratios"). i. For each month, depending upon the si?e of the moving seasona lity ratio computed in step 6h, an average of the modified ratios yielded by step 6c is computed, as specified in the table on the next page. 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 pre ceding 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 29 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 30 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 she 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 31 change in the seasonal component for a particular month and is referred to as Sy. 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 Ty. 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 IOC, "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. Moving seasonality ratio step llh Average of 11c values “No. oFTTeginnihg or ending 11c values average to extend the moving 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 0 to 1.49 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis — 32 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 llj. 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 llj 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 Xn + 1 is estimated by the equation Xn + 1 ~ 3Xn ~ _ 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 33 "Stable-Seasonal Factors" and "Stable-Seasonally 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 speci fied. 3. It is suggested that two additional features be incorporated into the X-10 program. The first is to allow the parameters of the moving sea sonality ratios in steps 6i and Hi to be changed easily if additional re search suggests their change or if individual users want to modify the program to their own needs. The second is to program an option which allows the moving average used for each month to be specified in advance by the user instead of being selected on the basis of the moving seasonality ratio; this step will reduce revisions when series are brought up to date. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 34 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 Number" reference preceding a description refers to the table in the print-out provided by the electronic computer program (IBM 650 basic installation). 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 begin six months later than the original series. However, the difference has been reduced to three *"BLS Seasonal Factor Method," 1960 Proceedings of the Business and Economic Statistics Section, Washington, D. C. American Statistical Association, 1960, pp. 8-11. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 35 months by the following series of steps. (Corresponding operations are applied at the end of the series. All operations in the entire procedure are symmetrical with respect to the time scale.) Table 102: 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). Table 103: https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 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 36 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 104: 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 37 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. Hie 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 orig inal 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 (fina1 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 38 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 irregular (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 +2.8(5" and are designed to provide a probability of about 50 percent that all "good" values will fall inside the limits. The 2.8<f 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 encountered, a discriminating test is needed that will detect the non-belonging observation without rejecting 39 too many acceptable values. The 2.8d is the point in the distribution which will, in 50 percent of the cases, reject no values; in the other 50 per cent, it will reject one or more (usually one) values. Initially, different sigma limits based on the length of the series were calculated. How ever, since our computer program handled series of from 6-12 years, and the limits varied only 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 de signated 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 orig inal 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 unforced seasonal (table 503) is multiplied by an ad justment 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 irregulars (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 40 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 of 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 adjust ment 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 original values are divided 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 component. 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 41 *Table 708: 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, *Table 709: Original series. values. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis This is the monthly series of original 42 Listing of Tables Prepared by Bureau of Labor Statistics Method* Table Number 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 approximation (extremes replaced) Irregular, first approximation (extremes replaced) 601 602 603 604 605 Moving average, modified once (extremes replaced) Seasonal-irregulars, 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 (extremes replaced) Seasonally adjusted series 709 Original series *"Tables Resulting from BLS Computer Program for Securing Seasonal Factors, 1960 Proceedings of the Business and Economic Statistics Section, Abe Rothman, American Statistical Association, Washington 25, D. C., p. 11. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 43 Format for All Tables, Except Table 407 Prepared by Bureau of Labor Statistics Method* Series Year Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. Format for Table 407* 407 Extreme Values - Tests and Replacement Values Series Month Year Irregular Lower Upper Original Trend Seasonal Replacement *1960 Proceedings of the Business and Economic Statistics Section, Abe Rothman, American Statistical Association, Washington, 25. D. C., p. 11. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 44 THE REGRESSION METHOD OF DEUTSCHE BUNDESBANK The regression method as adapted by the Deutsche Bundesbank develops seasonal measures by a regression between the actual data and a moving average of actual data. The Deutsche Bundesbank reports that reliable factors can be calculated from a relatively short series. The following quotation gives the rationale of this technique: "The 'classical methods’ of seasonal adjustment (e.g., adjustment by means of a seasonal index or the link relative method) cannot produce useful results in analyzing statistical time series unless the course of the seasonal movements is proportional to the general trend or the ratio of monthly values to the values of the corresponding month of the pre ceding year remains almost constant. The seasonal fluctuations of many statistical time series, however, are not proportional to the trend. For this reason, results of seasonal adjustment by means of the existing procedures are frequently rather un satisfactory. The fact that under certain conditions useful results have been obtained by such methods should not obscure the truth that all these methods contain a basic methodical error. If the various kinds of seasonal behaviour are to be computed by means of one method, this can be achieved only by a comprehensive mathematical/statistical approach applicable to proportional, additive, or any other seasonal behaviour (e.g., proportional and additive mixed). Since all linear relations between two variables can be covered by a re gression equation including one multiplicative (proportional) and one additive parameter, the idea suggests itself to solve the problem of seasonal fluctuations in analyzing statistical time series by means of the correlation calculus in that the seasonal movement is conceived as a correlated connection between series value and trend value. On the basis of the correlation calculus, seasonal adjustments cannot only be made in the case of purely proportional or purely additive behaviour, but also in all other cases where the seasonal movements are composed of an additive and a proportional component. It goes without saying that these components are the result of numerous single factors."* *"The Problem," Application of the Regression Method to the Analysis of Statistical Time Series, Frankfurt (Main), Germany, Deutsche Bundes bank, 1959, p. 1. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 45 BUREAU OF THE CENSUS SEASONAL ADJUSTMENT TECHNIQUE (METHOD III)* Method III will have further refinements and also utilize somewhat different methods of dealing with our principal problems. The principal improvement, however, will be the increased generality, flexibility, and adaptability of the new program. The new program will be constructed by utilizing a family of subroutines instead of a fixed set of consecutive instructions. For example, one subroutine of this program will consist of a generalized trend-cycle moving average with a period from 3 to 45 months and any weight pattern. Thus different moving averages can be used for series with different characteristics. The program will also be prepared in such a way that it will be easy to make fairly large modifications by replacing one subroutine by an entirely different one or by adding a new subroutine. Additional tests and controls will be built into the program to determine what are the appropriate moving averages, control limits, and weights to use for the particular series being run. Because of its adaptability, this new program will be an even more powerful tool for experimental work than SAG**. The Census Bureau has recently expanded its studies of mathematical and statistical methods which could be applied to the decomposition of economic time series. When improved or additional techniques are developed by these studies, they will be used in Method III. The new program will be prepared for our new computers, the Univac 1105's. These computers have about the same speed and capacity as the large scale IBM machines-- the 704 and 709. These machines compute much faster than our Univac I's. Since a 10-year series will probably require about one minute's running time on the 1105, it will matter little whether the short or long run is requested. Consequently, the distinction between the long run and the short run will probably be eliminated. It may also be possible to reduce the costs. Some of the major improvements under consideration for Method III are summarized below: 1. A working or trading-day adjustment based upon the internal evidence of the series. 2. A wider range of moving averages to measure the trend-cycle factor, selected on the basis of the I/C ratio. 3. A third iteration based upon original observations modified for extreme irregularities. *"Census Method III," Tests and Revisions of Bureau of the Census Methods of Seasonal Adjustments, Technical Paper No. 5, Washington, D.C., U.S. Bureau of the Census, 1961, p. 32. **Seasonal Adjustment Generator program. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 46 4. A wider range of moving averages to measure the seasonal factor, selected for individual months on the basis of the I’/S’ ratio. 5. An iterative technique for identifying extreme S-I ratios; i.e., a set of control limits and trend line would be computed without the preliminary extremes affecting the computations. 6. An alternative method of identifying extremes based upon the distribution of the complete series of irregular factors. 7. A moving amplitude adjustment. 8. Additional summary measures including CI/O, annual values for I, C, S, quadratic means of the irregular, cyclical, and seasonal factors and measures of the seasonal and irregular factors for each of the 12 months (i.e., I’, S'). 9. A technique for adjusting first for additive seasonality and subsequently for moving multiplicative seasonality (if this proves feasible). 10. To the extent that our research warrants it, statistical tests of significance will be added at appropriate sections of the program. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 47 Listing of Tables Prepared by Census Method III* Table Number 1 2 3. 4 5 6 7 8 9 10 10A 10B 11 12 12A 12B 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Title of Table Original series--to be adjusted 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 Preliminary seasonal adjustment factors Preliminary seasonally adjusted series Weighted 15-month moving average of preliminary seasonally adjusted series Ratios of original to weighted 15-month moving average Modified ratios of original to weighted 15-month moving average Stable-seasonal adjustment factors Stable-seasonally adjusted series Centered ratios of original to weighted 15-month moving average Final seasonally adjusted factors, 3x5-month moving averages Estimated seasonal factors, one year ahead Monthly seasonal factors as percent of year Estimated seasonal factors, one year ahead Quarterly seasonal factors as percent of year Estimated seasonal factors, one year ahead Final seasonally adjusted series Ratios of final seasonally adjusted series to preceding and following Averages Uncentered 12-month moving average of final seasonally adjusted series Ratios of 12-month moving averages--final seasonally adjusted series to original Ratios of each month to preceding January--final seasonally adjusted series Weighted 15-month moving average of final seasonally adjusted series Percent change from preceding month, original series Percent change from preceding month, seasonal Percent change from preceding month, final seasonally adjusted series Percent change from preceding month, seasonal-irregular ratios Irregular component Percent change from preceding month, irregular component Percent change from preceding month, cyclical component Moving average with term of MCD, of final seasonally adjusted series *”Sample Print-out Tables of the X-3 Program--Number of Private Nonfarm Dwelling Units Started, United States, 1939-1959," Tests and Revisions of Bureau of the Census Methods of Seasonal Adjustments, Technical Paper No. 5, Washington, D. C., U.S. Bureau of the Census, 1961, pp. 33-51. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 48 Table Number 26A 26B 26C 27 28 29 30 https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis Title of Table Symmetrical percent change from preceding month in table 26 Standard symmetrical percent change from preceding month in table 26 Standard short-term moving average index Irregular, cyclical, and seasonal components, their relation ship and average duration of run Ratios of 12-month moving averages of irregular and cyclical amplitudes Modified original observations (T| x T^q/Tq) Modified seasonally adjusted series (T.Z9/T.12) 49 BIBLIOGRAPHY Census Method II Electronic Computers and Business Indicators, Occasional Paper No. 57, New York, National Bureau of Economic Research, 1957, pp. 248-57. Seasonal Adjustment on Electronic Computers, Paris, France, Organization for Economic Cooperation and Development, 1961, pp. 110, 138-39, and 391-98. 4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (ShiskinCensus II Method), Long program-Complete Shiskin Method, Philadelphia, Research Department, Federal Reserve Bank, December 1961. 4-K 1401 Program, Seasonal Adjustment of Monthly Time Series (ShiskinCensus II Method), Short program with Point-Chart, Philadelphia, Research Department, Federal Reserve Bank, December 1961. The X-9 Version of Census Method II ’’Specifications for the X-9 Version of the Census Method II Seasonal Ad justment Program," Washington, D. C., Bureau of the Census, Office of Chief Economic Statistician, March 6, 1962. 1410 Program, Actual Print-out for Non-manufacturing Employment in Georgia, Washington, D. C., Board of Governors of the Federal Re serve System, 1962. The X-10 Version of Census Method II "Specifications for the X-10 Version of the Census Method II Seasonal Ad justment Program," Washington, D.C., Bureau of the Census, Office of Chief Economic Statistician, March 6, 1962. The Seasonal Adjustment Method of the Bureau of Labor Statistics 1960 Proceedings of the Business and Economic Statistics Section, Wash ington, D. C., American Statistical Association, 1960, pp. 8-11. The Regression Method of Deutsche Bundesbank "Application of the Regression Method to the Analysis of Statistical Time Series,"Frankfurt (Main), Germany, Deutsche Bundesbank, 1959. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis 50 "Experience in the Application of Regression Computing to the Seasonal Adjustment of Statistical Time-Series," reprint from: Monthly Reports of the Deutsche Bundesbank, Vol. 13, No. 8, Frankfurt (Main), Germany, Deutsche Bundesbank, August 1961, pp. 19 et. seq. "The Practice of Seasonal Adjustment with Regression Equations," Frankfurt (Main), Germany, Deutsche Bundesbank, 1960. Census Method III Tests and Revisions of Bureau of the Census Methods of Seasonal Adjustments, Technical Paper No. 5, Washington, D. C., U. S. Bureau of the Census, 1961,pp. 32-51. https://fraser.stlouisfed.org Federal Reserve Bank of St. Louis