measurement of seasonal variation

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311 T he Measurement of Seasonal Variat ion 167

THE MEASUREMENT O F SEASONAL VARIATION

BY HELEND. FALKNER, heaton College

The determination of seasonal variation in an economic series isrendered difficult by the fact that such series are usually subject tothree other m a i n type s of Jluctuation. We are assuming th at the changesin any economic series may be classified under four heads as suggestedby Professor W. M. Persons.2 The four components of the totalfluctuation are : seasonal variation, secular trend, cyclical fluctua-tion, and irregular deviations. Seasonal variation is tha t part of t hefluctuation due to the persistent tendency for certain months of eachyear t o be regularly higher than certain other months of the year,irrespective of the disturbing influence of cyclical fluctuation and irregu-la r deviations. The amount of seasonal variation, for a particularmonth, is measured by an i ndex of seasonal variation. This index isthe per cent ratio which tends, on the average over a considerabletime interval, t o hold between the actual value and the normal value inth at month. Thus, if there were no cyclical fluctuation and no irregu-lar items, this ratio would hold exactly between the actual value forthe month and the normal value (due to secular trend alone). Seculartrend is the long-time tendency of the items of the series to grow ordecline regardless of the temporary seasonal and cyclical and irregularmovements. This tendency is generally assumed to be constant inamount over a considerable interval of years. Cyclical fluctu atio n isthe wave-like variation which reflects the succession of prosperity anddepression in business conditions. This movement is no t periodic ina mathematical sense; the cycles are of varying duration in time, havediffering maximum deviations from the normal, and vary widely inshape. Irregular deviations are the residual elements of change no tincluded in the three other components of t he total fluctuation. Theyare usually abrup t in occurrence, of short duration, and of large magni-tude; and they do not occur regularly at an y particular time of t heyear or a t any specific stage of the cycle. These deviations affect someeconomic series very notably, and have almost no part in the fluctua-tion of certain other series.

Received by the Editor, January 14,1924.Professor W. L. Cnun of Harvard University examined the original draft of this paper and made

SRm E m bt., Prel. Vol. I, p. 8.m a n y valuable suggestions3

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168 American Statistical Associafion [32

The statistical data which constitute a n economic record a re subjecttherefore to three types of fluctuation other than seasonal variation,and most of these component fluctuations are not strict ly uniform inlength or size or shape. The serious obstacle to the measurement ofseasonal variation rests in the fact that each actual item is affectedin an unknown manner and to an unknown extent by the other influ-ences. The statistical scheme used for measuring seasonal move-ment should be designed to eliminate, so far as possible, the effectsof these other component fluctuations and to present an index whichreflects seasonal variation alone.

I1

Among the numerous devices which have been suggested for thepurpose there are three to which we shall give especial consideration.

It is the object of the present section to discuss them, in the light ofthe above description of the peculiar difficulties of the problem.1. One of the simplest schemes for determining seasonal variation

is the so-called method of monthly means.l According to this plan theseasonal movement is indicated by twelve averages: the average ofall the January items, the average of all the February items, and so onu p t o the average of all the December items. The resulting numbersmay be expressed in percentage form by dividing each of them by themean of the twelve. The chief advantage of the method is the

extreme ease of computation, bu t this is offset by the failure of thescheme to make adequate adlowance for the effects of the three othertypes of fluctuat ion.

The allowance which the method makes for cyclical fluctuation con-sists in covering a sufficiently long time interval to warrant the expec-tation that cyclical influences will tend to cancel each other. N oexact cancellation, however, can be relied upon.

The method admits of an extension, developed by Professor G. R.Davies Zoc. cit. p. 119), which eliminates the effect of secular trend.

When secular trend is not eliminated its influence is likely to be ap-parent as in the case of the December to January drop in the sea-sonal curves of many of Professor Kemmerers charts (for example,Chart IV, Eoc. cit. p. 24).

The method makes no provision for the influence of irregular devia-tions, and i t is here tha t we find the chief objection to its use. Thearithmetic average is peculiarly subject to extreme items, and it isfor that reason tha t a monthly seasonal index obtained by this method

Recently discussed JOUR. A M E R .TAT. asN, Sept.. 1922) by W. L. Hart; but already widely used,notably by E. W . Kemmerer in Seasonal variations in the relalive demand for m o n q and caplal in theu 8 ; nd presented and illustrated by G. R. Davies in his Economic Stafist ics.

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331 The Measurement of Seasonal Variation 169

may be governed more by an exceptional irregular deviation than bythe systematic seasonal movement. Moreover, even if there are noclearly irregular items, the average for an y particular month may notbe truly typical of tha t month. Thus, the average of nine January

items (assuming a series with a nine year tota l interval) may not be agood measure of the statist ical ar ray composed of those items, unlessthey are fairly well clustered about their average. The method, asgenerally used, affords no device for testing whether the individualmonthly means are tru ly typical.

Professor Kemmerer has att empted loc. cit., p. 13) to eliminate theeffects of the irregular items by the use of indexes, but it is doubtfulif this device is successful. The arbi trary selection of the highestand lowest items (for any one month, say January) as the 100 and 0

of his per cent scale makes all the indexes of that month depend onthese possibly irregular extreme items, and it amounts to selecting asa basis of comparison that most unstable of dispersion measures, therange.

2. The essence of the median-link-relative method is as follows: thecalculation of month-to-month link relatives from the original da ta ,the verification of the existence and estimate of the general nature ofthe seasonal movement by use of a multiple frequency table, the selec-tion of the median of the link relatives for each month, the adjus tmentof these medians to allow for trend and other discrepancies, and thederivation of the final seasonal index expressed as a percentage of theaverage for the year. The scheme does not have ease of applicationto commend it, as does the method of monthly means.

The method contemplates no direct allowance for the effects ofcyclical fluctuation, but requires merely that the interval covered bythe series be sufficiently long to insure that the individual medians besatisfactorily typical. There is an indirect tendency to minimize theinfluence of cycles because of t he comparison, in the link-relative opera-tion, of each month with that immediately preceding. This fac ttends in general to free the link-relatives from cyclical disturbances.

The first adjustment, the logarithmic correction, of the system aimsto remove the effects of secular trend. This adjustment is occasionallyquite different from its theoretical value, estimated from trend alone,and contains therefore other elements of discrepancy than the trend;bu t the introduction of the adjus tment appears in most cases to furnishan adequate correction of the seasonal index for secular trend.

The use of the median, ra ther than the arithmetic mean, is primarilyOriginated by Prof. W. M. Persons, in Rev. Econ Sta t . . Prel. Val. I ; and discussed and reconsidered

in various articles: W L. Crum in J O T R A M E R .STAT. SS for March, 1923; A . Fisher, ibid., June,1923; W. M. Persons. ;b id . June, 1923; E. B. Wilson, i b d . September, 1923.

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170 American Statistical Association t34

for the purpose of avoiding the effects of the irregular items. Theseitems are likely to appear, in the table of link relatives, as extremelyhigh or extremely low ratios; and the value of the median will be inde-pendent of these extreme ratios. Prof. W. L. Crum Zoc. cit., p. 614)maintains that the frequency distribution in an economic series willoften have such a form th a t the median is more stable than the mean,and suggests a slight modification of the median in cases in which thecentral concentration is no t well defined. Mr. Arne Fisher (Zoc. cit.,p. 793) and Professor E. B. Wilson (Zoc. cit. p. 841 have discussedthis question further, giving respectively unfavorable and favorableviews of the median.

The median-link-relative method appears to make a satisfactoryallowance for the disturbances due to the components, other thanseasonal, of the fluctuat ion; and, except for t he lengthy computat ionsinvolved in its application, i t appears superior to other methods nowin use.

Th e scheme in its ori-ginal form, in which th e elimination of seasonal variation is at temptedby replacing the actual item for a particular month by the average ofthe twelve monthly items of which i t is the center (with an ad justmentto insure exact centering), is now seldom used. What i t really assumesis tha t any one interval of twelve months is as good a picture of theseasonal swing as any other. It makes no direct allowance for any ofthe three disturbing influences, but indirectly makes some allowancefor all of them. If the secular trend is a straight line the methodmakes adequate provision for th at influence. If there are no irregulardeviations (within the twelve-months interval) the tac it assumptionis made tha t the cyclical element in a particular monthly item is idcn-tical with the average cyclical fluctuation over the correspondingtwelve months. This is scarcely true in general, although i t mayfrequently happen to be ,true. If, however, the twelve-months inter-val does contain one or more irregular deviations (none of which oc-curs in the month under examination) the above assumption requiresa modification: the cyclical element in the particular month is as-sumed identical with the twelve-months average of cyclical fluctuationsand irregular deviations. If an irregular deviation occurs in the monthin question, a corresponding different modification is necessary. Themodified forms of the assumption, which the very general occurrenceof irregular deviations necessitates, are obviously objectionable ; andthe method is not a satisfactory device for eliminating seasonalvariation.

There has recently been developed a method which makes use of the

3. A third method is tha t of moving averages.

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172 American Statistical Association [36TABLE I

THE SERIES TESTED AND THE ELEMENTS OF TH EIR RESPECTIVE SECULAR TRENDSFOR THE INTERI'AL 1903-1913

Source of Mon thly Average (forSymbol Name of series I da ta * increment July 1. 1908

ab

c

d

e

f

Bank Clearings of New York City. . . . . . 64Tonnage of Pig Iron pro duc ed in the

United States.. . . . . . . . . . . . . . . . . . . . . . 66Bank Clearings of the United States Out-

side New York C it y. . . . . . . . . . . . . . . . . 68Values of Building Per mit s Issued for

Twenty Leading Cities. . . . . . . . . . . . . . 74Ra te of Inter est o n 60-90 Day Coinmercial

Paper in New York . . . . . . . . . . . . . . . . . . 99Dividend Payments by Industrial Cor-

porations . . . . . . . . . . . . . . . . . . . . . . . . . . 164

1 . 7

7 . 9 4

2 . 2 6

1 . 2 0

- 0016

1 . 3 5

748.

1980.9

491.8

408.9

4 .895

273.9

*The page number in Rev. Econ. Slat., Prrl . Vol. I where the original monthly items are given.Th e month-to-month iink relatives used in this study, Gebruary, 1903. to January, 1914, will be foundon he same or adja cent pages of th at volume.

to the series; the present method merely requires tha t t he proper formof trend be used in finding the ratios.

The next step is the construction of the multiple frequency table,consisting of twelve adjacent columns, each giving the frequency ofthe ratios for a particular month in the pear ; an dthe result is shown inTable 11. Although the concentration in any one column of this tab leis not close, the distributions in pairs of adjacent columns a re on thewhole displaced considerably relative to each other. This evidentrelative displacement, although not as satisfactory a confirmation of

TABLE 11*

FREQUENCY DISTRIBUTIONS, MONTHLY, O F RATIOS O F ACTUAL ITEMS TO THERESPECTIVE ORDINATES OF SECULAR TREND, FOR BANK CLEARINGS OF NEWYORK CITY

* I n Tables 11, V. VI. VII. V III t he left-hand marginal number ia the lower limit of t he one per centclass interval, for each row of th e table .

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I n view of these observations, i t seems unwise to base the index on agroup of less than three items or more than seven items. In a particu-lar problem the multiple frequency table (Table 11) will usually givethe clue: if the ratios ar e closely concentrated the middle three itemsar e sufficient; if the scatte r is quite considerable, five items niay berequired; and in rare cases seven items will furnish the most typicalindex. Since the labor of calculating indexes for several groupings isvery alight, doubtful problems may be handled by trial. If, as in thepresent case, the changes in the indexes from the three-item columnto the five-item column are less significant than those from the five-item to the seven-item column, the results of the five-item or the three-item computation may be selected as final. When i t is decided whichgrouping to use, the crude indexes are found, and the adjusted indexesbased upon them are taken as measuring seasonal variation.

It is obvious that this scheme of finding seasonal indexes makesadequate allowance for secular trend, for the trend is actually elimi-nated from th e original da ta before the computation of the seasonalvariation. Moreover, i t provides against the influence of the irregulardeviations by using a group of middle items, in any one month, fromwhich the irregular extremes are omitted. The provision agains t theeffects of cyclical fluctuation consists chiefly in the insistence upon aseries covering a long time interval, bu t the proper selection of thegrouping used in getting the crude indexes tends to minimize the dis-turbing effects of non-uniformity in the cycles.

Viewed by itself, this scheme appears to have the great advantageof ease in computat ion, which is claimed for the method of monthlymeans, and t o retain also those safeguards against the influence of theother components of tota l fluctuation, which are characteristic fea tures

Since the development of th e analysis presented above, the me thod used by Dr. Lincoln W . Hallin A Study of the Cyelical Fluctuations Occuring in the h atiunul Banking System during the Years 1903 to

92 has been noted and examined. On page 22 of that work, he states under the third headingof his "summary of th e various steps" th a t his "seasonal variation is obtamed by taking an arith-metic mean of the residuals for each season." He presents on he succeeding pages the d at a and com-

putations for loans and discounts for the two intervals 1903-1914 and 1915-1921. In thi s analysis hehas allocated the dates of call on a quarterly basis, and t re at s each such time interval as a regular seasonalsegment of the year. The seasonal variations are found for th e two intervals as: -19, 30, 32, -16 ;and -5, -143. -lo 11. respectively ( at tem pt to r ecompute these average residuals yields identicalresul ts in all cases except th e first item of the earlier interva l, which' appe ars to be -36 instead of -19 .To fac ilitate examination, these average residuals are throw n on a percentage basis by dividing them bythe corresponding ordinates of trend for a n annual segment a t the middle of each of the two intervals.and increasing the resulting quotients by 100 per cent. Expressed in this manne r, the indexes ofseasonal variation according to Dr. Hall's process are: 99.2, 100.6, 100.6, 99.6; and 100.5, 99.2, 99.5,100.8, espectively.

It may be suggested th a t Dr . Hall's method would be considerably improved by using per cent ratiosof t he actu al items to the tre nd, instead of t he ac tual residuals from trend. If we first convert each ofhis residuals to a per ce nt basis by dividing i t by the corresponding ordi nate of trend , and th en tak earith metic means of the per cent ratios fo; each of the four seasons, we get as the seasonal indexes:

99.2, 100.5, 100.6, 99.6; and 100.5, 99.0, 99.5. 101.0, respectively, for the two intervals 1903-1914 and1915-1921. The differences between these results and those obtained by the direct app ication of his

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of the median-link-relative method. It remains, however, to subjectthe scheme to varying practical tests to determine the extent of itsreliability a nd t o reveal what limitations mus t be placed upon i ts use.

MLR

97739693919093

101109113111

114

IV

To test the method, i t was applied t o each of the series describedin Table I, and the results appear in the columns headed RTO (ratio-to-ordinate) of Table I V. T o facil itate comparison the corresponding

TABLE IV

COMPARISON OF TH E INDE XES OF SEASONAL VARIATION FOR T HE SIX TEST SERI ESAS OBTAINED BY T H E M L R AND RT O METHODS. USING I N EACH CASE TH E MEAN.

RTO~9

959891878695

100108111114

113

O F TH E MIDDLE THRE E ITEMS I N EACH MONTHLY DISTRIBUTION, INSTEAD O FTHE MEDIAN

January. .February. .March. .April.. . .

Series a l b c

11591

101102

J u n e . . . . .J u l y. . . . . .August. . .Septenibcr .October.November.

December..

96959089

107102

108

1139299

103100

9G968889

10910511

10093

105102103

97959898

104100

101

9893

1061061091031001031 11 1

90

90

11092

102999796989193

108105

108

TO

10891

102100

999s999293

108105

105~

~

LR

6870

117139124122107101

959487

76~

_ _

d

TO

7072

123133117121106105

959386

80

LR

12877

1171217093

1296893

12771

109

f

TO

12875

1131147198

1336990

12771

112

indexes obtained by the link relative method, the average of threemiddle links being used instead of the median, are shown in columnsheaded MLR.

The explanation will be based upon a comparison of cases b and c;and Tables V, VI, VII, VIII furnish the material which is needed.The ratios for any one month are much more closely concentrated inTable V than in Table VI, and i t is therefore t o be expected t ha t case

A graphic comparison is afforded by Charts A-F.

method are scarcely significant in this case; but in general it is to be expected that the ratio schemewould be the more satis facto ry, particularly in all cases in which the tre nd is fairly stee p.

If now the indexes for the two intervals are obtained by the ratio-to-ordinate method, using theaveragcs of the middle six ratios in the 1903-1914 interval and t he averages of the middle three ratiosin the 1915-1921 interval, the resulting indexes are: 99.2, 100.5, 100.7, 99.6; and 99.2, 98.0, 99.8,103.0, respectively. The differences in th e earlier interval are slight, and indeed there the examinationof the multiple frequency table would probably have led to the conclusion tha t th e deter mination ofthe seasonal indexes could not be precise in th is case. Th e differences in the la ter interval are quitemarked, an d here an examination of the multiple frequency table reveals the reason for the discrepancieu.Dr. Hall's method yields indexes which are controlled largely by the extreme items, and the few verylarge residuals in the la te r int erval ar c really decisive in determining his indexes. It is this blind use oftnc a rithme tic avera ge which is exceedingly dangerous; and i t is th e purpose of the ratio-to-brdinate

method dcvcloped in the present p aper, with it s presentation of the ratios in the mu ltiple frequencytable and i ts elimina tion of the extreme de viations, to gua'rd against these very dangers.

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.

J F n A n J J A S O N D

f 1

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TABLE VFREQUENCY DISTRIBUTIONS, MONTHLY,

O F RATIOS OF ACTUAL ITEMS TO THERESPECTIVE ORDINATES OF SECULARTREND, FOR OUTSIDE BANK CLEAR-INGS

TABLE VIFREQUENCY DISTRIBUTIONS, MONTHLY,

O F RATIOS OF ACTUAL ITEMS T O THERESPECTIVE ORDINATES OF SECULARTRE ND. FOR PIG IRON PRODUCTION

c will afford a more precise seasonal index than case b. Moreover,examination of Tables V I I and VIII indicates that the seasonal in-dexes obtained by the MLII, method are probably more reliable forseries c than for b. Comparative study of Tables VI and V I I I sug-gests that the M L I i index is probably somewhat more typical thanthe RT O index, but there is no sure conclusion. Neither methodyields a highly precise index in a case in which the frequency tablesshow such grea t dispersion. I t is true th at Table VI has hardly greaterdispersion than Table I1 (considered above), but the essential differ-ence between them is th at the relative displacement between adjacentcolumns is more clearly apparent in Table I1 than in Table VI. Theconclusion appears then that a multiple frequency table showing thewide dispersion and the slight relative displacement between adjacentcolumns, which are the striking features of Table VI, is an indicationtha t the RTO method does not yield a satisfactory seasonal index. Asubsequent trial of t he MLR method may show that the particularseries does admit of a seasonal correction by that scheme; but, unlessthe corresponding frequency table is better formed than Table VII I ,

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178 American Statistical Association [42TABLE VII

FREQUENCY DISTRIBUTIONS, MONTHLY,O F T H E MONTH-TO-MONTH LI NK RELA-TIVES, FOR OUTSIDE BANK CLEARINGS

TABLE VIII

FREQUENCY DISTRIBUTIONS, MONTHLY,

TIVES, FOR PIG IRON PRODUCTIONOF THE MONTH-TO-MONTH LINK RELA-

even the results of this method must be accepted as merely approxi-mate measures of the seasonal swing.

Investigation of the frequency tables, according to both methods,for series e confirm the above finding. I n this case also the consider-able discrepancy between the two results is explicable in the light ofthe wide dispersion in the frequency tables; and here also the MLRresults are probably the better, although even they are not highlyprecise.

The RTO methodis designed to remove t he influence of other-than-seasonal fluctuationsin calculating seasonal indexes, and is nevertheless exceedingly simpleand brief in application. The method can be used in cases in whichthe multiple frequency table has high concentration within each col-umn (series c and f above), and in cases in which concentration withineach column is slight but relative displacement between adjacentcolumns is well marked (series a and d above). It should not be usedin those doubtful cases in which the concentration is poor and therelative displacement slight (series b above), and may be used only as

Th e general results of this s tudy are as follows. D o w n l o a d e d

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an approximation in borderline cases (series e above). I n a11 th edoubtful cases, however, i t is well to t ry also the MLR method; some-times its results will be more precise than those of the RTO scheme,and sometimes they will not . Th e very large number of practicalproblems in economic analysis for which the method is satisfactoryjustifies the at tempt to use i t in all cases. I n the few cases in which itmust be abandoned the only labor lost is th a t of constructing themultiple frequency table, and in the many cases to which i t is applicablei t represents a great economy of effort.

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measurement of seasonal variation

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