Frederick R. Macaulay

English

Research Papers in Economics

This PDF is a selection from an out-of-print volume from the National
Bureau of Economic Research
Volume Title: The Smoothing of Time Series
Volume Author/Editor: Frederick R. Macaulay
Volume Publisher: NBER
Volume ISBN: 0-87014-018-3
Volume URL: http://www.nber.org/books/maca31-1
Publication Date: 1931
Chapter Title: The Whittaker-Henderson Method of Graduation
Chapter Author: Frederick R. Macaulay
Chapter URL: http://www.nber.org/chapters/c9366
Chapter pages in book: (p. 89 - 99)(:IIAP'FlR VI
'in ; \Vii VITA KI R-H F. NIRSON NI FT 11011) OF G RAI)UATION.
In smoothing data somedeparture from 01)-
served facts is, of course, necessary.In any par-
ticular case, the questionnaturaflarises as to
how far it is desirable todepart from the obscrved
facts in order to obtainsmoothness. If departure
from the observed factsbe meaSUFe(I by means of
the sum of the squares ofthe deviations of the data
from the graduated curve,and if roughness or lack
of smoothness bemeasured by the sum of the
squaresof the third differences ofthe graduated
curve, theproblem is to decide at what stageit is
undesirable to increase the sumof the squares of
the deviations of the datafrom the graduated curve
in order to decrease the sumof the squares of the
third differences of thegraduated curve itself.
Such a problem might beformulated as follows
How shall k times the sumof the squares of the
deviations of the data fromthe graduated curve
plus the sum of the squaresof the third differences
of that graduated CUrVe l)emade a miflifllUln? The
larger the value chosen fork the more importance
would he attached to closenessof fit as compared
with smoothness. As thevalue of k ap1)rOached
I/
90 THE SMOOT LuNG OF 'I'IME SERIES
infinity,the lesUlti flIil(Ill1tC(1 CUIVe voit1d
pr0aI1 the data points.As the value of k ap-
proachedy.ero,theresulting graduated (ur\e
ould a1)1)rOaCh a second degree parabola fitted
to the observatioiis in' themetho(1 of least s(ltlares.1
In 1923,Professor L. T. WThittakcr drew atten-
tion to the fact that if data be smoothed in such
a manner thatk times the sum of the 5(jUarCs of
the deviations of the data from the smooth curve
plus the sum of tic squares of the thi ni (liffCrCflCes
of the smooth curve itself be made a minimum.
tile smooth curve will be such that each of Its ordi-
nates will equal a corrcsj )On(Ii ng or(li nate of the
data plustimes a sixth difference of the smoot1
curve: He developed a lilethO(l tor computing
such a smooth curve
In1924,Robert Henderson pul)lished an article
in which he showed that if
'Ehe form of the smooth curve in thisI uniting case is the result
of makins the suni of the squares of the third differences vanish
and the sum of the squares of the deviations a minimum. If the
suni of the squares ot the seeoui ditterences hr Tilade to vanish, tue
limiting case will give a straight line as the graduated curve.
2A complete set of sixth differences is obtained by extent!ng the
graduated curve liv means of seioiid-dcgrte l)1raholap,It thruuli
the I rst three and last three graduated pointS.
See E. '1'. Whittaker :111(1 (. Robinson, 7hc (af(u/uof()/cC'-
ta/ions, pp. 303 to 31 ç, mci usve. lotan even earl icr sogge stiorl of
the method, see Henderson and Sheppard, C radna/zonoJ .11oita!i/
and Of her Tables (1919), page 6. eleven hues from bottom ofpage
and onwards,4Q000
26.060 -
TIlE SMOO'I'IIING OF TIME SERIES 91
6 (2n + 3Y
n (n+ 1 )'(n + 2 )(n + 3)
the numerical calculation of the ordinates of the
smooth curve Can he arranged in a simple form
for all values of k corresponding to integral values
CHARTIII
ILLUSTRATION OF'
ROBERT HENDERSON'S
METHOD OF GRADU-\T!ON
60,000 - 60,000
a,000
- 20,000
ofn.1'While Henderson's calculation scheme is
quite simple, the readerwill probably understand
it more easily if it is illustrated than hewould if
For thetheoryback of Ilendersons nietliod of 6tting, seehis
article, A .Vew Method of Graduation, 1'ransactionsof The Actu-
arial Socictv of America,May 1924,and Robert Henderson, Some
Points ini/o' General Theory of G,-adieation,Proceedings of the
Internatioiial Mathematical CongressToronto1 (24VoluflIe II,
pages 815-820, inclusive.
0 0-FilE sMooiilIN(; OF TIME SERIES
t were merely(kscfllwd. Aj)j)efl(I1X 'VI Contains
a numericalpaIL(ligL1Iv-itli complete iflstructioiis
for COI'flj)UtltIOfl. Chart IIIillustrates the ft ob-
tained in A1)1)endiX \J.
'Though Rol)crt Hcndcrsons method of com-
puting theWhittakcr-Henderson graduation is a
highly elegant COfltFil)Utiofl to mathematical tech-
nique, and appears extremely simple, the actual
computation is not without some SCFIOUS ptfalls.
In the hrst place, the reader must remember that
the three points from which the eonlI)utation in the
paradigm in Appendix VI began, were selected
after the exact graduation was aT ready known.
The were selected in such a manner as to give
immediately accurate results. In prtc:tiCethere-
/;m;flclil' operatu)fl (see A1)1)CfldiX VT) which is
concerned with ohtaining three points whcIi shall
closely approximate three points at one end of
the true smooth curve, is often Iengthvparticu-
lady if the three points from\Vhich the prelimi-
nary operation commences are taken tOo close to
the end or are at all balls' chosen.'
'Though each operation is extremely simple, l)C-
lug merely a multiplication by a small number or
an addition or a subtraction, the operations have
to he performed seriatim. In actual cornpiitatioii,
this leads to errors being ma(le with great ease. In
any summation formula, each operation isCoil]-
See Note 1, I 52.TIlE SMOOThiNG OF 'I'JME SERIES 93
pleted l)cf ore the fl&Xt One ISl)CUfl. For example.
the first operationconnected with the 434erm ap-
proximately fifth-degreetiaioiic formula is the
taking of ai 2-months moving total of all the
data. The next stepis the taking of an 8-months
movingtotal of the first moving totalfor all the
data. Each typeof operation is completed before
the next is begun. Computers are much less likely
to makemistakes if they do flOt have to change the
nature oftheir operations momentarily.
Furthermore, while there are checks on the ac-
curacy ofthe Henderson computation, such checks
cannot be useduntil after the entire graduation
has been calculatedunless we exceptthe impor-
tant check ofrepeating the preliminary operation
backwards and forwards the entire length of the
data until three identical, orpractically identical,
end figures have been arrived at.From the nature
of the Henderson computationscheme errors are
not easy to spot on achart, as they tend to be dis-
tributed smoothly over a considerable range.
rçhis
is not so with errors in some ofthe other gradua-
tions which have already beendiscussed. For ex-
ample, errors in the computation of theparabolic
summation formulas nearly always occur inthe
weight multiplications or in the finaladditions
and subtractions of such results.1 Now,such errors
generally stand out as cusps on a chart.Moreover,
The moving totals are automatically checked.S
94 TilE SMOOThING OF TIME SERIES
when discovered they (U1 cisilvl)Ccorrected
whereas an error in the Henderson computation
involves, at least theoretically, a re-computation
of the whole S('CO/I(I /iciif.
If n equals,as in the paradigm in Appendix
VI, the value of
4(211 + 3)
(n + i) (n + 2)(n +3)
is exactly .o6. However, if 11 equals 2 or 4 or ç, for
example, the value of the above function isflO
longer a simple decimal of few terms. The third
figure of each column in the computation will then
have to ie obtained from a calculating machine.
It may well be that, for any Particular set of
observations, no integral value of Ilvih1 give the
exact graduation desired. If a fractional vahie of
n be required, tile computation becomes consider-
ably lengthened.1
A Whittaker-Ilenderson graduation needsno
extrapolation; it covers the entirerange of the
In 1925 Mr. Henderson 1)ul)liSlIed another article in whichhe
described a method of applying the Same graduatmonto a series of
weiahfrd observations. 'hiiis second niethod he designated''Method
B." Though ''Method B'' is primarily designed foruse where the
observations are weighted, it can be appliedto unweighted observa-
tions by assuming a uniform weight of unity. However,when ap-
plied to unweighted observations,it has the (lisa(lvantage of being
considerably more laborious incomputation titan Mr. Henderson's
earlier method. On the other hand,it is mure sd f-checking, and the
difficulties connected with obtaining the threeend points are taken
care of in the niethod itself without involvingany guessing as in theI
'FIlE SMOOTHING OF TIME SERIES 9
data. Thisis a distinct element of mathematical
elegance atidsometimes an important practical
consideration.' However, graduation of the ends
of almost any seriesis necessarily extremely hypo-
thetical unless facts outside the range covered by
the graduation areused in obtaining the gradua-
tion. This is as trueof the Whittaker-Henderson
graduation as of any other type. Ifit were not so,
we shouldhave the philosophers stone which turns
all things to gold. Agraduation could be applied
to the courseof stock or commodity prices, and
successful SJ)ecUlatiofl be l)aSed on the slopeof
the curve at its end point.Though mathematically
ineiegant the most desirableprocedure in a major-
ity of the cases ofgraduation is to graduate not
only the actual data, butextrapolated data which
sometimes maybe extremely crude estimates.
However, we usually know.comct/eing about the
general nature of possible data outsideof the range
for which we have definitefigures. It is highly un-
desirable not to make some use ofsuch knowledge.
earlier method. Perhaps its outstalI(lIng superiorityto the earlier
method is that it is not restricted to any s1)ecIticvalues of k. As in
the earlier method, the graduation cannotbe extended to include
new data without re-calculation.
See Robert Henderson, Fun/icrReinark onC ,iulua!io n, Transac-
tions of The Actuarial Society of America,92ç, pages 52 to 57
inclusive.
1Though there are simple mathematicalmethods of extrapolating
aiiv graduation in such a maniser as to coverthe entire range of the
data, the labor involved in computations, in mI)st cases,large. See
Chapter VIII.
-*
'I'! IISMOOTI I iNOF TI Mi: rs
i TidChlii I C though imay he.
I\Vhi Ics ucti a
dure niav sonwtimes give a poorer fit, iftest((I I))'
Iflatliematical criteria based merckoii the (lata
COVCrC(l by t hc graduat ion, it will tendto givea
distinctly more l'ltI()nal ht. It willnot allow an
mere irregularity of the 1)1l'tiCLIlar sampleto dc-
terminc the j)OSi tion of time graduationin the end
regions.
Professor \'Vhittaker stresses the fact thatin oh-
taming the graduation all observationsare use(1.
Hit position of each (latluTlJ)0i lit afiects the po-
Sltjt)fl of every point On tile snioothicurve. \YhiI
this is theoretically true, itis not of great practical
importance. 'flic alteration in the shape ofthe
smooth curve atany pOint vhicli would he caused
bY a change ina far distant (latuni point is negli-
gible. The effectupon the smooth curve of chaiig-
ing a datumpointis most pronounced inthe
immediate ViCillity of the (latillilpoint. The re-
sults of sucha change are less and lessas the dis-
tance from the (latuinipoint increases. Moreover,
itis highly desirable thatsuch should be the
case. It would be highly undesirable thata change
in the position ofa datum point should seriously
affect the position of (listantparts of the smooth
IOne of the reasons why theHendersonraIIIIations of die ('all
Money data, January 886to .January894, loiik aittie different
from the other graduationsat the ends isthat, in eoi1iutwg the
other graduations, data outsideof tin- lange, ,Jiinu:irv i886 to Janu-
ary 1894, were used. Sec ApjiendjV 111.i'iil:SMOiY1'II f\(IF 1'! M :SERIES
curvc. For example. one of the great disadvantagc
of harmonic analysis is that tile conhguration of
the data in one section niay seriously affect the
shape of the htte(1 curve in a far (listant section.
'I'he \'Vhittaker-Hcndcrson graduation is free from
this objection.
The Whittaker-Henderson graduation does not
necessarily ci iminate seasonal fluctuations. only
iifl be taken SUff1('iefltlVlarge vill the percentage
of seasonal fluctuation renlaining he negligible.
For example. if the seasonal fluctuation were a
i 2-months sine curve and if n eouaied ç, over 9;
per cent of the seasonal fluctuation would 1)e elimi-
nated. However, if n equaled 4. less than 89 per
cent would be eliminated, and if n equaled 3. less
than 69 per cent.
If k equal infinit(ii equal 'zero) the gradua-
tion will coincide with the data no matter how
irregular they may be. For all other values of k,
the graduation 'damj)ens' the data, no matter
how smooth such data may l)eWitll certain para-
holic exceptions.1
If a Whittaker-Henderson graduation be ap-
1)licd to a regular and indefinitely extended sine
series with complete period M the resulting smooth
Strict1speaking a \Vhittaker-Henderson graduation in which n
doeS not equal zeri %Vill not fall exactlon even a niatheniatical
curve unless that curve he a second.degreeparabola or a straight
line.I
'Fl IF S MOO'I'I I ING OF 'i'iMI: SERI ES
curve xviii 1w a sine curVewhose amplitude will be
the amplitude of theoriginal sine curve divided by
4n(fl*
)3(a + 2(a
(2n - i\J
The dampening eftect on sinecurves of various
periods when n3, 4, or ç mahe seen from col-
umns 2.26 and 27 of the table inA1)j)CfldiX VII.
It will be seen fromthat table that if we wish to
eliminate the major portionof aI 2-illOfltlIS sine
seasonal we must make n aslarge as ç. Now if
ii5all cycles whose periO(I isless than '6 months
are veryinadequately ñtted. A comparisonof the
figures for the 4-terIngraduation in column 24
of the same table Wi11 illustratethe point better
than any lengthy discussion.'I'he Whittaker-Hen-
derson graduation is notparticularly vell adapted
to saving the major portionof the amplitude of
both short and long cycles, and atthe same time
eliminating seasonal fluctuations. This maybe an
important consideration in graduatingsuch a series
as monthlyCall \lonev Rates.
Some of the purely practical disadvantagesof
the method have already been suggeste(I.nilecorn-
putation is laboriouswhen accountis taken of
the difficulty of ol)taimng three accurate end points
to begin i/icrst /aif arid the very considerable
See Robert HendersonD,scussiou----'I'ransattion of 'I'he Actu-
arial Society of America, pages 306 and 307.THE SMOOTHING OF' TIME SERIES 99
chance of error at all stages of the work. In using
Meihod A, thc computcr does not have automatic
checks as he 1)roceeds with the work; if lie use
Method B, the advantage of checks is somewhat
offset by the increased amount of computation in
volved. Upon the discovery of an error, the short-
est procedure isusually to do the whole job over
again. The graduation cannot be extended to in-
dude new data without re-calculating alarge pirt
of the scoiul i,a/ftheoretically thewhole ofI/ic
sceomi halt. However, none of the above criticisms
should be interpreted as in anway suggesting
that the Hen(Ierson method is not a reallysuperla-
tive achievement in the theoryand practice of
graduation.
I wish to take this opportunitY tothank Mr.
Robert Henderson for his great kindnessand cour-
tesy in explaininghis method of computation, an-
swering numerous questions, andlending me illus-
trations of actual fittings.

The Smoothing of Time Series

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