A Significance Test for Time Series Analysis

A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp. 401409 Published by: American Statistical Association Stable URL: . Accessed: 19/02/2013 17:28 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@. .

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A SIGNIFICANCE TEST FOR TIME SERIES ANALYSIS*

BY W. ALLEN WALLIS AND GEOFFREY H. MooRE

NO KNOWN

StanfordUniversitaynd RutgersUniversity NationalBureauofEconomicResearch

SIGNIFICANCE TEST is entirelyappropriateto economic

timeseries.One shortcomingof testsin commonuse is that they

ignoresequentialor temporalcharacteristicst;hat is, theytake no ac-

countoforder.The standarderrorofestimate,forexample,implicitly

throwsall residualsinto a singlefrequencydistributionfromwhichto

estimatea variance. Furthermoret,he usual tests cannot be applied

when series are analyzed by movingaverages, free-handcurves,or

similardevices frequentlyresortedto in economicsforwant of more

adequate tools.This paper presentsa testofan oppositekind,one de-

pending solely on order. Its principal advantages are speed and

simplicitya,bsence ofassumptionsabout the formofpopulation,and

freedomfromdependenceupon "mathematicallyefficient"methods,

such as least squares. This test is based on sequences in directionof

movement,that is, upon sequences of like sign in the differencebse-

tween successiveobservations(or some derivedquantities,e.g., resid-

uals froma fittedcurve). In essence, it tests the randomnessof the

distributionofthesesequencesby length.

Each point at which the seriesunder analysis (eitherthe original

or a derivedseries) ceases to declineand starts to rise,or ceases to

riseand startsto decline,is called a turningpoint.A turningpointis a

"peak" ifit is a (relative)maximumor a "trough"ifa (relative)mini-

mum. The interval between consecutiveturningpoints is called a

"phase." A phase is an "expansion"or a "contraction"accordingto

whetheritstartsfroma troughand endsat a peak,orstartsfroma peak

and ends at a trough.For the purposesof the presenttest, the in-

completephase precedingthefirstturningpointand thatfollowingthe

last turningpointare ignored.The lengthor durationofa phase is the

numberofintervals(hereaftereferredto as "years,"thoughtheymay

representany systemof denotingsequence) betweenits initial and

terminalturningpoints.Thus, a seriesofN observationsmay contain

as fewas zerooras manyas N- 2 turningpoints;and a phase maybe as

short as one year (when two consecutiveobservationsare turning

* Presented(in slightlydifferenwtording)beforetheNineteenthAnnualConferenceofthePacific Coast EconomicAssociation,StanfordUniversityD, ecember28, 1940,and based on researchcarried out at the National Bureau ofEconomicResearch,1939-40,underResearchAssociateshipsprovided bytheCarnegieCorporationofNew York.A fulleraccountofthemethodand itsuseswillbe published soon by the National Bureau ofEconomicResearchas the firstof its new seriesof TechnicalPapers.

401

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402

AMERICASNTATISTICAALSSOCIATION

points)oras longas N -3 years (whenonlythe secondand penultimate

observationsare turningpoints). The greaterthe numberof consecutiverises in a series drawn at

randomfroma stable population,the less is the probabilityof an additionalrise; forthe higherany observationmay be the smalleris the chance of drawingone whichexceedsit. To calculate the expectedfrequencydistributionofphase durations,onlyone weak assumptionneed be made about the population fromwhich the observationscome, namely that the probabilityof two consecutiveobservationsbeing identical is infinitesimal-a conditionmet by all continuouspopulations,hencebyvirtuallyall metricdata.

Withoutfurtherpostulates about the formof the population,it is possible to conceivea mathematicaltransformationof it leading to a knownpopulation,but leavingunalteredthe patternof risesand falls of the originalobservations.For example, if each observationis replaced by its rankaccordingto magnitudewithinthe entireseries,the rankshave exactlythe same patternofexpansionsand contractionsas the originalobservations;and theirdistributionis simpleand definite, each integerfrom1 to N havinga relativefrequencyof 1/N. The distributionof phase durationsexpectedamong randomarrangementsof the digits 1 to N is, therefore,comparable with the distribution observedin any set ofdata. A littlemathematicalmanipulationreveals thatin randomarrangementosfN differenittemstheexpectednumber

2(d2+3d+1)(N-d-2)

of completed phases of d iS-

(d+3)!

. The expected

mean durationofa phase 'is

2N

6 7

-

essentially12.

To test the randomnessof a serieswithrespectto phase durations, the firststep is to list in order the signs of the differencebsetween successiveitems.Thus the sequence 0, 2, 1, 5, 7, 9, 8, 7, 9, 8 becomes +,-, +, +, +, -, -, +, -. The signs are, of course, one fewerthan the observations.The second step is to make a frequencydistribution ofthelengthsofrunsin thesigns.There are fourcompletedrunsin the examplejust given (the firstand last beingignoredas incomplete),of lengths,1, 3, 2, and 1. The frequencydistributionthus shows two phases of one year, one of two years,and one of threeyears. In case consecutiveitems are equal but it can be assumed that sufficiently refinedmeasurementwould reveal at least a slightdifference(an assumptionvalid wheneverthe testis applicable), the phase lengthsare

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*A SIGNIFICANCE TEST FOR TIME SERIES ANALYSIS

403

tabulated separatelyforeach possible sequence of signs of differences betweentied items;and the resultantdistributionsare averaged,each beingweightedby the probabilityofsecuringthat distributionifeach differencoebservedas zero is equally likelyto be positiveor negative. Third, the expectedfrequencyforeach lengthof phase is calculated fromthe formulaabove, takingas N the numberof itemsin the sequence beingtested-in thiscase, 10. Next,the observedand expected frequencydistributionsare comparedby computingchi-squarein the usual way fortestinggoodness-of-fit:hat is, by squaringthe differences between actual frequenciesand correspondingtheoreticalfrequencies, dividing these squares by the respective theoreticalfrequencies,and summingthe resultantratios. In nearlyall applications ofthepresenttest,avoidance ofexpectedfrequenciesthatare too small necessitatesrestrictingthe distributionof phase durations to three frequencyclasses,namelyone year'sduration,twoyears'duration,and over two years' duration,the theoreticalfrequenciesforthese classes being5(N-3)/12, ll(N-4)760, and (4N-21)/60, respectively.

The sum ofthe threeratiosofsquared deviationsto expectationsis, then,similarto chi-squarefortwo degreesof freedom,one degreeof freedombeinglost because a singlelinearconstraintis imposedon the theoreticalfrequenciesby takingthevalue ofN fromthe observations. It is advisable, however,to distinguishchi-squareforphase duratiolns by a subscriptp (denotingphase), because it does not quite conformto the Pearsonian distributionfunctionordinarilyassociated with the symbol X2. The phase lengthsin a singleseriesare not entirelyindependentofone another;as a result,verylargeand verysmallvalues of xp2 are a littlemorelikelythanis shownby the x2 distributiona,nd the mean and variance of Xp2 generallyexceed those of X2. We have not determinedthe samplingdistributionof Xp2mathematicallyb, ut have securedempiricallya substitutethat appears satisfactoryI.n the first place, a recursionformulaenabledus to calculatetheexact distribution of xp2 forsmall values ofN. Table I gives the exact probabilityof obtaininga value as largeas or largerthan each possiblevalue of xp2 for values ofN from6 to 12,inclusive.As a secondsteptowarddetermining the samplingdistributiona,n empiricaldistributionof xp2 was secured from700 random series,200 forN = 25, 300 forN = 50, and 200 for N = 75. The threedistributionfsorseparatevalues ofN werenothomogeneous with one anothernor with the exact distributionforN=12; but the differencesamong them were unimportantfor the present purposes,occurringchieflyat thehigherprobabilitiesratherthanat the tail (the importantregionfora test of significance)a,nd representing

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404

AMERICAN STATISTICAL ASSOCIATION-

TABLE I

DISTRIBUTIONS OF xP2: EXACT FOR N=6 TO 12, AND APPROXIMATE FOR LARGER VALUES OF N

(P representstheprobabilitythatan observedXp2willequal or exceedthespecifiedvalue)

N=6

N=9

N=11

N=12

N>12

Xp2

.467 .867 1.194 1.667 2.394 2.867 19.667

P

1.000 .869 .675 .453 .367 .222 .053

N=7

Xp2 .552 .733 .752 .933 1.733 2.152 2.333 3.933 5.606 7.504 8.904

P 1.000

.789 .703 .536 .493 .370 .302 .277 .169 .117 .055

N=8

Xp2 .284 .684 .844 .920 1.320 1.480 2.364 2.680 2.935 3.000 4.375 4.455 4.935 5.000 5.819 6.455

P 1.000

.843 .665 .590 .560 .506 .495 .471 .392 .299 .293 .235 .194 .133 .064 .033

Xp2

.358 1.158 1.267 1.630 2.067 2.430 2.758 3.158 3.267 3.667 4.030 4.067 4.758 5.667 6.067 7.485 15.666

P

1.000 .798 .631 .605 .489 .452 .381 .374 .321 .215 .164 .144 .110 .078 .064 .020 .005

N=10

Xp2 .328 .614 .728 1.055 1.341 1.419 1.585 1.705 1.772 1.814 1.819 2.313 2.577 2.676 2.743 2.863 2.905 2.977 3.242 3.834 3.970 4.333 4.400 4.676 4.858 5.128 5.491 6.515 7.133 11.308 12.965

P 1.000

.941 .917 .813 .693 .606 .601 .594 .592 .526 .419 .407 .374 .327 .327 .274 .242 .220 .181 .179 .165 .158 .158 .139 .107 .072 .059 .054 .042 .014 .006

Xp2

.479 .579 .817 .917 .979 1.088 1.279 1.317 1.588 1.700 1.800 2.079 2.200 2.309 2.409 2.417 2.500 2.579 2.688 2.809 3.026 3.109 3.213 3.300 3.779 3.800 3.909 4.117 4.313 4.388 4.726 5.000 5.609 5.700 6.013 8.200 8.635 9.468 9.735 10.214 11.435

P

1.000 .980 .934 .844 .730 .723 .655 .576 .537 .473 .472 .468 .467 .466 .440 .403 .392 .384 .304 .274 .261 .230 .201 .147 .147 .147 .133 .128 .126 .099 .091 .077 .077 .076 .055 .050 .032 .022 .018 .009 .004

Xp2

0.615 0.661 0.748 0.794 0.837 0.971 1.015 1.061 1.415 1.461 1.637 1.683 1.933 1.948 2.067 2.156 2.203 2.289 2.333 2.556 2.615 2.661 2.733 2.837 2.870 2.883 2.956 3.267 3.415 3.489 3.933 4.070 4.156 4.348 4.394 4.571 4.616 4.733 5.667 5.803 5.889 6.025 6.733

6.842 6.956 7.504 7.622 8.576 8.822 9.237 9.267 10.556 19.667

P

1.000 .984 .896 .891 .850 .786 .720 .685 .585 .583 .569 .533 .487 .486 .428 .427 .407 .344 .333 .331 .303 .303 .300 .300 .287 .246 .216 .211 .207 .149 .127 .127 .114 .113 .113 .112 .109 .101 .092 .092 .090 .090 .085 .072 .060 .050 .041 .029 .026 .019 .014 .003 .000

Xp2

6.448 5.50

6.674 5.75

5.927 6.00

6.163 6.25 6.50

6.541 6.75

6.898 7.00 7.25

7.401 7.50 7.75 8.00 8.009 8.25 8.50 8.75

8.886 9.00 9.25 9.50 9.75 10.00 10.25 10.312 10.50 10.75 11.00 11.25 11.50

11.756 12.00 13.00 14.00

15.085

P

.10 .098

.09 .087 .08 .077 .07 .069 .061 .06 .054 .05 .048 .043

.04 .038 .034 .030 .03 .027 .024 .021

.02 .019 .017 .015 .013 .012 .010 .01 .009 .008 .007 .006 .006

.005 .004 .003 .002

.001

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