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. 401- 409 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/2279616 . Accessed: 19/02/2013 17:28 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . 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 [email protected]. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association. http://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions A SIGNIFICANCE TEST FOR TIME SERIES ANALYSIS* BY W. ALLEN WALLIS AND GEOFFREY H. MooRE StanfordUniversity and RutgersUniversity National Bureau of EconomicResearch NO KNOWN SIGNIFICANCE TEST is entirelyappropriate to economic time series. One shortcomingof tests in commonuse is that they ignoresequential or temporalcharacteristics; that is, they take no ac- count of order.The standard errorof estimate,for example, implicitly throwsall residualsinto a singlefrequency distribution from which to estimate a variance. Furthermore,the usual tests cannot be applied when series are analyzed by moving averages, free-handcurves, or similardevices frequentlyresorted to in economicsfor want of more adequate tools. This paper presentsa test of an oppositekind, one de- pending solely on order. Its principal advantages are speed and simplicity,absence of assumptionsabout the formof population,and freedomfrom dependence upon "mathematicallyefficient" methods, such as least squares. This test is based on sequences in directionof movement,that is, upon sequences of like sign in the differencesbe- tween successive observations(or some derivedquantities, e.g., resid- uals froma fittedcurve). In essence, it tests the randomnessof the distributionof these sequencesby length. Each point at which the series under analysis (either the original or a derived series) ceases to decline and starts to rise, or ceases to rise and startsto decline,is called a turningpoint. A turningpoint is a "peak" if it is a (relative) maximumor a "trough"if a (relative)mini- mum. The interval between consecutive turningpoints is called a "phase." A phase is an "expansion" or a "contraction"according to whetherit startsfrom a troughand ends at a peak, or startsfrom a peak and ends at a trough.For the purposes of the presenttest, the in- completephase precedingthe firstturning point and that followingthe last turningpoint are ignored.The lengthor durationof a phase is the numberof intervals(hereafter referred to as "years,"though they may representany system of denotingsequence) between its initial and terminalturning points. Thus, a seriesof N observationsmay contain as fewas zeroor as manyas N- 2 turningpoints; and a phase may be as short as one year (when two consecutive observationsare turning * Presented(in slightlydifferent wording) before the NineteenthAnnual Conference of the Pacific Coast Economic Association,Stanford University, December 28, 1940, and based on researchcarried out at the National Bureau of Economic Research,1939-40, under ResearchAssociateships provided by the CarnegieCorporation of New York. A fulleraccount of the methodand its uses willbe published soon by the National Bureau of Economic Researchas the firstof its new seriesof TechnicalPapers. 401 This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions 402 AMERICANSTATISTICAL ASSOCIATION points)or as longas N -3 years (when only the second and penultimate observationsare turningpoints). The greater the number of consecutive rises in a series drawn at random froma stable population,the less is the probabilityof an ad- ditional rise; forthe higherany observationmay be the smalleris the chance of drawingone whichexceeds it. To calculate the expectedfre- quency distributionof phase durations,only one weak assumptionneed be made about the population from which the observations come, namely that the probabilityof two consecutive observations being identical is infinitesimal-a conditionmet by all continuous popula- tions,hence by virtuallyall metricdata. Without furtherpostulates about the formof the population,it is possible to conceive a mathematicaltransformation of it leading to a knownpopulation, but leaving unalteredthe patternof rises and falls of the original observations.For example, if each observationis re- placed by its rank accordingto magnitudewithin the entireseries, the ranks have exactlythe same patternof expansionsand contractionsas the originalobservations; and theirdistribution is simple and definite, each integerfrom 1 to N having a relativefrequency of 1/N. The dis- tributionof phase durationsexpected among random arrangementsof the digits 1 to N is, therefore,comparable with the distribution observedin any set of data. A littlemathematical manipulation reveals that in randomarrangements of N differentitems the expectednumber 2(d2+3d+1)(N-d-2) of completed phases of d iS- (d+3)! . The expected - 6 mean durationof a phase 'is 2N 7 - essentially1 2. To test the randomnessof a series with respectto phase durations, the firststep is to list in order the signs of the differencesbetween successiveitems. Thus the sequence 0, 2, 1, 5, 7, 9, 8, 7, 9, 8 becomes +,-, +, +, +, -, -, +, -. The signs are, of course, one fewer than the observations.The second step is to make a frequencydistribution of the lengthsof runsin the signs.There are fourcompleted runs in the example just given (the firstand last being ignoredas incomplete),of lengths, 1, 3, 2, and 1. The frequencydistribution thus shows two phases of one year, one of two years, and one of three years. In case consecutive items are equal but it can be assumed that sufficiently refinedmeasurement would reveal at least a slight difference(an as- sumptionvalid wheneverthe test is applicable), the phase lengthsare This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions *A SIGNIFICANCE TEST FOR TIME SERIES ANALYSIS 403 tabulated separatelyfor each possible sequence of signs of differences betweentied items; and the resultantdistributions are averaged, each being weightedby the probabilityof securingthat distributionif each differenceobserved as zero is equally likelyto be positive or negative. Third, the expected frequencyfor each length of phase is calculated fromthe formulaabove, taking as N the number of items in the se- quence being tested-in this case, 10. Next, the observedand expected frequencydistributions are compared by computingchi-square in the usual way for testinggoodness-of-fit: that is, by squaring the differ- ences between actual frequenciesand correspondingtheoretical fre- quencies, dividing these squares by the respective theoretical fre- quencies, and summingthe resultantratios. In nearlyall applications of the presenttest, avoidance of expectedfrequencies that are too small necessitates restrictingthe distributionof phase durations to three frequencyclasses, namely one year's duration,two years' duration,and over two years' duration,the theoreticalfrequencies for these classes being 5(N-3)/12, ll(N-4)760, and (4N-21)/60, respectively. The sum of the threeratios of squared deviationsto expectationsis, then, similar to chi-squarefor two degrees of freedom,one degree of freedombeing lost because a single linear constraintis imposed on the theoreticalfrequencies by takingthe value of N fromthe observations. It is advisable, however,to distinguishchi-square for phase duratiolns by a subscriptp (denotingphase), because it does not quite conformto the Pearsonian distributionfunction ordinarilyassociated with the symbol X2. The phase lengthsin a single series are not entirelyinde- pendentof one another;as a result,very large and very small values of xp2 are a littlemore likely than is shownby the x2 distribution,and the mean and variance of Xp2 generallyexceed those of X2. We have not determinedthe samplingdistribution of Xp2mathematically, but have secured empiricallya substitutethat appears satisfactory.In the first place, a recursionformula enabled us to calculate the exact distribution of xp2 forsmall values of N. Table I gives the exact probabilityof ob- taininga value as large as or largerthan each possible value of xp2 for values ofN from6 to 12, inclusive.As a second step towarddetermining the samplingdistribution, an empiricaldistribution of xp2 was secured from700 random series, 200 for N = 25, 300 for N = 50, and 200 for N = 75. The threedistributions for separate values of N werenot homo- geneous with one another nor with the exact distributionfor N=12; but the differencesamong them were unimportantfor the present purposes,occurring chiefly at the higherprobabilities rather than at the tail (the importantregion for a test of significance),and representing This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions 404 AMERICAN STATISTICAL ASSOCIATION- TABLE I DISTRIBUTIONS OF xP2: EXACT