First consider the spectral density This is the function such that ef 1 ir As long as the autocovariances xx do not go to zero too slowly we can invert this to give hhh exp ir so that the spectral density is the Fourier transform of the autocovar ID: 22555
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THEBISPECTRUMANDTESTSFORNONLINEARITYTheBispectrumhereasthespectrumprovidesadecompositionofthesecondmomentintocontributionsfromindividualcomponentfrequencies,thebispectrumprovidesasimilardecompositionofthethiroment.First,considerthespectraldensity.Thisisthefunction)suchthat xxxassumednottodependon.ThebispectrumisdeÞnedbeef1212Thiscanbeinverted(undersuitableconditions)togivsothatthebispectrumisthe(twodimensional)Fouriertransformof).Usingthespectralrepresenation,wehavehavexxxEedZedZedZ1212eeeEEdZ()dZ()dZ()]incethelefthandsidedoesnotdependon,neitherdoestherighthandside,sosodZ()dZ()dZ()]012u12nless0,andthereforethereforedZ((1212Comparisonwith(2)revealsthatthatdZ(()],12121212212sothatisthejointthirdmomentofthecomponentsoffromthreefrequenciehichsumtozero,twoofwhichare.Sincefrom(2)wehave12122x]f(,)dd,swecanalsothinkofasthecontributiontotox]fromfrequencies().If{1212Gaussian,then0forall,andthebispectrumwillbeidenticallyequaltozero. SubbaRaosTestforNonlinearityandNon-Gaussianitlthoughmanytimeseriesobservedinpracticemaybenonlinear,itwouldbeunrealistictoassumethatthedataweregeneratedexactlybyoneoftheknownparametricmodels.ThesemodelBilinear,TAR,GARCH,etc.)aremerelyapproximationswhichwillsometimesbeuseful.Thequestionofwhethertheseriesisnonlinear,however,seemsmorebasic.Byconsideringthebispectrum,anerhapshigherorderspectra,itispossibletoconstructtestsfornonlinearitywithoutspecifyinganyÞnite-parametermodels,eitherunderthenullhypothesisorthealternativehypothesis.Thenulypothesis(oflinearity)isthat,wherethe{}are.Itcanbeshown(homework)that}islinear,thenthequantityhhhhhhhhhhhhhhhhhhhhhhh,wherewheree].Furthermore,if{}isGaussian,thenitihhhwillbeconstant,inearwith0,so)(and))willbeidenticallyequaltozero.1212heÞrststageofSubbaRaoÕstestexamineswhethertheseriesisGaussian,bytestingwhethertheestimatedbispectrumoveragridoffrequenciesissigniÞcantlydifferentfromzero.Ifitis(asmeasredbyateststatisticrelatedtoHotellingÕs),thentheseriesisdeclaredtobenonGaussian,andwecanaskwhethertheprocessislinearornot.If{}islinear,then)willbeconstant.SothecondstageofSubbaRaoÕstestexamineswhetherthesampleversionof)issigniÞcantlynon-constant,overagridoffrequencies.Ifitis,wedeclaretheseriestobenonlinearTheadvantageofSubbaRaoÕstestisthatitisnonparametric.Adrawbackofthetestisthat,dependingonexactlyhowweestimatethespectrumandbispectrum,ourconclusionsaboutGaussianityndlinearityofagivenobservedseriesmaychange.Amoreseriousproblemisthatnotallnonlinearprocesseswillhaveanonzeroandnonconstantbispectrum.Forexample,thebispectrumofanARCHrocessisidenticallyzero.ThusSubbaRaoÕstestwillhavenochanceofdetectingnonlinearity(orevennonGaussianity)inthiscase.Ontheotherhand,wearenotlikelytoeverseeatimeserieswhichsexactlyARCH.Furthermore,itispossibletogeneralizeSubbaRaoÕstesttohigherorderspectra.For xample,thethird-orderspectrum("trispectrum")ofanARCHprocessisnonzero(seeMilhoj),whilethethird-orderspectrumofaGaussianprocessisidenticallyzero.SubbaRaoÕstestrevealsthattwoofthedatasetswhichtraditionallyhavebeenanalyzedinthetimeseriesliteratureusingARMAmodels(thesunspotseriesandtheCanadianLynxseries)actuallyeemtobenonlinear.