THREE IMPORTANT CONCEPTS IN TIME SERIES ANALYSIS: STATIONARITY, CROSSING RATES, AND THE WOLD REPRESENTATION THEOREM

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1 THR IMPORTANT CONCPTS IN TIM SRIS ANALYSIS: STATIONARITY, CROSSING RATS, AND TH WOLD RPRSNTATION THORM Prof. Thoms B. Fomb Deprmen of conomics Souhern Mehodis Universi June 8 I. Definiion of Covrince Sionri A ver imporn concep in ime series nlsis is he concep of he sionri of ime series. Consider he ime series process,,, L, T. This process is clled covrince sionr or simpl sionr if he following condiions hold: i μ for ll ii μ Vr for ll iii μ γ for ll nd given vlue of μ Cov, {,, L} Condiion i ses h he ime series hs consn men irrespecive of he ime period which i is observed. Condiion ii ses h hs consn vrince irrespecive of he ime period which i is observed. Condiion iii ses h he covrince beween observions nd is onl funcion of nd no. The covrince beween wo observions h re -periods pr is consn no mer when he -period pr observions re observed. For exmple consider he so-clled AR model where < is ssumed nd he error erm hs he so-clled whie noise properies i for ll ii for ll 3

2 iii s for ll s. Thus he error erm is whie noise becuse i hs consn zero men, consn vrince,, nd he errors re uncorreled wih ech oher s long s heir ime subscrips do no mch. A Mone Crlo relizion of his AR process for., 5.,., nd T is reproduced below in Figure. Figure For he AR process of we show in he Appendix h, Vr, nd Cov,. For he bove Mone Crlo d his implies, Vr, nd Cov, Visull, one cn see he consnc of he men nd possibl he consnc of he vrince bu i is difficul o visull guge he consnc of he covrince s prescribed b model. Neverheless, he d does seem o hve genle rolling nure h seems o repe iself s would be implied b he posiive covrinces of he series.

3 The bove ime series is o be compred o grph where for o 5 he model is he AR model.8 wheres for 5 o he AR model is.8. See Figure below. The mens of boh hlf processes re equl o zero nd heir vrinces re boh / while heir covrinces re differen in sign bu equl in mgniude for odd vlues of bu equl o ech oher for even vlues of. For exmple, for he firs pr of he d cov, while for he second hlf of he d, cov, Th is, for he firs hlf of he d, he covrince of he observions one period pr is posiive. When ls period s observion is bove he men, his period s observion ends lso o be bove he men nd when ls period s observion is below he men, his period s observion ends o be below he men. In conrs, for he second hlf of he d, he covrince of he observions one period pr is negive. When ls period s observion is bove he men, his period s observion ends lso o be below he men nd when ls period s observion is below he men, his period s observion ends o be bove he men. This covrince-schizophrenic behvior is refleced in he slow-rolling nure of he firs hlf of he d nd he sw-ooh pern of he d in he second hlf. Obviousl his d is no covrince sionr lhough he d hs consn men nd consn vrince. I us doesn hve consn covrince funcion. Figure Anoher cse h urns up someimes is he Chnging Men cse s presened in Figure 3 below. 3

4 Figure 3 In his cse he d mus lso be spli ino wo groups wih he groups being modeled seprel. One of he subsnil benefis of hving sionr ime series is h wih one relizion of he ime series which is usull ll we ever hve!, one cn consisenl esime he populion men, vrince, nd covrinces of he ime series process using he following smple sisics: T T smple men 4 s T T smple vrince 5 c T T smple covrince 6 r T c,,, L. 7 T s smple uocorrelion. 4

5 The ls smple sisic, r, is he smple sisic for esiming he populion uocorrelion lg, nmel, ρ Cov, / Vr. The displ of ll of he uocorrelions ρ on he -xis wih he lg on he x-xis of grph consiues wh is clled he populion uocorrelion funcion. The displ of ll of he smple uocorrelions on he -xis wih he lg on he x-xis of grph consiues wh is r clled he smple uocorrelion funcion. B he erm consisenc we men h, s he number of observions goes o infini, he following probbili semens hold: Pr ob μ < ε s T 8 Pr ob s < δ s T 9 Pr ob < τ s T c γ Pr ob < κ s T r ρ for rbirril smll vlues of ε >, δ >, τ > nd κ >. Thus, in reli he condiions 8 men h s he smple size on sionr ime series goes o infini, he exc vlue of he populion prmeers, μ,, γ nd ρ will become known wih probbili one. Bu for smple sizes h re lrge he probbili will be high h he corresponding smple sisics will be in smll neighborhood of he cul populion vlues. Deermining hese populion prmeers wih consisenc is imporn becuse he pl crucil pr in providing forecsing formuls h ield ccure forecss. II. Trnsforming Non-Sionr Time Series o Sionr Form Time series h hve non-consn mens, non-consn vrinces, nd nonconsn covrinces re exceeding difficul o forecs. However, s we will see, nonsionr ime series cn ofen be rnsformed o sionr ime series, s b king he nurl logrihm of he d nd/or differencing he d. In he bove non-consn covrince cse represened in Figure bove, ll we hve o do is spli he d ino wo prs, pr I being observions 5 nd pr II being observions 5 nd hen reing hem seprel s wo disinc sionr ime series. The firs pr of course would be of ineres o he economic or business hisorin while he second pr would be more relevn for he forecser. The following hree subsecions will discuss hree sionri rnsformions h ofen used o rnsform non-sionr ime series ino sionr ones. The firs wo rnsformions re ofen recommended in he Box- Jenkins pproch o building ime series forecsing models. 5

6 Differencing s Soluion A ver ineresing specil cse of he firs order uoregressive AR model of equion is when he firs-order uoregressive coefficien is equl one. Consider he Rndom Wlk wihou drif model where we le Figure 4.. A Mone Crlo relizion of his process is ploed below in Figure 4 B repeed subsiuion i is es o show h ields L 3 where nd is fixed zero. I follows h L L, 4 Vr Vr L Vr Vr L Vr, 5 he nex-o-ls equli following from he independence of he ' s. Also 6

7 Cov, L L L 6 he nex-o-ls equli following from he independence of he h he uocorrelion lg is given b ' s. Finll i follows ρ Corr, Cov, / Vr /. 7 Obviousl, he Rndom Wlk wihou drif process is non-sionr. Alhough he process hs consn men of zero, is vrinces is ever-chnging from one ime period o he nex nd, in fc, i pproches infini s he smple size goes o infini. See 5. Moreover, he covrince lg is funcion of ime s well. See 6. Also for fixed lg, he correlion beween observions h re periods pr pproches one s he smple size goes o infini. This is rher peculir behvior bu h is wh is implied b his rndom wlk uni roo cse. For fixed smple size i should be noed h he uocorrelion funcion will be ver slowl dmping sring for nd onl slowl pproching zero he re of / s goes o infini. In fc, one migh expec he smple uocorrelion funcion o behve similrl when he uoregressive process hs uni roo. I will end o be ver slowl dmping s well. Forunel, he problem of non-sionri cn be remedied simpl b * differencing he d nd insed focusing on he ime series Δ. Th is becuse, from, we hve Δ 8 nd, herefore, Δ 9 Vr Δ Vr nd Cov, for ll nd. We cn see h he rnsformed differenced series, Δ, is sionr becuse i hs consn men, consn vrince, nd he covrince funcion is no funcion of. The 7

8 differenced Rndom Wlk of he previous plo is presened below in Figure 5. Noice is roughl conn men, consn vrince, nd consn covrince. Figure 5 Trnsforming xponenil Growh D o Percenge Chnges Anoher frequenl occurring non-sionr ime series in business nd economic d is depiced below. See Figure 5 below. This is clled exponenil growh d such h king he nurl logrihm of he d nd hen differencing he logrihmicll rnsformed d produces sionr ime series. See he below wo grphs, where he originl d is ploed in Figure 6 nd in Figure 7 we hve ploed he rnsformed series Δ log. This ler rnsformion cn be hough of percenge chnge rnsformion since, for smll percenge chnges in, Δ log. 8

9 Figure 6 Figure 7 Ler on we will lern how o use he SAS Mcro %loges o disinguish beween he cses where we cn simpl difference he d versus siuions where we need o firs ke he logrihms of he d nd hen difference he logrihms of he d herefer. In fc ou cn see h if one is creful in he inspecion of he proposed rnsformed series one cn deec wheher n pproprie rnsformion o sionri hs been pplied or no. See Figure 8 below where he simple differences Δ of he exponenil growh d hve been ploed. 9

10 Figure 8 Obviousl, in his cse he rnsformed d re no sionr s he d rends o drif up over ime nd consn men nd consn vrince re no minined in he d. Deerminisic Trend versus Sochsic Trend Rndom Wlk d cn lso hve drif in i. Consider he d ploed in Figure 9. I hs been genered b he process 3 where nd 4. If one were o drw srigh line hrough his rending d one would noice h he crossing re of h line would be low hence hining of he uni roo in he d. Agin, he sionri rnsformion of his d is he firs differencing operion Δ since Δ represens consn men, consn vrince, nd consn covrince process.

11 Figure 9 See Figure below for he resul of ppling he difference operion o he d in figure 9. The d hs been rendered sionr b king firs differences. Figure However, here exis ime series in business nd economics h migh be beer chrcerized s following wh is clled deerminisic ime rend s in

12 β 4 β where is he ime index,, L,T ploed below in Figure below.. Such rend sionr TS ime series is Figure Here ou would see mn crossing of he rend should ou pu rend line hrough he bove d. In ler sud we will see h Crossing-re nd Dicke-Fuller uni roo ess cn be used o disinguish beween hese cses. An imporn poin o noe here is h in he sionr AR of Figure, he d revers o he men quie ofen wheres he Rndom Wlk d of Figure 3 rrel does. Likewise he Rndom Wlk wih drif d of Figure 9 rrel revers o rend line one would plce hrough he d wheres he deerminisic rend d revers ofen o he rend line. To he exen h ime series d revers frequenl o men or rend we cll such d men-revering nd hving deerminisic rend men. In conrs, o he exen h ime series d revers infrequenl o rend we cll such d non-men revering nd hving sochsic rend. To moive he erm sochsic rend, consider he following rndom-wlk-like model clled he Rndom Wlk srucurl model b Hrve 989. μ μ ε μ 5 We will sud more bou such models when we sud Unobserved Componens models ler. Here we specif h he men of he series μ is sochsic nd, in fc, follows rndom wlk wihou drif. We cn rewrie his model s

13 ν 6 h is lmos like he Rndom Wlk wihou drif model excep now ν ε 7 is so-clled MA error erm insed of being whie noise. However, he poin is mde. Rndom Wlk d behve much like d h hs sochsic rndom wlk men in i, hence he erm sochsic men s compred o deerminisic men. The quesion migh be, Wh mke such big del over wheher ime series hs sochsic rend in i which cn be reed b differencing or deerminisic rend h cn be modeled s consn men or deerminisic ime rend? The shor nswer is h when forecsing wih models h hve sochsic rends in hem he predicion confidence inervls become ever wider he furher ou ino he fuure one predics. In conrs, in he deerminisic men or rend models he forecs confidence inervls pproch limi equl o he vrince of he d round he men or rend. Thus he concepion of rend h one chooses o dop is n imporn decision o mke nd cn hve subsnil effec on how one conves he uncerin of predicing ime series in he fuure. We will see his poin more clerl ler when we seprel exmine he confidence limis of he deerminisic rend model nd he Box-Jenkins model. III. The Wold Decomposiion Represenion Theorem In 954, Hermn Wold proved ver imporn heorem concerning sionr ime series d. Here is forml semen of Wold s heorem wihou proof: Wold Decomposiion Represenion Theorem: An sionr process cn be uniquel represened s he sum of wo muull uncorreled processes D nd where D is linerl deerminisic nd Z is MA process. Th is, Z where D Z 8 Z ψ ψ ψ L 9 wih ψ nd he psi weighs ψ,, L being bsoluel summble in h ψ ψ < nd he ' s re uncorreled errors h hve zero men nd consn vrince. ******* 3

14 As we shll see, D in he Box-Jenkins models is equl o he men μ of he sionr form of he ime series. However, in he Wold Theorem cn be nhing rnging from deerminisic ime rend D β β, o deerminisic ccle like D cos ω θ 3 where is he mpliude of he ccle, θ is he phse, nd he period of he ccle p such h p π / ω nd deerminisic sesonl s in D D δ L S δ S δ sss 3 where S k is seson dumm vrible king he vlue of during seson k nd zero oherwise wih s number of sesons during he er. The ver useful resul h he Wold Represenion provides us is h ll sionr Box-Jenkins models of sionr ime series cn be wrien s MA process s in μ ψ ψ L. 3 This form will be ver helpful o use, especill s i reles o deriving he properies of he models nd heir forecs funcions nd confidence inervls. 4

15 APPNDIX Consider he AR model where < is ssumed nd he error erm hs he so-clled whie noise properies i for ll ii for ll 3 iii s for ll s. We re o show h A B Vr C Cov, D Corr, Resul A: h B bckwrd subsiuion we cn confirm he Wold Represenion heorem resul L L L. A. In geing A. we used he resul h if <, he geomeric series L converges o /. I follows from A. h 5

16 L L s desired. Resuls B: From A. nd he independence of he i follows h s ' L Vr Vr L 4 Vr Vr Vr Vr L 4 s desired. Resul C: Using A. we hve, μ μ Cov L L L L L 6

17 s desired. Resul D: Vr Cov Corr,, s desired. nd of Appendix. 7