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StochasticandDeterministicTrendModels

Inthissection,weconsidermodelsofnonstationarytimeseries,i.e.,series{yt}whosefirstandsecondmoments(meansandcovariances)arefunctionsoftime.Theseincludeallserieswithatrend.Trends,whichcanbeeitherdeterministic(likeatimetrend)orstochastic,willobviouslyproducenonstationarities.Classicalestimationmethods,however,arevalidforstationaryseries.Therefore,trendsmustberemovedfromnonstationaryseries(thusmakingthemstationary)beforeapplyingthemethodsofBoxandJenkins.Removal,however,dependsonidentifyingthetypeoftrendfirst.Generally,stationarityisachievedthroughdifferencingtheseries{D.yt}–differencestationary–orthroughremovalofadeterministictrendbyfirstestimatingthattrendinaseparateregression–trendstationarity.Wewilllookatbothmethods.First,however,let’sconsiderwhatnonstationaritymeans.

RandomWalk

SomeStatacodewasprovidedinthenotesondifferenceequationsandtheirsolutionsthatsimulatedrandomwalks.Youshouldrunthatprogramseveraltimestogetafeelforwhatarandomwalklookslike.Basically,arandomwalkisatimeserieswhosechangeisrandom.Specifically,thechangeiswhitenoise,viz.,

yt=yt-1+tinlevelsor

yt=twheret~N(0,1).

Solvingtheequationinlevelsbackwards(therearetperiods)

yt=y0+iwherethesummationisfromi=1,…,tforstartingvaluey0.

Whatarethestatisticalpropertiesofarandomwalk?Theunconditionalmeanistheexpectedvalue:

Eyt=y0+E(i)=y0.Theunconditionalvarianceis

E(i2)=t2whichisafunctionoftime.Let’snowconstructthe“forecastfunction”forarandomwalk.

Etyt+1=Et[yt+t+1]=ytThisistheconditionalmean.Noticealsothatthes-periodaheadforecastisthesame:

Etyt+s=Et[yt+s-1+t+s]=yt.(Substitutesuccessivelyforyt+s-1).Basically,theconditionalforecast(conditionaloninformationattimet)isthelastrealization.Thisshouldmakesensehoweverbecausechangesinyarewhitenoise.

Howaboutthevarianceofyt+s?(Thisisthesameasthevarianceforyt-s).Weknow

Var(yt)=Var(t+t-1+…+1)=t2.Similarly,

Var(yt-s)=Var(t-s+t-s-1+…+1)=(t-s)2.Alsoafunctionoftime.Ingeneral,wecanconcludethenthatthestandarddeviation(usedtoconstructconfidenceintervalsforforecasts)is2t.

Covariances?Theunconditionalmeanisy0.Thus,

E[(yt-y0)(yt-s-y0)]=(t-s)andsincey0isaconstant,

E[(t+t-1+…+1)(t-s+t-s-1+…+1)]=(t-s)2.

Thecorrelationcoefficientisthisnumberdividedbytheproductofthestandarddeviations:

s=(t-s)2/(t2(t-s)2)?=[(t-s)/t]?.Alsoafunctionoftime.Forlarget,onecaneasilyobservethepatternovers.Thepointhereisthatcorrelationsdonotgotozeroastheywouldinastationaryseries.Thereasonisthattheimpactofashockonfuturevaluesofyispermanent.Youcanreadilyseethatbylookingatthedifferencesolution,ytisreallyjusttheaccumulatedsumofpastshocks.Thus,ihaspermanenteffectsandiconstituteapermanentrandomchangeintheconditionalmean.

Thusythasastochastictrend.

RandomWalkplusDrift

yt=yt-1+a0+twherea0istheconstant“drift”.Solvingthedifferenceequation–

yt=y0+a0t+iwhere,again,thesummationisovert.Thetermsa0t+iarebothnonstationary.Now,wehaveadeterministicplusastochastictrend.Bytheway,

yt-yt-1=yt=a0+tisstationary.

TheunconditionalexpectationisEt(yt+s)=y0+a0(t+s).Theforecastfunction(whichisconditionalonpastyt)is:

Andthishasexpectedvalueequaltoyt+a0s.Youshouldconvinceyourselfthatthisisindeedtrue.

RandomWalkplusNoise

yt=t+t t~N(0,2)andE(tt-s)=0.

t=t-1+t.

Soytisarandomwalkt-1+tplusnoiset.Thisisaseriesthathasastochastictrendplusanoisecomponent.Therandomwalkcomponentisafirstorderdifferenceequationthatcanbesolvedas:

t=0+iwhere,again,thesummationisovert.Therefore:

yt=0+i+t.

Now,att0,solvey0=0+0implyingthat0=y0-0.Substituting,

yt=(y0-0)+i+t.

Theunconditionalmeanis

E(yt)=E(yt+s)=y0-0whichisconstant.Shocks,however,haveapermanenteffect.Noticethatnoiseistransitory,i.e.,ithasatemporaryeffectonytbutnotonyt+s.Furthermore,Var(yt+s)=t2+2which,naturally,includesthenoisecomponent.

Thenoisecomponentwillreducethecorrelationcoefficientbetweenytandyt-srelativetotherandomwalkimplyingthatthecorrelogramwillhavesomewhatfasterdecay(dependingonthenoise).

Howabouttheforecastfunction?

Thishasexpectedvalue(conditionalmean)yt-t.Noticethetransitorynatureofthenoise.Moreover,differencingyields

yt=yt-1+t+t-t-1.(thislookslikeanARMA(1,1)doesn’tit?)

NoiseandDrift

Wecancombinenoiseanddrifteasilyenough.

t=t-1+a0+t.

t=0+a0t+iwhere,again,thesummationisovert.Therefore:

yt=0+a0t+{i+t}.

Deterministictrend Stochastictrend

Bothtrendsarepermanent.TheStochastictrendincludesatransitorynoisecomponentaswell.

Imposingtheinitialconditiony0=0+0again,

yt=(y0-0)+a0t+i+t.Summationovert.

yt+s=(yt-t)+a0s+i+t.Summationovers.

LocalLinearTrendModel

Finally,weconsiderageneralformforwhichalloftheabovearespecialcases.

yt=t+t

t=t-1+at+t randomwalkplusdrift(thetrend)

at=at-1+t randomwalk

If{at}=0,thenwehavetherandomwalkplusnoise.Witht=0foralltime,thenit’sjustarandomwalk.If,ontheotherhand,Var()=0,thenitmustbethattheatareequalforalltime.Thus,ifatisnonzero,thetrendisarandomwalkplusdriftandytisanoisyrandomwalkplusdrift.

Wecansolvethesedifferenceequationstogettheparticularsolution:

at=a0+i.

t=t-1+a0+i+t

=0+i+t(a0+1)+(t-1)2+(t-2)3+…+1.Sincey0=0+0,then

yt=y0+(t-0)+i+t(a0+1)+(t-1)2+(t-2)3+…+1

irregular stochas tic notentirelydeterministic

term trend

Youcanupdatethistosolvefortheforecastfunction:

Etyt+s=(yt-t)+s(a0+1+2+…+t).

Thefirsttermistransitoryandthesecondtermisthetrend.

Removingthetrend

Theforegoingwasmeanttoillustratesomeofthepropertiesofnonstationaryseries.Inordertoestimatethese,wemustmakethemstationaryandthatisachievedbyremovingthetrend.Todothiscorrectly,wemustfirstknowwhetherthetrendisdeterministic(trendstationaryafterthetrendisremoved)orstochastic(differencestationaryafterthetrendisremoved).Thatisn’teasy.NelsonandPlosserwroteaninterestingseminalpaperin1992onthistopicandarguedthatmostmacroeconomicseriesaredifferencestationary(meaningtheyhavestochastictrendsandremovalusingtheestimatedtimetrendwouldhaveresultedinseriousandspuriousspecification).

So,whatdowemeanbytrendstationarity.Ifaserieshasadeterministictimetrend,thenwesimplyregressytonaninterceptandatimetrend(t=1,…,T)andsavetheresiduals.Theresidualsarethedetrendedseries.But,iftheseriesythasastochasticinsteadofdeterministictrendthenwedon’tnecessarilygetastationaryseries.Considertherandomwalkagain.Theresidualsfromthistimetrendregressionassumethatytgrowsataconstantrate.Itdoesnot.Rather,itgrowsatastochasticrate.Thus,theincorrectlydetrendedseriesmaydisplayspuriousbehavior,i.e.,itmaystillbenonstationary.Canyouprovideasimulatedexampleofthis?

Ifthetrendisstochastic,thendifferencing