time series components.pdf

8/22/2019 time series components.pdf

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TimeSeriesComponents

Recallthattheoptimalpointforecastofaseriesyt+h

isitsconditionalmean

Itisusefultodecomposethismeaninto

components

Tt=Trend St=Seasonal

Ct=Cycle

tttt CST ++=

( )thtt y = + |E


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Components

Trend Verylongterm(decades)

Smooth

Seasonal Patternswhichrepeatannually

Maybeconstantorvariable

Cycle Businesscycle

Correlationover27years Itisusefultoconsiderthecomponentsseparately

WestartwiththeTrend


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TrendForecasting

Apuretrendmodelhasnoseasonalorcycle

Inapuretrendmodel,theoptimalpointforecastfor yt+h ist=Tt.

Anactualforecastisanestimateof Tt.

tt T=


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ModelingTrend

Mosttrendmodelsareverysimple

Simplestpossibletrendisaconstant

Thismightseemoverlysimple,butisappropriateforstationary timeseries

Aseriesnotgrowingorchangingovertime Manyseriesreportedaspercentagechanges

0=tT


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U.S.PersonalConsumption(Quarterly)

MonthlyPercentageChange


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Estimation

If E(yt+h | t)=t=Tt=0 thentheoptimalforecast

isthemean 0 =E(yt+h) Theestimateof 0 isthesamplemean

Thisistheestimateoftheoptimalpointforecast

whent= 0 b0 isalsotheleastsquaresestimateinan

interceptonlymodel

=+

=T

tht

yT

b10

1


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InSTATA,usetheregress command

SeeSTATAHandoutonwebsite Samplemeanisestimatedconstant

Estimation


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FittedValues

Fittedvaluesarethesamplemean

InSTATAusethepredict command

Thiscreatesavariableypoffittedvalues

0 by tt ==


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Plotactualagainstfitted


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OutofSample

Pointforecastsarethesamplemean

InSTATA,usetsappend toexpandsample,and

predict togeneratepointforecasts.

0 by hT =+


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OutofSample


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ForecastErrors

Theforecasterror et isthedifference

betweentherealizedvalueandtheconditionalmean.

orequivalently

Wecall et theforecasterror.

thtt ye = +

ttht ey +=+


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Residuals

Theresidualsaretheinsamplefittederrors.

Thedifferencebetweentherealizedvalueandtheinsampleforecast.

Ingeneral,itisusefultoplottheresidualsagainsttime,toseeifanytimeseriespattern

remains.

0

by

yeht

thtt

==

+

+


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CalculateandPlotResiduals


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EstimationUncertainty

Thesamplemean

isanestimateof 0 =E(yt+h)

Theestimationerroris

=+=

T

thtyTb 10

1

( )

=

=+

=+

=

=

=

T

t

t

T

t

ht

T

t

ht

e

T

yT

yT

b

1

10

01

00

1

1

1


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EstimationVariance

Underclassicalconditions,

where 2=var(et) Thestandarderrorforb0 isanestimateofthe

standarddeviation

( )T

b2

0var

=

( )T

bsd2

0

=


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ForecastVariance

Whenthesamplemean b0 isusedasthe

forecastfor yT+h thenthepredictionerroris

whichisthesumoftheforecasterror eT+h andtheestimationuncertainty 0b0.

Theforecastvarianceis

000 beby hThT += ++

( ) ( ) ( )

2

22

000

11

varvarvar

+=

+=

+= ++

T

T

beby hThT


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StandardDeviationofForecast

Thestandarddeviationoftheforecastisthe

estimate

Thisisslightlylargerthantheregressionstandarddeviation

21

1

+=+T

s hT


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NormalForecastIntervals

LetT+h beaforecastforyT+h

Thepredictionerroris yT+h T+h Let sT+h bethest.deviationoftheforecast

Ifthepredictionerrorsarenormallydistributed,

the(1)%forecastintervalendpointsare

wherez/2 andz1/2arethe /2and1 /2quantiles ofthenormaldistribution

e.g. T+h1.64 sT+h fora90%interval

2/1

2/

+++

+++

+=

+=

zsyU

zsyL

hThThT

hThThT


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DeficiencyofNormalIntervals

Thenormalforecastintervalisbasedonthe

assumption thatthepredictionerrorsarenormallydistributed.

Thisrequiresthattheconditionaldistributionof yT+h benormal,whichisrarelyvalid.

Instead,wecancomputeforecastintervals

basedontheempiricaldistributionofthe

forecastresiduals.


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Empircal ForecastIntervals

Lett+h befittedvaluesforyt+h withresiduals

Letq/2 andq1/2bethe /2and1 /2

quantiles oftheresiduals.

The(1)%forecastintervalendpointsare

2/1

2/

++

++

+=

+=

qyU

qyL

hThT

hThT

hthtt yye ++ =


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EmpiricalForecastIntervals

Thebasicmethodtoobtainforecastintervals

isthesameforanyregressionmodel

The(1)%forecastintervalendpointsare

where q/2 andq1/2arethe /2and1 /2

quantiles ofthedistributionof et

.

ttht ey +=+

2/1

2/

+

+

+=

+=

qU

qL

thT

ThT


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Quantiles

Thexth quantile ofasetofnumbersisthe

value qx suchthatx%aresmallerthanqxand(1x)%arelargerthan qx.

Youcanfindqx bysortingthedata. InSTATA,usetheqreg command

(forquantile regresion)


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OutofSample


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MeanShifts

Sometimesthemeanofaserieschangesover

time Itcandriftslowly,orchangequickly

Possiblyduetoapolicychange Inthiscase,forecastingbasedonaconstant

meanmodelcanbemisleading


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StateandLocalGovernmentSpending

PercentageGrowthRate(Quarterly) Averagefor19472009: 3.6%

Butthishasnotbeenthetypicalrateinrecentyears.


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Alternatives

Subsampleestimation Estimatethemeanonsubsamples

Forecastsarebasedonthemostrecent DummyVariableformulation

isthebreakdate Thedatewhenthemeanshifts

Thecoefficient 0 isthemeanbefore t=

Thecoefficient 1 istheshiftat t=

Thesum 0+1 isthemeanaftert=

( )

=

+=

td

d

t

tt

1

10


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Forecast

LinearRegression yt+h ondt Example

StateandLocalGovernmentPercentageGrowth

Meanbreaksin1970q1and2002q1


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Fitted

Outofsampleforecastfallsfrom3.6%to0.6%!


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ShouldyouuseMeanShifts?

Onlyaftergreathesitationandconsideration.

Shoulduseshiftsandbreaksreluctantlyandwithcare.

Doyouhaveamodelorexplanation?

Whatistheforecastingpowerofameanshift? Iftheyhavehappenedinthepast,willtherebemore

inthefuture?

Yet,iftherehasbeenanobviousshift,asimpleconstantmeanmodelwillforecastterribly.


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HowtoSelectBreakdates

Judgmental Datesofknownpolicyshifts

Importantevents

Economiccrises

Informaldatabased Visualinspection

Formaldatabased

Estimateregressionformanypossiblebreakdates Selectonewhichminimizessumofsquarederror

Thisistheleastsquaresbreakdate estimator

time series components.pdf

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