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TrendModels

Atrendmodelis

where Timet isthetimeindex.

InSTATA,Timet isanintegersequence,normalizedto

bezeroatfirstobservationof1960. Mostcommonmodels

LinearTrend

ExponentialTrend QuadraticTrend

TrendswithChangingSLope

)( tt TimegT =

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Warning:

BeskepticalofTrendModels Whileinsomecases,trendforecastingcanbe

useful. Inmanycases,itcanbehazardous.

WewillexaminesomeofthetrendexamplesinChapter5ofDieboldstext

Theydidnotforecastwelloutofsample. Aconstructivealternativeistoforecastgrowth

rates,aswedidforconsumptionexpenditure.

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Example1

LaborForceParticipationRate FromBLS

Monthly,19482009,Seasonallyadjusted

MenandWomen,ages25+

Percentageofpopulationinlaborforce(employedplusunemployeddividedbypopulation)

Dieboldestimateson19481992

Wewillestimateon19481992,forecast19932009

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WomensLaborParticipationRate

19481992

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MensLaborParticipationRate

19481992

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LinearTrendModel

Thelaborforceparticipationrateshavebeen

smoothlyandlinearlyincreasing(forwomen)andsmoothlyandlinearlydecreasing(formen)over19481992

Thissuggestsalineartrend

Inthismodel, 1 istheexpectedperiodtoperiodchangeinthetrendT

t

tt TimeT 10 +=

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Example2

RetailSales,CurrentDollars FromCensusBureau

Monthly,19552001,seasonallyadjusted Thisparticularseriesdiscontinuedafter2001

Dieboldestimatesupto1991

Wewillforecast1992current

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RetailSales

19551993

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QuadraticTrends

Theretailsalesserieshasbeenincreasing

smoothlyover19551993,butnotlinearly. Tomodelthiswewilluseaquadratictrend

2

210 ttt TimeTimeT ++=

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Example3

TransactionVolume,S&PIndex FromYahooFinance

(SimilartoNYSEseriesinDiebold)

Weekly,1950current

Dieboldestimateson19551993,forecasts1994

Wewillforecast19942001

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TransactionVolume

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ExponentialTrend

Tomodelthiswewilluseanexponentialtrend

Theexponentialtrendislinearaftertaking

(natural)logarithms

Thisistypicallyestimatedbyalinearmodelaftertakinglogsofthevariabletoforecast

tTimet eT 10 +=

( ) tt TimeT 10ln +=

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Ln(Volume)

Inlogarithms,trendisroughlylinear.

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ExponentialTrends

Mosteconomicserieswhicharegrowing

(aggregateoutput,suchasGDP,investment,consumption)areexponentiallyincreasing

Percentagechangesarestableinthelongrun

Theseseriescannotbefitbyalineartrend

Wecanfitalineartrendtotheir(natural)

logarithm

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LinearModels

Thelinearandquadratictrendsarebothlinear

regressionmodelsoftheform

or

where x1t=Timet

x2t=Timet2

ttt xx 22110 ++=

tt x110 +=

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Example4

RealGDP FromBEA

Quarterly,19472009

Wewillestimateon19471990,forecast1991

2009 Alsouseanexponentialtrend

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RealGDP

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Ln(RealGDP)

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LinearForecasting

Thegoalistoforecastfutureobservations

givenalinearfunctionofobservables Inthecaseoftrendestimation,these

observablesarefunctionsofthetimeindex

Inothercases,theywillbeotherfunctionsofthedata

Inthemodeltheforecastfor yt+h is t+h=b0+b1xt where b0and b1 areestimates

tt x10 +=

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Estimation

Howshouldweselect b0 and b1 ?

Thegoalistoproduceaforecastwithlowmeansquareerror(MSE)

Thebestlinearforecastisthelinearfunction

0+1xt thatminimizestheMSE

WedonotknowtheMSE,butwecanestimateitbyasampleaverage

( ) ( )2102

thththt xyEyyE = +++

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SumofSquaredErrors

Sampleestimateofmeansquareerroristhesumofsquarederrors

Thebestlinearforecastisthelinearfunction 0+1xtthatminimizestheMSE,orexpectedsumofsquarederrors.

Oursampleestimateofthebestlinearforecastisthe

linearfunctionwhichminimizesthe(sample)sumofsquarederrors.

Thisiscalledtheleastsquaresestimator

( ) ( )=

+=

n

tthtn xy

nS

1

2

10101,

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LeastSquares

Theleastsquaresestimates(b0,b1)arethe

valueswhichminimizethesumofsquarederrors

Thisproducesestimatesofthebestlinear

predictor thelinearfunction 0+1xt that

minimizestheMSE

( ) ( )=

+=

n

tthtn xy

nS

1

2

1010

1,

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MultipleRegressors

Therearemultipleregressors

Forexample,thequadratictrend

Thebestlinearpredictoristhelinearfunction

0+1x1t+2x2t thatminimizestheMSE

2

210 ttt TimeTimeT ++=

ttt xx 22110 ++=

( ) ( )2221102

tthththt xxyEyyE = +++

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MultipleRegression

Thesampleestimateofthebestlinear

predictorarethevalues(b0,b1,b2)whichminimizethesumofsquarederrors

InSTATA,usetheregress command

( ) ( )=

+ =

n

ttthtn xxynS 1

2

22110210

1,,

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Example1

WomensLaborForceParticipationRate

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RegressionEstimation

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InSampleFit

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Residuals

Residualsaredifferencebetweendataand

fittedregressionline

tht

thtt

Timebby

Tye

10

=

=

+

+

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ResidualPlot

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InSampleFit

Computewithpredict command

Fitlooksgood

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Forecast

Forecastisthelinearfunctionwithestimated

coefficients

Computewithpredict command

hThT TimebbT ++ += 10

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ForecastIntervals

Computeresiduals

Computequantiles ofresiduals

Theseareconstantovertime

Addtopredictedvalues

Identicaltoconstantmeancase

tht

thtt

Timebby

ye

10

=

=

+

+

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OutofSampleForecast

Outofsamplepredictionmightbetoolow.

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OutofSample

WomensLaborForceParticipation

No:Predictionwaswaytoohigh!

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MensLaborForceParticipationRate

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Estimation

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InSampleFit

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Residuals

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Forecast

EndofSamplelooksworrying

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OutofSample

MensLaborForceParticipation

LinearTrendTerrible

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Example2

RetailSales

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LinearandQuadraticTrend

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LinearandQuadraticTrend

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Forecast

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Residuals

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ActualValues

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Example3:Volume

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EstimatingLogarithmicTrend

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FittedTrend

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Residuals

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Forecast

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OutofSample

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ForecastingLevelsfromaForecastofLogs

Let Yt beaseriesandyt=ln(Yt)itslogarithm

Supposetheforecastforthelogisalineartrend:E(yt+h | t)=Tt= 0+ 1 Timet

Thenaforecastfor Yt is exp(Tt)

If[LT ,UT]isaforecastintervalfor yT+h Then[exp(LT),exp(UT)] isaforecastintervalforYT+h

Inotherwords,justtakeyourpointandintervalforecasts,andapplytheexponentialfunction. InSTATA,usegenerate command

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ForecastinLevels

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OutofSample

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Example4:RealGDP

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Ln(RealGDP)

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Estimation

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FittedTrend

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Residuals

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Forecastofln(RGDP)

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ForecastofRGDP(inlevels)

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OutofSample

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ProblemswithTrendForecasts

Trendforecastsunderstateuncertainty Actualuncertaintyincreasesatlongforecast

horizons.

Shorttermtrendforecastscanbequitepoorunlesstrendlinedupcorrectly

Longtermtrendforecastsaretypicallyquitepoor,astrendschangeoverlongtimeperiods

Itispreferredtoworkwithgrowthrates,andreconstructlevelsfromforecastedgrowthrates(moreonthislater).

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TrendModels

IhopeIveconvincedyoutobeskepticalof

trendbasedforecasting. Theproblemisthatthereisnoeconomic

theoryforconstanttrends,andchangesin

thetrendfunctionarenotapparentbefore

theyoccur.

Itisbettertoforecastgrowthrates,andbuildlevelsfromgrowth.

Final Trend Forecast

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FinalTrendForecast

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