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Forecasting

Aforecastisaguessaboutanunknown.

Aneconomicforecastisaforecastaboutan

economicvariable,event,outcome,or

duration.

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LetsMakeaForecast

Supposewetakearandomhouseholdinthe

UnitedStates. Letsforecastthewage(hourly)oftheheadof

household. Whatisyourforecast?

Willyourforecastbecorrect?Why?

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WageDensity

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WageDistribution

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WageForecast

Wageshaveadistributioninthepopulation.

Itisimpossibletocorrectlyforecastan

individualswage.

Ifweforecastthewagewillbe$17.87itisclosetoimpossiblethatagivenpersonswage

willbeexactly$17.87.

Themostcorrectandaccurateforecastisthe

entiredistribution(ordensity).

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ForecastDistribution

Supposewearetryingtoforecastan

economicvariable y Forexample,arandompersonswage.

y hasadistribution F(y) whichismathematicallydefinedas

Visually,werepresentdistributionsthrough

theirdensityfunctions

)()( uyPuF =

)()( yF

dy

dyf =

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ForecastDistribution

Acompleteforecastfor y isitsdistribution F(y)

ordensityf(y). Either F(y) orf(y) summarizesallthatisknown

andunknownaboutthepotentialvaluesfor y.

Wecall F theforecastorpredictivedistribution.

Canyouforecastapersonswage?

Wecannotknowwithcertaintythewage

Weknow(orcanestimate)thedistribution:

Therangeandlikelihoodofpossiblewages.

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Whatmightthismatter?

Supposeyourcompanyunderwrites

unemploymentinsurancewhichpayapersonswage y iftheybecomeunemployed.

Supposearandompersonlosestheirjob. Whatisthecosttothecompany?

Wecannotknowwithcertainty,butwemayknowthedistributionofthepotentialcosts.

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PointForecast

Apointforecast isafunctionofthe

predictivedistribution F Itcanbeviewedasasummaryof F.

Whichfunctionshouldbeused?

Whatisourbestguessfor y basedon F?

Itturnsoutthattheanswerdependsonourlossfunction howwemeasurethecostsdue

topotentialforecasterror.

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ForecastError

Ifweforecastarandomvariable y witha

forecastf wesaythattheforecasterroris

Theforecasterroristhedifferencebetweenthe

actualandtheforecast.

Forexample,ifweforecastedthatanindividuals

wagewouldbe$18,butitturnsoutthatitis$24,

thentheerroris2418=6.Iftheirwagewas

actually$14,thentheerrorwouldbe1418=4.

yye =

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Forecasterror

Solongasthevariable y israndom(not

perfectlyforecastable)thentherewillalwaysbeforecasterror.

Thiscannotbeavoided. However,errorsarecostly.

Ausercanassigncoststoaforecasterror.

Wecallthisthelossfunction

FunctionLosseL =

)(

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LossFunctions

CommonmathematicalchoicesQuadraticLoss

AbsoluteLoss

BotharesymmetricTreatpositiveandnegativeforecasterrors

symmetrically

Quadraticlosspenalizeslargeerrorsmuchmorethansmallerrors.

AsymmetricLossFunctionsalsopossible.

2)( eeL =

eeL =)(

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ManyLossFunctionsare

(Approximately)Quadratic ConsideramonopolistsellingaproductQatapriceP,

withlineardemandandzerocost.

ThemonopolistsetspricePandthensellsQ.

Thedemandequationis Q=2aP

Theprofitfunctionis (P)=2aPP2

TheoptimalpriceisP*=a,optimalprofit*=(P*)=a2.

Letbeaforecastofawitherrore=a.

ThemonopolistsetsP= TheLossisL=*()=a22a+2=(a)2=e2

Thisisquadraticloss.

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Risk

TheRiskofaforecastisitsexpectedloss.

Mathematically,

Forquadraticloss

Forabsoluteloss

)(E)(E)( yyLeLyR ==

( )2E)( yyyR =

yyyR E)( =

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OptimalPointForecast

Theoptimal(best)pointforecastisthe

function ofthepredictivedistribution Fwhichminimizestherisk (minimizesthe

expectedloss).

Inthequadraticcase

whichisaquadraticin

( )22

2

E2E

E)(

yyyy

yyyR

+=

=

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OptimalPointForecast QuadraticLoss

The whichminimizestheRiskisfoundby

differentiation

Whichhasthesolution

Theoptimalpointforecastisthemeanofthe

predictivedistribution

yy 2E20 +=

yy E =

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OptimalForecastUnderQuadraticLoss

istheMean Theoptimalpointforecastunderquadratic

lossisthemean. Forexample,toforecastthewageofarandom

person,ouroptimalpointforecastis$17.87

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OptimalPredictionUnderAbsoluteLoss

istheMedian Theriskofaforecastis

Thisisminimizedbythemedian

Theoptimalforecastunderabsolutelossis

Forexample,toforecastthewageofarandomperson,theoptimalpointforecastis$14.76

yyyR E)( =

)( yMediany =

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ChoiceofLossFunction

Wehavelearnedthattheoptimalpointforecastdependsuponthelossfunction

Themean Ey minimizestheexpectedsquarederror Themedianminimizestheexpectedabsoluteerror.

Otherlossfunctionsleadtodifferentsolutions.

Inmostcases,wedonothaveanexplicitlossfunction.Sowetakethesimplestapproachandusethemean,whichisequivalenttosquaredloss.

However,inarealworldapplication,youmightbeabletoarticulatetheexplicitlossduetoforecastingerror.Inthiscase,itwouldbebesttousethelossfunctionexplicitly,leadingtospecializedestimatorsandforecasts.

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IntervalForecast

Wehavesaidthatacompleteforecastforthe

unknownwage y isitsdensityf,butforsimplicityusersoftenwantapointforecast

Anintermediatesolutionistoreportaforecast

interval C=[L,U].

Aforecastintervalissimilartoaconfidence

intervalinstatistics. Thegoalisfortheunknownwage y tolieinthe

forecastintervalwithaprespecifiedprobability.

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IntervalForecast

Thusanx%forecastinterval C satisfies

Commonchoicesfor x include

x=.90 (90%)

x=.80 (80%)

x=.50 (50%)

50%intervalshavethesimplepropertythat

theycontaintheunknown y withevenodds.

( ) xCyP =

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Quantiles

Theendpointsof C=[L,U] arequantiles ofthedistribution F of y.

Definition:Theth quantile of y isthenumberq

whichsatisfies

Theyarefoundbyinvertingthedistributionfunction

Forax%interval,youneedthex/2and1x/2quantiles

Forexample,the25%and75%quantiles fora

50%forecastinterval.

( )

qF=

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Quantiles ofWageDistribution

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NormalRule

Ifthevariable y isnormallydistributedN(,2)

Thepointforecastis

Theforecastintervalsare[z/2,+z /2] wherez/2

arequantiles fromthenormaldistribution table.

Forexample,fora90%interval,z.05=1.645,orfora50%

interval,z.25=0.675

Allyouneedtoknowisthestandarddeviation

Buteconomicdataareoftenfarfromnormal,so

thisrulemaybeinaccurate.

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UseForecastIntervals!

Forecastintervalsaresimple,yetnotwidely

used. Apointforecastbyitselfdoesnot

communicatetheuncertaintyintheforecast Aforecastintervaliseasiertointerpretthan

theentiredistribution

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Summary UnconditionalForecasts

Acompleteforecastofarandomvariable y isthedistribution F ordensityf ofthevariable.

Apointforecastisasinglenumber tosummarizethedistribution.

Theoptimalchoicedependsonthelossfunction.

Whenlossisquadratic,theoptimalpointforecastisthemean.

Forecastintervalsarequantiles oftheforecastdistribution,andconveyusefulinformationabouttheuncertainin y.

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ConditionalForecast

Wehadconsideredforecastingthewageofa

randomperson. Thedistributionisquitediffuseasitincludes

allwageearners.Weknownothingaboutthepersonbeingforecast.

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ConditionalForecast

Nowsupposeweknowthatthepersonisaman(orawoman).

Theinformationimprovestheforecast.

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ConditionalonSex,Race,Education

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ConditionalForecasts(Means)

Men Women

White High School $17 $13

College $27 $20

Graduate $32 $26

Black High School $14 $11

College $21 $21Graduate $29 $23

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RoleofConditioning

Byconditioningonavailableinformation,we

canmakeforecastsmoreaccurate. Conditioningreducestheriskoftheforecast.

Ignoringestimation,conditioningonmoreinformationisalwaysbetterinthesenseof

reducingrisk.