Statisticsfor

BusinessandEconomics(14e)

MetricVersionAnderson,Sweeney,Williams,Camm,Cochran,Fry,Ohlmann?2020CengageLearning?2020Cengage.Maynotbescanned,copiedorduplicated,orpostedtoapubliclyaccessiblewebsite,inwholeorinpart,exceptforuseaspermittedinalicensedistributedwithacertainproductorserviceorotherwiseonapassword-protectedwebsiteorschool-approvedlearningmanagementsystemforclassroomuse.1Chapter17-TimeSeriesAnalysisandForecasting17.1–TimeSeriesPatterns17.2–ForecastAccuracy17.3–MovingAveragesandExponentialSmoothing17.4–TrendProjection17.5–SeasonalityandTrend17.6–TimeSeriesDecomposition2ForecastingMethods:QualitativeForecastingmethodscanbeclassifiedasqualitativeorquantitative.Qualitativemethodsgenerallyinvolvetheuseofexpertjudgmenttodevelopforecasts.Suchmethodsareappropriatewhenhistoricaldataonthevariablebeingforecastareeithernotapplicableorunavailable.Wewillfocusexclusivelyonquantitativeforecastingmethodsinthischapter.3ForecastingMethods:Quantitative(1of5)Quantitativeforecastingmethodscanbeusedwhen:Pastinformationaboutthevariablebeingforecastisavailable,Theinformationcanbequantified,andItisreasonabletoassumethepatternofthepastwillcontinueInsuchcases,aforecastcanbedevelopedusingatimeseriesmethodoracausalmethod.4ForecastingMethods:Quantitative(2of5)Quantitativemethodsarebasedonananalysisofhistoricaldataconcerningoneormoretimeseries.Atimeseriesisasetofobservationsmeasuredatsuccessivepointsintimeoroversuccessiveperiodsoftime.Ifthehistoricaldatausedarerestrictedtopastvaluesoftheseriesthatwearetryingtoforecast,theprocedureiscalledatimeseriesmethod.Ifthehistoricaldatausedinvolveothertimeseriesthatarebelievedtoberelatedtothetimeseriesthatwearetryingtoforecast,theprocedureiscalledacausalmethod.5ForecastingMethods:Quantitative(3of5)TimeSeriesAnalysisTheobjectiveoftimeseriesanalysisistodiscoverapatterninthehistoricaldataortimeseriesandthenextrapolatethepatternintothefuture.Theforecastisbasedsolelyonpastvaluesofthevariableand/orpastforecasterrors.6ForecastingMethods:Quantitative(4of5)CausalMethodsCausalforecastingmethodsarebasedontheassumptionthatthevariableweareforecastinghasacause-effectrelationshipwithoneormoreothervariables.Lookingatregressionanalysisasaforecastingtool,wecanviewthetimeseriesvaluethatwewanttoforecastasthedependentvariable.Ifwecanidentifyagoodsetofrelatedindependentorexplanatoryvariables,wemaybeabletodevelopanestimatedregressionequationforforecastingthetimeseries.7ForecastingMethods:Quantitative(5of5)RegressionAnalysisBytreatingtimeastheindependentvariableandthetimeseriesasadependentvariable,regressionanalysiscanalsobeusedasatimeseriesmethod.Time-seriesregressionreferstotheuseofregressionanalysiswhenthesoleindependentvariableistime.Cross-sectionalregressionreferstotheuseofregressionanalysiswhentheindependentvariable(s)is(are)somethingotherthantime.8ForecastingMethods9TimeSeriesPatternsAtimeseriesisasequenceofobservationsonavariablemeasuredatsuccessivepointsintimeoroversuccessiveperiodsoftime.Thepatternofthedataisanimportantfactorinunderstandinghowthetimeserieshasbehavedinthepast.Ifsuchbehaviorcanbeexpectedtocontinueinthefuture,wecanuseittoguideusinselectinganappropriateforecastingmethod.10TimeSeriesPlot

(1of2)Ausefulfirststepinselectinganappropriateforecastingmethodistoconstructatimeseriesplot.Atimeseriesplotisagraphicalpresentationoftherelationshipbetweentimeandthetimeseriesvariable.Timeisonthehorizontalaxis,andthetimeseriesvaluesareshownontheverticalaxis.11TimeSeriesPlot

(2of2)Example:RoscoDrugsSalesofComfortbrandheadachetonic(inbottles)forthepast10weeksatRoscoDrugsareshownbelow.RoscoDrugswouldliketoidentifytheunderlyingpatterninthedatatoguideitinselectinganappropriateforecastingmethod.WeekSales1110211531254120512561207130811591101013012TimeSeriesPatterns

(1of6)Thecommontypesofdatapatternsthatcanbeidentifiedwhenexaminingatimeseriesplotinclude:HorizontalTrendSeasonalTrendandSeasonalCyclical13TimeSeriesPatterns(2of6)HorizontalPatternAhorizontalpatternexistswhenthedatafluctuatearoundaconstantmean.Changesinbusinessconditionscanoftenresultinatimeseriesthathasahorizontalpatternshiftingtoanewlevel.Achangeinthelevelofthetimeseriesmakesitmoredifficulttochooseanappropriateforecastingmethod.14TimeSeriesPatterns(3of6)TrendPatternAtimeseriesmayshowgradualshiftsormovementstorelativelyhigherorlowervaluesoveralongerperiodoftime.Trendisusuallytheresultoflong-termfactorssuchaschangesinthepopulation,demographics,technology,orconsumerpreferences.Asystematicincreaseordecreasemightbelinearornonlinear.Atrendpatterncanbeidentifiedbyanalyzingmultiyearmovementsinhistoricaldata.15TimeSeriesPatterns(4of6)SeasonalPatternSeasonalpatternsarerecognizedbyseeingthesamerepeatingpatternofhighsandlowsoversuccessiveperiodsoftimewithinayear.Aseasonalpatternmightoccurwithinaday,week,month,quarter,year,orsomeotherintervalnogreaterthanayear.Aseasonalpatterndoesnotnecessarilyrefertothefourseasonsoftheyear(spring,summer,fall,andwinter).16TimeSeriesPatterns(5of6)TrendandSeasonalPatternSometimeseriesincludeacombinationofatrendandseasonalpattern.Insuchcasesweneedtouseaforecastingmethodthathasthecapabilitytodealwithbothtrendandseasonality.Timeseriesdecompositioncanbeusedtoseparateordecomposeatimeseriesintotrendandseasonalcomponents.17TimeSeriesPatterns(6of6)CyclicalPatternAcyclicalpatternexistsifthetimeseriesplotshowsanalternatingsequenceofpointsbelowandabovethetrendlinelastingmorethanoneyear.Often,thecyclicalcomponentofatimeseriesisduetomultiyearbusinesscycles.Businesscyclesareextremelydifficult,ifnotimpossible,toforecast.Inthischapterwedonotdealwithcyclicaleffectsthatmaybepresentinthetimeseries.18SelectingaForecastingMethodTheunderlyingpatterninthetimeseriesisanimportantfactorinselectingaforecastingmethod.Thus,atimeseriesplotshouldbeoneofthefirstthingsdevelopedwhentryingtodeterminewhatforecastingmethodtouse.Ifweseeahorizontalpattern,thenweneedtoselectamethodappropriateforthistypeofpattern.Ifweobserveatrendinthedata,thenweneedtouseamethodthathasthecapabilitytohandletrendeffectively.19ForecastAccuracy(1of8)Measuresofforecastaccuracyareusedtodeterminehowwellaparticularforecastingmethodisabletoreproducethetimeseriesdatathatarealreadyavailable.Measuresofforecastaccuracyareimportantfactorsincomparingdifferentforecastingmethods.Byselectingthemethodthathasthebestaccuracyforthedataalreadyknown,wehopetoincreasethelikelihoodthatwewillobtainbetterforecastsforfuturetimeperiods.20ForecastAccuracy(2of8)Thekeyconceptassociatedwithmeasuringforecastaccuracyisforecasterror.ForecastError=ActualValue–ForecastApositiveforecasterrorindicatestheforecastingmethodunderestimatedtheactualvalue.Anegativeforecasterrorindicatestheforecastingmethodoverestimatedtheactualvalue.21ForecastAccuracy(3of8)MeanError(ME)Asimplemeasureofforecastaccuracyisthemeanoraverageoftheforecasterrors.Becausepositiveandnegativeforecasterrorstendtooffsetoneanother,themeanerrorislikelytobesmall.Thus,themeanerrorisnotaveryusefulmeasure.MeanAbsoluteError(MAE)Thismeasureavoidstheproblemofpositiveandnegativeerrorsoffsettingoneanother.Itisthemeanoftheabsolutevaluesoftheforecasterrors.22ForecastAccuracy(4of8)MeanSquaredError(MSE)Thisisanothermeasurethatavoidstheproblemofpositiveandnegativeerrorsoffsettingoneanother.Itistheaverageofthesquaredforecasterrors.MeanAbsolutePercentageError(MAPE)ThesizeofMAEandMSEdependuponthescaleofthedata,soitisdifficulttomakecomparisonsfordifferenttimeintervals.Tomakesuchcomparisonsweneedtoworkwithrelativeorpercentageerrormeasures.TheMAPEistheaverageoftheabsolutepercentageerrorsoftheforecasts.23ForecastAccuracy(5of8)Todemonstratethecomputationofthesemeasuresofforecastaccuracy,wewillintroducethesimplestofforecastingmethods.Thenaiveforecastingmethodusesthemostrecentobservationinthetimeseriesastheforecastforthenexttimeperiod.24ForecastAccuracy(6of8)Example:RoscoDrugsSalesofComfortbrandheadachetonic(inbottles)forthepast10weeksatRoscoDrugsareshown.IfRoscousesthena?veforecastmethodtoforecastsalesforweeks2–10,whataretheresultingMAE,MSE,andMAPEvalues?WeekSales1110211531254120512561207130811591101013025ForecastAccuracy(7of8)WeekSalesNa?veforecastForecastErrorAbsoluteErrorSquaredErrorAbsolutePercentError1110EmptyCellEmptyCellEmptyCellEmptyCellEmptyCell211511055254.35312511510101008.004120125Negative

55254.17512512055254.006120125Negative

55254.17713012010101007.698115130Negative

151512513.049110115Negative55254.5510130110202040015.38TotalEmptyCellEmptyCellEmptyCell8085065.3626ForecastAccuracy(8of8)NaiveForecastAccuracy27MovingAveragesandExponentialSmoothingNowwediscussthreeforecastingmethodsthatareappropriateforatimeserieswithahorizontalpattern:MovingAveragesWeightedMovingAveragesExponentialSmoothingTheyarecalledsmoothingmethodsbecausetheirobjectiveistosmoothouttherandomfluctuationsinthetimeseries.Theyaremostappropriateforshort-rangeforecasts.28MovingAverages(1of7)Themovingaveragesmethodusestheaverageofthemostrecentkdatavaluesinthetimeseriesastheforecastforthenextperiod.Eachobservationinthemovingaveragecalculationreceivesthesameweight.29MovingAverages(2of7)Thetermmovingisusedbecauseeverytimeanewobservationbecomesavailableforthetimeseries,itreplacestheoldestobservationintheequation.Asaresult,theaveragewillchange,ormove,asnewobservationsbecomeavailable.30MovingAverages(3of7)

31MovingAverages(4of7)Example:RoscoDrugsIfRoscoDrugsusesa3-periodmovingaveragetoforecastsales,whataretheforecastsforweeks4-11?WeekSales1110211531254120512561207130811591101013032WeekSales3MAForecast1110EmptyCell2115EmptyCell3125EmptyCell4120116.75125120.06120123.37130121.78115125.09110121.710130118.311EmptyCell118.333MovingAverages(5of7)MovingAverages(6of7)WeekSales3MAForecastForecastErrorAbsoluteErrorSquaredErrorAbsolutePercentError1110EmptyCellEmptyCellEmptyCellEmptyCellEmptyCell2115EmptyCellEmptyCellEmptyCellEmptyCellEmptyCell3125EmptyCellEmptyCellEmptyCellEmptyCellEmptyCell4120116.73.33.310.892.755125120.05.05.025.004.006120123.3Negative3.33.310.892.757130121.78.38.368.896.388115125.0Negative10.010.010.008.709110121.7Negative11.711.7136.8910.6410130118.311.711.7136.899.00TotalEmptyCellEmptyCell3.335.33489.4544.2234MovingAverages(7of7)3-MAForecastAccuracyThe3-weekmovingaverageapproachprovidedmoreaccurateforecaststhanthenaiveapproach.35WeightedMovingAverages(1of2)Tousethismethodwemustfirstselectthenumberofdatavaluestobeincludedintheaverage.Next,wemustchoosetheweightforeachofthedatavalues.Themorerecentobservationsaretypicallygivenmoreweightthanolderobservations.Forconvenience,theweightsshouldsumto1.36WeightedMovingAverages(2of2)Anexampleofa3-periodweightedmovingaverage(3WMA)is:3WMA=0.2(110)+0.3(115)=119Inthisexampletheweightsare0.2,0.3,and0.5(whichsumto1).125isthemostrecentofthethreeobservations.37ExponentialSmoothing

(1of6)Thismethodisaspecialcaseofaweightedmovingaveragesmethod;weselectonlytheweightforthemostrecentobservation.Theweightsfortheotherdatavaluesarecomputedautomaticallyandbecomesmallerastheobservationsgrowolder.Theexponentialsmoothingforecastisaweightedaverageofalltheobservationsinthetimeseries.Thetermexponentialsmoothingcomesfromtheexponentialnatureoftheweightingschemeforthehistoricalvalues.38ExponentialSmoothing(2of6)ExponentialSmoothingForecast39ExponentialSmoothing(3of6)ExponentialSmoothingForecastWithalgebraicmanipulation,wecanrewrite

40ExponentialSmoothing(4of6)

WeekSales1110211531254120512561207130811591101013041ExponentialSmoothing(5of6)

42ExponentialSmoothing(6of6)

43

WeekSalesalphaequals.1ForecastForecastErrorAbsoluteErrorSquaredErrorAbsolutePercentError1110EmptyCellEmptyCellEmptyCellEmptyCellEmptyCell2115110.005.005.0025.004.353125110.5014.5014.50210.2511.604120111.958.058.0564.806.715125112.7612.2412.24149.949.796120113.986.026.0236.255.027130114.5815.4215.42237.7311.868115116.12Negative1.121.121.260.979110116.01Negative6.016.0136.125.4610130115.4114.5914.59212.8711.22TotalEmptyCellEmptyCellEmptyCell82.95974.2266.9844

ForecastAccuracy

45ExponentialSmoothing(α=0.8)

(1of2)WeekSalesalphaequals.8ForecastForecastErrorAbsoluteErrorSquaredErrorAbsolutePercentError1110EmptyCellEmptyCellEmptyCellEmptyCellEmptyCell2115110.005.005.0025.004.353125114.0011.0011.00121.008.804120122.80Negative2.202.207.841.835125120.564.444.4419.713.556120124.11Negative4.114.1116.913.437130120.829.189.1884.237.068115128.16Negative13.1613.16173.3011.449110117.63Negative7.637.6358.266.9410130111.5318.4718.47341.2714.21TotalEmptyCellEmptyCellEmptyCell75.19847.5261.6146ExponentialSmoothing(α=0.8)(2of2)ForecastAccuracy

47TrendProjectionIfatimeseriesexhibitsalineartrend,themethodofleastsquaresmaybeusedtodetermineatrendline(projection)forfutureforecasts.Leastsquares,alsousedinregressionanalysis,determinestheuniquetrendlineforecastwhichminimizesthemeansquareerrorbetweenthetrendlineforecastsandtheactualobservedvaluesforthetimeseries.Theindependentvariableisthetimeperiodandthedependentvariableistheactualobservedvalueinthetimeseries.48LinearTrendRegression

(1of5)Usingthemethodofleastsquares,theformulaforthetrendprojectionis:

49LinearTrendRegression

(2of5)ForthetrendprojectionequationTt

=b0+b1t

50LinearTrendRegression(3of5)ThenumberofplumbingrepairjobsperformedbyAuger'sPlumbingServiceinthelastninemonthsislistedontheright.ForecastthenumberofrepairjobsAuger'swillperforminDecemberusingtheleastsquaresmethod.MonthJobsMarch353April387May342June374July396August409September399October412November40851LinearTrendRegression(4of5)(month)ttminustoverbarleftparenthesistminustoverbarrightparenthesissquaredysubscripttbaselineleftparenthesisYsubscripttbaselineminusYoverbar)leftparenthesistminustoverbarrightparenthesisleftparenthesisYsubscripttbaselineminusYoverbarrightparenthesis(Mar.)1-416353-33.67134.68(Apr.)2-393870.33-0.99(May)3-24342-44.6789.34(June)4-11374-12.6712.67(July)5003969.330(Aug.)61140922.3322.33(Sep.)72439912.3324.66(Oct.)83941225.3375.99(Nov.)941640821.3385.32Sum45EMPTYCELLEMPTYCELL3480EMPTYCELL444.0052LinearTrendRegression(5of5)53TrendProjection

(1of3)

MonthJobsMarch353April387May342June374July396August409September399October412November40854TrendProjection

(2of3)Three-MonthWeightedMovingAverageTheforecastforDecemberwillbetheweightedaverageoftheprecedingthreemonths:September,October,andNovember.TrendProjection:F10=422.27(fromearlierslide)55TrendProjection

(3of3)Conclusion:Duetothepositivetrendcomponentinthetimeseries,thetrendprojectionproducedaforecastthatismoreinlinewiththetrendthatexists.Theweightedmovingaverage,evenwithheavy(0.6)weightplacedonthecurrentperiod,producedaforecastthatislaggingbehindthechangingdata.56NonlinearTrendRegression

(1of3)Sometimestimeserieshaveacurvilinearornonlineartrend.Avarietyofnonlinearfunctionscanbeusedtodevelopanestimateofthetrendinatimeseries.Oneexampleisthisquadratictrendequation:Anotherexampleisthisexponentialtrendequation:57NonlinearTrendRegression

(2of3)Example:CholesterolDrugRevenueTheannualrevenueinmillionsofdollarsforacholesteroldrugforthefirst10yearsofsalesisshown.Acurvilinearfunctionappearstobeneededtomodelthelong-termtrend.YearRevenue123.1221.3327.4434.6533.8643.2759.5864.4974.21099.358NonlinearTrendRegression

(3of3)Example:CholesterolDrugRevenue59SeasonalitywithoutTrend(1of6)Totheextentthatseasonalityexists,weneedtoincorporateitintoourforecastingmodelstoensureaccurateforecasts.Wewillfirstlookatthecaseofaseasonaltimeserieswithnotrendandthendiscusshowtomodelseasonalitywithtrend.60SeasonalitywithoutTrend(2of6)Example:UmbrellaSalesSometimesitisdifficulttoidentifypatternsinatimeseriespresentedinatable.Plottingthetimeseriescanbeveryinformative.YearQuarter1Quarter2Quarter3Quarter41125153106882118161133102313814411380410913712510951301651289661SeasonalitywithoutTrend(3of6)UmbrellaSalesTimeSeriesPlot62SeasonalitywithoutTrend(4of6)Thetimeseriesplotdoesnotindicateanylong-termtrendinsales.However,closeinspectionoftheplotdoesrevealaseasonalpattern.Thefirstandthirdquartershavemoderatesales,thesecondquarterthehighestsales,andthefourthquartertendstobethelowestquarterintermsofsales.63SeasonalitywithoutTrend(5of6)

64SeasonalitywithoutTrend(6of6)GeneralFormofEstimatedRegressionEquationis:

65SeasonalityandTrend(1of4)Wewillnowextendtheregressionapproachtoincludesituationswherethetimeseriescontainsbothaseasonaleffectandalineartrend.Wewillintroduceanadditionalindependentvariabletorepresenttime.Example:Terry’sTieShopBusinessatTerry'sTieShopcanbeviewedasfallingintothreedistinctseasons:(1)Christmas(NovemberandDecember);(2)Father'sDay(lateMaytomidJune);and(3)allothertimes.Averageweeklysales($)duringeachofthethreeseasonsduringthepastfouryearsareshownonthenextslide.66SeasonalityandTrend(2of4)Example:Terry’sTieShopDetermineaforecastfortheaverageweeklysalesinyear5foreachofthethreeseasons.YearSeason1Season2Season311856201298521995216810723224123061105422802408112067SeasonalityandTrend(3of4)Therearethreeseasons,sowewillneedtwodummyvariables.Seas1=1ifSeason1,0otherwiseSeas2=1ifSeason2,0otherwiseGeneralFormofEstimatedRegressionEquationis:EstimatedRegressionEquationis:Sales=797.0+1095.43(Seas1)+1189.47(Seas2)+36.47(t)68SeasonalityandTrend(4of4)Theforecastsofaverageweeklysalesinthethreeseasonsofyear5are:Seas.1:Sales=797+1095.43(1)+1189.47(0)+36.47(13)=2366.5Seas.2:Sales=797+1095.43(0)+1189.47(1)+36.47(14)=2497.0Seas.3:Sales=797+1095.43(0)+1189.47(0)+36.47(15)=1344.069TimeSeriesDecomposition

(1of5)Timeseriesdecompositioncanbeusedtoseparateordecomposeatimeseriesintoseasonal,trend,andirregular(error)components.Whilethismethodcanbeusedforforecasting,itsprimaryapplicabilityistogetabetterunderstandingofthetimeseries.Understandingwhatisreallygoingonwithatimeseriesoftendependsupontheuseofdeseasonalizeddata.70TimeSeriesDecomposition(2of5)Decompositionmethodsassumethattheactualtimeseriesvalueatperiodtisafunctionofthreecomponents:trend,seasonal,andirregular.Howthesethreecomponentsarecombinedtogivetheobservedvaluesofthetimeseriesdependsuponwhetherweassumetherelationshipisbestdescribedbyanadditiveoramultiplicativemodel.71TimeSeriesDecomposition(3of5)Anadditivemodelfollowstheform:Anadditivemodelisappropriateinsituationswheretheseasonalfluctuationsdonotdependuponthelevelofthetimeseries.72TimeSeriesDecomposition(4of5)Amultiplicativemodelfollowstheform:Amultiplicativemodelisappropriate,forexample,iftheseasonalfluctuationsgrowlargerasthesalesvolumeincreasesbecauseofalong-termlineartrend.73TimeSeriesDecomposition(5of5)Example:Terry’sTieShopDetermineaforecastfortheaverageweeklysalesinyear5foreachofthethreeseasons.YearSeason1Season2Season311856201298521995216810723224123061105422802408112074CalculatingtheSeasonalIndexes

(1of9)Step1.Calculatethecenteredmovingaverages.Therearethreedistinctseasonsineachyear.Hence,takeathree-seasonmovingaveragetoeliminateseasonalandirregularfactors.Forexample:1stCMA=(1856+2012+985)/3=1617.672ndCMA=(2012+985+1995)/3=1664.00Etc.75CalculatingtheSeasonalIndexes

(2of9)Step2.CentertheCMAsoninteger-valuedperiods.Thefirstcenteredmovingaveragecomputedinstep1(1617.67)willbecenteredonseason2ofyear1.Notethatthemovingaveragesfromstep1centerthemselvesoninteger-valuedperiodsbecausenisanoddnumber.76CalculatingtheSeasonalIndexes(3of9)YearSeasonDollarSalesleftparenthesisYsubscripttbaselinerightparenthesisMovingAverage111856EMPTYCELL1220121617.67139851664.002119951716.002221681745.002310721827.003122411873.003223061884.003311051897.004122801931.004224081936.00431120EMPTYCELL77CalculatingtheSeasonalIndexes

(4of9)Thecenteredmovingaveragevaluestendto“smoothout”boththeseasonalandirregularfluctuationsinthetime