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Time!Series!Forecasts!of!Renewable!Energy!
Consumption!in!the!United!States!
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MSc!thesis!Marketing!and!Consumer!Behavior!(MCBB80433)!
Author:!Yao!Lu!
Registration!number:!910629530070!
Supervisor:!Dr.!Andres!Trujillo!Barrera! !
CoGreader:!Dr.!Ivo!van!der!Lans! !
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Abstract!
Renewable!energy,!as!one!rising!star! in!energy! industry,! its!amount!of!consumption! in! the!U.S.!has!increased! quickly! in! last! few! decades.! This! paper! explored! several! time! series! models,! namely!exponential!smoothing,!seasonal!ARIMA,!seasonal!ARCH!models!with!the!historical!data!of!renewable!energy!consumption!in!the!U.S.!to!compare!their!forecasting!performances.!Using!data!from!Google!Trends,!an!augmented!ARIMA!model!was!built!with!the!purpose!for!testing!whether!it!improves!the!fitness! and! forecasting! accuracy! of! the! baseline!ARIMA!model.! The! result! revealed! that! renewable!energy!consumption!in!the!U.S!could!be!almost!precisely!predicted!by!time!series!models.!Moreover,!it! is! proved! that! including! the! Google! Trends! data! could! improve! the! outGofGsample! forecasting!performance.! The! empirical! study! revealed! that! web! search! volume! data! would! be! a! resource! to!capture!the!changes!in!real!consumption.!
Keywords!
Time!series!models;!renewable!energy;!forecast!accuracy;!Google!Trends!
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List!of!abbreviations!
ACF Autocorrelation!function ADF Augmented!DickeyGFuller AIC Akaike’s!Information!Criterion AR Autoregressive!model ARCH Autoregressive!conditional!heteroskedasticity ARIMA Autoregressive!integrated!moving!average BIC! Schuwarz!Bayesian!Information!Criterion Btu British!thermal!units GARCH! Generalized!autoregressive!conditional!heteroskedasticity!LS Least!squared MA Moving!average!model MAE Mean!absolute!error MAPE Mean!absolute!percentage!error PACF Partial!autocorrelation!function RMSE Root!mean!squared!error SVI Web!search!volume!index
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Table!of!contents!1.! Introduction!......................................................................................................................!1!
1.1! Background!..............................................................................................................!1!1.2! Problem!statement!..................................................................................................!2!1.3! Research!questions!..................................................................................................!2!1.4! Thesis!outline!...........................................................................................................!3!
2.! Literature!study!................................................................................................................!4!2.1! Forecasting!models!for!energy!demand!and!consumption!.....................................!4!2.2! Forecasting!with!Google!trends!...............................................................................!5!
3.! Theoretical!and!methodological!framework!....................................................................!7!3.1! Exponential!smoothing!............................................................................................!7!3.2! ARIMA!......................................................................................................................!9!
3.2.1! Phase!1G!Identification!.......................................................................................!10!3.2.2! Phase!2G!Estimation!and!Testing!........................................................................!11!3.2.3! Phase!3G!Application!..........................................................................................!12!
3.3! Seasonal!dummy!and!ARCH!model!........................................................................!12!3.4! Web!search!index!construction!.............................................................................!13!3.5! Forecasting!error!measurement!............................................................................!16!
4.! Empirical!analysis!............................................................................................................!17!4.1! Exponential!Smoothing!..........................................................................................!18!4.2! ARIMA!....................................................................................................................!20!4.3! Seasonal!dummy!and!ARCH!model!........................................................................!24!4.4! ARIMA!with!web!search!index!...............................................................................!27!
5.! General!discussion!..........................................................................................................!30!5.1! Summary!of!data!analysis!results!..........................................................................!30!5.2! Conclusion!and!implications!..................................................................................!32!5.3! Limitations!and!further!study!................................................................................!33!
Reference!................................................................................................................................!36!
Appendix!.................................................................................................................................!41!
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Chapter!1!
1.! Introduction!
1.1! Background! !
Energy! is!vitally! interrelated! to! the!development!of!any!nation! in! todays!world.!Under! the! trend!of!globalization! and! industrialization,! energy! demand!has! increased! exponentially! in! the! past! decades!(Suganthi!&!Samuel,!2012).!Taking!fossil!fuel,!one!of!mostly!known!energy!resources,!as!an!example,!it!has!been!wildly!used! in!most!nations! (Ediger!et!al.,!2006).!For! instance,! in! the!United!States,! the!total!consumption!of! fossil! fuels!accounts!for!more!than!90%!of!the!total!energy!consumption!(U.S.!Energy!Information!Administration,!2015).!However,!large!scale!production!and!consumption!of!fossil!fuels! will! lead! to! the! degradation! of! environment! and! depletion! of! resources.! With! the! dramatic!increasing!concerns!about!the!environment!and!sustainability,!renewable!energy!gradually!comes!to!be!known!by!the!public.!In!2014,!the!consumption!of!renewable!energy!sources!in!the!United!States!totaled!up!to!9.6!quadrillion!British!thermal!units!(Btu),!which!took!about!10%!of!the!total!U.S.!energy!consumption.!The!increasing!consumption!of!renewable!energy!already!drew!attention!of!researchers.!Up!to!now,!there!are!many!researches!focusing!on!the!future!demand!of!renewable!energy!(Sadorsky,!2009;!Banos!et!al.,!2011;!Lee!&!Shih,!2011).! !
Generally! speaking,! an! effective! and! timely! forecasting! could! be! used! as! a! reliable! tool! for! policy!makers! and! for! companies! to! decide! their! production! and! price! planning.! Lots! of! forecasting!researches!have!been!done!with!various!methods! in!different! fields! (Mokhtarian!&!Cao,!2004;!Ahn,!2008;! Suganthi,! &! Samuel,! 2012).! Considering! that! renewable! energy! is! a! kind! of! environmental!friendly!energy!source!and!it!could!be!regarded!as!a!substitute!source!for!fossil!fuel,!forecasting!the!future!consumption!of!renewable!energy!becomes!an!important!topic.! !
With! the! popularity! of! Internet! and! the! development! of! wireless! applications,! some! researchers!believe! that! Internet!plays!an! influential! role! in! consumers’!daily! life! (Molesworth!&!Suortti,!2002).!Through!the!Internet,!consumers!could!easily!collect!new!information!and!learn!new!stuff.!Compared!with!conventional!ways,!Internet!provides!a!more!convenient!and!easier!access!to!the!mostly!updated!information!for!consumers!to!get!familiar!with!a!new!idea!or!new!concept.!Feng!and!Liu!(2013)!stated!that! in! the! information! explosion! age,! consumers’! attention! is! a! scarcity! but! important! resource.!Although! consumers’! searching! behaviors! only! represent! the! possibility! of! consumption,! it! is!reasonable! to! assume! that! there! does! exist! some! relationship! between! consumers’! attention! and!final!demand.!
Along!with!the!Internet,!both!scientists!and!the!Internet!companies!start!to!realize!the!importance!of!data!mining! from! their! users’! searching!behaviors.! For! example,! as! the!world’s!most! visited! search!engine! website 1 ,! Google! Inc.! provides! one! public! web! facility! named! Google! Trends!(https://www.google.com/trends/)! which! presents! searching! frequency! with! linear! graphs.! Within!
1! Source:!http://www.alexa.com/siteinfo/google.com! !
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Google!Trends,!users!can!see!the!searching!frequency!of!a!particular!term!or!keywords!entered!into!the! search! engine! relative! to! the! total! search! volume! all! over! the!world! or! in! one! specific! region.!Searching! frequency! can! be! interpreted! as! an! indicator,! which! represents! the! proportion! of! the!searched! amount! of! a! given! keyword! among! the! total! number! of! searches! on! Google.! So! if! the!searching! frequency! on! particular! terms! or! keywords! increased! in! a! given! period,! people’s! search!interests!on!the!keywords!in!that!period!has!increased!as!well.! !
1.2! Problem!statement!
A! plenty! of! researches! have! focused! on! the! energy! consumption! forecasting! and! various!methods!have! been! applied! in! prediction.! However,! up! to! now,! few! researches! provide! the! quantitative!forecasting! results! on! renewable! energy! consumption! with! the! investigation! on! the! forecasting!performance.!Besides,!a!comparison!test!of!renewable!energy!consumption!among!different!methods!is!still!lacking.!
Moreover,!with! the!popularity!of! Internet,!more!and!more!people!would! like! to!collect! information!through!searching!engine.!Hence,!whether!consumers’! searching!keywords!could!predict! the! future!has!become!a!frontier!of!nowadays!researches!(Markey!&!Markey,!2013).!Researchers!believed!that!searching! records! could! represent! consumers’! intention! and! concerns! (Vosen! &! Schmidt,! 2011).!Hence,! different! from! using! conventional! forecasting! methods! which! only! use! historical! data,!researchers!are!considering!to!combine!the!web!search!volume!such!as!the!data!from!Google!Trends!into!forecasting!models!to!test!whether!there!is!a!more!accurate!forecasting!result.! !
So!far,!some!researches!have!linked!the!web!search!volume!to!forecasting!on!private!goods!such!as!cars! and! houses,! also! the! conclusion! of! those! researches! indicated! that! including! the! web! search!volume!could!provide!a!better!forecasting!result.!However,! few!researches!focused!on!goods!which!are!not!so!directly!related!to!consumers’!purchasing!decisions!such!as!natural!resources!and!energy!consumption.! It! is! unaware! of! attempts! to! test! the! link! between! web! search! volume! and!consumptions!in!the!energy!market.! !
Hence,! along! with! aforementioned! statements,! this! study! aims! to! investigate! and! compare! the!forecasting! performance! among! various!models! and! then! to! test!whether! including! search! volume!could! improve! the! forecasting! result! of! renewable! energy! consumptions.! To! be!more! specific,! this!study! firstly! uses! different! models! to! forecast! the! consumption! of! renewable! energy;! then! after!including! the! web! search! volume! from! Google! Trends! into! conventional! forecasting! model,! the!evaluation!of!how!much!improvement!could!be!gained!is!investigated.! !
1.3! Research!questions!
The!general!research!question!addressed!in!this!study!is:!•! To!compare!the!forecasting!performance!of!various!methods!in!the!case!of!predicting!the!future!
consumption!of! renewable!energy! in! the!U.S.! and!evaluate!how!much! improvement! could!be!gained!when!including!the!web!search!volume!from!Google!Trends!into!forecasting!models.!!
Based!on!this!general!research!question!the!following!specific!research!questions!will!be!considered:!
•! to! select! and! elaborate! the! models! that! could! be! applied! in! forecasting! renewable! energy!
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•! to!compute!and!compare!the!prediction!performance!of!the!selected!models! �•! to!build!the!model!which!includes!the!web!search!volume!and!evaluate!whether!the!forecasting!
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1.4! Thesis!outline!
This! paper! is! proceeded! in! the! following! way.! In! Chapter! 2,! an! overview! of! current! studies! about!energy! consumption! forecasting!with! various!methods! is! summarized! and! a! literature! study! about!forecasting!based!on!the!application!of!Google!Trends!data!is!presented.!Then,!Chapter!3!introduces!three!selected!forecasting!methods!which!applied!to!forecast!renewable!energy!consumption!in!the!empirical!study;!meanwhile!the!method!to!evaluate!the!performance!of!models!is!elaborated!in!this!chapter!as!well.!In!Chapter!4,!the!empirical!study!is!presented!and!the!results!from!selected!models!are! compared! and! analyzed! along! with! the! evaluation! measures.! Last! but! not! least,! the! main!conclusion,!recommendations,!limitations!and!suggestions!for!further!study!are!elaborated!in!the!last!Chapter.!
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Chapter!2!
2.! Literature!study!
2.1!Forecasting!models!for!energy!demand!and!consumption!
Forecasting!has!been!a!popular!topic!in!diverse!fields!and!up!to!now!various!methods!have!been!used!in!forecasting!(Ediger!&!Akar,!2007;!Ahn!et!al.,!2008;!Adams!&!Shachmurove,!2008;!Banos!et!al.,!2011;!Coshall! &! Charlesworth,! 2011;! Choi! &! Varian,! 2012).! From! the! energy! perspective,! Suganthi! and!Samuel! (2012)! summarized! that! an! appropriate! forecasting! will! help! in! planning! the! future!requirement!and!identifying!conservation!planning!of!energy.!Therefore,!a!literature!study!of!energy!forecasting!and!energy!models!is!presented!in!the!following!paragraphs.! !
Regression)models)
Regression! model! is! a! popular! method! used! in! forecasting.! For! example,! Harris! and! Liu! (1993)!investigated! the! dynamic! relationship! between! electricity! consumption! and! price,! weather! and!income! in! south! east! USA! through! building! regression! models.! Yumurtaci! and! Asmaz! (2004)! used!linear! regression!method! to! forecast! the!electricity!demand!of! Turkey!during! the!period!of! 2003! to!2050! based! on! the! population! and! per! capita! consumption! rates.! Usually,! this! method! is! used! to!ascertain! the! causal! effect! of! one! variable!upon!another! (Sykes,! 1993).! Regression!model! describes!how!the!value!of!the!dependent!variable!changes!when!one!of!the!independent!variables!changes.!So!it!has!been!widely!used!to!estimate!the!relationship!among!different!variables.
Grey)prediction)models)
Grey!prediction! is!a!newly!popular!prediction!model! since!a! few!decades!ago.!Suganthi!and!Samuel!(2012)! explained! that! its! increasing! popularity! is! because! the! simplicity! and! ability! to! characterize!unknown! system! by! using! a! few! data! points.! When! applying! this! theory,! Deng! (1999)! stated! that!recent!data!could!represent!the!information!on!reality!and!the!data!for!past!years!were!not!sufficient!to!describe!the!information!on!reality.!That!means,!Grey!theory!emphasizes!the!information!on!reality!by!using!recent!data!and!the!newest! information!plays!the!most! important!role!in!forecasting!(Lin!&!Hsu,! 2002).! It! is! proved! that! some! factors! such! as! income,! Gross! Domestic! Product! (GDP)! and!population! can! influence! the!energy!demand!but!how!exactly! the! interactions! among! those! factors!are!not! clear,! Suganthi!and!Samuel! (2012)! concluded! that!energy!demand! forecasting!could!also!be!regarded! as! grey! system! problem.! Therefore,! the! grey! prediction! has! been!widely! used! in! demand!forecasting.! !
Econometric)models)
Econometric!models! provide! a!method! to! correlate! the!energy!demand!with!other!macroeconomic!variables!such!as!economic!growth,!energy!price,!and!technology!(Suganthi!&!Williams,!2000;! Iniyan!et! al.,! 2006).! Moreover,! some! researchers! take! weather! condition! as! an! important! variable! when!
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doing! the!energy!demand! forecasting!especially! in!predicting!electricity! consumption! (Bianco!et! al.,!2009;! Zachariadis,! 2010).! In! general,! researchers! take! the! GDP,! energy! price! and! population! as!essential!variables!in!energy!demand!prediction!at!macro!level.! !
Time)series)models)
Time! series!models!have!been!described!as! the! simplest! forecasting!models!which!uses! time! series!trend! analysis! to! deduce! the! future! demand! (Suganthi! &! Samuel,! 2012).! Time! series! models! can!provide!the!dynamic!path!of!a!series!to!improve!forecasts!by!including!the!predictable!components!of!series! which! can! be! extrapolated! into! the! future! (Enders,! 2008).! For! example,! Bargur! and!Mandel!(1981)! investigated! the! relationship!between! the!energy! consumption!and! the!economic!growth!by!using! the! trend! analysis.! Aras! and!Aras! (2004)! used! firstGorder! autoregressive!model! to! predict! the!natural!gas!demand!in!Turkey.!The!aim!of!using!time!series!data!in!forecasting!is!to!estimate!how!the!sequence!of!historical!data!will!continue!into!the!future!(Hyndman!&!Athanasopoulos,!2014).!Hence,!time!series!model!is!a!useful!model!to!predict!the!future!based!on!previous!observations.! !
2.2!Forecasting!with!Google!trends!
According!to!Nielsen!Net!Ratings,!Google!has!consistently!been!the!most!widely!used!search!engine,!which!takes!more!than!65%!of!the!total!market!share!in!the!US!(Nielsen,!2010).!Although!the!records!of! the! searching! behavior! through! the! search! engines! seems! disorganized,! researchers! still! believe!that!there!must!be!some!useful!information!behind!such!a!big!database.!Therefore,!they!are!trying!to!mine! the! information! by! including! the! Internet! search! queries! into! nowcasting! and! forecasting!through!various!methods!in!different!fields.! !!Shimshoni! et! al.! (2009)! claimed! that! Google! search! data! provides! a! method! to! observe! people’s!interests!and!to!know!what!is!currently!topGofGmind.!They!investigated!several!categories!and!found!that! some! of! those! categories! have! particularly! high! fraction! with! predictable! queries.! Moreover,!Internet! search! data! has! been! included! in! forecasting! study! in! many! fields! such! as! epidemiology,!macroeconomics,! consumer! behavior,! price! volatility! and! returns! in! financial! markets! (Eysenbach,!2006;!Choi!&!Varian,!2009;!Vosen!&!Schmidt,!2011;!Hamid!&!Heiden,!2015).!!Polgreen! et! al.! (2008)! and!Ginsberg! et! al.! (2009)! concluded! the! clicks! on!web! search! results! could!detect!the! influenza!epidemics.!Wilson!and!Brownstein!(2009)! investigated!the!correlation!between!web! search!queries!and! listeriosis!outbreak! in!Canada.!Base!on! their! result,! the! listeriosis!outbreak!could!be!predicted!one!month!ahead!of!the!federal!official’s!announcement.!Their!studies!show!the!evidence!that!search!queries!could!be!a!good!predictor!especially!with!a!strong!timeliness!for!some!sudden!affairs.! !
Researches! done! by! Choi! and! Varian! (2009),! Askitas! and! Zimmermann! (2009)! and! Suhoy! (2009)!proved! that! aggregated! search! trends! from! Google! Trends! can! also! be! used! as! extra! indicators! in!several!econometrics!prediction!models!in!the!US,!Germany!and!Israel,!respectively.!Kholodilin!et!al.!(2009)! made! a! comparison! on! the! forecasting! accuracy! between! the! conventional! consumer!confidence!index!indicators!and!Google!indicators.!Their!results!show!that!the!forecasting!precision!of!conventional!model!is!lower!than!the!model!with!Google!data.!Later,!Vosen!and!Schmidt!(2011)!did!a!
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similar! study!which! also! included! the! search! query! provided! by!Google! trends! in! their! study.! They!compared!a!simple!autoregressive!model!(AR)!of!consumption!growth!as!a!baseline!model!with!a!new!model! which! was! formulated! via! adding! search! index! from! Google! into! the! baseline! model.! The!results! of! their! study! show! that! the! search! data! could! be! used! as! a! new! indicator! for! private!consumption!forecasting!and!the!Google!indicators!have!stronger!predictive!power!than!conventional!indicators.! Therefore,! combining! information! from!Google! trends! could!be! the!promising! source!of!data!when!predicting!personal!consumptions.!
Wu!and!Brynjolfsson! (2014)!estimated! the! relationship!between!housing!market! indicators!and! the!search!index!from!Google!website!with!a!simple!seasonal!autoregressive!model!by!adding!the!search!index!as!an!explanatory!variable!in!the!AR!model.!Based!on!their!results,!Google!search!is!correlated!with! some!housing!market! indicators! such!as!housing! sales!and! the!house!price! index.!Hence,! they!believed! that! it! is! creditable! to! assume! that! web! search! data! could! be! used! predict! the! future!economic!activities.!
CarrièreGSwallow!and!Labbé!(2013)!used!autoregressive!moving!average!(ARMA)!model!to!investigate!the!relationship!between!the!data!from!Google!Trends!and!the!sales!of!automobile!in!Chile.!By!adding!the!search!index!as!an!extra!independent!variable!into!the!benchmark!framework,!the!results!indicate!that! including! the! Google! index! could! provide! a! better! prediction! result! with! a! lower! root! mean!square! error! (RMSE).! Feng! and! Liu! (2013)! applied! CarrièreGSwallow! and! Labbé’s! forecasting!model!into! the! automobile! market! in! China.! Their! results! supported! CarrièreGSwallow’s! finding! and! they!concluded!that!web!search!volumes!could!capture!consumers’!attention!of!their!daily!life.!
In! stock!market,! web! search! queries! also! have! been! studied! in! different! ways.! Hamid! and! Heiden!(2015)! used! Google! Trends! data! in! Empirical! Similarity! model! to! investigate! the! relation! between!investor’s! behavior! and! stock! market! reaction.! Basistha! et! al.! (2015)! employed! a! vector!autoregressive!(VAR)!model!and!Granger!causality!tests!with!Google!search!volume.!They!investigated!the! volatility! in! six! kinds! of! commodities! (gold,! silver,! copper,! crude! oil,! natural! gas! and! corn)! by!including! the! Google! search! volume,! continuously! compounded! futures! return,! realized! standard!deviation!of!futures!volatility,!and!futures!trading!volume!into!the!model.!The!results!show!that!the!web!search!data!from!Google!could!predict!the!price!volatility!for!these!commodities.! !
Based!on!those!aforementioned!studies,! it!has!been!a!trend!to! include!the!web!search!volume!into!forecasting.!Many!researches!already!proved!that!the!data!from!Google!Trends!could!provide!timely!information!for!some!sudden!affair!and!could!indicate!consumers’!attention!in!their!daily!life.!By!using!the!data! from!Google!Trends,! the! forecasting!results!would!be!more!accurate!and!reasonable!since!the!web!search!data!generated! the!most!upGtoGdata! information! in!nowadays!world.!Hence,! in! this!research,!including!the!web!search!volume!into!forecasting!would!be!a!reasonable!attempt.!
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Chapter!3!
3.! Theoretical!and!methodological!framework!
Time!series!forecasting!is!an!important!part!of!forecasting.!It!collects!and!analyzes!past!observations!and!then!develops!models!based!on!previously!observed!values.!After! that,! the!models!are!used!to!extrapolate! the! future.! In!other!words,! time!series! forecasting!could!be! regarded!as!a!method! that!extract!meaningful! statistics! and! regularity! from! the!historical! data.!Moreover,!because! time! series!models! only! require! historical! data! as! a! variable,! it! is! easy! and! less! costly! in! collecting! data! and!estimating!models! (Song!&!Li,!2008).!This! is!another!good!reason!of!choosing!time!series!models! in!this! research.!Up! to!now,!plenty!of! efforts!have!been!devoted! to!develop!and! improve! time! series!forecasting!models!in!past!decades!(Zhang,!2003).! !
As! summarized! by! Kahforoushan! et! al.! (2010),! exponential! smoothing,! autoregressive! integrated!moving! average! (ARIMA),! autoregressive! conditional! heteroskedasticity! (ARCH)! are! commonly!employed!methods!in!univariate!time!series!models.!Exponential!smoothing!has!been!proved!as!one!of! leading! forecasting!strategies! in!various! fields!such!as!electricity!demand,! tourism!prediction!and!agricultural! product! price! (Lim!&!McAleer,! 2002;! Pao,! 2009;! Assis! et! al.,! 2010).! As! for! ARIMA,! Pao!(2009)! summarized! that! it! has! been! widely! applied! for! modeling! in! medical,! environmental,! and!financial.! Moreover,! Chavez! et! al.! (1999)! implemented! ARIMA! models! to! predict! the! energy!production! and! consumption! in! Northern! Spain.! Ediger! and! Akar! (2007)! forecasted! the! primary!energy!demand! in!Turkey!by!using!ARIMA!and!seasonal!ARIMA!(SARIMA).!Their! research!confirmed!the!suitability!of!ARIMA!models!in!energy!consumption!prediction.!As!for!autoregressive!conditional!heteroskedasticity! (ARCH),! it! has! been! successfully! applied! in! describing! the! volatile! variance! and!predicting!the!energy!demand!as!well!(Pao,!2009;!Bakhat!&!Rosselló,!2011).! !
Hence,! in! this! research,! aforementioned! time! series! methods! are! applied! in! empirical! study! to!investigate!the!forecasting!performances!of!renewable!energy!consumption!in!the!U.S.!
3.1!Exponential!smoothing!
Exponential! smoothing! is! a! very! popular! scheme! to! do! forecasting!with! time! series! data! (Natrella,!2010).! Forecasting! conducts! with! exponential! smoothing! methods! exponentially! decreasing! the!weights! as! the! observations! got! older! (Hyndman!&! Athanasopoulos,! 2014).! That!means,! the!more!recent! observations,! the! higher! value! will! be! considered! when! using! the! past! observations! in!forecasting.!Level,!trend!and!seasonal!are!three!elements!that!need!to!be!considered!before!applying!the!exponential!smoothing!method.!Simple!exponential!smoothing!is!suitable!for!data!with!no!trends!or! seasonal! pattern;! Holt’s! linear! trend! method! could! apply! to! data! which! only! with! trends;!HoltGWinters! seasonal! method! takes! the! level,! trend! and! seasonal! components! into! account! with!smoothing!parameters.!Within!HoltGWinter!seasonal!method,!there!are!two!variations!in!this!method,!named! HoltGWinter! additive! method! and! HoltGWinter! multiplicative! method.! When! the! seasonal!variations!could!be!roughly!considered!as!constant!through!the!series,!the!additive!method!would!be!a!good!choice!to!build!the!model.!However,! if! the!seasonal!variations!changes!with!a!proportion!to!
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the!level!of!a!series,!then!the!multiplicative!method!is!preferred!(Hyndman!&!Athanasopoulos,!2014).!For! both! additive! method! and! multiplicative! method,! there! are! three! smoothing! equations! and!smoothing!parameters:!one!equation!for!the!level! !",!one!for!trend! #",!one!for!seasonal!component!$"! and!smoothing!parameters! %, '! and! (.!The! )*represents!the!seasonality!and! +"*represents!the!observed!value!at!time! ,.!
For!additive!method,!the!revised!estimates!are:!
!" = % +" − $"/0 + 1 − % !"/3 + #"/3 !
#" = ' !" − !"/3 + 1 − ' #"/3!
$" = ( +" − !"/3 − #"/3 + 1 − ( $"/0!
and!the!formula!for!forecasting!is:!
+"45 " = !" + ℎ#" + $"/04578 !
where! ℎ09 = ℎ − 1 *mod*) + 1,! and! ℎ ! represents! the! forecasting! horizon.! As! for! the! error!correction!form!of!the!smoothing!equation!is:!
!" = !"/3 + #"/3 + %="!
#" = #"/3 + %'="!
$" = $"/0 + (="!
where! =" = +" − !"/3 + #"/3 + $"/0 ! represents!the!oneGstep!training!estimate!errors.! !
For!multiplicative!method,!the!revised!estimates!are:!
!" = %+"$"/0
+ 1 − % !"/3 + #"/3 !
#" = ' !" − !"/3 + 1 − ' #"/3!
$" = (+"
!"/3 + #"/3+ 1 − ( $"/0!
and!the!formula!for!forecasting!is! !
+"45 " = !" + ℎ#" ∗ $"/04578 !
and!the!error!correction!form!of!the!smoothing!equation!is:!
!" = !"/3 + #"/3 + %="$"/0
!
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#" = #"/3 + %'="$"/0
!
$" = $"/0 + (="
!"/3 + #"/3!
where! =" = +" − !"/3 + #"/3 ∗ $"/0.!
In! Eviews,! the! estimation! value! and! forecasting! results! could! be! generated! automatically.! The!forecasting! performance! could! be! evaluated! through! the! forecast! error! measurement! which! is!elaborated!in!following!paragraphs.!
3.2!ARIMA!
As! one! of! the!most! popular! models! for! time! series! forecasting! analysis,! autoregressive! integrated!moving!average!(ARIMA)!is!originated!from!the!autoregressive!model!(AR),!the!moving!average!model!(MA)!and!the!combination!of!AR!and!MA!(Blanchard!&!Desrochers,!1984;!Ho!et!al.,!2002;!Ediger!et!al.,!2006;!Ediger!&!Akar,!2007).!Ediger!et!al.!(2006)!summarized!that!the!ARIMA!could!be!considered!as!a!stepGbyGstep!procedure!for!ARMA.!It!combines!the!AR!coefficients!which!are!multiplied!by!historical!value!of!the!time!series!data!and!MA!coefficients!which!are!multiplied!by!historical!random!error!term.!When! applying! the! ARIMA! analysis,! time! series! should! be! stationary.! That! means,! the! mean! and!variance!of!the!series!are!supposed!to!be!constant!throughout!time!and!the!value!of!covariance!only!depends!on!the!gap!between!two!time!periods.! !
Generally,!the!full!ARIMA!model!could!be!written!as:!
+"9 = ? + @3+"/39 + ⋯+ @B+"/B9 + C3="/3 + ⋯+ CD="/D + =")
where! +"9! is!the!differenced!series,! E! is!the!order!of!the!autoregressive!part,! F! is!the!order!of!the!moving! average! part,! and! G! represents! the! degree! of! the! differencing! involved.! Box! and! Jenkins!(1976)! stated! that!ARIMA! technique! contains! identification,! estimation! and!diagnostic! checking.! To!make!a!clear!description!for!the!process!of!building!ARIMA!model,!this!research!refers!the!procedure!stated!by!Sato!(2013)!with!the!following!schematic!diagram.!
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!
Figure!3.2!Scheme!for!the!use!of!ARIMA!model!
3.2.1! Phase!1G!Identification! !
•! Stationary)and)unit)root)test)
When!applying!the!ARIMA!model,!the!raw!data!needs!a!stationary!transforming.!In!this!research,!the!concept! of! “weak! stationary”! is! adopted.! “Weak! stationary”! means! the! mean,! variance,! and!covariance!of!the!time!series!are!independent!of!time!and!could!be!well!predicted!by!sufficiently!long!time!averages!based!on!present!value!(Ediger!et!al.,!2006;!Enders,!2008).! !
Unit!root!test!is!used!to!examine!the!stationarity!of!a!time!series.!If!there!is!a!unit!root!in!a!time!series,!then! it!means!the!series! is!nonstationary!and! it!needs!to!be!transformed!into!a!stationary!series!by!
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removing!the!unit!root!before!applying!the!ARIMA!model.!Normally!it!could!be!achieved!by!replacing!the!time!series!with!a!first!or!second!difference.!In!this!research,!to!test!whether!there!is!a!unit!root!in!time!series,!the!most!notable!and!widely!used!test!named!the!augmented!DickeyGFuller!(ADF)!test!is!employed!(Chen!et!al.,!2009).! )
•! Autocorrelation)function)and)partial)autocorrelation)function) )
The!autocorrelation! function! (ACF)!and! the!partial! autocorrelation! function! (PACF)!are! two! tools! to!explore! time! series! data.! ACF! provide! a! way! to! identify! the! seasonality,! cycles! and! other! patterns!within!a!time!series!which!helps!researchers!to!understand!the!association!of!the!information!from!a!prior!period!and!the!sequential!observation!(Sato,!2013).!The!ACF!at!lag! H,!denoted!by! IJ! could!be!defined!as:!
IJ =(J(K!
where! (J! represent!the!covariance!at!lag! H,! (K! is!the!variance!and! IJ! is!a!number!lies!between!G1!and!+1!(Erdogdu,!2007).!
PACF!is!a!tool!to!identify!the!extent!of!lags!in!an!AR!model!which!enables!the!estimation!of!the!fitness!degree!of!the!variable!with!its!previous!values.!In!a!time!series,!the!correlation!between! L"! and! L"/J!are! mainly! from! the! correlation! with! the! intervening! lags,! i.e.,! L"/3, L"/M, … , L"/J43.! The! partial!autocorrelation! measures! the! correlation! between! L" ! and! L"/J ! after! eliminating! the! effect! of!intermediate! L’s.! !
If! the! time! series! data! has! been! made! as! a! stationary! series,! then! based! on! the! ACF,! PACF! and!resulting!correlogram,!the! E, G*and*F! of!ARIMA!could!be!approximately! identified.!By!using!Eviews,!the!ACF!and!PACF!values!are!automatically!calculated!and!presented!in!correlogram!table.! !
3.2.2! Phase!2G!Estimation!and!Testing! !
•! Estimation) )
In! this! step,! the! estimation! starts! with! the!models! which! is! built! on! the! results! in! the! first! phase.!Because! the! identification! process! of! the! E, G*and*F! of! ARIMA! is! based! on! a! subjective! judgment,!hence,!usually!there!are!more!than!one!ARIMA!model!that!could!work!for!a!data!series.!To!conduct!the!parameter!estimation!in!this!research,!least!squared!(LS)!is!applied!in!Eviews.!Moreover,!Akaike’s!Information!Criterion!(AIC)!and!Schuwarz!Bayesian! Information!Criterion!(BIC)!are!used!as!criterions!to!select!the!predictors!for!regression!and!to!determine!the!fittest!model.! !
•! Akaike’s!Information!Criterion!(AIC)!
The!AIC!is!calculated!with!the!following!formula:!AIC = −2 log W + 2(E + F + H + 1)*
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where! W! is!the!likelihood!of!the!data,! H = 1! if!the!constant!of!ARIMA!(?)!is!unequal!to!0!and! H = 0!if! ?! equals!to!0.!
•! Schwarz!Bayesian!Information!Criterion!(BIC)!
The!BIC!can!be!written!as:!
BIC = AIC + (log \ − 2)(E + F + H + 1)!
where! \! is!the!length!of!the!time!series.! !
To!choose!a!better!model,!either!the!AIC!and!BIC!are!supposed!to!be!minimized.!In!other!words,!the!lower! value! of! AIC! and! BIC! are! desirable! within! a! model.! The! value! of! AIC! and! BIC! could! be!automatically!calculated!through!the!Eviews!software.!
•! Testing)
After! selecting! the! fittest! model,! the! analysis! of! residual! needs! to! be! performed.! In! a! wellGfitted!ARIMA!model!with!purely! random!residues,! there! is! supposed! to!have!no!existence!of!a! significant!autocorrelation! or! a! partial! autocorrelation! between! residues.! However,! if! there! are! statistically!significant! in! the!model,! the!process!of! identification!needs!to!repeat!again! to!assess!other!existing!patterns!until!the!residuals!present!a!“white!noise”!pattern!(Sato,!2013).!Once!the!ACF!and!PACF!of!the!residual! in! the!model!show!nonGsignificance,! that!means! this!ARIMA!model!could!be!applied! to!conduct!further!study.! !
3.2.3! Phase!3G!Application! !
In!this!stage,!the!selected!ARIMA!(E, G, F)!model!is!employed!to!do!the!inGsample!test.!Moreover,!the!outGofGsample!forecasting!is!conducted!and!the!adequacy!of!model!is!evaluated!as!well.!The!detailed!evolution!criterion!is!elaborated!in!following!paragraphs.! !
3.3!Seasonal!dummy!and!ARCH!model!
Dummy!variables!are!widely!used!in!measuring!average!differences!when!comparing!discrete!groups!(Gujarati,!2009).!However,!it!could!also!be!used!to!deal!with!data!which!has!a!seasonality!(Tang!et!al.,!2008;!Pao,!2009).!By!using!dummy!variables,! the!seasonal!variation! in!the!data!could!be!picked!out!and!controlled.!It!is!achieved!by!including!a!set!of!dummy!variables!for!each!interval!which!will!then!net!out!the!average!change! in!the!series!resulting!from!any!seasonal! fluctuations.! If! the!time!series!has!the!seasonality,!the!seasonal!dummy!model!could!be!considered!to!deal!with!seasonal!effects.!
The!seasonal!frequency!could!be!represented!as! $.!When! $! equals!to!4,!it!means!the!time!series!has!a!quarterly!seasonal!frequency.!If!the!series!consists!of!monthly!data,!the! $! is!equal!to!12.!Then!let!D3^, DM^, D_^, … , D`^! be! seasonal! dummies.! For! example,! D3^! =! 1! if! s! is! the! first! period,! otherwise!D3^! =!0.!At!any!time!period,!one!of!the!seasonal!dummies! D3^, DM^, D_^, … , D`^! is!set!equal!to!1!and!all!the!others!equals!to!0.!
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After! setting! seasonal! dummies,! the! model! could! be! estimated! by! LS! regression! like! following!expression:!
+"45 = % + 'bcb"
d/3
be3
+ f")
In! Eviews,! to! test! a! monthly! data,! 11! seasonal! dummy! variables! are! needed! to! build! a! seasonal!dummy!model.!After!building!the!model,! the!ARCH!effect!should!be!conducted!to!test!whether!the!model! could! be! improved! or! not.! ARCH! is! the! acronym! of! Autoregressive! conditional!heteroscedasticity! which! was! introduced! by! Engle! (1982).! Bollerslev! (1986)! and! Taylor! (1986)!generalized!the!models!as!Generalized!ARCH!(GARCH).!The!GARCH!model!has!been!described!as!an!approach! to!modeling! time! series!with! heteroscedastic! errors! (Pao,! 2009).! Both! ARCH! and!GARCH!models! are! widely! used! to! describe! volatile! variance.! Moreover,! Cheong! (2009)! summarized! that!ARCH! models! performed! successfully! in! the! applications! of! financial! markets! such! as! measuring!investment! risk! and! pricing! financial! derivatives.! In! the! field! of! energy! consumption,! Pao! (2009)!applied!this!model!into!forecasting!the!electricity!consumption!and!petroleum!consumption!in!Taiwan.!His!research!reveals!that!GARCH!model!could!also!have!a!good!performance!in!energy!consumption!prediction.! !
The!standard!form!of!GRACH!(F, E)!models!is!presented!as!following!equations:!
Y^ = X^9θ + ϵ^*
k"M = l + %b
B
be3
f"/bM + 'm
D
me3
k"/bM !
where! F! is!the!order!of!the!autoregressive!GARCH!terms!and*E! is!the!order!of!the!moving!average.!The!mean!equation!is!written!as!a!function!of!exogenous!variables!with!an!error!term!as!presented!in!the! first! equation.! The! σ^M! is! called! the! conditional! variance!which! represents!oneGperiod!ahead!of!forecast! variance!based!on!past! information.!As! presented! in! the! second!equation,! the! conditional!variance!equation!is!consisted!of!three!terms!which!are! l,!a!constant!term;! f"/bM ,!information!about!volatility! from! the! previous! periods,! measured! as! the! lag! of! the! squared! residual! from! the! mean!equation;!and! k"/mM ,!the!last!period’s!forecast!variance.! !
To!sum!up,!when!building!a!ARCH!model!for!time!series!data,!there!are!four!steps!to!follow.!Firstly,!specify!a!mean!equation!by!investigating!the!linear!dependence!in!the!series.!Secondly,!do!the!ARCH!effects!test!based!on!the!residuals!of!the!mean!equation!and!specify!the!order!of!ARCH!to!be!tested!against.! Then,! the! volatility! model! is! specified! if! ARCH! effects! are! statically! significant! and! a! joint!estimation!of!the!mean!and!volatility!equation!is!performed.!At!last,!after!checking!the!fitted!model,!it!could!be!applied!into!forecasting!and!the!performance!need!to!be!evaluated.! !
3.4!Web!search!index!construction!
In! order! to! include! the! web! search! index! into! forecasting! model,! the! data! series! generated! from!Google! Trends! needs! to! be! transformed! into! web! search! volume! index! (SVI).! On! the! website! of!Google!Trends,! it!provides!the!database!with!keyword’s!searching!frequency!as!a!CSV!file!which!can!
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be!download!directly! from!the!website.!Considered!this! research! is!based!on!the!monthly!data!and!the! availability! of! database,! the!web! search! volume! data! from!Google! Trends! is! collected!monthly!between! the! period! of! January,! 2004! to! December,! 2015.! Besides,! Google! Trends! centralizes!searching! queries! made! in! different! countries.! That! means,! when! download! the! data! from! the!website,!the!searching!area!could!be!narrowed!down!to!one!specific!region!or!country.! It!will!make!the!web!search!volume!data!more!representative!in!the!analysis!(Fondeur!&!Karamé,!2013).! !
To!build!the!web!search!volume!index,!the!first!thing!is!to!collect!the!web!search!volume!data.!At!this!stage,! the! keywords!need! to!be! selected! and! then! the!web! search! volume!data! could!be! collected!through!Google!Trends.! !
As!for!keywords!selection,!Liu!et!al.!(2014)!summarized!currently!there!is!no!unified!theory!or!method!could! apply.! Based! on! tons! of! literature! studies,! they! concluded! that! there! are! generally! three!methods.!One!is!to!build!an!informationGseeking!system!and!include!all!possible!searching!keywords!into! the! system.! After! that,! researchers! need! to! define! the! selection! criterions! and! reduce! the!dimensionality! as! programming! language! to! pick! out! the! most! suitable! keywords.! In! fields! of!sentiment! analysis,! it! is! a! popular! method! to! conduct! data! mining.! For! example,! Pak! &! Paroubek!(2010)!built!an!informationGseeking!system!with!a!corpus!of!300000!text!posts!from!Twitter.!Also,! it!was!applied!by!Ginsberg!et!al.!(2009)!who!selected!keywords!through!50!million!search!queries!on!all!possible!topics!about!influenza!epidemics!with!Google!Trends.!This!method!has!a!high!accuracy!but!it!is!also!with!a!high!threshold!and!large!workload!since!it!requires!researchers!have!a!strong!technical!knowledge!on!programming.! !
The!second!method!is!selecting!keywords!based!on!the!categories!provided!by!Google!Trends.!When!applying! this!method,! researchers! need! to! define! a! general! scope!of! possible! factors! in! researches!and!find!the!most!relevant!queries!based!on!the!categories!in!Google!Trends.!For!example,!Choi!and!Varian! (2012)! included! two! categories! named! Trucks! &! SUVs! and! Automotive! Insurance! into!investigation!when!analyzing!the!series!of!motor!vehicles.!Their!results!proved!that!the!performance!of! inGsample!test!has!been!significantly! improved.!The!workload!of!this!method!is!bearable!but!to!a!great!extent,!it!depends!on!the!categories!provided!by!Google!Trends.!If!the!categories!are!omitted!or!not!listed!separately,!it!is!would!be!difficult!to!use!this!method.! !
The! last!method! is! to! select! the! keywords! directly! based! on! researchers’! experience.! Because! this!method!has!the!advantages!of!the!least!workload!and!the!simplest!operation,!it!has!been!employed!by! researchers! in! fields! of! unemployment! predication! with! Google! Trends! in! Germany! (Askitas! &!Zimmermann,!2009;!D'Amuri,! 2009).!After! that,!D’Amuri! and!Marcucci! (2010)! simply!used!keyword!“jobs”! in! their! study! of! unemployment! in! the! United! States.! Their! researches! indicate! that! it! is!acceptable!to!select!keywords!directly!based!on!researchers!experience.!While!when!choosing!words!subjectively,! there! are! still! some! suggestions! to! follow.! Researches! investigated! the! relationship!between!the!number!and!quality!of!searching!keywords!when!doing!forecasting!with!search!queries!(Hulth!et!al.,!2009;!Sun,!2012).!They!summarized!that!keywords!with!a!higher!searching!volume!would!contribute! more! in! forecasting! than! the! keywords! with! a! relatively! lower! searching! volume.! That!means,! it! would! be! reasonable! to! firstly! consider! the! keywords! with! a! high! search! volume! when!selecting!the!searching!keywords.!Moreover,!Sun!(2012)!also!summarized!that!people!prefer!simple!phrase! than! complicated! sentences! in! web! search,! for! example,! when! people! want! to! know!
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something! about! “stock”,! usually! they! would! type! “stock”! into! search! engine! instead! of! “what! is!stock”.!Hence,!when!selecting!the!keywords,!simple,!concise!and!precise!words!or!phrases!with!high!search!volume!is!preferred.! !
In! this! research,! the! empirical! study! focuses! on! “renewable! energy”.! Considered! the! limitation! on!knowledge,! it! is! difficult! to! select! the! keywords! by! using! the! informationGseeking! programming!method.!Besides,!there!is!no!specific!category!with!focuses!on!renewable!energy!in!Google!Trends,!so!selecting! keywords! based! on! the! categories! is! not! suitable! in! this! case.! Hence,! the! keywords! are!selected! by! experience! in! this! study.! In! the! beginning,! several! keywords! related! to! the! renewable!energy! are! tried,! they! are! “renewable”,! “renewable! energy”,! “renewable! resource”,! “sustainable!energy”!and!“sustainable! resource”.!Because!energy,! resource,! renewable,!and!sustainable!are! four!common!words!that!used!in!energy!industry,!the!selection!firstly!started!with!all!of!the!combinations!of!these!keywords.!Among!those!five!keywords,! the!first! two!keywords!are!considered!the!simplest!and! popular! expression! of! renewable! energy! and! they! are! supposed! to! have! a! higher! searching!volume.!This!assumption!has!been!confirmed!by!the!graph!from!Google!Trends!as!presented!in!Figure!3.4.!
!
Figure! 3.4! Search! volume! for! keywords:! renewable! (blue),! renewable! energy! (red),! renewable! resource!
(yellow),!sustainable!energy!(green)!and!sustainable!resource!(purple).!
According! to! the! figure! from!Google! Trends,! “renewable”! and! “renewable! energy”! are! on! the! top!among!five!lines!which!means!they!have!a!considerable!amount!of!search!volume!compared!to!other!three! keywords.! Since! the! searching! keywords! with! high! search! volume! is! preferred,! the! keyword!would!be!considered!between!“renewable”!and!“renewable!energy”.!As!presented!above,!the!search!volume!of! “renewable”! and! “renewable! energy”! has! the! similar! trends! at!most! of! time.!Compared!with!keyword!“renewable”,!“renewable!energy”!seems!to!be!the!best!choice!for!the!purpose!of!this!research.!On! the!one!hand,! in! the!empirical! study,! the! series!data! is! focused!on! renewable!energy!which! exactly! matches! the! keyword! ”renewable! energy”.! On! the! other! hand,! Google! search!associated!with!this!keyword!is!expected!to!be!directly!connected!to!the!introductions!of!renewable!energy!because! it! is! the!most!precise!and!simplest!way!to!find!relevant!knowledge!about! it.!Hence,!with!aforementioned!discussion,!the!keyword!in!this!empirical!study!is!selected!as:!renewable!energy.! !
After!gathering!the!web!search!volume!data!from!website,!the!next!thing!is!to!transform!the!raw!data!into!applicable!data!which!could!be!used!into!building!model.!Taking!the!ARIMA!model!as!an!example,!if!we!want!to!include!the!web!search!volume!data!as!an!explanatory!variable!into!model,!those!data!is!
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required! to!be!stationary!by!doing!differencing!or!other! transforming.! In! this! research,! the!building!process!of!SVI!is!elaborated!in!the!empirical!study!part.! !
3.5!Forecasting!error!measurement! !
To! test! the!performance!of! the! forecasting!model,! the!adjusted! RM,! the!mean!absolute!percentage!error! (MAPE),! the!mean! absolute! error! (MAE)! and! the! root!mean! squared! error! (RMSE)! are! useful!tools!to!evaluate!the!results.!GoodnessGofGfit!(RM)!is!the!calculation!result!of!the!differences!between!the!predicted!values!for!the!variable!obtained!and!the!observed!values.!It!has!been!considered!as!the!most! important! concept! in! regression!analysis! (Wei,!1994).!Adjusted! RM! adjusts! the! statistic!of! RM!based! on! the! number! of! independent! variables! in! the!model.! A! high! adjusted! RM! value! indicate! a!good!correlation!between!variables.!Therefore,!by!examining!adjusted! RM,! the!quality!of! regression!analysis!could!be!determined!(Ediger!et!al.,!2006).!As! for!MAPE,!MAE!and!RMSE,!the!value!of! them!could!be!calculated!respectively!with!following!formula:!
MAPE =1n
y^ − y^y^
s84t
^es843
*
MAE=1n
y^ − y^
s84t
^es843
*
RMSE =1n
y^ − y^ M
s84t
^es843
*
where! \9! denotes!the!sample!size!of!model!estimation,! w! is!the!size!of!outGof!sample!in!forecasting!period,! +"*! is! the! value! from! the! forecasting!model! and! +"! represents! the! value! of! observations.!Usually!the!statistical!software!calculates!value!of!adjusted! xM,!MAPE,!MAE!and!RMSE!automatically.!To!judge!the!adequacy!of!the!model,!the!lower!value!of!MAPE,!MAE!and!RMSE!are!desired.!
!
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Chapter!4!
4.! Empirical!analysis! !
In! first! three! sections,! three! types! models! are! built,! named! HoltGWinter’s! model,! seasonal! ARIMA!model,! Seasonal! GARCH! model.! The! performance! of! the! alternative! modeling! approaches! is!compared!by!using!monthly! time! series! data:! renewable! energy! consumption! in! the!United! States.!This!time!series!data!of!renewable!energy!consumption! is!collected!through!the!website!of!the!U.S.!Energy!Information!Administration.!In!the!last!section,!the!performance!of!the!ARIMA!model!with!the!web! search! volume! index! from!Google! Trends! is! presented.! The! series! of!web! search! volume!with!keyword:! renewable! energy! is! gathered! from! Google! Trends! and! the! region! of! data! collection! is!narrowed!down!to!“the!U.S.”.!
In!this!empirical!analysis,!the!period!extended!from!January!2004!to!December!2015!with!a!total!of!144! observations! in! each! series.! The! forecasting! horizon! in! this! research! is! determined! as!monthly!which! is! consistent! with! the! frequency! of! the! series! of! data.! The! period! from! January! 2004! till!December! 2014! is! used! as! the! training! period! for! the! models.! The! period! from! January! 2014! to!December!2014!is!treated!as!inGsample!testing!period!and!January!2015!to!December!2015!is!treated!as!outGofGsample!testing!period.! !
In!this!study,!the!series!of!renewable!energy!consumption! is!denoted!by! y".!As!for!the!data!of!web!search!volume,!it!is!denoted!by!SVI.!
As! shown! in! Figure! 4.1,! the! time! series! data! of! the! United! States’! renewable! energy! consumption!shows!growth!trends!and!seasonality.! !
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!
Figure!4.1!Total!renewable!energy!consumption! !
4.1!Exponential!Smoothing!
The! renewable! energy! consumption! series! y",! as! presented! in! Fig.! 4.1,! assumes! the! growth! trends!and!seasonality!will! last! to! the! future!with! the!same!pattern.!HoltGWinters!additive!seasonal!model!and! multiplicative! seasonal! model! consist! with! three! estimated! parameters! as! presented! in! Table!4.2.1! and! Table! 4.2.2.! The! zero! value! for! Beta! and! Gamma! in! the! table!mean! that! the! trend! and!seasonal!components!are!estimated!as!fixed!and!not!changing.!The!tables!also!display!the!level,!trend!and!seasonal!factors.!
Table!4.2.1!Eviews!output!of!HoltBWinters!additive!seasonal!model!
Method:!Holt+Winters!Additive!Seasonal! !
Parameters:! Alpha! ! 0.60!
! Beta! ! 0.00!
! Gamma! ! 0.00!
Sum!of!Squared!Residuals! ! 55742.01!
Root!Mean!Squared!Error! ! 20.55!
End!of!Period!Levels:! Mean! 819.20!
! !Trend! 2.47!
! !Seasonals:! 2014M01! 24.86!
! ! ! 2014M02! +45.87!
! ! ! 2014M03! 24.55!
! ! ! 2014M04! 25.65!
! ! ! 2014M05! 60.34!
! ! ! 2014M06! 51.37!
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! ! ! 2014M07! 23.16!
! ! ! 2014M08! +18.94!
! ! ! 2014M09! +73.30!
! ! ! 2014M10! +47.61!
! ! ! 2014M11! +34.54!
! ! ! 2014M12! 10.36!
! ! ! ! !
!Table!4.2.2!Eviews!output!of!the!HoltBWinters!multiplicative!seasonal!model!
Method:!Holt+Winters!Multiplicative!Seasonal!
Parameters:! Alpha! ! 0.53!
! Beta! ! 0.00!
! Gamma! ! 0.00!
Sum!of!Squared!Residuals! ! 51420.68!
Root!Mean!Squared!Error! ! 19.74!
End!of!Period!Levels:! Mean! 823.63!
! ! Trend! 2.47!
! ! Seasonals:! 2014M01! 1.04!
! ! ! 2014M02! 0.93!
! ! ! 2014M03! 1.04!
! ! ! 2014M04! 1.05!
! ! ! 2014M05! 1.09!
! ! ! 2014M06! 1.08!
! ! ! 2014M07! 1.04!
! ! ! 2014M08! 0.97!
! ! ! 2014M09! 0.89!
! ! ! 2014M10! 0.93!
! ! ! 2014M11! 0.94!
! ! ! 2014M12! 1.02!
! ! ! ! !
The!actual!values!(renewable)!and!the!smoothed!forecasts!with!HoltGWinters!additive!seasonal!model!(esmoothingad)!and!HoltGWinters!multiplicative!seasonal!model!(esmoothingmu)!are!generated!on!a!single!graph:!
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!Figure! 4.2! Actual! renewable! energy! consumption! (the! green! line),! forecasting! value! on! renewable! energy!
consumption!with!HoltBWinters!additive!seasonal!model!(the!red!line)!and!HoltBWinters!multiplicative!model!
(the!blue!line)!
!Judging!from!the!Figure!4.2,!both!the!HoltGWinters!additive!seasonal!model!and!multiplicative!model!performed! well! of! tracking! the! seasonal! movements! with! the! actual! series.! The! best! model! of!HoltGWinters! models! is! selected! based! on! the! lowest! value! of! MAPE,! MAE! and! RMSE.! The!performance!results!are!presented!in!Table!5.1!in!Chapter!5.!According!to!the!table,!the!result!shows!that! the! HoltGWinters! additive! model! did! a! better! job! of! forecasting! the! renewable! energy!consumption!in!the!future!with!lower!values!of!MAPE,!MAE!and!RMSE,!hence,!the!best!model!applied!in!this!series!is!HoltGWinters!additive!model.! !
4.2!ARIMA!
The!first!step!to!apply!ARIMA!model!is!to!make!sure!the!time!series!data!is!stationary.!It!is!tested!by!using!Augmented!DickeyGFuller! test!with!hypothesis! zK:! data!contains!a!unit! root.! In! this! research,!the! total! renewable! energy! consumption! is! denoted! as! renewable! in! Eviews.! The! rest! starts! at! a!rather!general! level!by! including!a!high!number!of! lags!(12)!and!a!deterministic!trend.!The!ADF!test!output!is!shown!in!Table!4.3.12.! !
Table!4.3.1!ADF!statics!results!for!series!renewable!
Null!Hypothesis:!RENEWABLE!has!a!unit!root! !
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Exogenous:!Constant,!Linear!Trend! !
! ! ! t+Statistic! !!Prob.*!
Augmented!Dickey+Fuller!test!statistic! +2.26! !0.4537!
Test!critical!values:! 1%!level! ! +4.03! !
! 5%!level! ! +3.44! !
! 10%!level! ! +3.15! !
*MacKinnon!(1996)!one+sided!p+values.! !
!At! this! point! the! ADF! test! statistic! is! larger! than! all! critical! values,! which! means! the! zK ! of!nonGstationary! is! accepted.! In! other! words,! the! ADF! test! indicates! that! the! untransformed! data! is!nonGstationary.!To!solve!this!problem,!firstly!we!do!a!natural!logarithm!processing!of!this!series!data!(lrenew).! Since! the!data! shows!a! strong! seasonal! regularity!with!12!months’! lag,! the! lrenew! series!then! be! dealt! with! a! seasonal! difference! with! 12! lags! (llrenew).! After! the! first! difference! of! the!seasonal!difference,!the!ADF!test!indicate!that!the!firstGdifferenced!llrenew!series!is!stationary!(Table!4.3.23),! so! the! time! series! of! logarithm! value! for! renewable! energy! consumption! (lrenew)! is!integrated!of!order!one.!
Table!4.3.2!ADF!statics!results!for!series!llrenew!
Null!Hypothesis:!D(LLRENEW)!has!a!unit!root! !
Exogenous:!Constant,!Linear!Trend! !
! ! ! t+Statistic! !!Prob.*!
Augmented!Dickey+Fuller!test!statistic! +5.30! !0.0001!
Test!critical!values:! 1%!level! ! +4.05! !
! 5%!level! ! +3.45! !
! 10%!level! ! +3.15! !
*MacKinnon!(1996)!one+sided!p+values.! !
!To!investigate!the!specification!of!the!ARIMA!model,!the!next!step!is!to!consider!the!autocorrelation!(AC)! and! partial! autocorrelation! (PAC)! of! series! llrenew.! In! Eviews,! by! doing! the! series! statistic! of!correlogram,!the!AC!and!PAC!can!be!calculated!as!presented!in!the!following!Table!4.3.3.!!!!!!!!!!!!!
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Table!4.3.3!Correlogram!QBstatistics!test!results!for!series!llrenew! !
!According! to! the! correlogram! of! the! series! llrenew! presented! in! Table! 4.3.3,! the! autocorrelation!breaks! off! after! the! first! autocorrelation! and! the! partial! correlations! also! cuts! off! after! first!autocorrelation.!Such!a!pattern!in!the!correlogram!suggests!the!model!could!be!AR(1)!AR(2)!and!MA(1)!MA(2)!with!seasonal!differencing!SAR(12)!SMA(12)!model.!After!a!few!comparison!and!selection,!the!best!model!is!selected!which!with!the!lower!AIC,!BIC!values!and!the!higher!adjustedGR2!value.!!Table!4.3.3!Correlogram!QBstatistics!test!results!for!series!lrenew! !
! ARIMA!(1,1,1)! ARIMA!(2,1,1)! ARIMA!(1,1,2)! ARima(2,1,2)!
AdjustedBR2! 0.52! 0.54! 0.52! 0.54!
AIC! G4.10! G4.13! G4.05! G4.11!
BIC! G4.00! G3.99! G3.93! G3.96!
!
As!presented!in!Table!4.3.3,!the!ARIMA!(1,1,1)!model!has! lowest!BIC!value!in!the!four!models!while!the!ARIMA!(2,1,1)!model!has!both!the!highest!value!of!AdjustGR2!and!lowest!AIC!value!among!all!of!the!models.!However,!the!values!are!just!slightly!differenced!with!each!other!and!normally!models!with!the!smallest!possible!number!of!terms!are!preferred,!hence,!in!this!situation!the!ARIMA!(1,1,2)!model,!ARIMA! (2,1,1)! model! and! the! ARIMA! (2,1,2)! model! are! dropped! in! this! selection.! After! selection!model,!the!residual!diagnostics!is!conducted!to!prove!that!there!is!no!autocorrelation!existed!in!the!residual.! The! residual! diagnostics! result! shows! that! the! residual! of! ARIMA! (1,1,1)! is! a! whiteGnoise!series 4 ,! therefore,! this! model! could! be! applied! in! forecasting.! The! estimation! of!ARIMA(1,1,1)×(1,1,1)12!is!presented!below
5:!
1 − 0.24L 1 + 0.07L3M z^ = (1 − 0.66L)(1 − 0.90L3M)e^*
where! L! denoted!the!lag!operator,! Ltz^ = z^/t! and! Lte^ = e^/t.!
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or!equivalently,!to:!
z^ = 0.24z^/3 − 0.07z^/3M + 0.0168z^/3_ + e^ − 0.66e^/3 − 0.9e^/3M + 0.594e^/3_!
It!should!be!noted!that!the!ARIMA!(1,1,1)*×! (1,1,1)12!model!is!build!on!the!series!of!natural!logarithm!value!of!actual!value.!Hence,!here!the! y"! represents!the!natural! logarithm!value!and!it!needs!to!be!transformed!by!the!inverse!function!when!doing!forecasting.! !
In! the! phase! of! application,! the! selected! ARIMA! (1,1,1)*×! (1,1,1)12!model! is! applied! in! forecasting.!After! gathering! the! estimation! and! transforming,! the! forecasting! is! conducted! as! present! in! Figure!4.3.1.!
!Figure! 4.3.1! Actual! renewable! energy! consumption! (the! blue! line),! forecasting! on! renewable! energy!
consumption!with!ARIMA!(1,1,1)*×! (1,1,1)12!model!(the!red!line)!
!As!presented! in! this! figure,! the! forecasting!value!of! renewable!energy!consumption! in!year!2014! is!close!to!the!real!value.!In!2015,!the!forecasting!value!in!second!quarter!is!higher!than!the!actual!value,!while!in!first!quarter!and!fifth!quarter!the!forecasting!value!is!quite!close!to!the!actual!value.!To!get!a!precise! estimation! of! the! performance! of! this! model,! the! evaluation! of! MAPE,! MAE! and! RMSE! is!calculated!and!presented!in!Table!5.1!in!Chapter!5.!
!
!
!
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4.3!Seasonal!dummy!and!ARCH!model!
According!to!the!performance!of!both!HoltGWinter’s!model!and!ARIMA!model,!it!is!proved!that!there!is!a!strong!seasonality!existed!in!the!time!series!data!of!renewable!energy!consumption.!Hence,!in!this!section,!the!seasonal!dummy!variables!are!included!in!this!analyzing!to!control!the!seasonality!of!the!series.!As!presented! in! Figure!4.2,! this! series! shows!an! increasing! trend,! therefore,! the!model!with!trends!and!seasonal!dummy!variables!is!expressed!as!follows:!
z^ = αK + α3trend + D3x`3,^ + DMx`M,^ + ⋯+ D3Kx`3K,^ + D33x`33,^ + ϵ^!
and! !
ãd3," =10**if*period*,*is*January…
otherwise!
ãd33," =10**if*period*,*is*November
otherwise!
!To!build!this!model,!the!constant,!trend!and!all!of!the!seasonal!dummy!variables!are!entered!into!the!estimate!equation.!The!output!from!Eviews!is!presented!in!Table!4.4.1.!
Table!4.4.1!Eviews!output!of!seasonal!dummy!model!
Dependent!Variable:!RENEWABLE! ! !
Method:!Least!Squares! ! !
Sample:!2004M01!2014M12! ! !
Included!observations:!132! ! !
Variable! Coefficient! Std.!Error! t+Statistic! Prob.!!!
C! 485.59! 7.70! 63.09! 0.0000!
@TREND! 2.69! 0.07! 40.33! 0.0000!
D1! 9.92! 10.75! 0.92! 0.3578!
D2! +61.03! 10.74! +5.68! 0.0000!
D3! 9.18! 10.74! 0.85! 0.3946!
D4! 10.04! 10.74! 0.94! 0.3516!
D5! 44.55! 10.74! 4.15! 0.0001!
D6! 35.36! 10.74! 3.29! 0.0013!
D8! +35.38! 10.73! +3.30! 0.0013!
D9! +89.96! 10.73! +8.38! 0.0000!
D10! +64.48! 10.73! +6.01! 0.0000!
D11! +51.62! 10.73! +4.81! 0.0000!
!!!!Akaike!info!criterion!(AIC)! 9.66!
!!!!Schwarz!criterion!(BIC)! 9.93!
! ! ! ! !
After!gathering!the!estimation!of!linear!regression!model,!then!the!residual!diagnostic!is! !conducted! to! see!whether! there! exists!ARCH!effect! in! the! residual.! It! is! achieved!by! the! doing! the!serial!correlation!LM!test!and!heteroskedasticity!test.!The!results!of!two!tests!are!presented!in!Table!4.4.2!and!Table!4.4.3.!The!null!hypothesis!of!these!two!tests!is!that!there!is!no!ARCH!effect!exists!in!
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the! serial.! However,! the! results! of! the! tests! indicate! that! the! null! hypothesis! is! rejected! since! the!EGvalue!of!both!test!are!small!than!0.05.!Therefore,!both!of!these!tests!reveal!that!there!exist!error!variance!and!it!is!reasonable!to!building!a!(G)ARCH!model!in!this!case.! !
Table!4.4.2!Eviews!output!of!serial!correlation!LM!test!
Breusch+Godfrey!Serial!Correlation!LM!Test:! !
F+statistic! 54.32!!!!!Prob.!F(2,118)! 0.0000!
Obs*R+squared! 63.27!!!!!Prob.!Chi+Square(2)! 0.0000!
!Table!4.4.3!Eviews!output!of!heteroskedasticity!test!
Heteroskedasticity!Test:!ARCH! ! !
F+statistic! 64.15!!!!!Prob.!F(1,129)! 0.0000!
Obs*R+squared! 43.51!!!!!Prob.!Chi+Square(1)! 0.0000!
! ! ! ! !
Based! on! aforementioned! seasonal! dummy! model,! the! (G)ARCH! model! is! built.! By! doing! the!comparisons!and!selections,!GARCH!(1,1)!model!has!the!best!performance!among!all!tested!models.!However,!the!parameter!estimation!shows!that!the!first,!third,!fourth!and!seventh!seasonal!dummy!variables! of! the!GARCH!model! are! not! significant6.! After! a! few! attempts,! it! is! decided! to! drop! out!those!four!seasonal!dummies.!The!residual!diagnostics!of!selected!model!shows!that!the!residual!is!a!whiteGnoise!series7.!Hence,!the!final!GARCH!(1,1)!model! is!build!up!on!7!seasonal!dummies!and!the!estimation!out!is!in!following!Table!4.4.6.! !
!
Table!4.4.6!Eviews!output!of!GARCH!(1,1)!model!
Method:!ML!+!ARCH! ! !
Sample:!2004M01!2014M12! ! !
Convergence!achieved!after!12!iterations! !
Presample!variance:!backcast!(parameter!=!0.7)!
GARCH!=!C(10)!+!C(11)*RESID(+1)^2!+!C(12)*GARCH(+1)!
Variable! Coefficient! Std.!Error! z+Statistic! Prob.!!!
C! 489.80! 3.45! 141.80! 0.0000!
@TREND! 2.64! 0.04! 63.45! 0.0000!
D2! +65.93! 4.50! +14.66! 0.0000!
D5! 41.25! 8.58! 4.81! 0.0000!
D6! 33.39! 7.55! 4.42! 0.0000!
D8! +42.42! 7.97! +5.32! 0.0000!
D9! +95.66! 7.20! +13.29! 0.0000!
D10! +64.05! 6.59! +9.72! 0.0000!
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D11! +47.40! 5.26! +9.01! 0.0000!
! Variance!Equation! ! !
C! 358.37! 121.66! 2.95! 0.0032!
RESID(+1)^2! 0.79! 0.23! 3.37! 0.0007!
GARCH(+1)! +0.20! 0.10! +1.96! 0.0501!
!!!!Akaike!info!criterion! 9.34!
!!!!Schwarz!criterion! 9.60!
!
Based!on!the!output!above,!the!estimation!of!GARCH!(1,1)!is!presented!below:!
z^ = 489.80 + 2.64t − 65.93x`M,^ + 41.25x`ó,^ + 33.39x`ò,^ − 42.42x`ô,^ − 95.66x`ö,^ − 64.05x`3K,^− 47.40x`33,^ + ϵ^!
σ^M = 358.37 + 0.79ϵ^/õM − 0.20σ^/õ
M *
and*
x`3,^ =10**if*period*t*is*January
otherwise*
x`33,^ =10**if*period*t*is*November
otherwise*
In!the!phase!of!application,!this!GARCH!(1,1)!model!is!employed!to!forecast!the!amount!of!renewable!energy!consumption.!
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!Figure! 4.4.1! Actual! renewable! energy! consumption! (the! blue! line),! forecasting! on! renewable! energy!
consumption!with!GARCH!(1,1)!model!(the!red!line)!
!
As!presented!in!this!figure,!the!forecasting!value!of!renewable!energy!consumption!in!2014!is!near!to!the!real!value.!Especially!in!the!second!and!third!quarter,!the!forecasting!value!could!capture!the!peak!and!valley!of!the!real!consumption.!As!for!outGofGsample!period,!in!2015,!the!forecasting!value!of!first!quarter! and! fifth! quarter! are! close! to! the! actual! value.! But! in! the! second! and! third! quarter,! the!forecasting!value!seems!fail!to!capture!the!drop!of!real!consumption.!To!get!a!precise!estimation!of!the!performance!of!this!model,!the!evaluation!of!MAPE,!MAE!and!RMSE!is!needed!and!the!results!are!presented!in!Table!5.1!in!Chapter!5.!
4.4!ARIMA!with!web!search!index!
Different!with!aforementioned!time!series!forecasting!methods,!a!new!dependent!variable!named!the!web!search!volume!index!(SVI)!is!included!in!this!section.!Analyzing!the!web!search!volume!index!by!a!yearGoverGyear!percentage!change!has!been!investigated!by!CarrièreGSwallow!and!Labbé!(2013)!with!focuses!on!car!demand!in!Chili!and!Feng!&!Liu!(2013)!with!focuses!on!automobile!industry!in!China.!Their!researches!proved!that!it!is!a!feasible!way!to!do!the!index!construction,!hence,!the!construction!of! SVI! in! this! research! refers! to! the! way! they! did.! In! this! empirical! study,! the! SVI! denotes! the!monthGoverGmonth!percentage! change! in! the!popularity!of! keywords:! renewable!energy!on!Google!Trends.! Before! using! SVI,! the! ADF! test! is! needed.! Test! results! indicated! that! the! firstGdifference! of!series! SVI! is! stationary8.! As! for! the! series! of! renewable! energy! consumption,! the! natural! logarithm!
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processing!is!conducted!as!well!(lrenew).!According!to!the!ADF!test!of!this!series,!the!firstGdifferenced!is!stationary9! which!can!be!directly!used!in!building!the!ARIMA!model.! !
Based! on! the! result! from! section! 4.3,! the! baseline!ARIMA!model! is! defined! as! AR(1)! AR(12)!MA(1)!MA(12)!and!the!parameter!estimation!result!is!presented!below:!
Table!4.5.3!Parameter!estimation!for!baseline!ARIMA!
Method:!Least!Squares! ! !
Variable! Coefficient! Std.!Error! t+Statistic! Prob.!!!
AR(1)! +0.04! 0.02! +1.67! 0.0972!
AR(12)! 0.99! 0.02! 45.49! 0.0000!
MA(1)! +0.31! 0.07! +4.31! 0.0000!
MA(12)! +0.69! 0.07! +10.34! 0.0000!
!!!!Akaike!info!criterion!(AIC)! +3.94!
!!!!Schwarz!criterion!(BIC)! +3.85!
!
Moreover,! the! result! of! residual! diagnostics! proved! that! the! residual! is! a!whiteGnoise! series!which!indicated!that!the!baseline!ARIMA!model!is!suitable10.! !Then!to!build!the!augmented!model,!the!search!volume!index!is!added!to!this!baseline!model.!After!numbers! of! tests,! the! SVI!with! second! and! third! lags! have! the! best! performance! among! all! tested!combinations.!Hence,! in!the!augmented!model,! the!SVI! is! included!with!two!and!three! lags!and!the!parameter!estimation!is!presented!in!Table!4.5.5.!
Table!4.5.5!Parameter!estimation!for!augmented!ARIMA!
Method:!Least!Squares! ! !
Variable! Coefficient! Std.!Error! t+Statistic! Prob.!!!
DSVI(+2)! 0.07! 0.03! 2.76! 0.0067!
DSVI(+3)! 0.08! 0.03! 2.93! 0.0041!
AR(1)! +0.01! 0.03! +0.49! 0.6270!
AR(12)! 0.94! 0.03! 36.16! 0.0000!
MA(1)! +0.32! 0.07! +4.64! 0.0000!
MA(12)! +0.68! 0.06! +11.48! 0.0000!
!!!!Akaike!info!criterion!(AIC)! +3.96!
!!!!Schwarz!criterion!(BIC)! +3.81!
! ! ! ! !
As!for!the!augmented!model,!although!the!variable!AR!(1)!is!not!significant,!after!trying!dropping!this!variable,! there! is! no! extremely! improvement! about! the! performance! of! the!model.! Hence,! at! this!stage,! the!AR!(1)!variable! is!kept! in!the!augmented!model!as!presented!above.!To!make!sure! it! is!a!suitable!model,!the!residual!diagnostics!is!conducted!and!the!result!of!the!residual!diagnostics!shows!
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that! the! residual! is! a! whiteGnoise! series11,! therefore,! the! augmented!model! could! be! applied! into!forecasting.! !
Compared!the!augmented!model!with!the!baseline,!both!the!adjusted!RGsquared!and!AIC!value!has!been!slight!improved.!In!order!to!know!is!there!any!improvement!after!including!SVI!into!the!model,!the!performance!of!forecasting!has!been!investigated.!
!
Figure! 4.5! Actual! renewable! energy! consumption! (the! green! line),! forecasting! value! on! renewable! energy!
consumption!with!baseline!ARIMA!model!(the!red!line)!and!augmented!ARIMA!model!(the!blue!line)!
In!Figure!4.5,!both!the!baseline!model!and!augmented!model!could!capture!the!increasing!and!drop!of!the!actual!renewable!consumption!at!most!of!time!expect!the!second!quarter!in!2015.!Moreover,!the!forecasting!value!of!baseline!model!seems!have!a!better!performance!in!the!year!of!2014!since!it!is!closer!to!the!actual!renewable!energy!consumption!in!the!first,!second!and!third!quarter.!However,!in!2015,! for! the! suddenly!drop! in! second!quarter,! the! forecasting! value!of! augmented!model! looks!closer! to! the! real! consumption!data.!To!know!a!more!accurate!evaluation!of! these! two!model,! the!value!of!MAPE,!MAE!and!RMSE!is!calculated!in!Table!5.1!in!Chapter!5.!
! !
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Chapter!5!
5.! General!discussion! ! !
5.1!Summary!of!data!analysis!results!
Results!of! the!empirical! study! show! that!exponential! smoothing,! seasonal!ARIMA,! seasonal!dummy!with!ARHC!model! and!ARIMA!with!web! search! volume!give! good! results! for! forecasting! renewable!energy!consumption!in!the!U.S.!Although!the!performances!of!all!models!are!different!but!the!curves!of!forecasting!results!indicated!that!the!models!are!suitable.! !
Figure!5.1!Actual! renewable!energy! consumption,! forecasting!value!on! renewable!energy! consumption!with!
exponential!smoothing!additive!and!multiplicative!model,!seasonal!ARIMA,!seasonal!GARCH,!baseline!ARIMA!
model!and!augmented!ARIMA!model.!
!
!
!
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Table&5.1&Goodness/of/fits&and&forecasting&performance&evolution&for&various&models&
&
! MODEL&TYPES& &
! Exponential&
smoothing/additive&
Exponential&smoothing/&
multiplicative&
ARIMA&(1,1,1)& ×&(1,1,1)12&
GARCH&(1,1)& ARIMA&baseline&model& ARIMA&augmented&
model&
INFORMATION&CRITERIONS& & & & & & &
AIC! /& /& %4.10!!
9.34! %3.94!!
%3.96!!
BIC! /& /& %4.00!!
9.60!!
%3.85!!
%3.81!!
IN/SAMPLE&TEST& & & & & & &
MAPE! ! 0.03& ! 0.03& 0.03! 0.02! 0.02! 0.03!
MAE! ! 24.07& ! 24.87& 20.46! 17.28! 13.47! 26.49!
RMSE! ! 28.63& ! 27.73& 22.67! 27.05! 17.63! 29.90!
OUT/OF/SAMPLE&TEST& & & & & & &
MAPE! ! 0.04& ! 0.05& 0.06! 0.04! 0.03! 0.03!
MAE! ! 33.19& ! 37.77& 44.50! 29.97! 25.79! 21.78!
RMSE! ! 42.31& ! 51.22& 59.71! 40.35! 33.86! 27.09!
!
!
&
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Information*criterions*of*all*models*with*behavior*of*the*curve*and*the*error*measurement*show*the*reliability* and* accuracy* of* the* forecast* models.* The* in8sample* and* out8of8sample* forecasting*performance*of*each*model*is*evaluated*by*three*statistical*measures:*MAPE,*MAE,*RMSE.*All*of*the*models*have*highly*accurate*forecasts*because*most*of*the*MAPE*results*are*less*than*5%*(Pao,*2009).*Among*the*six*models,*the*baseline*ARIMA*has*the*best*performance*in*the*in8sample*test*because*of*the*lowest*value*for*all*three*error*measurements.*It*indicates*that*baseline*ARIMA*model*has*a*good*fit* to* this* series* data.* However,* the* out8of8sample* performance* results* show* that* the* augmented*ARIMA*is*the*best*choice.*That*means,*after*including*the*web*search*volume*into*forecasting*model,*the* prediction* accuracy* has* been* improved.* Judging* from* the* value* of* error* measurements,*augmented*ARIMA*has*the*lowest*value*among*all*six*models.*Moreover,*in*Figure*5.1,*the*amount*of*renewable*energy*consumption*presents*a*suddenly*drop*between*June*and*July,*2015*and*the*curve*of*the*augmented*ARIMA*model*is*the*closest*one*to*the*curve*of*real*consumption.*Compared*with*the* performance* of* other* five* models* in* June* 2015,* it* is* reasonable* to* conclude* that* some*information* has* been* captured* by* investigating* the* web* search* volume* even* though* it* is* not* so*obviously.*
5.2!Conclusion*and*implications*
To*answer*the*research*question*raised*in*the*beginning*of*this*research,*there*are*four*contributions*to*remark*in*this*study.*Firstly,*this*research*proves*that*the*time*series*models*play*a*vital*role*in*the*renewable*energy* consumption*prediction.* There* are*different* forecasting*models*proposed* in* this*research*for*renewable*energy*consumption*in*the*United*States.*Those*models*include*exponential*smoothing* additive*model,* exponential* smoothing*multiplicative*model,* seasonal* ARIMA,* seasonal*dummy* with* GARCH* and* ARIMA* models,* which* are* very* common* in* time* series* forecasting.*Theoretically,* time* series* models* excavate* the* features* existed* within* the* series* itself* (Hamilton,*1994).*If*the*series*shows*a*regularity*pattern*at*equally*spaced*time*intervals*and*time*series*models*could*more* or* less* capture* the* trends* and* changes* existed* in* the* series* (Bakhat*&* Rosselló,* 2011;*Suganthi*&*Samuel,*2012).*This*theory*has*been*confirmed*in*this*research.*The*amount*of*renewable*energy* consumption* shows* a* trend* and* seasonality* and* the* time* series* models* could* successful*capture* them.* Although* there* are* slightly* differences* on* the* performance* among* all* models,* this*research* still* proves* that* those* time* series* models* could* forecast* the* future* renewable* energy*consumption*with*a*good*performance.*
Secondly,* from* a* practical* perspective,* this* research* indicates* that* exponential* smoothing* additive*model,* seasonal* dummy* with* GARCH* and* ARIMA* models* are* feasible* approaches* to* predict*renewable* energy* consumption* in* the* future.* Among* those* models,* ARIMA* model* has* the* best*performance* in* the* case* of* renewable* energy* consumption* in* the* United* States.* Therefore,*when*predicting* the* future* renewable* energy* consumptions,* this* research* could* be* considered* as* a*reference* for* consultants* and* governments.* Because* of* the* importance* of* renewable* energy,* an*accuracy*forecasting*would*have*lots*of*benefits*especially*in*fields*of*energy*security,*job*offers*and*even* the* economy* (Turner,* 1999;* Dincer,* 2000;* Lund,* 2007).* For* instance,* an* accuracy* forecasting*would*be*useful* for*consultants*and*policy*makers*when* they*are*deciding* investments*on*building*and*maintaining*facilities.*Apart*from*this,*an*accuracy*prediction*on*renewable*energy*consumption*could*help*governments* to*make*production*plans* in*advance.*A*beforehand*preparation* in*energy*planning* would* be* useful* especially* in* case* of* energy* crisis.* Furthermore,* this* research* provides*
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quantitative*forecasting*results*and*comparison*rests*on*renewable*energy*consumption*which*filled*the*blank*in*present*energy*researches.* *
Thirdly,* this* research* demonstrates* that* including*web* search* volume* series* into* forecasting* could*improve*the*forecasting*results.*This*conclusion*is*consistent*with*previous*researches*conducted*by*Choi* and* Varian* (2009),* Kholodilin* et* al.* (2009)* and* Vosen* and* Schmidt* (2011).* The* comparison*between* baseline* and* augmented*model* shows* the* improvement* of* including*web* search* volume*data* into* forecasting.* Therefore,* this* research* contributes* to* a* new* consideration* on* predicting*energy*consumption*with*web*search*volume*series.*Apart*from*this,*in*this*empirical*study,*the*best*in8sample*result* is*obtained*by*baseline*ARIMA*model*and*the*best*out8of8sample*result* is*obtained*by*augmented*ARIMA*model.*The*fittest*model*for*in8sample*test*and*out8of8sample*test*is*different.*This* result* supports* the* statement* that* the* fittest* model* is* not* necessarily* the* one* with* best*forecasting*result*(Geurts*&*Ibrahim,*1975;*Colino,*2011).*Therefore,*when*doing*model*selections,*it*is* suggested* to* start*with* a* broad* scope* of* selection* rather* than* only* focusing* on* the* information*criterions*and*performance*of*error*measurements.* *
At* last,* including*web* search* volume* from*Google* Trends* is* supposed* to* significantly* improve* the*performance* of* baseline* model,* however,* this* research* indicates* that* the* improvement* of*performance*may*depend*on*the*type*of*products.*Researchers*already*proved*that*including*Google*Trends*could*significantly*improve*the*forecasting*accuracy*in*fields*of*epidemiology*(Polgreen*et*al.,*2008;* Ginsberg* et* al.,* 2009),* car* consumptions* (Carrière8Swallow* &* Labbé,* 2013)* and* consumer*confidence* index* (Vosen* &* Schmidt,* 2011).* Their* studies* reveal* that* Google* Trends* is* extremely*predictable*with*sudden*events*such*as*flu,*and*products*that*directly*linked*to*consumers’*intentions.*As* for* renewable* energy,* it* is* hardly* to* link* with* sudden* events* and* it* is* not* directly* related* to*consumers’*purchase*behavior.*Besides,*lots*of*energy*consumption*occurs*at*the*industrial*level*and*that* might* make* it* difficult* to* capture* through* Google* Trends.* To* some* extent,* the* industries*determine*the*amount*of*renewable*energy*consumption.*Even*though*consumers*are*interested*in*renewable*energy,* it* is*still*powerless*to*make*some*changes* in*the*amount*of*consumptions.*With*those* ideas,* it* makes* sense* that* including* web* search* volume* into* forecasting* renewable* energy*consumption*only*slightly*improved*the*prediction*accuracy*in*out8of8sample*testing.* *
5.3!Limitations*and*further*study*
There*are*some*limitations*with*regard*to*this*research,*further*studies*are*suggested*based*on*these*limitations.*
Firstly,* the*web* search* volume* data* is* added* as* an* explanatory* variable* in* the* augmented*model,*however,*the*selection*of*keywords*is*based*on*a*subjective*experience.*When*selecting*the*keywords,*some*information*may*be*omitted*or*missed*due*to*the*limitation*of*knowledge.*It*might*be*a*reason*for*the*performance*of*the*augmented*model*is*not*improved*significantly*as*expected*as*proved*by*other* researchers* (Choi* &* Varian,* 2009;* Carrière8Swallow* &* Labbé,* 2013).* Therefore,* in* further*studies,* a*more*accurate*and*convincible*keywords* selection*procedure* could*be* conducted*before*building*models.*
Secondly,*the*sample*size*of*historical*data*could*be*extended*if*the*data*is*available.*In*this*research,*totally*132*observations*were*used*to*build*the*model.*However,*if*there*are*more*observations*could*
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be*used* in* training*models,* the* forecasting* accuracy*might*be* improved.*Besides,* some* researches*proved* that* hybrid* models* could* provide* a* better* forecasting* result* (Pao,* 2009;* Colino,* 2011;*Suganthi* &* Samuel,* 2012).* Hence,* the* combinations* of* time* series*models* could* be* considered* in*future*researches.* *
Thirdly,* the* seasonal* ARIMA*model*was* applied* in* this* research.* But* according* to* the* information*criterions*and*the*forecasting*performance,*the*result*is*not*as*good*as*expected.*Although*numbers*of*comparisons*have*been*done* in* the*empirical* studies,* there* is*still* room*for* improvement* in* the*phase*of*selecting*model.*Hence,*a*broader*selection*and*estimation*of*parameters*could*be*done*in*further*research.*
Last* but* not* least,* monthly* forecasting* horizon* and* monthly* data* sets* are* used* in* this* research,*however,* the* data* frequency* could* be* altered*when* data*with* a* shorter* time* interval* is* available,*especially*when*doing* forecasting*with* the*web*search*volume.*A*short* time* interval* could*capture*the*subtle*changes*in*the*data*and*provide*more*information*when*estimating*the*parameters.*
* * *
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Acknowledgments.
I* would* like* to* express* my* special* thanks* to* Dr.* Andres* Trujillo* Barrera* for* the* supervision* and*comments*on*this*paper,*and*to*Dr.*Ivo*van*der*Lans*for*the*helpful*comments*on*the*earlier*versions*of* this* paper.* Also* to* my* parents* and* friends,* thanks* for* giving*me* your* support* and* sharing* my*frustration*and*happiness*during*the*whole*process.* *
*
* *
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� ���
Appendix. .
Table.4.3.1.ADF.statics.results.for.series.?.renewable.
Null!Hypothesis:!RENEWABLE!has!a!unit!root! !
Exogenous:!Constant,!Linear!Trend! !
Lag!Length:!12!(Automatic!C!based!on!AIC,!maxlag=12)!
! ! ! tCStatistic! !!Prob.*!
Augmented!DickeyCFuller!test!statistic! C2.257444! !0.4537!
Test!critical!values:! 1%!level! ! C4.029595! !
! 5%!level! ! C3.444487! !
! 10%!level! ! C3.147063! !
*MacKinnon!(1996)!oneCsided!pCvalues.! !
! ! ! ! !
Augmented!DickeyCFuller!Test!Equation! !
Dependent!Variable:!D(RENEWABLE)! !
Method:!Least!Squares! ! !
Sample!(adjusted):!2005M02!2015M12! !
Included!observations:!131!after!adjustments! !
Variable! Coefficient! Std.!Error! tCStatistic! Prob.!!!
RENEWABLE(C1)! C0.275050! 0.121841! C2.257444! 0.0259!
D(RENEWABLE(C1))! C0.032608! 0.137780! C0.236671! 0.8133!
D(RENEWABLE(C2))! C0.025914! 0.133833! C0.193626! 0.8468!
D(RENEWABLE(C3))! C0.065346! 0.127967! C0.510644! 0.6106!
D(RENEWABLE(C4))! C0.096304! 0.119919! C0.803078! 0.4236!
D(RENEWABLE(C5))! C0.056627! 0.111354! C0.508527! 0.6120!
D(RENEWABLE(C6))! C0.130631! 0.106699! C1.224285! 0.2233!
D(RENEWABLE(C7))! C0.093832! 0.101140! C0.927742! 0.3555!
D(RENEWABLE(C8))! C0.168907! 0.096333! C1.753369! 0.0822!
D(RENEWABLE(C9))! C0.188013! 0.089703! C2.095960! 0.0383!
D(RENEWABLE(C10))! C0.159671! 0.086660! C1.842494! 0.0680!
D(RENEWABLE(C11))! C0.137258! 0.083284! C1.648081! 0.1020!
D(RENEWABLE(C12))! 0.575898! 0.078542! 7.332342! 0.0000!
C! 131.2749! 55.10595! 2.382228! 0.0188!
@TREND(2004M01)! 0.733551! 0.330506! 2.219478! 0.0284!
RCsquared! 0.705897!!!!!Mean!dependent!var! 2.567366!
Adjusted!RCsquared! 0.670402!!!!!S.D.!dependent!var! 46.87158!
S.E.!of!regression! 26.90926!!!!!Akaike!info!criterion! 9.530219!
Sum!squared!resid! 83996.56!!!!!Schwarz!criterion! 9.859440!
Log!likelihood! C609.2293!!!!!HannanCQuinn!criter.! 9.663996!
FCstatistic! 19.88715!!!!!DurbinCWatson!stat! 2.119635!
Prob(FCstatistic)! 0.000000! ! ! !
. .
� ���
Table.4.3.2.ADF.statics.results.for.series.–.llrenew.
Null!Hypothesis:!D(LLRENEW)!has!a!unit!root! !
Exogenous:!Constant,!Linear!Trend! !
Lag!Length:!12!(Automatic!C!based!on!AIC,!maxlag=12)!
! ! ! tCStatistic! !!Prob.*!
Augmented!DickeyCFuller!test!statistic! C5.297219! !0.0001!
Test!critical!values:! 1%!level! ! C4.046925! !
! 5%!level! ! C3.452764! !
! 10%!level! ! C3.151911! !
*MacKinnon!(1996)!oneCsided!pCvalues.! !
! ! ! ! !
Augmented!DickeyCFuller!Test!Equation! !
Dependent!Variable:!D(LLRENEW,2)! !
Method:!Least!Squares! ! !
Sample!(adjusted):!2006M03!2014M12! !
Included!observations:!106!after!adjustments! !
Variable! Coefficient! Std.!Error! tCStatistic! Prob.!!!
D(LLRENEW(C1))! C2.613174! 0.493311! C5.297219! 0.0000!
D(LLRENEW(C1),2)! 1.304702! 0.463565! 2.814498! 0.0060!
D(LLRENEW(C2),2)! 1.154075! 0.444238! 2.597872! 0.0109!
D(LLRENEW(C3),2)! 1.045893! 0.417694! 2.503971! 0.0141!
D(LLRENEW(C4),2)! 1.061025! 0.390908! 2.714257! 0.0079!
D(LLRENEW(C5),2)! 1.018116! 0.359593! 2.831304! 0.0057!
D(LLRENEW(C6),2)! 0.978129! 0.331050! 2.954623! 0.0040!
D(LLRENEW(C7),2)! 0.967434! 0.301198! 3.211959! 0.0018!
D(LLRENEW(C8),2)! 0.811470! 0.267311! 3.035677! 0.0031!
D(LLRENEW(C9),2)! 0.820555! 0.228876! 3.585144! 0.0005!
D(LLRENEW(C10),2)! 0.731214! 0.189930! 3.849909! 0.0002!
D(LLRENEW(C11),2)! 0.770675! 0.141916! 5.430521! 0.0000!
D(LLRENEW(C12),2)! 0.342922! 0.096321! 3.560180! 0.0006!
C! C3.08EC05! 0.009999! C0.003084! 0.9975!
@TREND(2004M01)! C1.82EC06! 0.000119! C0.015317! 0.9878!
RCsquared! 0.749611!!!!!Mean!dependent!var! C0.000146!
Adjusted!RCsquared! 0.711090!!!!!S.D.!dependent!var! 0.069076!
S.E.!of!regression! 0.037129!!!!!Akaike!info!criterion! C3.618415!
Sum!squared!resid! 0.125447!!!!!Schwarz!criterion! C3.241513!
Log!likelihood! 206.7760!!!!!HannanCQuinn!criter.! C3.465655!
FCstatistic! 19.45962!!!!!DurbinCWatson!stat! 1.978485!
Prob(FCstatistic)! 0.000000! ! ! !
! ! ! ! !
!! !
� ���
Table.4.3.4.Residual.diagnostics.of.AR(1).MA(1).SAR(1).SMA(1).
!
! !
� ���
Table.4.3.5.Estimation.of.AR(1).MA(1).SAR(12).SMA(12).
Dependent!Variable:!D(LRENEW,1,12)! !
Method:!Least!Squares! ! !
Sample!(adjusted):!2006M03!2014M12! !
Included!observations:!106!after!adjustments! !
Convergence!achieved!after!9!iterations! !
MA!Backcast:!2005M02!2006M02! ! !
Variable! Coefficient! Std.!Error! tCStatistic! Prob.!!!
AR(1)! 0.239868! 0.186833! 1.283868! 0.2021!
SAR(12)! C0.074149! 0.098047! C0.756265! 0.4512!
MA(1)! C0.658840! 0.141990! C4.640038! 0.0000!
SMA(12)! C0.902884! 0.019025! C47.45670! 0.0000!
RCsquared! 0.533040!!!!!Mean!dependent!var! C0.000443!
Adjusted!RCsquared! 0.519306!!!!!S.D.!dependent!var! 0.044055!
S.E.!of!regression! 0.030544!!!!!Akaike!info!criterion! C4.102267!
Sum!squared!resid! 0.095162!!!!!Schwarz!criterion! C4.001760!
Log!likelihood! 221.4201!!!!!HannanCQuinn!criter.! C4.061531!
DurbinCWatson!stat! 1.870861! ! ! !
!
.
!
� ���
Table.4.4.4.Eviews.output.of.GARCH.(1,1).model.with.11.seasonal.dummy.variables.
Dependent!Variable:!RENEWABLE! ! !
Method:!ML!C!ARCH!(Marquardt)!C!Normal!distribution!
Sample:!2004M01!2014M12! ! !
Included!observations:!132! ! !
Failure!to!improve!Likelihood!after!40!iterations!
Presample!variance:!backcast!(parameter!=!0.7)!
GARCH!=!C(14)!+!C(15)*RESID(C1)^2!+!C(16)*GARCH(C1)!
Variable! Coefficient! Std.!Error! zCStatistic! Prob.!!!
C! 486.9397! 6.422392! 75.81905! 0.0000!
@TREND! 2.675464! 0.044747! 59.79071! 0.0000!
D1! C0.386272! 8.163909! C0.047315! 0.9623!
D2! C64.97663! 7.926757! C8.197126! 0.0000!
D3! 6.289813! 7.908738! 0.795299! 0.4264!
D4! C0.792522! 7.116659! C0.111361! 0.9113!
D5! 42.35780! 10.50427! 4.032436! 0.0001!
D6! 33.18109! 7.383428! 4.493995! 0.0000!
D7! 5.384500! 8.560577! 0.628988! 0.5294!
D8! C39.44469! 9.393808! C4.199009! 0.0000!
D9! C94.14768! 8.926311! C10.54721! 0.0000!
D10! C62.82291! 8.628967! C7.280467! 0.0000!
D11! C46.91655! 6.986141! C6.715660! 0.0000!
! Variance!Equation! ! !
C! 354.5282! 136.7257! 2.592989! 0.0095!
RESID(C1)^2! 0.755777! 0.249670! 3.027103! 0.0025!
GARCH(C1)! C0.198045! 0.113429! C1.745989! 0.0808!
RCsquared! 0.937710!!!!!Mean!dependent!var! 645.5141!
Adjusted!RCsquared! 0.931429!!!!!S.D.!dependent!var! 112.9082!
S.E.!of!regression! 29.56623!!!!!Akaike!info!criterion! 9.392762!
Sum!squared!resid! 104025.2!!!!!Schwarz!criterion! 9.742193!
Log!likelihood! C603.9223!!!!!HannanCQuinn!criter.! 9.534755!
DurbinCWatson!stat! 0.630718! ! ! !
! ! ! ! !
!
.
� ���
Table.4.5.1.Eviews.output.of.ADF.statics.results.for.series.–.SVI.
Null!Hypothesis:!D(SVI)!has!a!unit!root! !
Exogenous:!Constant! ! !
Lag!Length:!12!(Automatic!C!based!on!SIC,!maxlag=12)!
! ! ! tCStatistic! !!Prob.*!
Augmented!DickeyCFuller!test!statistic! C8.950818! !0.0000!
Test!critical!values:! 1%!level! ! C3.487046! !
! 5%!level! ! C2.886290! !
! 10%!level! ! C2.580046! !
*MacKinnon!(1996)!oneCsided!pCvalues.! !
! ! ! ! !
Augmented!DickeyCFuller!Test!Equation! !
Dependent!Variable:!D(SVI,2)! ! !
Method:!Least!Squares! ! !
Sample!(adjusted):!2005M04!2014M12! !
Included!observations:!117!after!adjustments! !
Variable! Coefficient! Std.!Error! tCStatistic! Prob.!!!
D(SVI(C1))! C17.61733! 1.968237! C8.950818! 0.0000!
D(SVI(C1),2)! 15.37584! 1.900466! 8.090563! 0.0000!
D(SVI(C2),2)! 13.94595! 1.767193! 7.891581! 0.0000!
D(SVI(C3),2)! 12.46632! 1.617710! 7.706153! 0.0000!
D(SVI(C4),2)! 10.93035! 1.463162! 7.470362! 0.0000!
D(SVI(C5),2)! 9.435104! 1.296437! 7.277718! 0.0000!
D(SVI(C6),2)! 7.935883! 1.121012! 7.079213! 0.0000!
D(SVI(C7),2)! 6.464146! 0.941907! 6.862827! 0.0000!
D(SVI(C8),2)! 5.006110! 0.760330! 6.584128! 0.0000!
D(SVI(C9),2)! 3.541105! 0.582497! 6.079181! 0.0000!
D(SVI(C10),2)! 2.039771! 0.411306! 4.959251! 0.0000!
D(SVI(C11),2)! 0.724574! 0.246196! 2.943076! 0.0040!
D(SVI(C12),2)! 0.212560! 0.102492! 2.073931! 0.0406!
C! 0.003365! 0.008792! 0.382679! 0.7027!
RCsquared! 0.922064!!!!!Mean!dependent!var! 0.000235!
Adjusted!RCsquared! 0.912227!!!!!S.D.!dependent!var! 0.320543!
S.E.!of!regression! 0.094965!!!!!Akaike!info!criterion! C1.758738!
Sum!squared!resid! 0.928897!!!!!Schwarz!criterion! C1.428222!
Log!likelihood! 116.8862!!!!!HannanCQuinn!criter.! C1.624553!
FCstatistic! 93.73829!!!!!DurbinCWatson!stat! 1.995713!
Prob(FCstatistic)! 0.000000! ! ! !
! ! ! ! !
.
! !
� ��
Table.4.5.2.Eviews.output.of.ADF.statics.results.for.series.–.lrenew.
Null!Hypothesis:!D(LRENEW)!has!a!unit!root! !
Exogenous:!Constant! ! !
Lag!Length:!11!(Automatic!C!based!on!SIC,!maxlag=12)!
! ! ! tCStatistic! !!Prob.*!
Augmented!DickeyCFuller!test!statistic! C4.238535! !0.0009!
Test!critical!values:! 1%!level! ! C3.486064! !
! 5%!level! ! C2.885863! !
! 10%!level! ! C2.579818! !
*MacKinnon!(1996)!oneCsided!pCvalues.! !
! ! ! ! !
Augmented!DickeyCFuller!Test!Equation! !
Dependent!Variable:!D(LRENEW,2)! !
Method:!Least!Squares! ! !
Sample!(adjusted):!2005M02!2014M12! !
Included!observations:!119!after!adjustments! !
Variable! Coefficient! Std.!Error! tCStatistic! Prob.!!!
D(LRENEW(C1))! C3.428533! 0.808896! C4.238535! 0.0000!
D(LRENEW(C1),2)! 2.101217! 0.749467! 2.803614! 0.0060!
D(LRENEW(C2),2)! 1.831729! 0.684108! 2.677543! 0.0086!
D(LRENEW(C3),2)! 1.554775! 0.617081! 2.519563! 0.0132!
D(LRENEW(C4),2)! 1.250157! 0.551052! 2.268674! 0.0253!
D(LRENEW(C5),2)! 1.013595! 0.486880! 2.081816! 0.0398!
D(LRENEW(C6),2)! 0.734810! 0.421198! 1.744570! 0.0840!
D(LRENEW(C7),2)! 0.499959! 0.354831! 1.409006! 0.1618!
D(LRENEW(C8),2)! 0.191847! 0.285344! 0.672336! 0.5028!
D(LRENEW(C9),2)! C0.124710! 0.218602! C0.570488! 0.5696!
D(LRENEW(C10),2)! C0.363265! 0.152384! C2.383874! 0.0189!
D(LRENEW(C11),2)! C0.552701! 0.084773! C6.519816! 0.0000!
C! 0.013000! 0.004901! 2.652296! 0.0092!
RCsquared! 0.877873!!!!!Mean!dependent!var! 0.000454!
Adjusted!RCsquared! 0.864047!!!!!S.D.!dependent!var! 0.111860!
S.E.!of!regression! 0.041245!!!!!Akaike!info!criterion! C3.435783!
Sum!squared!resid! 0.180320!!!!!Schwarz!criterion! C3.132182!
Log!likelihood! 217.4291!!!!!HannanCQuinn!criter.! C3.312500!
FCstatistic! 63.49581!!!!!DurbinCWatson!stat! 2.084668!
Prob(FCstatistic)! 0.000000! ! ! !
! ! ! ! !
.
� ��
Table.4.5.4.Residual.diagnostics.of.baseline.ARIMA.model.
!!
� ���
Table.4.5.6.Residual.diagnostics.of.augmented.ARIMA.model.
