Forecasting models for economic and environmental applications

Transcription

1 Universiy of Souh Florida Scholar Commons Graduae Theses and Disseraions Graduae School 008 Forecasing models for economic and environmenal applicaions Shou Hsing Shih Universiy of Souh Florida Follow his and addiional works a: hp://scholarcommons.usf.edu/ed Par of he American Sudies Commons Scholar Commons Ciaion Shih, Shou Hsing, "Forecasing models for economic and environmenal applicaions" (008). Graduae Theses and Disseraions. hp://scholarcommons.usf.edu/ed/493 This Disseraion is brough o you for free and open access by he Graduae School a Scholar Commons. I has been acceped for inclusion in Graduae Theses and Disseraions by an auhorized adminisraor of Scholar Commons. For more informaion, please conac scholarcommons@usf.edu.

2 Forecasing Models for Economic and Environmenal Applicaions by Shou Hsing Shih A disseraion submied in parial fulfillmen of he requiremens for he degree of Docoral of Philosophy Deparmen of Mahemaics and Saisics College of Ars and Science Universiy of Souh Florida Major Professor: Chris P. Tsokos, Ph.D. Gangaram Ladde, Ph.D. Kandehody Ramachandran, Ph.D. Wonkuk Kim, Ph.D. Marcus McWaers, Ph.D. Dae of Approval: April 3, 008 Keywords: Time Series, Global Warming, Sock, S&P Price Index, Temperaure, Carbon Dioxide Copyrigh 008, Shou Hsing Shih

3 Acknowledgemen I would like o express deep appreciaion o my major professor Dr. Chris P. Tsokos for his paien advice, generous suppor, friendship, and he many counless hours of personal ime he has spen assising me hroughou my graduae career. Wih his guidance, i has fosered an environmen ha has helped me complee his work diligenly and wih greaer ease han I would have hough possible oherwise. I would like o hank Professor Ladde for his assisance and advisemen on he heoreical aspecs of his work. I would also like o express my appreciaion o Professors Ramachadran, McWaers, and Kim, for heir suppor and guidance o complee his work.

4 Table of Conens Lis of Tables Lis of Figures Absrac iii v viii Chaper Review and Basic Conceps.0 Inroducion. Lieraure Review. Fundamenal Conceps 4..Sochasic Process 5..Whie Noise Process 7..3Esimaions on Single Realizaion 8.3 The Saionary AR, MA, and ARMA Models 9.3. The Auoregressive (AR) Models 0.3. The Moving Average (MA) Models The General Mixed ARMA Models.4 Aims of Presen Sudy.5 Conclusion 3 Chaper Analyical Formulaions of Modified Time Series 4.0 Inroducion 4. The General ARIMA Model 4. The k-h Moving Average Time Series Model 8.3 The k-h Weighed Moving Average Time Series Model 0.4 The k-h Exponenial Moving Average Time Series Model.5 The Muliplicaive ARIMA Model 4.6 Conclusion 6 Chaper 3 Proposed Economerics Forecasing Models Inroducion 8 3. Forecasing Models on Sock XYZ Daa Preparaion The General ARIMA Model The k-h Moving Average Time Series Model The k-h Weighed Moving Average Time Series Model The k-h Exponenial Moving Average Time Series Model Forecasing Models on S&P Price Index Daa Preparaion 49 i

5 3.. The General ARIMA Model The k-h Moving Average Time Series Model The k-h Weighed Moving Average Time Series Model The k-h Exponenial Moving Average Time Series Model Evaluaions on Proposed Models VS. Classical Mehod Conclusion 67 Chaper 4 Global Warming: Amospheric Temperaure Forecasing Model Inroducion Analyical Procedure Developmen of Forecasing Models Comparison Conclusion 79 Chaper 5 Global Warming: Carbon Dioxide Proposed Forecasing Model Inroducion Carbon Dioxide Emission Modeling Amospheric Carbon Dioxide Modeling Conclusion 9 Chaper 6 Global Warming: Temperaure & Carbon Dioxide Predicion Modeling Inroducion 9 6. Relaionship beween Carbon Dioxide & Temperaure 9 6. Carbon Dioxide & Temperaure Modeling Carbon Dioxide & Temperaure Modeling Temperaure Model Conclusion 04 Chaper 7 Fuure Research 06 References 07 Appendices 4 Appendix A: Daily Closing Price of Sock XYZ 5 Appendix A: MA3 Series on Daily Closing Price of Sock XYZ 7 Appendix A3: WMA3 Series on Daily Closing Price of Sock XYZ 9 Appendix A4: EWMA3 Series on Daily Closing Price of Sock XYZ Appendix B: Daily Closing Price of S&P Price Index 3 Appendix B: MA3 Series on Daily Closing Price of S&P Price Index 5 Appendix B3: WMA3 Series on Daily Closing Price of S&P Price Index 7 Appendix B4: EWMA3 Series on Daily Closing Price of S&P Price Index 9 Appendix C: Monhly Temperaure for (Version Daase) 3 Appendix C: Monhly Temperaure for (Version Daase) 35 Appendix D: Monhly CO Emission Appendix E: Monhly CO in he Amosphere ii

6 Lis of Tables Table 3. Basic Evaluaion Saisics for Classical ARIMA 33 Table 3. Acual and Prediced Price for Classical ARIMA 34 Table 3.3 Basic Evaluaion Saisics for MA3 ARIMA 38 Table 3.4 Acual and Prediced Price for MA3 ARIMA 38 Table 3.5 Basic Evaluaion Saisics for WMA3 ARIMA 4 Table 3.6 Acual and Prediced Price for WMA3 ARIMA 43 Table 3.7 Basic Evaluaion Saisics for EWMA3 ARIMA 46 Table 3.8 Acual and Prediced Price for EWMA3 ARIMA 47 Table 3.9 Basic Evaluaion Saisics for Classical ARIMA 5 Table 3.0 Acual and Prediced Price for Classical ARIMA 5 Table 3. Basic Evaluaion Saisics for MA3 ARIMA 56 Table 3. Acual and Prediced Price for MA3 ARIMA 56 Table 3.3 Basic Evaluaion Saisics for WMA3 ARIMA 60 Table 3.4 Acual and Prediced Price for WMA3 ARIMA 6 Table 3.5 Basic Evaluaion Saisics for EWMA3 ARIMA 64 Table 3.6 Acual and Prediced Price for EWMA3 ARIMA 65 Table 3.7 Ranking Comparison on sock XYZ 66 Table 3.8 Ranking Comparison on S&P Price Index 67 Table 4. Basic Evaluaion Saisics (Version Daase) 76 iii

7 Table 4. Basic Evaluaion Saisics (Version Daase) 76 Table 4.3 Original VS. Forecas Values (Version Daase) 77 Table 4.4 Original VS. Forecas Values (Version Daase) 78 Table 5. Basic Evaluaion on CO Emissions Model 83 Table 5. Original VS. Forecasing Values on CO Emissions Model 84 Table 5.3 Basic Evaluaion on Amospheric CO Mode 89 Table 5.4 Original VS. Forecasing Values on Amospheric CO Model 90 Table 6. Comparison beween Two Series 96 Table 6. Residual Analysis for Combine Model- 99 Table 6.3 Comparison beween Acual and Forecas for Model- 00 Table 6.4 Residual Analysis for Combine Model- 0 Table 6.5 Comparison beween Acual and Forecas for Model- 0 Table 6.6 Basic Evaluaion Saisics for he Temperaure Mode- 04 iv

8 Lis of Figures Figure 3. Daily Closing Price for Sock XYZ 9 Figure 3. Comparisons on Classical ARIMA Model VS. Original Time Series 3 Figure 3.3 Time Series Plo of he Residuals for Classical Model 33 Figure 3.4 Classical ARIMA Forecasing on he Las 5 Observaions 35 Figure 3.5 MA3 Series VS. The Original Time Series 35 Figure 3.6 Comparisons on MA3 ARIMA Model VS. Original Time Series 36 Figure 3.7 Time Series Plo of he Residuals for MA3 Model 37 Figure 3.8 MA3 ARIMA Forecasing on he Las 5 Observaions 39 Figure 3.9 WMA3 Series VS. The Original Time Series 40 Figure 3.0 Comparisons on WMA3 ARIMA Model VS. Original Time Series 4 Figure 3. Time Series Plo of he Residuals for WMA3 Model 4 Figure 3. WMA3 ARIMA Forecasing on he Las 5 Observaions 44 Figure 3.3 WMA3 Series VS. The Original Time Series 44 Figure 3.4 Comparisons on EWMA3 ARIMA Model VS. Original Time Series 45 Figure 3.5 Time Series Plo of he Residuals for EWMA3 Model 46 Figure 3.6 EWMA3 ARIMA Forecasing on he Las 5 Observaions 48 Figure 3.7 Daily Closing Price for S&P Price Index 48 Figure 3.8 Comparisons on Classical ARIMA Model VS. Original Time Serie 50 Figure 3.9 Time Series Plo of he Residuals for Classical Model 5 v

9 Figure 3.0 Classical ARIMA Forecasing on he Las 5 Observaions 53 Figure 3. MA3 Series VS. The Original Time Series 53 Figure 3. Comparisons on MA3 ARIMA Model VS. Original Time Series 54 Figure 3.3 Time Series Plo of he Residuals for MA3 Model 55 Figure 3.4 MA3 ARIMA Forecasing on he Las 5 Observaions 57 Figure 3.5 WMA3 Series VS. The Original Time Series 58 Figure 3.6 Comparisons on WMA3 ARIMA Model VS. Original Time Series 59 Figure 3.7 Time Series Plo of he Residuals for WMA3 Model 60 Figure 3.8 WMA3 ARIMA Forecasing on he Las 5 Observaions 6 Figure 3.9 WMA3 Series VS. The Original Time Series 6 Figure 3.30 Comparisons on EWMA3 ARIMA Model VS. Original Time Series 63 Figure 3.3 Time Series Plo of he Residuals for EWMA3 Model 64 Figure 3.3 EWMA3 ARIMA Forecasing on he Las 5 Observaions 66 Figure 4. Time Series Plo for Monhly Temperaure for (Version ) 69 Figure 4. Time Series Plo for Monhly Temperaure for (Version ) 70 Figure 4.3 Acual VS. Prediced Values for Version Daase 74 Figure 4.4 Acual VS. Prediced Values for Version Daase 74 Figure 4.5 Residual Plo for Monhly Temperaure (Version Daase) 75 Figure 4.6 Residual Plo for Monhly Temperaure (Version Daase) 75 Figure 4.7. Acual VS. Prediced Values for he Las Observaions (Version ) 78 Figure 4.8 Acual VS. Prediced Values for he Las Observaions (Version ) 79 Figure 5. Time Series Plo on CO Emission Figure 5. Acual VS. Prediced Values for CO Emission vi

10 Figure 5.3 Residuals Plo for CO Emissions 83 Figure 5.4 Monhly CO Emission VS. Forecas Values for he Las Observaions 85 Figure 5.5 Geographical Locaion of Mauna Loa 86 Figure 5.6 Time Series Plo for Monhly CO in he Amosphere Figure 5.7 Acual VS. Prediced Values for Amospheric CO Figure 5.8 Residuals Plo for Amospheric CO 89 Figure 5.9 Acual VS. Prediced Values for CO in he Amosphere 90 Figure 6. Monhly Amospheric Carbon Dioxide from 965 o Figure 6. Monhly Temperaure of he Coninenal Unied Saes from 965 o Figure 6.3 Firs Order Differencing Monhly Amospheric CO Series 95 Figure 6.4 The Temperaure Series VS. The Differencing CO Series 95 Figure 6.5 Comparison beween Boh Series Afer Transformaion 97 Figure 6.6 Cross Correlaion a Differen Lag 98 vii

11 Forecasing Models for Economic and Environmenal Applicaions Shou Hsing Shih ABSTRACT The objec of he presen sudy is o inroduce hree analyical ime series models for he purpose of developing more effecive economic and environmenal forecasing models, among ohers. Given a sochasic realizaion, saionary or nonsaionary in naure, one can uilize exciing mehodology o develop an auoregressive, moving average or a combinaion of boh for shor and long erm forecasing. In he presen sudy we analyically modify he sochasic realizaion uilizing (a) a k-h moving average, (b) a k-h weighed moving average and (c) a k-h exponenial weighed moving average processes. Thus, we proceed in developing he appropriae forecasing models wih he new (modified) ime series using he more recen mehodologies in he subjec maer. Once he proposed saisical forecasing models have been developed, we proceed o modify he analyical process back ino he original sochasic realizaion. The proposed mehods have been successfully applied o real sock daa from a Forune 500 company. A similar forecasing model was developed and evaluaed for he daily closing price of S&P Price Index of he New York Sock Exchange. The proposed forecasing model was developed along wih he saisical model using classical and mos recen mehods. The effeciveness of he wo models was compared viii

12 using various saisical crieria. The proposed models gave beer resuls. Amospheric emperaure and carbon dioxide, CO, are he wo variables mos aribuable o GLOBAL WARMING. Using he proposed mehods we have developed forecasing saisical models for he coninenal Unied Saes, for boh he amospheric emperaure and carbon dioxide. We have developed forecasing models ha performed much beer han he models using he classical Box-Jenkins ype of mehodology. Finally, we developed an effecive saisical model ha relaes CO and emperaure; ha is, knowing he amospheric emperaure we can a he specific locaion esimae he carbon dioxide and vice versa. ix

13 Chaper Lieraure Review and Fundamenal Conceps.0 Inroducion Time series analysis is one of he major areas in saisics ha can be applied o many realisic problems. In he presen chaper, we begin wih summarizing he developmen of ime series modeling and inroduce some mehodologies ha have been developed recenly. We hen inroduce some fundamenal conceps ha are essenial for dealing wih ime series models. Mos real-world ime series are nonsaionary in naure. Thus, before we deal wih hose nonsaionary phenomena in he laer chaper, we shall firs define sochasic process and whie noise process, hen survey some of he mos popular saionary ime series forecasing mehodologies, finally give a brief ouline of he basic srucure of each model.. Lieraure Review During he preparaion of he presen sudy, we have surveyed lieraure in he area of ime series forecasing. Alhough he presen sudy is concerned mainly wih economic and environmenal applicaions, our lieraure survey was exended o he forecasing in general.

14 In 960, Muh presened a mehod for forecasing by an exponenially weighed moving average. Muh s sudy included he simple, seasonal effecs and seasonal effecs wih linear rend. Acual sales daa is used in he sudy o examine his proposed modeling scheme. D. W. Trigg iniiaed a sudy of auomaically monioring a forecasing process o assure ha he forecas remains in conrol. Trigg uilizes a firs order exponenial model wih daa ha conained jumps. A sudy for shor erm sales forecasing was carried ou by Harrison in 965. The sudy examines several mehods for shor erm sales predicions; he mehod of Box and Jenkins in 970 and Brown in 965. I is argued ha muli-parameer procedures for shor erm forecasing of sales do no give significanly beer resuls. Harrison recommends he one parameer exponenial procedure for sales daa for nonseasonal forecasing. Some mehods are recommended for shor erm sales process wih seasonal effecs. A wo sage exponenial model wih exponenial smoohing and muliple regression was inroduced by Crane and Eealy in 967. They firs applied he exponenial process forecass along wih oher independen variables ino developing a muliple regression model. This combined forecasing procedure was applied in modeling some economic daa series relaed o bank deposis. This mehod produced good forecasing resuls of he economic series. In 97, Tsokos sudied some forecasing models for shor erm forecasing of economic daa. He sudied he auoregressive, moving average and mixure of

15 auoregressive moving average process as shor erm forecasing models for commodiy conracs. Several models were developed for soybean, silver and New York Time Daily averages of fify combined socks for he firs 00 business days. Their shor erm forecass were evaluaed using he minimum residual variance crieria. The invesigaion of ouliers in ime series was he aim of Fox in 97. Two ypes of ouliers in he ime series were considered. Gross error in he sample of observaions and a single innovaion are exreme, bu hey were included in he sudy. The second ype of oulier does no only effec he presen observaions bu also subsequen observaions. A likelihood raio crierion was developed o sudy he problem of he wo ouliers. In 994, Box and Jenkins inroduced he muliplicaive ARIMA model on airline daa. This ype of model addresses he issue of seasonal variaion ha was no idenifiable by he classical ARIMA model. Thus, once we idenify he period of a ime series, we can use he muliplicaive ARIMA model o forecas his series much beer han he classical mehod. We shall invesigae his model furher in Chaper 4 and 5. In 98, Engle inroduced he auoregressive condiional heeroskedasiciy (ARCH) model ha considers he variance of he curren erm o be a funcion of he variances of he previous ime period s error erms. ARCH relaes he error variance o he square of a previous period s error. In 986, he generalized auoregressive condiional heeroskedasiciy (GARCH) model was firs proposed by Bollerslev, and 3

16 i is employed commonly in modeling financial ime series ha exhibi ime varying volailiy clusering. For furher discussion on GARCH modeling, see (Enders 995, Engle 00, Gujarai 003, and Nelson 99). The sae space represenaion of a sysem is relaed o he Kalman Filer and was originally developed by conrol engineers Kalman 960, Kalman and Bucy 96, and Kalman, Falb, and Arbib 969. The sysem is also known as a sae space model and is defined o be he minimum se of informaion from he presen and pas such ha he fuure behavior of he sysem can be compleely described by he knowledge of he presen sae and fuure inpu. For furher discussion and developmen on he sae space model, see (Chen 999, Khalil, Nise 004, Hinrichsen & Prichard 005, Sonag 999, and Durbin & Koopman 00). Fundamenal Conceps A ime series can be hough of as comprising of a sequence of measuremens, almos cerainly inercorrelaed, represening some phenomena in differen areas. Each of he measuremens is associaed wih a momen of ime, wih some measuremens incorporaing oher parameers. A ime series can be classified as coninuous or discree depending upon wheher he sequence is coninuous or discree. In he presen sudy, we shall be only concerned wih finie discree ime series which are measured a equal-disan ime inervals or hose coninuous ime series ha have been digiized o a finie discree ime series. 4

17 .. Sochasic Process A sochasic process is a family of ime indexed random variables X ( ω, ), where ω belongs o a sample space and belongs o an index se. For a fixed, X ( ω, ) is a random variable. For a given ω, X ( ω, ), as a funcion of, is called a sample funcion or realizaion. The populaion ha consiss of all possible realizaions is called ensemble in sochasic processes and ime series analysis. Thus, a ime series is a realizaion or sample funcion from a cerain sochasic process. Consider a finie se of random variables { X, X,..., X n } from a sochasic process { X ( ω, ) : = 0, ±, ±,...}. The n-dimensional disribuion funcion is defined by F ( x, x,..., xn) = P{ ω : X x,..., X xn} (..) X X,, X n n where x i, i =,,..., n are any real numbers. A process is said o be firs-order saionary if is one-dimensional disribuion funcion is ime invarian. i.e., if F F x ) = F ( x ) for any inegers k, and + k ; second-order saionary if ( X X + k, X X + k + k ( x, x) = FX, X ( x, x) for any ineger,, k, + k and + k ; and nhorder saionary if F ( x, x,..., x ) = n FX, X,, X ( x, x,..., xn) (..) X, X,, X n + k + k n + k for any n-uple,,..., ) and k inegers. A process is said o be sricly saionary if ( n (..) is rue for n =,,... The erms srongly saionary and compleely saionary are also used o denoe a sricly saionary process. Suppose (..) is rue for some n = m, i would also be rue for n m because he mh-order disribuion funcion deermines all disribuion funcions of lower order. Therefore, a higher order of 5

18 saionariy always implies a lower order saionariy. For a real-value process { : = 0, ±, ±,...}, we defined he mean funcion of he X process he variance funcion of he process μ = E X ) (..3) ( σ = E[( X μ ) ] (..4) he covariance funcion γ (, ) = E( X μ )( ) X μ (..5) and he correlaion funcion γ (, ) ρ (, ) = (..6) σ σ Since he disribuion funcion for a sricly saionary process is he same for all, he mean funcion μ = μ is a consan, provided E ( X ) <. This implies E( X ) <, and hence σ = σ for all is also a consan. In addiion, since F, X ( x, x) = FX, X ( x, x) for any ineger, and k, we have X + k + k and Leing = k and =, we ge γ (, ) = γ ( + k, + k ρ (, ) = ρ( + k, + k γ = ) ) (, ) = γ ( k, ) = γ (, + k) γ k (..7) and ρ = (, ) = ρ( k, ) = ρ(, + k) ρk (..8) Thus, for a sricly saionary process wih firs wo momens finie, he covariance 6

19 and correlaion beween X and depend only on he ime difference k. X + k A process is said o be nh-order weakly saionary if all of is join momens up o he nh-order exis and are ime invarian, hence a second-order weakly saionary process will have consan mean and variance, and he covariance and correlaion funcions depend only on he ime difference. Moreover, a sricly saionary process wih he firs wo momens being finie is also a second-order weakly saionary or covariance saionary process. However, a sricly saionary process does no always have finie momens. When his occurs, i may no be covariance saionary. Therefore, i is possible for a sricly saionary process, wih no join momens exising, o no be weakly saionary of any order... Whie Noise Process A process is said o be a whie noise process { W } if i is a sequence of uncorrelaed random variables from a fixed disribuion wih consan mean E( ) = μ and consan variance Var ) = σ and γ Cov( W, W ) = 0 for all W W ( W W k = +k k 0. I is obvious o see ha a whie noise process W } is saionary wih he auocovariance funcion γ = when k = 0, γ = 0 when k 0; he auocorrelaion k σ W funcion ρ = when k = 0, ρ = 0 when k 0; and parial auocorrelaion funcion k φ = when k = 0, φ = 0 when k 0. kk kk k The basic phenomenon of he whie noise is ha boh is Auocorrelaion Funcion (ACF) and Parial Auocorrelaion Funcion (PACF) are equal o zero. In addiion, a whie noise is Gaussian if is join disribuion is normal. This concep plays an k { 7

20 imporan role in consrucing ime series models in our laer chapers...3 Esimaions on a Single Realizaion A saionary ime series is characerized by is mean μ, variance σ, auocorrelaion ρ k, and parial auocorrelaion φ kk. We can calculae he exac values of hese parameers if he ensemble of all possible realizaions is available. However, for mos ime series, only a single realizaion is available, hus making i impossible o calculae he ensemble average. In he following discussion, we discuss some saisical properies on hese esimaors by using ime averages, which are useful when we deal wih only a single realizaion. The mean μ can be esimaed by is sample mean x = n n x = (..9) which is he ime average of n observaions. To see wheher his is a valid or good esimaor, we ake he expecaion of x, we ge n E( x) = E( x ) = nμ = μ (..0) n n which indicaes ha x is an unbiased esimaor for μ. And since = Var( x) = n γ = n γ 0 = n n n = s= n 0 ( k = ( n ) n k = ( n ) γ 0 Cov( x, xs) = n n k ) ρ ( k k ) ρk n n n = s= ρ ( s) (..) Choose k = ( s), we know ha if lim n n k = ( n ) k ρ k n 8

21 is finie, hen Var ( x) 0 as n, and x is a consisen esimaor for μ. Tha is, Similarly, he variance average as follows σ n lim x = μ (..) n n = can be esimaed by is sample variance by using he ime n σ = ( x x) (..3) n = he sample auocovariance funcion γ k is given by n k = he sample ACF ρ k can be esimaed by γ = ( x x)( x x) (..4) k n k + k γ ( x x)( x x) ρ, k = 0,,,... (..5) k = + k k = = n γ ( x = x) 0 and he sample PACF φ kk can be esimaed by using a recursive mehod saring wih = φ ρ was published by Durbin(960) as follows: φ k +, k + ρ = k + k φ j= kj k + j k φ j = kj ρ ρ j (..6).3 The Saionary AR, MA, and ARMA Models In ime series analysis here are wo very useful represenaions o express as a ime series process, namely he auoregressive process and he moving average process. Before he discussion of hese models, i is imporan o define several useful noaions for simplifying purposes. The backshif operaor is defined as B j x = x j (.3.) 9

22 and le φ ( B) = ( φ B φ B p... φ B p p ) (.3.) and θ ( B) = ( θ B θ B q... θ B q q ) (.3.3) By defining (.3.) and (.3.3), we can reduce models ino compac forms in laer discussion..3. The Auoregressive (AR) Models The auoregressive process of order p, denoed as AR(p) was firs inroduced by Yule in 96. I is useful in describing siuaions in which he presen value of a ime series is explained by is pas values plus a random shock. The ph-order auoregressive process is defined as follows: x = φ x + φ x φp x p + ε (.3.4) or φ p( B) x = ε (.3.5) where { ε } is a zero mean whie noise process, and x = x μ. In addiion, he AR(p) process conains p + parameers, which are φ φ,..., φ p σ. As φ <,,, ε he process is always inverible. To be saionary, he roos of φ ( B) = 0 mus lie ouside of he uni circle. p p j = j.3. The Moving Average (MA) Models The moving average process of order q, denoed as MA(q) was firs presened 0

23 by Sluzky in 97. I is useful in describing phenomena ha produce an immediae effec ha only lass for shor periods of ime. The qh-order moving average process is defined as follows: x = ε θ ε θ ε... θ ε q q (.3.6) or x = θ q( B) ε (.3.7) The MA(q) process conains q + parameers, which are θ θ,..., θ σ. Moreover, a finie moving average process is always saionary as, q, ε q + θ + θ θ <, and he process is inverible if he roos θ ( B) = 0 of lie ouside he uni circle. q.3.3 The General Mixed ARMA Models The mixure of auoregressive and moving average process, denoed as ARMA(p, q), can be produced if we combine he auoregressive and moving average process. The process was pu ogeher by Wold in 938, and i is defined as follows: x = φ x + φ x φp x p + ε θε θε... θqε q (.3.8) or φ p( B) x = θq( B) ε (.3.9) where φ ( B) = ( φ B φ B p... φ B p p ) and θ ( B) = ( θ B θ B q... θ B The general mixed ARMA(p, q) process has p + q + parameers, which are q q )

24 φ, φ,..., φ p, θ, θ,..., θq, σ ε. In addiion, he process is inverible if he roos ofθ ( B) = 0 lie ouside he uni circle and he process is saionary if he roos of q φ ( B) = 0 lie ouside of he uni circle and under he assumpion ha θ ( B) = 0 and p φ ( B) = 0 share no common roos. p q.4 Aims of Presen Sudy The aim of he presen sudy is o propose several new ime series mehodologies ha can be applied o many economic and environmenal problems. We begin wih summarizing some fundamenal conceps ha are essenial for dealing wih ime series modeling and inroduce some preliminary processes ha are necessary for model building in ime series. Afer we define all he necessiies for saionary ime series, we shall inroduce he classical nonsaionary ime series modeling. Box and Jenkins mehodology is one he mos famous ime series approach ha is widely being used by many researchers working in many differen areas. The idea of our proposed models is o filer a nonsaionary ime series and smooh he edges of ha series so we can be beer forecas he phenomena. The smoohing componens ha we implemen ino he ime series shall be removed once we have obained he model. Due o he complexiy of he developmenal processes, we summarize he mehodologies for boh he classical approach and our proposed models by providing sep by sep procedures in chaper. In chaper 3, 4, 5 and 6, we inroduce several economic and environmenal applicaions by developing ime series forecasing models using boh he classical

25 approach and our proposed mehodologies. We shall show he qualiy and usefulness of our proposed models on hose applicaions by comparing hem wih he classical mehods. In each of he applicaions, we shall examine he proposed models by looking ino some essenial saisical properies numerically and idenify he usefulness under circumsances. Finally, we shall evaluae he proposed models by ranking heir efficiency under differen circumsances according o heir performance in forecasing he fuure phenomena..5 Conclusion In he presen chaper, we began wih surveying several major lieraures in he area of ime series forecasing and providing he references of he developmen of hose mehodologies in he subjec area. We hen defined some fundamenal conceps such as sochasic process, whie noise process and saionary ARMA process. Those processes are no only essenial for our proposed model building procedures bu also play a major role in ime series forecasing. Finally, we briefly discuss he approach of our proposed mehodologies and show he usefulness and effeciveness of our proposed models wih he applicaions in he presen sudy. 3

26 Chaper Analyical Formulaions of Modified Time Series.0 Inroducion In he presen chaper, we shall summarize he updaed sep-by-sep process on developing a forecasing model from a non-saionary sochasic realizaion. In addiion, we inroduce hree addiional models, namely simple moving average, weighed moving average, and he exponenial moving average. The basic srucure of hese models begins wih he acual non-saionary realizaions and we formulae a new ime series from he original daa based on he models we menioned above.. The General ARIMA Model The ime series processes we have discussed so far are all saionary processes, bu mos of he ime series in realiy are non-saionary. Therefore, he roos of heir AR polynomials do no lie ouside he uni circle. Hence, we will no be able o use he general mixed ARMA(p,q) as we discussed in he previous chaper on a non-saionary ime series. In order for us o resolve his issue, we mus firs inroduce he difference filer as follows: d ( B) (..) where B j x = x j, and d is he degree of differencing of he series. 4

27 Due o he reason ha he local behavior of such non-saionary ime series is independen of is level, we le π (B) be he auoregressive operaor which describes he behavior of his homogeneous non-saionary ime series, we have π ( B )( x + C) = π ( B) (..) x for any consan C. This equaion implies ha π (B) mus be of he form π ( B ) = φ( B)( B) d (..3) for some d > 0, where φ (B) is a saionary auoregressive operaor. Hence, we can reduce a non-saionary ime series o a saionary ime series afer aking a proper degree of differencing of he series. Afer we define he difference filer, we can now inroduce he famous ARIMA(p,d,q) auoregressive inegraed moving average model as follows: d φ p( B)( B) x = θq( B) ε (..4) where d is he degree of differencing, and φ ( B) = ( φ B φ B p θ ( B) = ( θ B θ B Consider he simples case, ARIMA(0,,), we have q ( B) x = ( θ B) ε... φ B p... θ B q p q ) ) or we can expand he model as x = x + ε θε which is a MA() model on he difference of he non-saionary ime series. Consider anoher simples case, ARIMA(,,0), we have 5

28 ( φ B)( B) = ε x or we can expand he model as x = ( + ) x x φ + ε which is a AR() model on he difference of he non-saionary ime series. In ime series analysis, someimes i is very difficul o make a decision in selecing he bes order of he ARIMA(p,d,q) model when we have several models ha all adequaely represen a given se of ime series. Hence, Akaike s informaion crierion (AIC), [], plays a major role when i comes o model selecion. AIC was firs inroduced by Akaike in 973, and i is defined as follows: AIC(M) = - ln [maximum likelihood] + M, (..5) where M is he number of parameers in he model and he uncondiional log-likelihood funcion suggesed by Box, Jenkins, and Reinsel in 994, [4], is given by n S( φ, μ, θ ) ln L( φ, μ, θ, σ ε ) = ln πσ ε, (..6) σ where S ( φ, μ, θ ) is he uncondiional sum of squares funcion given by n S( φ, μ, θ ) = [ E( ε φ, μ, θ, x)] (..7) = where E( ε φ, μ, θ, x) is he condiional expecaion of ε given φ, μ, θ, x. ε μ The quaniies φ,, and θ ha maximize (..6) are called uncondiional maximum likelihood esimaors. Since ln L( φ, μ, θ, σ ε ) involves he daa only hrough S ( φ, μ, θ ), hese uncondiional maximum likelihood esimaors are equivalen o he uncondiional leas squares esimaors obained by minimizing S ( φ, μ, θ ). In pracice, he summaion in (..7) is approximaed by a finie form 6

29 n S( φ, μ, θ ) = [ E( ε φ, μ, θ, x)] (..8) where M is a sufficienly large ineger such ha he backcas incremen = M E( ε φ, μ, θ, x) E( ε φ, μ, θ, x) is less han any arbirary predeermined small ε value for ( M +). This expression implies ha E( ε φ, μ, θ, x) μ ; hence, E( ε φ, μ, θ, x) is negligible for ( M +). Afer obaining he parameer esimaes φ,, and θ, he esimae of can hen be calculaed from μ σ ε S ( φ, μ, θ ) σ ε =. (..9) n For an ARMA(p,q) model based on n observaions, he log-likelihood funcion is ln n L = ln πσ ε S( φ, μ, θ ). (..0) σ We proceed o maximize (..0) wih respec o he parameers φ, μ, θ, and, from (..9), we have n n ln L = lnσ ε ( + ln π ). (..) Since he second erm in expression (..) is a consan, we can reduce he AIC o he following expression σ ε ε AIC(M) = nln + M. (..) Thus, we generae an appropriae ime series model and selec he saisical process wih he smalles AIC. The model ha we have idenified will possess he smalles average mean square error. In addiion, we summarize he developmen of he model as follows: 7 σ ε σ ε

30 Check for saionariy of he ime series by deermining he order of differencing d, where d = 0,,,... according o KPSS es [], unil we achieve saionariy Deciding he order m of he process, for our case, we le m = 5 where p + q = m Afer (d, m ) being seleced, lising all possible se of (p, q) for p + q m For each se of (p, q), esimaing he parameers of each model, ha is, φ,...,, φ,..., φ p, θ, θ θ q Compue he AIC for each model, and choose he one wih smalles AIC According o he crierion ha we menioned above, we can obain he ARIMA(p,d,q) model ha bes fi a given ime series, where he coefficiens are φ φ,..., φ, θ, θ,..., θ, p q.. The k-h Moving Average Time Series Model In his secion, we propose a model which is consruced based on he concep of he simple moving average. Before we acually discuss he proposed models, we shall firs define he simple moving average. In ime series analysis, he primary use for he k- h simple moving average is for smoohing a ime series. I is very useful in discovering shor-erm, long-erm rends and seasonal componens of a ime series. The k-h simple moving average process of a ime series { x } is defined as follows: k = x k + + j k j = 0 y (..) where = k, k +,..., n. I is obvious o see ha as k increases, he number of observaions of { y } decreases, and he series { y } ges closer and closer o he mean of he series x } as k increases. In { 8

31 addiion, when k = n, he series y } reduces o a single observaion, and equals o μ, because { y = n x j n j = = μ (..) On he oher hand, if we choose a fairly small k, we can smooh he edges of he series wihou losing oo much of he general informaion. We proceed o develop our proposed model by ransforming he original ime series { x } ino y } by applying (..). Afer esablishing he new ime series, usually { very nonsaionary, we begin he process of reducing i o saionary ime series by selecing he appropriae differencing order. We hen proceed wih he model building procedure of developing ime series model and a each sage calculae he AIC. The bes model will be he one wih he smalles AIC. Using he model ha we developed for forecas values of { y } he original phenomenon and subjec o he AIC crieria, we and proceed o apply he back-shif operaor o obain esimaes of { x }, ha is, { y } x = k y x k x... x +. (..3) In addiion, we summarize he developmen of he model as follows: Transforming he original ime series { x } ino y } by applying (..) Check for saionariy of he series { y } by deermining he order of differencing d, where d = 0,,,... according o KPSS es, unil we achieve saionariy { Deciding he order m of he process, for our case, we le m = 5 where p + q = m Afer (d, m ) being seleced, lising all possible se of (p, q) for p + q m 9

32 For each se of (p, q), esimaing he parameers of each model, ha is, φ,...,, φ,..., φ p, θ, θ θ q Compue he AIC for each model, and choose one wih he smalles AIC Solve he esimaes of he original ime series by applying (..3) The proposed model and he corresponding procedure discussed in his secion shall be illusraed wih real economic applicaion and he resuls will be compared wih he classical ime series model..3 The k-h Weighed Moving Average Time Series Model Compare o he model ha we proposed in he previous secion, he model we proposed in his secion is more useful o hose ime series which behave more aggressively compared wih some ime series which do no. Before we acually presen he model, we shall firs inroduce he weighed moving average process. The k-h weighed moving average process is also a very good smoohing ool. However, he srucure of i is slighly differen from he simple moving average process. I pus more weigh on he mos recen observaion, and he weigh consisenly decreases up o he firs observaion. In addiion, i capures he original ime series beer han he moving average process, which is suiable for hose analyss who believe he recen observaions should weigh more han he old ones. The k-h weighed moving average process of a ime series { x } is defined as follows: k = ( j + ) x k + + j ( + k) k / j = 0 z (.3.) 0

33 where = k, k +,..., n. Similar o he moving average process, as k increases, he number of observaions of he series { z } decreases, and as k n, from (.3.) he series z } becomes { z = ( + n) n / jx j j= n (.3.) If we choose a fairly small k, we can smooh he edges of he series, and he series { z } is closer o he series { x } compared wih he moving average mehod ha we discussed in he las secion. ino { z } Our firs proposed model begins wih ransforming he original ime series by applying (.3.). Afer esablishing he new ime series, we begin he process of reducing i o saionary ime series by selecing he appropriae differencing order. We hen proceed wih he model building procedure of developing ime series model and a each sage calculae he AIC. The bes model will be he one wih he smalles AIC. Using he model ha we developed for forecas values of { z } he original phenomenon { x } and subjec o he AIC crieria, we and proceed o apply he back-shif operaor o obain esimaes of { x }, ha is, { z } x [( + k) k / ] z ( k ) x ( k ) x... x k + = (.3.3) k In addiion, we summarize he developmen of he model as follows: Transforming he original ime series { x } ino z } by applying (.3.) Check for saionariy of he series z } by deermining he order of differencing { d, where d = 0,,,... according o KPSS es, unil we achieve saionariy {

34 Deciding he order m of he process, for our case, we le m = 5 where p + q = m Afer (d, m ) being seleced, lising all possible se of (p, q) for p + q m For each se of (p, q), esimaing he parameers of each model, ha is, φ,...,, φ,..., φ p, θ, θ θ q Compue he AIC for each model, and choose one wih he smalles AIC Solve he esimaes of he original ime series by applying (.3.3) To conclude he goodness of fi of he proposed model, we will illusrae in a laer chaper wih numerical examples..4 The k-h Exponenial Weighed Moving Average Time Series Models Similar o he las wo secions, in order for us o presen our proposed models, we mus firs inroduce he exponenial weighed moving average mehod. The k-days exponenial weighed moving average serves prey much he same purpose as hose wo average processes as we discussed in he las wo secions. Insead of decreasing weigh consisenly as he weighed moving average mehod, i decreases weigh exponenially. This fis perfec for hose analyss who care mos on he recen observaions and no pay oo much aenion o he old ones. In addiion, i is no difficul o see ha he exponenial weighed moving average caches he original series faser han boh average processes as we discussed earlier. Hence, i is very useful in dealing wih very aggressive ime series. The k-h exponenial weighed moving average process of a ime series is defined as follows: { x } v = k k j ( α) j = 0 j = 0 ( α ) k j x k + + j (.4.)

35 where = k, k +,..., n, and he smoohing facor α is defined as α =. If we le k + k k = n, we have α =. Moreover, n + j= 0 j ( α ) reaches is maximum when k = 3, and i ges closer and closer o as k increases. As k increases, he number of observaions he series { v } decreases, and i evenually reduces o a single observaion when k = n. As k n, he series v } becomes { v = n j= 0 n ( α) j j= 0 ( α) n j x j+ (.4.) From (.4.), i is obvious o see ha he exponenial weighed moving average process weighs heavily on he mos recen observaion and decreases he weigh exponenially as ime decreases. If we choose a fairly small k, we can smooh he edges of he series and he { v } would be he closes o he series x } compared wih boh moving average { processes ha we discussed in he previous wo secions. We proceed o develop our proposed model by ransforming he original ime series { x } ino v } by applying (.4.). Afer obaining he new ime series, we selec { he appropriae differencing order o reduce he series { v } ino a saionary ime series. We hen proceed he model building procedure of developing ime series model and a each sage calculae he AIC. The bes model will be he one wih he smalles AIC. Using he model ha we developed for { v } and subjec o he AIC crieria, we forecas values of { v } and proceed o apply he back-shif operaor o obain esimaes of he original phenomenon { x }, ha is, 3

36 x k = ( α) x ( )... ( ) α x α x k k j j = 0 v ( α ) (.4.3) In addiion, we summarize he developmen of he model as follows: Transforming he original ime series { x } ino v } by applying (.4.) Check for saionariy of he series v } by deermining he order of differencing { d, where d = 0,,,... according o KPSS es, unil we achieve saionariy { Deciding he order m of he process, for our case, we le m = 5 where p + q = m Afer (d, m ) being seleced, lising all possible se of (p, q) for p + q m For each se of (p, q), esimaing he parameers of each model, ha is, φ,...,, φ,..., φ p, θ, θ θ q Compue he AIC for each model, and choose one wih he smalles AIC Solve he esimaes of he original ime series by applying (.4.3) The proposed model and he corresponding procedure discussed in his secion shall be illusraed wih real economic applicaion and he resuls will be compared wih he classical ime series model..5 The Muliplicaive ARIMA Model In ime series analysis, seasonal variaions someimes dominae he variaions of he original nonsaionary ime series. I occurs very commonly on he environmenal applicaions, as mos of he environmenal forecasing problems, here exiss some ype of periodic rend ha is obvious for us o recognize. We can rea he seasonal ime series as a nonsaionary ime series ha varies along some kind of seasonal periodic rend. 4

37 Hence, addressing he seasonal variaions for he model becomes exremely useful when we deal wih hese ypes of problems. The muliplicaive seasonal auoregressive inegraed moving average, ARIMA model is defined by Φ P ( B Γ s d s D s ) φ p( B)( B) ( B ) x = θq( B) Q( B ) ε (.5.) where p is he order of he auoregressive process, d is he order of regular differencing, q is he order of he moving average process, P is he order of he seasonal auoregressive process, D is he order of he seasonal differencing, Q is he order of he seasonal moving average process, and he subindex s refers o he seasonal period. We shall denoe he s s subjec model by ARIMA ( p, d, q) ( P, D, Q), and φ ( B), θ ( B), Φ ( B ), Γ ( B ) defined as follows: φ ( B) = ( φ B φ B p θ ( B) = ( θ B θ B q s p... φ B p q... θ B q p q ) ) P q and s s s Φ P( B ) = ΦB ΦB... s s s Γ Q( B ) = Γ B Γ B... Φ The order of he muliplicaive ARIMA model deermines he srucure of he model, and i is essenial o have a good mehodology in erms of developing he forecasing model. Below we summarize he model idenifying procedure: Deermine he seasonal period s. Γ Q P B B Qs Ps. 5

38 Check for saionariy of he given ime series { x } by deermining he order of differencing d, where saionariy. d = 0,,,... according o KPSS es, unil we achieve Deciding he order m of he process, for our case, we le m = 5 where p + q + P + Q = m. Afer ( d, m) being seleced, lising all possible configuraions of ( p, q, P, Q) for p + q + P + Q m. For each se of ( p, q, P, Q), esimaes he parameers for each model, ha is, φ Φ Φ Φ Γ Γ,..., Γ., φ,..., φ p, θ, θ,..., θ q,,,..., P,, Q Compue he AIC for each model, and choose he one wih smalles AIC. Afer ( p, d, q, P, Q ) is seleced, we deermine he seasonal differencing filer by selecing he smaller AIC beween he model wih D = 0 and D =. Our final model will have idenified he order of ( p, d, q, P, D, Q ). To conclude he goodness of fi of he Muliplicaive ARIMA model, we will illusrae in laer chaper wih environmenal applicaions..6 Conclusion In he presen chaper, we began wih developing a sep-by-sep forecasing procedure for he classical ARIMA model. We hen inroduced our hree proposed models, namely he k-h Moving Average Time Series Model, he k-h Weighed Moving Average Time Series Model, and he k-h Exponenial Weighed Moving Average Time Series Model, of which we believe perform beer han he classical mehodology. In 6

39 addiion, we have developed a sep-by-sep procedure for each of our proposed models. This sep-by-sep procedure of each model shall be very helpful when one needs o develop model following our mehodologies. Finally, we inroduce he muliplicaive ARIMA model ha is able o address he seasonal variaion when we deal wih environmenal applicaions in laer chapers and a sep-by-sep procedure for he subjec model. 7

40 Chaper 3 Economerics Forecasing Models 3.0 Inroducion The objec of he presen chaper is o illusrae he resuls of our proposed forecasing models for wo nonsaionary sochasic realizaions. The subjec models are based on modifying boh given ime series ino a new k-h moving average, k-h weighed moving average, and k-h exponenial weighed moving average ime series o begin he developmen of he model. The sudy is based on he auoregressive inegraed moving average process along wih is analyical consrains. The analyical procedure of he proposed models is given in chaper wo. The firs series is a sock XYZ seleced from he Forune 500 companies and is daily closing price consiue he ime series. The second series is he closing price of he S&P price index. Boh he classical and proposed forecasing models were developed and a comparison of he accuracy of heir responses is given. 3. Forecasing Models on Sock XYZ In his secion, we firs begin wih collecing 500 observaions for a sock from he forune 500 companies and ry o forecas is daily closing value by using he radiional approach versus differen forecasing models ha we proposed in chaper wo. We hen examine each model by looking ino he properies of is plos and residuals and rank each model based on he resuls of our findings. Finally, we hide he las 5 8

41 observaions and forecas he nex 5 observaions wihou using he fuure informaion. Thus, he coefficiens of he models are updaed every ime when we ge new informaion. We hen examine he las 5 residuals so we can deermine he goodness of fi of hose models and draw conclusions based on hese resuls. 3.. Daa Preparaion In order for us o properly illusrae differen ypes of moving average ARIMA models, we need o firs creae anoher hree ime series, namely 3 days moving average ime series (MA3), 3 days weighed moving average ime series (WMA3), and 3 days exponenial weighed moving average ime series (EWMA3) by using he mehodology ha we discussed in chaper wo. Suppose we le { x } (see Appendix A) be he daily closing values of sock XYZ ha we collec from he forune 500 companies ha we menioned earlier. A plo of he acual informaion is given by Figure 3. Price Time Figure 3. Daily Closing Price for Sock XYZ 9

42 In order for us o apply he fiing procedure for our firs proposed model, we mus ransform { x } ino a 3 days moving average series y } (see Appendix A) by referring o (..). Hence, we have { y = y are no available, and y 3 = x + x + x3 3 y 4 = x + x3 + x 3 4 y x + x + x (3..) 3 = Similar o he las ransformaion, we need o creae anoher series by referring o (.3.), which is ransferring { x } ino a 3 days weighed moving average series z } (see Appendix A3). Hence, we have z = z are no available, and { z z 3 4 = = x x x + 3x 6 4 x3 + 3x 6 z x + x + 3x (3..) 6 = The las ransformaion we need o make is urning he original ime series { x } ino a 3 days exponenial weighed moving average series { v } (see Appendix A4) by referring o (.4.). ha is v = v are no available, and v 3. 5x +.5x + x3 =.75 30

43 v 4. 5x +,5x3 + x4 =.75 v. = 5x +.5x + x (3..3).75 Afer finishing all of he above ransformaions, we end up wih 4 se of differen ime series observaions, which are { x }, y }, z }, and v }. Now we are ready for he { model fiing procedures in he nex secion. In he following sub-secions, we shall follow hose sep by sep procedures as we discussed in chaper wo for boh he classical ime series approach and our proposed mehods. { { 3.. The General ARIMA Model Following he sep-by-sep procedure ha we inroduced in Secion., he classical forecasing model wih he bes AIC score is he ARIMA(,,). Tha is, a combinaion of firs order auoregressive (AR) and a second order moving average (MA) wih a firs difference filer. Thus, we can wrie i as (.963B)( B) x = (.053B +.058B ) ε. (3..4) Afer expanding he auoregressive operaor and he difference filer, we have (.963B +.963B ) x = (.053B +.058B ) ε and he model can be wrien as x =.963x.963x + ε.053ε ε 3

44 by leing ε = 0, we have he one day ahead forecasing ime series of he closing price of sock XYZ as x =.963x.963x.053ε ε. (3..5) Using he above equaion, we graph he forecasing values obained by using he classical approach on op of he original ime series, shown as Figure 3. Price Original Daa Classical ARIMA Time Figure 3. Comparisons on Classical ARIMA Model VS. Original Time Series The basic saisics ha reflec he accuracy of model (3..5) are he mean r, variance Sr, sandard deviaion S and sandard error r S r of he residuals. Figure 3.3 gives a n ime series plo of he residual and able 3. provides he basic saisics. 3

45 Price Time Figure 3.3 Time Series Plo of he Residuals for Classical Model Table 3. Basic Evaluaion Saisics for Classical ARIMA r S r S r S r n Furhermore, we resrucure he model (3..5) wih n = 475 daa poins o forecas he las 5 observaions only using he previous informaion. The purpose is o see how accurae our forecas prices are wih respec o he acual 5 values ha have no been used. Table 3. gives he acual price, prediced price, and residuals beween he forecass and he 5 hidden values. 33

46 Table 3. Acual and Prediced Price for Classical ARIMA N Acual Price Prediced Price Residuals The average of hese residuals is r = , and he Figure 3.4 is a graph presenaion of he forecasing resul. 34

47 Price Original Daa Classical ARIMA Time Figure 3.4 Classical ARIMA Forecasing on he Las 5 Observaions 3..3 The 3 Days Moving Average ARIMA Models Figure 3.5 shows he new ime series y } along wih he original ime series x }, ha { we shall use o develop he proposed forecasing model. { Price Original Daa MA Time Figure 3.5 MA3 Series VS. The Original Time Series 35

48 Following he procedure we discussed in secion., we found he bes model ha characerizes he behavior of { y } o be ARIMA(,,3). Tha is, (.896B.0605B )( B) y = ( B.0056B 3 B ) ε. (3..6) Expanding he auoregressive operaor and he firs difference filer, we have (.896B B B ) y = ( B.0056B B 3 ) ε. and he model can be wrien as y =.896y.8356y.0605y ε. 0056ε ε 3 ε. The final analyical form of he proposed forecasing model can be wrien as y =.896y.8356 y.0605y ε. 0056ε ε 3. (3..7) Using he above equaion, we graph he forecasing values obained by using he MA3 ARIMA approach on op of he original ime series, shown as Figure 3.6 Price Original Daa MA3 ARIMA Time Figure 3.6 Comparisons on MA3 ARIMA Model VS. Original Time Series Noe he very closeness of he wo plos ha reflec he qualiy of he proposed model. 36

49 Similar o he classical model approach ha we discussed earlier, we shall use he firs 475 observaions y, y,..., } o forecas. Then we use he observaions { y475 y 476 { y, y,..., y476} o forecas y 477, and coninue his process unil we obain forecass all he observaions, ha is, { y476, y477,..., y500}. From equaion (..3), we can see he relaionship beween he forecasing values of he original series { x } and he forecasing values of 3 days moving average series { y },ha is, x = 3 y x x. (3..8) Hence, afer we esimaed { y476, y477,..., y500}, we can use he above equaion, (3..8), o solve he forecasing values for. Figure 3.7 is he residual plo generaed by he MA3 ARIMA model, and followed by Table 3.3, ha includes he basic evaluaion saisics. { x } Price Figure 3.7 Time Series Plo of he Residuals for MA3 Model Time 37

50 Table 3.3 Basic Evaluaion Saisics for MA3 ARIMA r S r S r S r n Boh of he above displayed evaluaions reflec on accuracy of he proposed model. The acual daily closing prices of sock XYZ from he 476 h day along wih he forecased prices and residuals are provided in Table 3.4. Table 3.4 Acual and Prediced Price for MA3 ARIMA N Acual Price Prediced Price Residuals

51 The Resuls given above aes o he good forecasing esimaes for he hidden daa. The average of hese residuals is r = , and he Figure 3.8 is a graph presenaion of he forecasing resul. Price Original Daa MA3 ARIMA Time Figure 3.8 MA3 ARIMA Forecasing on he Las 5 Observaions 3..4 The 3 Days Weighed Moving Average ARIMA Models Figure 3.9 shows he new ime series y } along wih he original ime series x }, ha { we shall use o develop he proposed forecasing model. { 39

52 Price Original Daa WMA Time Figure 3.9 WMA3 Series VS. The Original Time Series Following he procedure we have saed in secion.3, we found he bes model ha characerizes he behavior of { z } o be ARIMA(,,3). Tha is, ( B)( B) z = ( B B B 3 ) ε (3..9) expand he auoregressive operaor and he difference filer, we have (.097B.9073B ) z = ( B B +.456B 3 ) ε and he model can be wrien as z =. 097z z + ε ε ε ε 3 by leing ε = 0, we have he one day ahead forecasing series as z =. 097z z ε ε ε 3 (3..0) Using he above equaion, we graph he forecasing values obained by using he WMA3 ARIMA approach on op of he original ime series, shown as Figure

53 Price Original Daa WMA3 ARIMA Time Figure 3.0 Comparisons on WMA3 ARIMA Model VS. Original Time Series Similarly, we shall use he firs 475 observaions z, z,..., } o forecas. { z475 z 476 Then we use he observaions z, z,..., } o forecas, and coninue his process { z476 unil we obain forecass all he observaions, ha is, { z 476, z477,..., z500}. From equaion (.3.3), we can see he relaionship beween he forecasing values of he original series z 477 { x } and he forecasing values of 3 days moving average series { z },ha is, solve he forecasing values for 6 z x x x =. (3..) 3 Hence, afer we esimaed { z 476, z477,..., z500}, we can use he above equaion, (3..), o { x }. Figure 3. is he residual plo generaed by our proposed model, and followed by Table 3.5, ha includes he basic evaluaion saisics. 4

54 Price Time Figure 3. Time Series Plo of he Residuals for WMA3 Model Table 3.5 Basic Evaluaion Saisics for WMA3 ARIMA r S r S r S r n A deail ranking comparison beween models will be illusraed in laer secion, and Table 3.6 shows a head o head comparison beween he acual and prediced price for WMA3 ARIMA model. 4

55 Table 3.6 Acual and Prediced Price for WMA3 ARIMA N Acual Price Prediced Price Residuals The average of hese residuals is r = , and he Figure 3. is a graph presenaion of he forecasing resul. 43

56 Price Original Daa WMA3 ARIMA Time Figure 3. WMA3 ARIMA Forecasing on he Las 5 Observaions 3..5 The 3 Days Exponenial Weighed Moving Average ARIMA Models Figure 3.3 shows he new ime series { y } along wih he original ime series x }, ha we shall use o develop he proposed forecasing model. { Price Original Daa EWMA Time Figure 3.3 WMA3 Series VS. The Original Time Series 44

57 Following he procedure we have saed in secion.4, we found he bes model ha characerizes he behavior of { v } o be ARIMA(3,,), ha is, 3 (.4766B.9045B )( B) v = (.436B.078B.907B ) ε (3..) expand he auoregressive operaor and he difference filer, we have (.4766B.479B B ) v = (.436B.078B.907B 3 ) ε and rewrie he model as v =.4766v +.479v.9045v 3 + ε.436ε.078ε. 907ε 3 by leing ε = 0, we have he one day ahead forecasing series as v =.4766v +.479v.9045v 3.436ε.078ε. 907ε 3 (3..3) Using he above equaion, we graph he forecasing values obained by using he EWMA3 ARIMA approach on op of he original ime series, shown as Figure 3.4 Price Original Daa EWMA3 ARIMA Time Figure 3.4 Comparisons on EWMA3 ARIMA Model VS. Original Time Series Similarly, we shall use he firs 475 observaions v, v,..., } o forecas. { v475 v 476 Then we use he observaions v, v,..., } o forecas, and coninue his process { v v 477

58 unil we obain forecass all he observaions, ha is, { v476, v477,..., v500}. From equaion (.4.3), he relaionship beween he forecasing values of he original series x } and he { forecasing values of 3 days moving average series { v } can be derived as x =.75v.5x. 5x. (3..4) Hence, afer we esimaed { v476, v477,..., v500}, we can use he above equaion, (3..4), o solve he forecasing values for { x }. Figure 3.5 is he residual plo generaed by our proposed model, and followed by Table 3.7, ha includes he basic evaluaion saisics. Price Figure 3.5 Time Series Plo of he Residuals for EWMA3 Model Time Table 3.7 Basic Evaluaion Saisics for EWMA3 ARIMA r S r S r S r n 46

59 A deail ranking comparison beween models will be illusraed in laer secion, and Table 3.8 shows a head o head comparison beween he acual and prediced price for WMA3 ARIMA model. Table 3.8 Acual and Prediced Price for EWMA3 ARIMA N Acual Price Prediced Price Residuals The average of hese residuals is r = , and he Figure 3.6 is a graph presenaion of he forecasing resul. 47

60 Price Original Daa EWMA3 ARIMA Time Figure 3.6 EWMA3 ARIMA Forecasing on he Las 5 Observaions 3. Forecasing Models on S&P Price Index In he following applicaion, we shall use he S&P Price Index, and consider is daily closing price for 500 days o consiue he ime series informaion is given by Figure 3.7. { x }. A plo of he acual Price Figure 3.7 Daily Closing Price for S&P Price Index Time 48

61 3.. Daa Preparaion To proceed wih he model building processes, we shall firs creae anoher hree ime series, namely 3 days moving average ime series (MA3), 3 days weighed moving average ime series (WMA3), and 3 days exponenial weighed moving average ime series (EWMA3) by using he mehodologies ha we discussed in chaper wo. Suppose we le { x } (see Appendix B) be he daily closing values of he S&P Price Index ha we menioned earlier. In order for us o proceed he fiing procedure for our firs proposed model, we mus ransform { x } ino a 3 days moving average series { y } (see Appendix B), a 3 days weighed moving average series (see Appendix B3), and a 3 days exponenial weighed moving average series v } (see Appendix B4) by referring o (3..), (3..), and (3..3) respecively. { { z } Afer finishing all of he above ransformaions, we end up wih 4 ses of differen ime series observaions, which are { x }, y }, z }, and v }. Hence, we are now ready { o proceed wih he model fiing procedures. { In he following sub-secions, we shall follow hose sep by sep procedures as we discussed in chaper wo for boh he classical ime series approach and our proposed mehods. { 3.. The General ARIMA Model Following he sep-by-sep procedure ha we inroduced in Secion., he classical forecasing model wih he bes AIC score is he ARIMA(0,,). Tha is, a second order moving average (MA) wih a firs difference filer. Thus, we can wrie i as ( B) x = (.033B.04B 49 ) ε. (3..)

62 Afer expanding he auoregressive operaor and he difference filer, we have x x = (.033B.04B ) ε and he model can be wrien as x = x + ε. 033ε. 04ε by leing ε = 0, we have he one day ahead forecasing ime series of he closing price of sock XYZ as x = x ε ε. (3..) Using he above equaion, we graph he forecasing values obained by using he classical approach on op of he original ime series, shown as Figure 3. Price Original Daa Classical ARIMA Time Figure 3.8 Comparisons on Classical ARIMA Model VS. Original Time Series The basic saisics ha reflec he accuracy of model (3..5) are he mean r, variance Sr, sandard deviaion S and sandard error r S r of he residuals. Figure 3.9 gives a n ime series plo of he residual and able 3.9 provides he basic saisics. 50

63 Price Time Figure 3.9 Time Series Plo of he Residuals for Classical Model Table 3.9 Basic Evaluaion Saisics for Classical ARIMA r S r S r S r n Furhermore, we resrucure he model (3..5) wih n = 475 daa poins o forecas he las 5 observaions only using he previous informaion. The purpose is o see how accurae our forecas prices are wih respec o he acual 5 values ha have no been used. Table 3.0 gives he acual price, prediced price and residuals beween he forecass and he 5 hidden values. 5

64 Table 3.0 Acual and Prediced Price for Classical ARIMA N Acual Price Prediced Price Residuals The average of hese residuals is r = , and he Figure 3.0 is a graph presenaion of he forecasing resul. 5

65 Price Original Daa Classical ARIMA Time Figure 3.0 Classical ARIMA Forecasing on he Las 5 Observaions 3..3 The 3 Days Moving Average ARIMA Models Figure 3. shows he new ime series { y } along wih he original ime series x }, ha we shall use o develop he proposed forecasing model. { Price Original Daa MA Time Figure 3. MA3 Series VS. The Original Time Series 53

66 Following he procedure we discussed in secion., we found he bes model ha characerizes he behavior of { y } o be ARIMA(0,,4). Tha is, ( B) y = ( B +.844B.56B 3 4.3B ) ε. (3..3) Expanding he auoregressive operaor and he firs difference filer, we have y 4 y.3b ) ε. 3 = ( B +.844B.56B and he model can be wrien as y = y ε +.844ε.56ε 3. 3ε 4 ε. The final analyical form of he proposed forecasing model can be wrien as y = y ε.56ε 3. 3ε 4 ε. (3..4) Using he above equaion, we graph he forecasing values obained by using he MA3 ARIMA approach on op of he original ime series, shown as Figure 3. Price Original Daa MA3 ARIMA Time Figure 3. Comparisons on MA3 ARIMA Model VS. Original Time Series Noe he very closeness of he wo plos ha reflec he qualiy of he proposed model. 54

67 Similar o he classical model approach ha we discussed earlier, we shall use he firs 475 observaions y, y,..., } o forecas. Then we use he observaions { y475 y 476 { y, y,..., y476} o forecas y 477, and coninue his process unil we obain forecass all he observaions, ha is, { y476, y477,..., y500}. From equaion (..3), we can see he relaionship beween he forecasing values of he original series { x } and he forecasing values of 3 days moving average series { y },ha is, x = 3 y x x. (3..8) Hence, afer we esimaed { y476, y477,..., y500}, we can use he above equaion, (3..8), o solve he forecasing values for. Figure 3.3 is he residual plo generaed by he MA3 ARIMA model, and followed by Table 3., ha includes he basic evaluaion saisics. { x } Price Figure 3.3 Time Series Plo of he Residuals for MA3 Model Time 55

68 Table 3. Basic Evaluaion Saisics for MA3 ARIMA r S r S r S r n Boh of he above displayed evaluaions reflec on accuracy of he proposed model. The acual daily closing prices of sock XYZ from he 476 h day along wih he forecased prices and residuals are provided in Table 3.. Table 3. Acual and Prediced Price for MA3 ARIMA N Acual Price Prediced Price Residuals

69 The Resuls given above aes o he good forecasing esimaes for he hidden daa. The average of hese residuals is r = , and he Figure 3.4 is a graph presenaion of he forecasing resul. Price Original Daa MA3 ARIMA Time Figure 3.4 MA3 ARIMA Forecasing on he Las 5 Observaions 3..4 The 3 Days Weighed Moving Average ARIMA Models Figure 3.5 shows he new ime series { y } along wih he original ime series x }, ha we shall use o develop he proposed forecasing model. { 57

70 Price Original Daa WMA Time Figure 3.5 WMA3 Series VS. The Original Time Series Following he procedure we have saed in secion.3, we found he bes model ha characerizes he behavior of { z } o be ARIMA(0,,). Tha is, ( B) z = ( B +.49B ) ε (3..9) expand he auoregressive operaor and he difference filer, we have z z = ( B +.49B ) ε and he model can be wrien as z = z + ε ε +. 49ε by leing ε = 0, we have he one day ahead forecasing series as z = z ε ε (3..0) Using he above equaion, we graph he forecasing values obained by using he WMA3 ARIMA approach on op of he original ime series, shown as Figure

71 Price Original Daa WMA3 ARIMA Time Figure 3.6 Comparisons on WMA3 ARIMA Model VS. Original Time Series Similarly, we shall use he firs 475 observaions z, z,..., } o forecas. { z475 z 476 Then we use he observaions z, z,..., } o forecas, and coninue his process { z476 unil we obain forecass all he observaions, ha is, { z 476, z477,..., z500}. From equaion (.3.3), we can see he relaionship beween he forecasing values of he original series z 477 { x } and he forecasing values of 3 days moving average series { z },ha is, solve he forecasing values for 6 z x x x =. (3..) 3 Hence, afer we esimaed { z 476, z477,..., z500}, we can use he above equaion, (3..), o { x }. Figure 3.7 is he residual plo generaed by our proposed model, and followed by Table 3.3, ha includes he basic evaluaion saisics. 59

72 Price Time Figure 3.7 Time Series Plo of he Residuals for WMA3 Model Table 3.3 Basic Evaluaion Saisics for WMA3 ARIMA r S r S r S r n A deail ranking comparison beween models will be illusraed in a laer secion, and Table 3.4 shows a head o head comparison beween he acual and prediced price for WMA3 ARIMA model. 60

73 Table 3.4 Acual and Prediced Price for WMA3 ARIMA N Acual Price Prediced Price Residuals The average of hese residuals is r = , and he Figure 3.8 is a graph presenaion of he forecasing resul. 6

74 Price Original Daa WMA3 ARIMA Time Figure 3.8 WMA3 ARIMA Forecasing on he Las 5 Observaions 3..5 The 3 Days Exponenial Weighed Moving Average ARIMA Models Figure 3.9 shows he new ime series { y } along wih he original ime series x }, ha we shall use o develop he proposed forecasing model. { Price Original Daa EWMA Time Figure 3.9 EWMA3 Series VS. The Original Time Series 6

75 Following he procedure we have saed in secion.4, we found he bes model ha characerizes he behavior of { v } o be ARIMA(3,,), ha is, (.967B +.96B B )( B) v = (.544B.946B 3 ) ε (3..) expand he auoregressive operaor and he difference filer, we have (.967B +.093B +.967B B ) v = (.544B.946B 3 ) ε and rewrie he model as v =.967v.093v.967v v 4 + ε.544ε. 946ε by leing ε = 0, we have he one day ahead forecasing series as v =.967v.093v.967v v ε ε (3..3) Using he above equaion, we graph he forecasing values obained by using he EWMA3 ARIMA approach on op of he original ime series, shown as Figure 3.30 Price Original Daa EWMA3 ARIMA Time Figure 3.30 Comparisons on EWMA3 ARIMA Model VS. Original Time Series Similarly, we shall use he firs 475 observaions v, v,..., } o forecas. { v475 v 476 Then we use he observaions v, v,..., } o forecas, and coninue his process { v v 477

76 unil we obain forecass all he observaions, ha is, { v476, v477,..., v500}. From equaion (.4.3), he relaionship beween he forecasing values of he original series x } and he { forecasing values of 3 days moving average series { v } can be derived as x =.75v.5x. 5x. (3..4) Hence, afer we esimaed { v476, v477,..., v500}, we can use he above equaion, (3..4), o solve he forecasing values for { x }. Figure 3.3 is he residual plo generaed by our proposed model, and followed by Table 3.5, ha includes he basic evaluaion saisics. Price Figure 3.3 Time Series Plo of he Residuals for EWMA3 Model Time Table 3.5 Basic Evaluaion Saisics for EWMA3 ARIMA r S r S r S r n 64

77 A deail ranking comparison beween models will be illusraed in laer secion, and Table 3.6 shows a head o head comparison beween he acual and prediced price for WMA3 ARIMA model. Table 3.6 Acual and Prediced Price for EWMA3 ARIMA N Acual Price Prediced Price Residuals The average of hese residuals is r =. 66, and he Figure 3.3 is a graph presenaion of he forecasing resul. 65

78 Price Original Daa EWMA3 ARIMA Time Figure 3.3 EWMA3 ARIMA Forecasing on he Las 5 Observaions 3.3 Evaluaions on Proposed Models VS. Classical Mehod In he previous secions, we have discussed he classical ARIMA, MA3 ARIMA, WMA3 ARIMA, and EWMA3 ARIMA models on boh sock XYZ and S&P Price Index. We shall now compare he performance of each model hrough examining heir basic saisical properies. The Following Table 3.7 shows he comparison beween models for sock XYZ. Table 3.7 Ranking Comparison on sock XYZ r S r S r S r Order n Classical ARIMA ,, MA3 ARIMA ,,3 WMA3 ARIMA ,,3 EWMA3 ARIMA ,, 66

79 According o Table 3.7 he MA3 ARIMA model performs bes on forecasing sock XYZ compare o oher models. Table 3.8 Ranking Comparison on S&P Price Index r S r S r S r RANK n Classical ARIMA ,, MA3 ARIMA ,,4 WMA3 ARIMA ,, EWMA3 ARIMA ,, According o Table 3.8 he EWMA3 ARIMA model performs bes on forecasing S&P Price Index among all models. In addiion, he MA3 ARIMA model also performs beer han he classical ARIMA model; i speaks ou he qualiy of our proposed models. 3.4 Conclusion In he presen chaper, we ry o show he usefulness and effeciveness of our proposed models by applying hem on wo differen economic ime series, namely he daily closing price of Sock XYZ and he daily closing price of S&P Price Index. In boh cases, once we obained our proposed models we compare hem wih he classical approach and rank heir efficiency by examining some basic saisical crieria of he residuals. In addiion, by hiding he las 5 observaions and rying o predic he fuure informaion, we are able o show ha our proposed models perform well wihou knowing he fuure informaion. The encouraging resuls speak ou he qualiy of he models. 67

80 Chaper 4 Global Warming: Amospheric Temperaure Forecasing Model 4.0 Inroducion Temperaure plays a very imporan role in Global Warming and is relaion wih Carbon Dioxide. The aim of he presen chaper is o develop a saisical forecasing model for he emperaure in he Coninenal Unied Saes. There are wo mehods being used in recording emperaures and we shall refer hem as Version (see Appendix C) and Version (see Appendix C) daa ses. Thus, an addiional aim in he presen sudy is o deermine if he wo mehods of recording emperaures are indeed differen. Version daa was colleced by he Unied Saes Climae Division, USCD, and Version daa by he Unied Saes Hisorical Climaology Nework, USHCN. The Version daase consiss of monhly mean emperaure and precipiaion for all 344 climae divisions in he coniguous U. S. from January 895 o June 007. The daa is adjused for ime of observaion bias. However, no oher adjusmens are made for inhomogeneiies. These inhomogeneiies include changes in insrumenaion, observer, and observaion pracices, saion and insrumenaion moves, and changes in saion composiion resuling from saions closing and opening over ime wihin a division. For addiional informaion concerning Version of he daa, see (Easerling & Peerson, 995; Karl e al., 986; Karl & Williams, 987; Karl e al., 988; Karl e al., 990; Peerson & Easerling, 994; Quayle e al., 99). A graphical presenaion of he Version daase is given by Figure

81 Temperaure Monh Figure 4. Time Series Plo for Monhly Temperaure for (Version ) The Version daase was firs become available in July 007, and i consiss of daa from a nework of 9 saions in he coniguous Unied Saes ha were defined by scieniss a he Global Change Research Program of he U. S. Deparmen of Energy a Naional Climae Daa Cener. A mehodology was developed and applied o es known saion changes for heir impac on he homogeneiy, and necessary adjusmens were made if he changes caused a saisically significan response in he ime series. They claim ha he daa se is a consisen nework hrough ime, which minimizes any biasing due o nework changes hrough ime. For informaion on Version of he ime series, see (Alexandersson & Moberg, 997; Baker, 975; Easerling e al., 996; Easerling e al., 999; Hughes e al., 99; Karl e al., 990; Karl e al., 988; Karl e al., 986; Karl & Williams, 987; Lund & Reeves, 00; Menne & Williams, 005; Quinlan e al., 987; Vose e al., 003; Wang, 003). A graphical presenaion of he Version daase is given by Figure

82 Temperaure Figure 4. Time Series Plo for Monhly Temperaure for (Version ) Monh 4. Analyical Procedure The muliplicaive seasonal auoregressive inegraed moving average, ARIMA model is defined by Φ P ( B Γ s d s D s ) φ p( B)( B) ( B ) x = θq( B) Q( B ) ε (4..) where p is he order of he auoregressive process, d is he order of regular differencing, q is he order of he moving average process, P is he order of he seasonal auoregressive process, D is he order of he seasonal differencing, Q is he order of he seasonal moving average process, and he subindex s refers o he seasonal period. We shall denoe he s s subjec model by ARIMA ( p, d, q) ( P, D, Q), and φ ( B), θ ( B), Φ ( B ), Γ ( B ) defined as follows: φ ( B) = ( φ B φ B p θ ( B) = ( θ B θ B q s p... φ B p q... θ B q p q ) ) P q s s s Φ P( B ) = ΦB ΦB Φ P B Ps

83 and s s s Γ Q( B ) = Γ B Γ B... Γ The order of he muliplicaive ARIMA model deermines he srucure of he model and i is essenial o have a good mehodology in erms of developing he forecasing model. In he presen sudy, we sar wih addressing he issue of he seasonal subindex s. Afer we examine he original daa, shown by Figure 4. and 4., we have reason o believe he monhly emperaure of he Coninenal Unied Saes behaves as a periodic funcion wih a cycle of monhs. Hence, we le he seasonal subindex s =. In ime series analysis, one canno proceed wih a model building procedure wihou confirming he saionariy of a given sochasic realizaion, hus, we es he overall saionariy of he series by using he mehod inroduced by Kwiakowski, D., Phillips, P. C. B., Schmid, P., and Shin, Y in 99, (Kwiakowski e al., 99). Once he order of he differencing is idenified, i is common for one ARIMA ( p, d, q) ( P, D, Q) model ha we have several ses of ( p, q, P, Q) ha are all s adequaely represening a given se of ime series. Akaike s informaion crierion, AIC, (Akaike, 974), was firs inroduced by Akaike in 974 plays a major role in our model Q B Qs. selecing process. We shall choose he se of ( p, q, P, Q) ha produces he smalles AIC. Anoher imporan aspec in our model selecion process is o deermine he seasonal differencing, D, he goal is o selec a smaller AIC wihou complicaing he seleced model. Hence, we only compue he AIC for boh D = 0 and D = based on our previous selecion of he orders ( p, d, q, P, Q ), and choose he model wih smaller AIC o be our final model. Below we summarize he model idenifying procedure: 7

84 Deermine he seasonal period s. Check for saionariy of he given ime series { x } by deermining he order of differencing d, where saionariy. d = 0,,,... according o KPSS es, unil we achieve Deciding he order m of he process, for our case, we le m = 5 where p + q + P + Q = m. Afer ( d, m) being seleced, lising all possible configuraions of ( p, q, P, Q) for p + q + P + Q m. For each se of ( p, q, P, Q), esimaes he parameers for each model, ha is, φ Φ,..., Γ., φ,..., φ p, θ, θ,..., θ q,, Φ,..., Φ P, Γ, Γ Q Compue he AIC for each model, and choose he one wih smalles AIC. Afer ( p, d, q, P, Q ) is seleced, we deermine he seasonal differencing filer by selecing he smaller AIC beween he model wih D = 0 and D =. Our final model will have idenified he order of ( p, d, q, P, D, Q ). 4.. Developmen of Forecasing Models The hisorical emperaure daa for he coninenal Unied Saes ha we shall use are shown by Figure 4. and 4.. A visual inspecion does no show any obvious rends being presen. Thus, we le he seasonal period s =. Following he sep-by-sep procedure we described above, we found ha he model bes characerizes he average monhly emperaure of he Coninenal Unied Saes for boh Version and is a ARIMA(,,) (,,) process, analyical given by 7

85 ( Φ B )( B φb )( B)( B ) x = ( θb)( Γ φ B ) ε (4..) Expanding boh sides of he above ARIMA, we have [ ( + φ ) B + ( φ φ ) B + ( φ + φ Φ + ( φ Φ φ Φ φ φ Φ ) B 6 ) B + φ Φ + φ B 4 B ( φ + φ Φ 7 3 ] x ( + Φ ) B ) B = ( θ B Γ B 5 + ( + φ + Φ + Φ B 4 ( φ + Φ + θ Γ B 3 + φ Φ ) ε ) B ) B 5 3 Simplify i, we ge x ( + φ ) x + ( φ Φ + ( φ + φ Φ + ( φ φ ) x φ Φ ) x 6 φ φ Φ ) x 4 + φ Φ x + φ x 7 3 ( φ + φ Φ ( + Φ = ε θ ε ) x 5 ) x + Φ Γ ε + ( + φ + Φ x 4 3 ( φ + Φ + θ Γ ε + φ Φ ) x ) x 5 3 Thus, he one-sep ahead forecasing model for Version daa is given by x =.094x x x x x ε 0.037x x Γ ε x x ε.089x x (4..) and he one-sep ahead forecasing model for Version daa is given by x =.095x x x x x ε x 0.974ε Φ x x x ε x x 6 + (4..3) Noe he closeness of he wo forecasing models. 4.3 Evaluaion of he Proposed Models We begin by forecasing for he las one hundred observaions he monhly average emperaure in he Coninenal Unied Saes for boh Version and, using he models given by expression 4.. and A graphical presenaion of he resuls is presened below by Figure 4.3 and

86 Temperaure Original Daa Prediced Value Monh Figure 4.3 Acual VS. Prediced Values for Version Daase Temperaure Original Daa Prediced Value Monh Figure 4.4 Acual VS. Prediced Values for Version Daase As can be observed ha boh models are similar and he one-sep ahead forecasing is quie good, excep he emperaure of January 006 ook an unexpeced urn. We idenify his inconsisency as a possible oulier. 74

87 We proceed o calculae he residuals esimaes, r = x x, for boh forecasing process given by (4..) and (4..3). The resuls are graphically presened below by Figure 4.5 and 4.6. Temperaure Monh Figure 4.5 Residual Plo for Monhly Temperaure (Version Daase) Temperaure Monh Figure 4.6 Residual Plo for Monhly Temperaure (Version Daase) 75

88 We observe ha he residuals are quie small and isolaing around he zero axis as expeced. I indicaes ha boh models are good models in predicing he Version and Version of he ime series. Nex, we evaluae he mean of he residuals, r, he variance, S r, he sandard deviaion, S r, sandard error, SE, and he mean square error, MSE. The resuls are presened below by Table 4. and 4., for Version and Version daa, respecively. Table 4. Basic Evaluaion Saisics (Version Daase) r S r S r SE MSE Table 4. Basic Evaluaion Saisics (Version Daase) r S r S r SE MSE We observe ha all evaluaion crieria suppor he qualiy of he proposed forecasing model. We can also conclude he similariy of he wo models. Thus, i raises he quesion is he effor o collec wo daa ses implemen wo differen procedures by wo agencies necessary? We have demonsraed ha our proposed models are capable of represening he pas monhly average emperaure of he Coninenal Unied Saes, i is also essenial o show ha hese models are also capable of forecasing he fuure values of he emperaure. Therefore, we hide he las monhs of he emperaure, resrucure he 76

89 models (4..) and (4..3) and ry o predic he following monhs only using he previous informaion. For example, we used he firs 334 observaions { x, x,..., x334} o forecas x. Then we use he observaions x, x,..., } o forecas, and coninue his 335 process unil we obain he forecasing values of he las observaions, ha is, subjec monhs. { x335 x 336 { x335, x336,..., x346}. Table 4.3, gives he acual, forecasing and residual daa for he Table 4.3 Original VS. Forecas Values (Version Daase) Original Values Forecas Values Residuals March April May June July Augus Sepember Ocober November December January February Figure 4.7 below gives a graphical presenaion of he informaion presened in Table 4.3 for Version observed ime series. 77

90 Temperaure Original Daa Prediced Value Monh Figure 4.7. Acual VS. Prediced Values for he Las Observaions (Version ) Similarly, for Version of he daa se, we have calculaed he esimaes presened by Table 4.4. Table 4.4 Original VS. Forecas Values (Version Daase) Original Values Forecas Values Residuals March April May June July Augus Sepember Ocober November December January February

91 A graphical presenaion of he resuls given in Table 4.4 are given below by Figure 4.8. Temperaure Original Daa Prediced Value Monh Figure 4.8 Acual VS. Prediced Values for he Las Observaions (Version ) We remark he similariy of he resuls of boh models and he good forecas values. 4.4 Conclusion In he presen sudy, we have developed wo seasonal auoregressive inegraed moving average models o forecas he monhly average emperaure in degrees Fahrenhei in he Coninenal Unied Saes using hisorical monhly daa from The wo saisical models are based on wo differen mehods of recording and weighing he subjec emperaure namely, USCD (Version ) and USHCN (Version ). Alhough he wo differen ses of daa are somewha similar, we believe from a saisical perspecive ha he one from USHCN is more appropriae o use. The wo developed saisical models were evaluaed using various saisical crieria and i was shown ha boh forecasing processes produced good esimaes of he subjec maer. 79

92 Chaper 5 Global Warming: Carbon Dioxide Proposed Forecasing Model 5.0 Inroducion Global Warming is one of he mos compelling and difficul problems facing our sociey. I is well undersood ha carbon dioxide (CO ), along wih emperaure are he primary causes of global warming. The presen sudy is concerned wih developing analyical saisical models o predic CO. Jim Verhuls, Perspecive Edior, S. Peersburg Times, wries, Carbon dioxide is invisible- no color, no odor, no ase. I pus ou fires, pus he fizz in selzer, and is o plans wha oxygen is o us. I is hard o hink of i as a poison. (Verhuls 007). The Unied Saes is emiing approximaely 5.9 billion meric ons of carbon dioxide in he amosphere, which makes us one of he World leaders. In addiion o CO in he amosphere, we have CO emissions ha are relaed o gas, liquid, and solid fuels along wih gas flares and cemen producion. The aim of he presen chaper is o develop wo differen saisical models for he carbon emissions and amospheric carbon dioxide in he Unied Saes using hisorical daa from he subjec maer. 5. Carbon Dioxide Emission Modeling The CO emissions daa se ha we used o develop he proposed model conains he monhly emissions daa from 98 o 003 (see Appendix D). I was published by Carbon Dioxide Informaion Analysis Cener (CDIAC), which is suppored by he Unied 80

93 Saes Deparmen of Energy. The CDIAC is a well known organizaion, which responds o daa and informaion requess from users worldwide invesigaing he greenhouse effec and global climae change. For deailed informaion, see (Unied Saes Environmenal Proecion Agency (EPA), 004; Marland e al., 003). A graphical presenaion of he emissions daa is given by Figure 5.. CO Emission Monh Figure 5. Time Series Plo on CO Emission In forecasing he CO emission, we sar wih addressing he issue of he seasonal subindex s. Afer we examine he original daase, shown by Figure 5., we noe ha he monhly CO emission behaves as a periodic funcion wih a cycle of monhs and conains a small upward rend. Thus, we le he seasonal subindex S =. Follow by he sep-by-sep procedure as we described in secion 4., we found he model ha bes characerizes he monhly emissions of he Unied Saes is an ARIMA(,,) (,,) process, analyically given by ( Φ B )( B)( B)( B ) x = ( θb θ B )( Γ φ B ) ε (5..) 8

94 Afer expanding boh sides of model (5.) and esimae is coefficiens. The final saisical model for CO emission, CO E, wih he appropriae esimae of he weighs are given by CO E =.503x 0.853ε x x x ε x x 0.057ε.57749x ε x ε (5..) Once we obained proper coefficiens, we shall proceed o evaluae he proposed model and illusrae he qualiy of he model. The forecasing values ha obained from he proposed saisical model, (5..), for CO emissions in he Unied Saes is graphically presened Figure 4.. CO Emissions Original Daa Prediced Value Monh Figure 5. Acual VS. Prediced Values for CO Emission As can be observed, he prediced values follow he acual values of CO Emission closely. I indicaes ha he overall qualiy of he model is good. We shall 8

95 proceed o calculae he residuals esimaes, r = x x, and he resuls are graphically presened below by Figure 5.3 CO Emissions Monh Figure 5.3 Residuals Plo for CO Emissions The residuals are quie small and isolaing around he zero axis as expeced. I indicaes ha he proposed model forecass he CO emissions closely in he Unied Saes. The mean of he residuals, r, he variance, S, he sandard deviaion, S, sandard error, SE, and he mean square error, MSE, are presened below by Table 5.. r r Table 5. Basic Evaluaion on CO Emissions Model r S r S r SE MSE

96 We observe ha all evaluaion crieria suppor he qualiy of he proposed forecasing model for CO emissions. We now proceed o furher evaluae model (5..) hiding he las monhs of he CO recordings and re-esimaing he coefficiens of he model (5..). Having resrucured he model (5..) we proceed o esimae he hidden recordings. For example, we used he firs 64 observaions x, x,..., } o forecas. Then we use he { x64 observaions x, x,..., } o forecas, and coninue his process unil we obain { x65 x 66 he forecasing values of he las observaions, ha is, x, x,..., x }. Table 5., x 65 { gives he acual, forecasing and residual daa for he subjec monhs. Table 5. Original VS. Forecasing Values on CO Emissions Model Original Values Forecas Values Residuals January February March April May June July Augus Sepember Ocober November December

97 Noe he closeness beween he original and forecas values. A graphical presenaion of he resuls given in Table 4. is given below by Figure 5.4. CO Emissions Original Daa Prediced Value Monh Figure 5.4 Monhly CO Emission VS. Forecas Values for he Las Observaions I can be seen ha he prediced values produced by our proposed model follow he acual values of CO Emissions closely. I no only shows ha our proposed model is capable of forecasing CO Emissions wihou using any fuure informaion bu also speaks ou he usefulness of he model. 5. Amospheric Carbon Dioxide Modeling The daa se ha we used o develop our second proposed model consiss of monhly CO concenraions in he amosphere from 958 o 004 (see Appendix E). The daa was colleced in Mauna Loa by Carbon Dioxide Research Group, Scripps Insiuion of Oceanography, Universiy of California. A map of geographical locaion of Mauna Loa is provided by Figure

98 Figure 5.5 Geographical Locaion of Mauna Loa A he earlier sage of our model building process, we spo several missing values in he early 960s. To address his problem, we decided o use he daa from 965 o 004, which is a period which conains no missing values. For addiional informaion concerning he daa se on CO concenraions in he amosphere, see (Bacasow, 979; Bacasow & Keeling, 98; Bacasow e al., 980; Bacasow e al., 985; Keeling, 960; Keeling, 984; Keeling, 998; Keeling e al., 976; Keeling e al., 98; Keeling e al., 989; Keeling e al., 996; Keeling e al., 995; Pales & Keeling, 965; Keeling e al., 00; Whorf & Keeling, 998). A plo of he acual CO concenraion in he amosphere is given by Figure 5.6. I provides a visual presenaion of he ime series plo of CO concenraions in he amosphere. 86