Analysing Trends and Volatility in Atmospheric Carbon Dioxide Concentration Levels

Transcription

1 Inernaional Congress on Environmenal Modelling and Sofware Brigham Young Universiy BYU ScholarsArchive 2nd Inernaional Congress on Environmenal Modelling and Sofware - Osnabrück, Germany - June 2004 Jul 1s, 12:00 AM Analysing Trends and Volailiy in Amospheric Carbon Dioxide Concenraion Levels Felix Chan Michael McAleer Follow his and addiional works a: hps://scholarsarchive.byu.edu/iemssconference Chan, Felix and McAleer, Michael, "Analysing Trends and Volailiy in Amospheric Carbon Dioxide Concenraion Levels" (2004). Inernaional Congress on Environmenal Modelling and Sofware hps://scholarsarchive.byu.edu/iemssconference/2004/all/190 This Even is brough o you for free and open access by he Civil and Environmenal Engineering a BYU ScholarsArchive. I has been acceped for inclusion in Inernaional Congress on Environmenal Modelling and Sofware by an auhorized adminisraor of BYU ScholarsArchive. For more informaion, please conac scholarsarchive@byu.edu, ellen_amaangelo@byu.edu.

2 Analysing Trends and Volailiy in Amospheric Carbon Dioxide Concenraion Levels Felix Chan a and Michael McAleer b a School of Economics and Commerce, Universiy of Wesern Ausralia (Felix.Chan@uwa.edu.au) b School of Economics and Commerce, Universiy of Wesern Ausralia Absrac: Amospheric carbon dioxide concenraion (ACDC) is a crucial variable for many environmenal simulaion models, and is regarded as an imporan facor for predicing emperaure and climae changes. However, he condiional variance of ACDC levels has no previously been examined. This paper analyses he rends and volailiy in ACDC levels using monhly daa from January 1965 o December The daa are a subse of he well known Mauna Loa amosphere carbon dioxide record obained hrough he Carbon Dioxide Informaion Analysis Cener. The condiional variance of ACDC levels is modelled using he generalised auoregressive condiional heeroscedasiciy (GARCH) model and is asymmeric variaions, namely he GJR and EGARCH models. These models are shown o be able o capure he dynamics in he condiional variance in ACDC levels and o improve he ou-of-sample forecas accuracy of ACDC. Keywords: Amospheric Carbon Dioxide Concenraion, Condiional Volailiy, Forecasing, GARCH, GJR, EGARCH. 1. Inroducion Amospheric carbon dioxide concenraion (ACDC) is a crucial variable for many environmenal simulaion models, and is regarded as an imporan facor for predicing emperaure and climae changes (Glaser (2000)). Many sudies in environmenal modelling have focused on he applicaion of ACDC as an indicor of he saus of he environmen (see, for example, Phillips e al. (1998)), while oher sudies have been ineresed in he impacs of rising ACDC on he ecological sysem (see, for example, Jones e al. (1998)). However, hese sudies have seldom modelled he level of ACDC direcly, while he condiional variance of ACDC has no previously been invesigaed. Alhough here are mahemaical models ha are designed o esimae he level of ACDC based on Carbon Dioxide (CO2) emissions from he environmen (Phillips e al. (1998)), hese simulaion models are ofen complicaed and compuaionally inensive. Moreover, hey do no generally provide a simple descripion of he dynamics in he level of ACDC, and i is difficul o evaluae heir forecas performance. This paper invesigaes he rends and volailiy in ACDC levels using he well known Mauno Lao daa se. There are wo moivaions for modelling he condiional variance of ACDC. Firs, modelling he condiional variance of ACDC would allow a more accurae confidence inerval o be consruced for he one-period ahead forecas. Consider he general regression model given by y = E(y x ) + ε, for which he variance of he forecas error, ( y ˆ T y T ), is given by ( ) Var( ˆ ) x0 x y 0 1 T yt x = σ + +, T T 2 ( x x) = 1 where he variance of he innovaion, σ 2, is ypically assumed o be consan. However, if σ 2 is ime varying, he forecas variance can be reduced by accommodaing he condiional variance of he ime series o permi a more accurae confidence inerval o be consruced for he one-period ahead forecas. The second moivaion for modelling he condiional variance of ACDC is relaed o he pricing of carbon dioxide emission quoas. In financial markes, he risk associaed wih a sock reurn is ypically measured by is (possibly ime-varying) volailiy. Therefore, he volailiy of ACDC should be an imporan indicaor of he risk in selling or buying emission righs, and would also be an imporan facor in deermining he marke value of such quoas. Furher deails of emissions rading can be found a hp:// Modelling he condiional variance, or volailiy, of a ime series has been a popular opic in he financial economerics lieraure. Three of he mos popular models o capure he ime-varying volailiy in financial ime series are he Generalised Auoregressive Condiional Heeroscedasiciy (GARCH) model of 1

3 Engle (1982) and Bollerslev (1986), he Glosen, Jagannahan and Runkle (1992) GJR model, and Nelson s (1991) Exponenial GARCH (EGARCH) model. This paper examines he dynamics of he condiional variance in he level of ACDC using he GARCH, GJR and EGARCH models. The forecas performance of each model will also be invesigaed, and he sandard errors of he one-day ahead forecass arising from each model compared. The plan of he paper is as follows. Secion 2 describes he daa used. The srucural and saisical properies of he hree condiional variance models, namely GARCH, GJR and EGARCH, are given in Secion 3. The empirical resuls are presened in Secion 4, and Secion 5 conains some concluding remarks. 2. Daa The level of ACDC has been closely moniored and documened for over 30 years. The daa used in his paper are a subse of he famous Mauna Loa monhly daa se, which can be downloaded from hp://cdiac.esd.ornl.gov/rends/co2/sio-mlo.hm. The scienific deails regarding he measuremen of he ACDC level can be found in Keeling, Bacasow and Whorf (1982). Due o missing observaions in 1958 and 1964, only he daa from January 1965 o December 2002 are used in his paper, giving a oal of 456 observaions. Figure 1 conains he plos of ACDC levels from January 1965 o December The daa exhibi cyclical paerns around a ime rend. Furhermore, he auocorrelaion funcion of ACDC suggess ha i is highly correlaed wih is pas and is highly persisen, as shown in Table 1. The high firs-order auocorrelaion coefficien migh sugges ha he series are non-saionary, bu he Phillips-Perron (1988) (PP) es for non-saionariy shows ha he ACDC level is rend saionary. Using he EViews 4 economeric sofware package wih a wide range of lags, he choice of he runcaed lag order did no seem o affec he es resuls. The moivaion for using he PP es over he convenional Augmened Dickey-Fuller (ADF) es is o accommodae he possible presence of ARCH/GARCH errors. While he ADF es accommodaes serial correlaion by specifying explicily he srucure of serial correlaion in he errors, he PP es does no assume he specific ype of serial correlaion or heeroscedasiciy in he disurbances, and can have higher power han he ADF es under a wide range of circumsances. The sample volailiy, v, of a ime series, y, wih a non-consan condiional mean is ypically calculaed as follows: v = (y E(y I 1 )) 2 = ε 2, (1) where I denoes he informaion se available o ime. Since he level of ACDC exhibied cyclical paerns, a ime rend, and srong auocorrelaion, i is reasonable o specify he condiional mean o be E(y I 1 ) = φ 0 y 1 + φ 1 + θ'd (2) where θ = ( θ1, θ 2,..., θ12 )' and D = (D 1,D 2,...,D 12 )' is he vecor of seasonal dummy variables, such ha D i = 1 in monh i, oherwise D i = 0, i = 1,..., 12. The plo of he volailiy of ACDC can be found in Figure 2. Table 1: Auocorrelaion of he ACDC level. Lag Auocorrelaion Figure 1. Amospheric Carbon Dioxide Concenraion, January 1965 December Amospheric Carbon Dioxide Concenraion The descripive saisics of he level, y, he esimaed residuals from (1), ε, and he volailiy, v, of ACDC are given in Table 2. As shown in Figure 1 and Table 2, he level of ACDC grew seadily over he las 35 years. The descripive saisics of he esimaed residuals, as given in equaions (1) and (2), indicae ha he error erm, ε, is 2

4 normally disribued. In fac, he Lagrange muliplier es for normaliy, LM(N), is wih a p-value 0.485, suggesing ha normaliy canno be rejeced. The p-values of boh he F and LM es saisics for he null hypohesis of no ARCH effecs wih one lag are 0.001, suggesing ha he null hypohesis can be rejeced a he 1% level of significance. Therefore, here is considerable evidence o sugges ha he condiional variance of ACDC is no consan over ime, so ha condiional volailiy models would seem o be an appropriae choice for capuring he ime-varying volailiy in he level of ACDC. Figure 2. Volailiy of Amospheric Carbon Dioxide Concenraion, January 1965 December Volailiy of Amospheric Carbon Dioxide Concenraion Table 2. Descripive saisics of he level, esimaed residuals and volailiy of ACDC Saisics y ε v Mean Median Maximum Minimum SD Skewness Kurosis Models Specificaions The primary empirical purpose of he paper is o model he volailiy in he level of ACDC. This approach is based on Engle s (1982) idea of capuring ime-varying volailiy (or uncerainy) using he auoregressive condiional heeroskedasiciy (ARCH) model, and subsequen developmens forming he ARCH family of models (see, for example, he recen survey by Li, Ling and McAleer (2002)). Of hese models, he mos popular has been he symmeric generalised ARCH (GARCH) model of Bollerslev (1986) and he asymmeric Glosen, Jagannahan and Runkle (1992) (GJR) model, especially for he analysis of financial daa. A number of furher heoreical developmens has been suggesed by Wong and Li (1997) and Ling and McAleer (2002a, 2002b, 2003). Consider a GARCH(p,q) model for he level of ACDC, y : y = E(y I 1 )+ε, (3) where I denoes he informaion se available o ime, and he shocks (or variaions in he level of ACDC) are given by ε = η h, η ~ iid(0,1) p h = ω + α i ε 2 i + β i h i, i=1 and ω > 0, α i 0 (i = 1,,p) and β i 0 (i = 1,,q) are sufficien condiions o ensure ha he condiional variance h > 0. The ARCH (or α ) effec capures he shor run persisence of shocks, while he GARCH (or β ) effec capures he conribuion of shocks o long run persisence (namely, α + β for p=q=1). Using resuls from Ling and Li (1997) and Ling and McAleer (2002a, 2002b) (see also Bollerslev (1986) and Nelson (1990)), he necessary and sufficien condiion for he exisence of he second momen of ε, or E(ε 2 ) <, for GARCH(1,1) is α + β < 1. q i=1 (4) Equaion (2) assumes ha a posiive shock (ε > 0) has he same impac on he condiional variance, h, as a negaive shock (ε < 0), bu his assumpion is ofen violaed in pracice. In order o accommodae he possible differenial impac on he condiional variance beween posiive and negaive shocks, Glosen, Jagannahan and Runkle (1992) proposed he following asymmeric GJR specificaion for h : p h = ω + (α i + γ i I(ε i ))ε 2 i + β h, (5) i i=1 where I(ε ) is an indicaor funcion such ha I(ε ) = 0, ε 0 1, ε < 0. When β = 0, GJR(1,1) is called he asymmeric ARCH(1), or AARCH(1), model. Furhermore, for GJR(1,1), ω > 0, α + γ > 0 and β > 0 are sufficien condiions o ensure ha he condiional variance h > 0. The shor run persisence of posiive (negaive) q i=1 3

5 shocks is given by α (α + γ ). Under he assumpion ha he condiional shocks, η, follow a symmeric disribuion, he average shor run persisence is α + γ 2, and he conribuion of shocks o average long run persisence is α + γ 2 + β. Ling and McAleer (2002a) showed ha he necessary and sufficien condiion for E(ε 2 ) < is α + γ 2 + β <1. The parameers in equaions (1), (2) and (3) are ypically esimaed by he maximum likelihood mehod o obain Quasi-Maximum Likelihood Esimaors (QMLE) in he absence of normaliy of η. The condiional log-likelihood funcion is given as follows: l = 1 2 log h + ε. 2 h Ling and McAleer (2003) showed ha he QMLE for GARCH(p,q) is consisen if he second momen is finie, ha is, E( ε 2 ) <. Furhermore, Jeanheau (1998) showed ha, when β 0, he following logmomen condiion E(log(αη 2 + β)) < 0 (6) is sufficien for he QMLE o be consisen for GARCH(1,1), while Boussama (2000) showed ha he QMLE is asympoically normal for GARCH(1,1) under he same condiion. I is imporan o noe ha (6) is a weaker condiion han he second momen condiion, namely α + β <1. However, he log-momen condiion is more difficul o compue in pracice as i is he expeced value of a funcion of an unknown random variable and unknown parameers. McAleer, Chan and Marinova (2002) esablished he log-momen condiion for GJR(1,1) when β 0, namely E(log((α + γ I(η ))η 2 + β)) < 0, (7) and showed ha i is sufficien for he consisency and asympoic normaliy of he QMLE for GJR(1,1). Furhermore, using Jensen s inequaliy, hey showed ha he second momen condiion, namely α + γ 2 + β <1, is also a sufficien condiion for consisency and asympoic normaliy of he QMLE for GJR(1,1). Therefore, he srucural and saisical properies of boh GARCH(1,1) and GJR(1,1) have been esablished (see Chan, Hoi and McAleer (2002) for he srucural and saisical properies of he mulivariae GJR(p,q) model). An alernaive model o capure asymmeric behaviour in he condiional variance is he Exponenial GARCH (EGARCH(1,1)) model of Nelson (1991), namely: log h ω α η γη β, β < 1. (8) = log h 1 When β = 0, EGARCH(1,1) becomes EARCH(1). There are some disinc differences beween EGARCH and he previous wo GARCH models, as follows: (i) EGARCH is a model of he logarihm of he condiional variance, which implies ha no resricions on he parameers are required o ensure h > 0 ; (ii) Nelson (1991) showed ha β < 1 ensures saionariy and ergodiciy for EGARCH(1,1); (iii) Shephard (1996) observed ha β < 1 is likely o be a sufficien condiion for consisency of QMLE for EGARCH(1,1); (iv) as he condiional (or sandardized) shocks appear in equaion (4), McAleer e al. (2002) observed ha is likely β < 1 is a sufficien condiion for he exisence of all momens, and hence also sufficien for asympoic normaliy of he QMLE of EGARCH(1,1). Furhermore, EGARCH capures asymmeries differenly from GJR. The parameers α and γ in EGARCH(1,1) represen he magniude (or size) and sign effecs of he condiional (or sandardized) shocks, respecively, on he condiional variance. However, α and α + γ represen he effecs of posiive and negaive shocks, respecively, on he condiional variance in GJR(1,1). As GARCH is nesed wihin GJR, a sandard asympoic es of H 0 : γ = 0 can be used o es he wo models agains each oher. However, as EGARCH is non-nesed wih regard o boh GARCH and GJR, he non-nesed models are no direcly comparable. Ling and McAleer (2000) proposed a simple non-nesed es o discriminae beween GARCH and EGARCH. Denoing GARCH as he null hypohesis and EGARCH as he alernaive, he opimal es saisic for H GARCH : δ = 0 is given by: h ˆ (9) 2 = w + αε 1 + βh 1 + δg where ĝ is he generaed one-period ahead condiional variance of EGARCH. For he reverse case, ha is, denoing EGARCH as he null hypohesis and GARCH as he alernaive, he opimal es saisic for H : δ = 0 is given by: EGARCH logg loghˆ (10) 1 = w+ α η 1 + γη 1 + βlogg + δ where ĥ is he generaed one-period ahead condiional variance of GARCH. Ling and McAleer (2000) showed 4

6 ha he QMLE of δ in boh (9) and (10) are asympoically normal under he respecive null hypoheses, and consisen under he respecive alernaive hypoheses. They also derived he power funcions of boh es saisics under he respecive hypoheses. A similar non-nesed es for esing GJR and EGARCH agains each oher was derived in McAleer e al. (2002). 4. Empirical Resuls 4.1 Full Sample Esimaes The parameer esimaes and heir Bollerslev- Wooldridge (1992) robus -raios of he ARCH(1), AARCH(1), EARCH(1), GARCH(1,1), GJR(1,1) and EGARCH(1,1) models, wih condiional means as defined in (2), are available on reques. These esimaes were obained from EViews 4.0 using he BHHH algorihm. The parameer esimaes in he condiional mean are no paricularly sensiive o he specificaion of he condiional variance equaion, which is due o he block-diagonaliy of he Hessian marix of he loglikelihood funcion. Moreover, he log-momen condiions are saisfied for boh GARCH(1,1) and GJR(1,1), and he second momen condiions are saisfied for he ARCH(1) and AARCH(1) models, hereby indicaing ha he QMLE are consisen and asympoically normal for each of hese models. Furhermore, ˆ β < 1 for EGARCH, and i is no significan in he oher wo cases, suggesing he absence of long run persisence. Ineresingly, γ is no significan in eiher AARCH(1) or GJR(1,1), bu i is significan in boh EARCH(1) and GARCH(1,1), indicaing he presence of asymmeric behaviour. Based on he significance of he parameer esimaes, ARCH(1) and EARCH(1) are empirically superior o he oher four specificaions. Subsequenly, non-nesed ess based on (9) and (10), wih β = 0 in boh equaions, are conduced in order o choose beween he wo remaining adequae specificaions. The es saisics are given in Table 3. As shown in Table 3, he es saisic rejecs ARCH(1) in favour of EARCH(1) a he 10% level of significance, bu does no rejec EARCH(1) in favour of ARCH(1) a any reasonable significance level. Table 3. Non-nesed Tess beween ARCH(1) and EARCH(1) Null H 0 H : ARCH(1) H 0 0 : EARCH(1) Alernaive H 1 H1 : EARCH(1) H1 : ARCH(1) Tes Saisics Forecasing This secion examines he forecas performance and forecas variance for he model as defined in equaion (2), wih hree differen condiional variance specificaions, namely he consan condiional variance, ARCH(1) and EARCH(1). The hree models are reesimaed using he sub-sample from January 1965 o December 2001, and he ou-of-sample one-period ahead forecas of ACDC is calculaed for January 2002 o December Three sandard forecas crieria, namely roo mean square error (RMSE), mean absolue error (MAE) and mean absolue percenage error (MAPE), for each model are repored in Table 4. Table 4. Forecas Performance of Three Condiional Variance Specificaions Performance crieria Consan condiional variance ARCH(1) EARCH(1) RMSE MAE MAPE As shown in Table 4, EARCH(1) has he bes forecas performance based on he hree forecas crieria. More imporanly, allowing dynamic condiional variances improves he accuracy of he parameer esimaes and also he ou-of-sample forecass. Table 5 gives he sandard errors of he one-period ahead forecass for each monh from he hree models. Table 5. Sandard Errors of he One-Period Ahead Forecass for Three Volailiy Models Monh Consan ARCH(1) EARCH(1) January February March April May June July Augus Sepember Ocober November December Apar from having he bes forecas performance, he one-day ahead forecass produced by EARCH(1) also have he smalles sandard errors, as shown in Table 5. This suggess ha he one-day ahead forecas produced by EARCH(1) will have he smalles confiden inervals, indicaing EARCH(1) is superior in erms of forecasing accuracy for he levels of ACDC. Moreover, he sandard errors of he one-day ahead forecass produced by ARCH(1) are smaller han hose from he 5

7 consan condiional variance model for eleven of welve monhs. These resuls show ha he accuracy in forecasing ACDC levels can be improved subsanially by accommodaing ime-varying condiional variance in modelling ACDC. 5. Concluding Remarks This paper examined he rends and volailiy in he level of ACDC. Six differen specificaions of he condiional variance, namely ARCH(1), AARCH(1), EARCH(1), GARCH(1,1), GJR(1,1) and EGARCH(1,1), have been esimaed and esed agains each oher. The es saisics suggesed ha EARCH(1) was superior o he oher five specificaions, having he bes ou-of-sample forecas performance in erms of hree differen forecas crieria, namely roo mean square error, mean absolue error and mean absolue percenage error. Moreover, he one-day ahead forecass produced by EARCH(1) also had he smalles sandard errors. 6. References Bollerslev, T. (1986), Generalised auoregressive condiional heeroskedasiciy, Journal of Economerics, 31, Bollerslev, T. and J.M. Wooldridge (1992), Quasimaximum likelihood esimaion and inference in dynamic models wih ime-varying covariances, Economeric Reviews, 11, Boussama, F. (2000), Asympoic normaliy for he quasi-maximum likelihood esimaor of a GARCH model, Compes Rendus de l Academie des Sciences, Serie I, 331, (in French). Chan, F., S. Hoi and M. McAleer (2002), Srucure and asympoic heory for mulivariae asymmeric volailiy: Empirical evidence for counry risk raings, paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, Ausralia, July Engle, R.F. (1982), Auoregressive condiional heeroskedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 50, Glaser, K. (2000), Global warming: Correlaion beween amospheric carbon dioxide concenraion and he emperaure, Technical Repor, Roche Colorado Corporaion, Boulder, Colorado, USA. Glosen, L., R. Jagannahan and D. Runkle (1992), On he relaion beween he expeced value and volailiy of nominal excess reurn on socks, Journal of Finance, 46, Jeanheau, T. (1998), Srong consisency of esimaors for mulivariae ARCH models, Economeric Theory, 14, Jones, T.H., L.J. Thompson, J.H. Lawon, T,M. Bezemer, R.D. Bardge, T.M. Blackburn, K.D. Bruce, P.F. Cannon, G.S. Hall, S.E. Harley, G. Howson, C.G. Jones, C. Kampichler, E. Kandeler and D.A. Richie (1998), Impacs of rising amospheric carbon dioxide on model erresrial ecosysems, Science, 280, Keeling, C.D., R.B. Bacasow and T.P. Whorf (1982) Measuremens of he concenraion of carbon dioxide a Mauna Loa Observaory, Hawaii, in W.C. Clark (ed.), Carbon Dioxide Review: 1982, Oxford Universiy Press, New York. Li, W.K., S. Ling and M. McAleer (2002), Recen heoreical resuls for ime series models wih GARCH errors, Journal of Economic Surveys, 16, Reprined in M. McAleer and L. Oxley (eds.), Conribuions o Financial Economerics: Theoreical and Pracical Issues, Blackwell, Oxford, 2002, pp Ling, S. and W.K. Li (1997), On fracionally inegraed auoregressive moving-average models wih condiional heeroskedasiciy, Journal of he American Saisical Associaion, 92, Ling, S. and M. McAleer, (2000), Tesing GARCH versus E-GARCH, in W.-S. Chan, W.K. Li and H. Tong (eds.), Saisics and Finance: An Inerface, Imperial College Press, London, pp Ling, S. and M. McAleer (2002a), Necessary and sufficien momen condiions for he GARCH(r,s) and asymmeric power GARCH(r,s) models, Economeric Theory, 18, Ling, S. and M. McAleer (2002b), Saionariy and he exisence of momens of a family of GARCH processes, Journal of Economerics, 106, Ling, S. and M. McAleer (2003), Asympoic heory for a vecor ARMA-GARCH model, Economeric Theory, 19, McAleer, M., F. Chan and D. Marinova (2002), An economeric analysis of asymmeric volailiy: heory and applicaion o paens, paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, July 2002, o appear in Journal of Economerics. Nelson, D.B. (1990), Saionariy and persisence in he GARCH(1,1) model, Economeric Theory, 6, Nelson, D.B. (1991), Condiional heeroscedasiciy in asse reurns: a new approach, Economerica, 59, Phillips, O.L., Y. Malhi, N. Higuchi, W. F. Laurance, P. V. Nunez, R.M. Vasquez, S.G. Laurance, L. V. Ferreira, M. Sern, S. Brown and J. Grace (1998), Changes in he carbon balance of ropical foress: Evidence from long-erm plos, Science, 282, Phillips, P. and P. Perron (1988), Tesing for a uni roo in ime series regression, Biomerika, 75(2), Shephard, N. (1996), Saisical aspecs of ARCH and sochasic volailiy, in O.E. Barndorff-Nielsen, D.R. Cox and D.V. Hinkley (eds.), Saisical Models in Economerics, Finance and Oher Fields, Chapman & Hall, London, pp

8 Wong, H. and W.K. Li (1997), On a mulivariae condiional heeroscedasiciy model, Biomerika, 4,