Modelling Financial Returns and Volatility Across Environmental Industry Sectors

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1 Inernaonal Congress on Envronmenal Modellng and Sofware Brgham Young Unversy BYU ScholarsArchve nd Inernaonal Congress on Envronmenal Modellng and Sofware - Osnabrück, Germany - June 4 Jul s, : AM Modellng Fnancal Reurns and Volaly Across Envronmenal Indusry Secors Jasslyn Yeo Follow hs and addonal works a: hps://scholarsarchve.byu.edu/emssconference Yeo, Jasslyn, "Modellng Fnancal Reurns and Volaly Across Envronmenal Indusry Secors" (4). Inernaonal Congress on Envronmenal Modellng and Sofware. 93. hps://scholarsarchve.byu.edu/emssconference/4/all/93 Ths Even s brough o you for free and open access by he Cvl and Envronmenal Engneerng a BYU ScholarsArchve. I has been acceped for ncluson n Inernaonal Congress on Envronmenal Modellng and Sofware by an auhorzed admnsraor of BYU ScholarsArchve. For more nformaon, please conac scholarsarchve@byu.edu, ellen_amaangelo@byu.edu.

2 Modellng Fnancal Reurns and Volaly Across Envronmenal Indusry Secors Jasslyn Yeo School of Economcs and Commerce, Unversy of Wesern Ausrala, Ausrala Absrac: In recen decades, he momenum of global envronmenal proecon has culmnaed n he Kyoo Agreemen of 998, placng he lmelgh on green ssues. Ths paper argues ha he proecon of envronmenal sysems nvolves a fragle balance beween he coss of envronmen preservaon and he prof movaons of ndusralss. In parcular, one of he ssues ha needs o be addressed s he rsk pressures envronmenal ndusres face n fnancal markes, where he hgher he rsk, he more pressure ndusres are under o explo naural resources. Therefore, n order o devse effecve envronmenally-frendly ye economcally vable polces, s crucal o analyse he rsks encounered by envronmenal ndusres n fnancal markes. The success of he auoregressve condonal heeroskedascy (ARCH) or generalsed ARCH (GARCH) models n explanng he sylsed facs of fnancal asse reurns has led o s wdespread use n he emprcal fnance leraure. By modellng he me-varaon n condonal varances or volaly, he unvarae ARCH model by Engle (98) and he GARCH model by Bollerslev (986) are able o capure he sylzed feaures of he perssence of volaly, volaly clusers and kuross, whle exensons of he GARCH model such as he asymmerc GARCH (GJR) model by Glosen, Jagannahan and Runkle (993) can accommodae he addonal sylzed fac ha posve and negave shocks have asymmerc effecs, whereby a negave shock has a greaer mpac on volaly han a posve shock. Ths paper models he me-varyng condonal varances of he reurns on a varey of envronmenal ndusry secors usng he unvarae ARMA(,)-GARCH(,) and he ARMA(,)-GJR (,) models. Our daase consss of daly reurns on seven Ausralan envronmenal ndusry secors ncludng Gold Mnng, Oher Mnng, Mnng Fnance, Ol & Gas, Farmng & Fshng, Foresry and Paper over her respecve me perods. The fndngs of hs paper sugges ha he rsks faced by envronmenal ndusres n fnancal markes are generally well-explaned by he ARMA(,)-GARCH(,); he ARMA(,)- GJR(,), on he oher hand, receved much less suppor due o he lack of asymmerc effecs. The log-momen and second momen condons were also sasfed emprcally, mplyng ha momens exs and he QMLE are boh conssen and asympocally normal. Therefore, nferences of he ARMA(,)-GARCH(,) esmaes can be used o ad n formulang new green and economcally vable envronmenal polces. Keywords: Unvarae GARCH; Asymmerc effecs. INTRODUCTION In recen decades, he momenum of global envronmenal proecon has culmnaed n he Kyoo Agreemen of 998, placng he lmelgh on green ssues. Ths paper argues ha he proecon of envronmenal sysems nvolves a fragle balance beween he coss of envronmen preservaon and he prof movaons of ndusralss. In parcular, one of he ssues ha needs o be addressed s he rsk pressures envronmenal ndusres face n fnancal markes, where he hgher he rsk, he more pressure ndusres are under o explo naural resources. Therefore, n order o devse effecve envronmenally-frendly ye economcally vable polces, s crucal o analyse he rsks encounered by envronmenal ndusres n fnancal markes. The success of he auoregressve condonal heeroskedascy (ARCH) or generalsed ARCH (GARCH) models n explanng he sylsed facs of fnancal asse reurns has led o s wdespread use n he emprcal fnance leraure. By modellng he me-varaon n condonal varances or volaly, he unvarae ARCH model by Engle (98) and he GARCH model by Bollerslev (986) are able o capure he sylzed feaures of he perssence of volaly, volaly clusers and kuross, whle exensons of he GARCH model such as he GJR asymmerc GARCH model by Glosen, Jagannahan and Runkle (993) can accommodae he addonal sylzed fac ha posve and negave shocks have asymmerc effecs, whereby a negave shock has a greaer mpac on volaly han a posve shock.. UNIVARIATE GARCH MODELS The man objecve of hs paper s o model he me-varyng condonal volaly of reurns usng unvarae GARCH models. The oulne and conen of he followng secon on GARCH and GJR models

3 s based on he heorecal presenaon n McAleer, Chan and Marnova (3). To esmae me-varyng condonal varance, Engle (98) proposed he Auoregressve Condonal Heeroskedascy (ARCH) model. The unvarae ARCH ( p ) process s as follows: ε = η h, where ε s he uncondonal shock, η s an ndependenly and dencally dsrbued sandardsed (or condonal) shock wh zero mean and un varance, and h s he condonal varance of ε, gven by: h p = ω + α ε, = wh ω >, α for =,..., p as suffcen condons o guaranee ha h s non-negave for all. In pracce, s usually assumed ha η s normally dsrbued such ha maxmzng he lkelhood funcon yelds he Maxmum Lkelhood Esmaor (MLE). If η s non-normal, however, hen maxmzng he lkelhood funcon wll lead o he Quas-MLE (QMLE). Bollerslev (986) exended ARCH ( p ) o he Generalsed ARCH (GARCH) model, GARCH ( p, q ), whch specfes he condonal varance as: p h = ω + α ε + β h = q j= j j where ω >, α for =,..., p and β for =,..., p are suffcen condons o ensure ha h s non-negave for all. The ARCH (or α ) effecs conrbue o he shor-run perssence of volaly shocks, whle he GARCH (or β ) effecs conrbue o he long-run perssence of volaly shocks, α + β. Lng and McAleer (a) esablshed he necessary and suffcen condons for he exsence of momens and he asympoc heory for QMLE of he unvarae GARCH( p, q ). They showed ha he QMLE of he GARCH( p, q ) s conssen f he second momen s fne. For he exsence of he second momen of ε for GARCH(,), he necessary and suffcen condon s α + β <. Jeanheau (998) showed ha he weaker logmomen condon s suffcen for conssency of he QMLE for he GARCH( p, q ) model. The suffcen condon for conssency and asympoc normaly of he QMLE of GARCH(,) s E [(log( αη + β )] <. The unvarae ARCH and GARCH models are aracve n ha hey are able o explan sylsed facs, or feaures, of fnancal asse reurns such as he perssence of volaly, volaly clusers and excess kuross. Anoher srkng feaure of fnancal asse reurns s ha posve and negave shocks have asymmerc effecs, whereby a negave shock has a greaer mpac on volaly han a posve shock. Ths feaure has led o several exensons of unvarae GARCH, one of whch s he Glosen, Jagannahan and Runkle s (993) asymmerc (or hreshold) GARCH (GJR). The unvarae GJR ( p, q ) model of Glosen e al. (99) ncorporaes asymmerc effecs no he condonal volaly process, and s gven as: p h = ω + α ε + γ I( ε ) ε + β h, = q j= j j where ω >, α γ for =,..., p and β + for =,..., p are suffcen condons for h o be non-negave for all, whle I( ε ) s an ndcaor varable defned by:, I( ε ) =, ε ε >, whch allows he sgn of he lagged uncondonal shock affec he condonal varance. The GJR (or γ ) effec measures he mpac of asymmerc condonal volaly and conrbues o boh he shor-run perssence of shocks, α + γ /, and he long-run perssence of shocks, α + β + γ /. Lng and McAleer (b) esablshed he necessary and suffcen condons for he exsence of momens and he asympoc heory for QMLE of he unvarae GJR( p, q ). For he exsence of he second momen of GJR(,) under symmery of η s α + β + γ / <. The weaker suffcen log momen for GJR(,) o ensure conssency and asympoc normaly of he QMLE of GJR(,) s E [(log(( α + γi ( η )) η + β )] <. A comprehensve survey of recen heorecal (ha s, srucural and sascal) developmens assocaed wh unvarae and mulvarae GARCH models ha are of neres o appled praconers n fnancal economcs and economercs s provded n L, Lng and McAleer ().

4 3. DATA Our daa sample conans daly (5-day weeks) share prce ndces on seven envronmenal ndusry secors - Gold Mnng (GOLDS), Oher Mnng (MINES), Mnng Fnance (MIFIN), Ol & Gas (OILEP), Farmng & Fshng (FMFSH), Foresry (FORST) and Paper (PAPER) - n Ausrala. The sample me perods are as follows: Gold Mnng, Oher Mnng, Mnng Fnance and Ol & Gas, //73 o 4//3; Farmng & Fshng, //73 o 4//3; Foresry, //96 o 4//3; and Paper, 5/6/99 o 4//3. Daly reurns are calculaed for each envronmenal ndusry secor on a connuouscompoundng bass, compued as he naural logarhm of he prce dfferences. All daa s obaned from Thompson Daasream Advance. Reurns and her volales for he seven envronmenal ndusry secors over her respecve sample perods are ploed n Fgure and Fgure respecvely. Volaly s defned as he squared devaon of each ndusry secor reurn observaon from s respecve mean reurn. Vsual observaon of Fgure shows ha envronmenal ndusry reurns, n general, flucuaed around a zero mean wh no apparen rends or seasonales over he sample perod. In Gold Mnng, here s a dramac change n he magnude of s reurns, where reurns soared n he 97s before sablsng. In Oher Mnng, here s a dsnc negave spke sgnfyng he Ocober 987 sock marke crash ouler, whle for Mnng Fnance and Ol & Gas, boh he second ol prce shock of 979 and he Ocober 987 oulers are noceable. There are also obvous negave spkes n 998 and 999 for he Foresry ndusry. For Farmng & Fshng, a hgh degree of varaon s presen from year onwards compared o he res of he sample perod, whle for Paper, he converse was rue, where a hgher degree of varaon was presen pror o year. There s dscernble volaly cluserngs for he envronmenal ndusry secors of Farmng & Fshng and Paper. Hgh volaly clusers were apparen n he early s for Farmng & Fshng, whle hgh volales were bunched up n 999 for Paper. The condonal volales of he Oher Mnng, Mnng Fnance, Ol & Gas and Foresry ndusry reurns are ypcal of fnancal me-seres daa, where volaly cluserng s no as noceable, excep for presence of oulers. As he reurn me-seres of he envronmenal ndusry secors show a consderable degree of perssence (see Fgure ), we choose o model he reurns usng he Auoregressve Movng Average (ARMA) processes of Box and Jenkns (976), whch assume ha a me seres s a lnear combnaon of s own pas values as well as curren and pas values of a random error erm For smplcy, we employ an ARMA(,) process, gven by: R φ R + δ + ε θ ε. = Table repors he resuls of he OLS regresson of he ARMA(,) for each of he seven envronmenal ndusry secors under he assumpon ha he error erm s ndependenly, dencally dsrbued (d) wh zero mean and un varance. The Newey-Wes (987) mehod s employed o correc for he poenal unspecfed deparures from homoskedascy and no seral correlaon. There s some suppor for he ARMA(,) model. The AR() and MA() erms are sascally sgnfcan for Oher Mnng, Mnng Fnance, Farmng & Fshng and Paper. The es for condonal heeroskedascy s he Lagrange mulpler es (LM) for auoregressve condonal heeroskedascy (ARCH) by Engle (98). The LM (ARCH) es sasc s asympocally dsrbued as χ ( p), where we choose p = o es for ARCH(). The LM (ARCH) p -values ndcae ha here s consderable condonal heeroskedascy n he reurn resduals, where he null hypohess of homoskedascy s rejeced n all seven of he envronmenal ndusry secors Table. OLS Esmaon of he ARIMA(,) Model and he ARCH(LM) of he ARIMA model resduals δ AR() MA() ARCH(LM) GOLDS MINES MIFIN OILEP FMFSH FORST PAPER Noes: The enres correspondng o he esmaes (n bold) for he consan, AR() and MA() are he Newey-Wes(987) correced -raos, whle he enres correspondng o he esmae (n bold) for he ARCH(LM) are p-values. In bref, a prelmnary nvesgaon of our daase has revealed ha he envronmenal ndusry reurns dsplay consderable ARCH/GARCH effecs. Accordngly, we model he condonal volales usng unvarae GARCH models. In addon, we also nvesgae unvarae GJR models, whch ncorporae asymmerc effecs such ha a negave shock has a greaer mpac on volaly han a posve shock.

5 Fgure Gold Mnng Oher Mnng Mnng Fnance VGOLDS VMINES VMIFIN Ol & Gas Farmng & Fshng Foresry VOILEP VFMFSH VFORST Paper VPAPER Fgure Gold Mnng Oher Mnng Mnng Fnance VGOLDS VMINES VMIFIN Ol & Gas Farmng & Fshng Foresry VOILEP VFMFSH VFORST Paper VPAPER

6 4. EMPIRICAL RESULTS Ths paper models he me-varyng condonal means and condonal varances of he daly reurns of he seven envronmenal ndusry secors over her respecve sample me perods usng he ARMA (,) GARCH(,) and he ARMA (,) GJR (,). Boh he ARMA (,) GARCH(,) and he ARMA (,) GJR (,) models are esmaed usng Evews Verson 4.. The Brend-Hall-Hall- Hausman (Bernd e al, 974) algorhm s used o maxmse he lkelhood funcon, wh he quasmaxmum lkelhood (QMLE) esmaes convergng n all cases. Tables and 3 presen he QMLE coeffcen esmaes for ARMA (,) GARCH(,) and he ARMA (,) GJR (,) for all seven envronmenal ndusry secors respecvely. The AR() esmaes for GARCH(,) and GJR(,) are hghly sgnfcan for Oher Mnng, Farmng & Fshng, Foresry and Paper, suggesng a consderable degree of perssence n he reurns of hese ndusres. The MA() esmaes, on he oher hand, are hghly sgnfcan for Oher Mnng, Mnng Fnance, Farmng & Fshng, Foresry and Paper, whch ndcae ha he uncondonal shock on he prevous day affecs reurns oday. I should be also noed ha he coeffcens of he AR() erm are negave for Oher Mnng and Farmng & Fshng, whle he coeffcens of he MA() erm are negave for Foresry and Paper. The esmaes of he condonal volaly for he GARCH(,) and he GJR(,) are hghly sasfacory. The suffcen condons ω >, α, β o ensure ha he condonal varance s nonnegave for all me perods are me for all seven envronmenal ndusry secors, excep for Gold Mnng. The log-momen condons and he second momen condons are also sasfed for all seven envronmenal secors, excep for Gold Mnng. Ths resul esablshes he exsence of momens and ensures ha he QMLE for GARCH(,) and GJR(,) are conssen and asympocally normal. Hence, nferences on hese esmaes can be used n he formaon of new envronmenal polces. The esmaes of he asymmerc effec n GJR(,), however, are nsgnfcan for all envronmenal ndusres excep for Paper, based on he Bollerslev- Wooldrdge (99) robus--raos, suggesng ha he GARCH(,) s preferred o GJR(,). 4. CONCLUDING REMARKS The fndngs of hs paper sugges ha he rsks faced by envronmenal ndusres n fnancal markes are generally well-explaned by he ARMA(,)- GARCH(,); he ARMA(,)-GJR(,), on he oher hand, receved much less suppor due o he lack of asymmerc effecs. The log-momen and second momen condons were also sasfed emprcally, mplyng ha momens exs and he QMLE are boh conssen and asympocally normal. Therefore, nferences of he ARMA(,)- GARCH(,) esmaes can be used o ad n formulang new green and economcally vable envronmenal polces. 5. ACKNOWLEDGEMENTS The auhor s very graeful o Mchael McAleer, Felx Chan and Suhejla Ho for her nvaluable commens and suggesons, and o Raz Shareef for hs knd help and paence. The auhor also hanks Felx Chan for he EVews codes used n hs paper. 6. REFERENCES Bernd, E.K., Hall, B.H., Hall, R.E. and Hausman, J. (974) Esmaon and Inference n Nonlnear Srucural Models, Annals of Economc and Socal Measuremen, 3, Bollerslev, T. (986) Generalsed Auoregressve Condonal Heeroskedascy, Journal of Economercs, 3, Bollerslev, T. and Wooldrdge, J.M. (99) Quasmaxmum Lkelhood Esmaon and Inference n Dynamc Models wh Tme-Varyng Covarances, Economerc Revews,, Engle, R.F. (98) Auoregressve Condonal Heeroskedascy wh Esmaes of he Varance of Uned Kngdom Inflaon, Economerca, 5, Glosen, L.J., Jagannahan, R. and Runkle,D. (99), On he Relaon Beween he Expeced value and Volaly of Nomnal excess Reurn on Socks, Journal of Fnance, 46, Jeanheau, T. (998) Song Conssency of Esmaors for Mulvarae ARCH Models, Economerc Theory, 4, L, W.K., Lng, S. and McAleer, M. (), Recen Theorecal Resuls for Tme Seres Models wh GARCH Errors, Journal of Economc Surveys, 6, Lng, S. and McAleer, M. (a) Necessary and Suffcen Momen Condons for GARCH(r,s) and Asymmerc Power of GARCH(r,s) Models, Economerc Theory, 9, Lng, S. and McAleer, M. (b) Saonary and he Exsence of Momens of a Famly of GARCH Processes, Journal of Economercs, 6, 9-7. McAleer, M., Chan, F. and Marnova, D. (3) An Economerc Analyss of Asymmerc Volaly: Theory and applcaon o paens, paper presened o he Ausralasan Meeng of he Economerc Socey, Brsbane, Ausrala, July, o appear n he Journal of Economercs.

7 Table : GARCH (,) Esmaes for he Daly Envronmenal Indusry Secors INDUSTRY SECTOR AR() MA() ω α β LOG- MOMENT SECOND MOMENT GOLDS MINES MIFIN OILEP FMFSH FORST PAPER Noes: The hree enres correspond o he esmae (n bold), he asympoc -rao and he Bollerslev- Wooldrdge (99) robus--rao respecvely. Table 3: GJR (,) Esmaes for he Daly Envronmenal Indusry Secors INDUSTRY SECTOR AR() MA() ω α γ β α + γ / LOG- MOMENT SECOND MOMENT GOLDS MINES MIFIN OILEP FMFSH FORST PAPER Noes: The hree enres correspond o he esmae (n bold), he asympoc -rao and he Bollerslev- Wooldrdge (99) robus--rao respecvel

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