Modelling the Asymmetric Volatility of Anti-pollution Technology Patents Registered in the USA

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1 Modelling he Asymmeric Volailiy of Ani-polluion Technology Paens Regisered in he USA Felix Chan a, Dora Marinova b, Michael McAleer a a Deparmen of Economics, Universiy of Wesern Ausralia (Michael.McAleer@uwa.edu.au) b Insiue for Susainabiliy and Technology Policy, Murdoch Universiy Absrac: The paper analyses he asymmeric volailiy of paens relaed o polluion prevenion and abaemen (ha is, ani-polluion) echnologies regisered in he USA. Ecological and polluion prevenion echnology paens have increased seadily over ime, wih he 1990's having been a period of inensive paening of echnologies relaed o he environmen. The ime-varying naure of he volailiy of ani-polluion echnology paens regisered in he USA is examined using monhly daa from he US Paen and Trademark Office for he period January 1975 o December Alernaive symmeric and asymmeric volailiy models, such as GARCH, GJR and EGARCH, are esimaed and esed agains each oher. Keywords: Ani-polluion paens, rends, volailiy, GARCH, GJR, EGARCH, asymmery, nesed models, nonnesed models. 1. INTRODUCTION Since he Indusrial Revoluion in he 18 h Cenury, echnology has dominaed he economic, social and environmenal fabric of sociey. From is earlies days, indusrial developmen was associaed wih polluion. For example, smoke, fumes and fog-laden air can be seen clearly in he beauiful ciy landscapes pained by impressioniss who depiced images direcly from naure. The general ones, effecs and vivid colours in painings such as Claude Mone s The Tames a Wesminser (pained in 1871), Georges Seura s neoimpressionis Bahing a Asnières ( , see Figure 1) and Enrance o he Por of Honfleur (1886), or Camille Pissaro s Pon Neuf A Winer Morning (pained near his deah in 1903), show ha polluion has long lef is ominous mark on he ecological environmen. Only in recen decades, however, has he serious environmenal damage caused by polluion been recognised. Figure 1. George Seura: Bahing a Asnières (Une Bagnade, Asnières), o he earh s amosphere each year (Dunn and Peerson, 001), wih he emission of CO alone oalling more han.6 billions in 1996 (World Bank, 001). This enormous polluion has caused global warming and climae change, and posed major hreas o he inhabians of he plane. Major expecaions for dealing wih he impac of polluion have been imposed on boh human creaiviy and he developmen of echnologies which can preven or abae he problem. For purposes of his paper, ani-polluion echnologies incorporae he range of echnical soluions o he problem of polluion. The paper examines he rends and volailiies in he paening of ani-polluion echnologies in he USA, he world s larges and echnologically mos advanced marke, using monhly daa from he US Paen and Trademark Office (PTO) from January 1975 o December The plan of he paper is as follows. Secion describes he daa used. Secion 3 discusses various symmeric and asymmeric volailiy models, including GARCH, GJR and EGARCH. The empirical resuls regarding rends and volailiies in ani-polluion paen regisraions are presened in Secion 4. Some concluding remarks are given in Secion 5.. DATA Billions of ons of carbon and oher harmful elemens produced hrough he burning of fossil fuels are added The US Paen and Trademark Office paen daabase (URL hp:// /neahml/search-adv.hm), which is organised according o boh he US and inernaional classificaions of echnologies, does no include a special class for ani-polluion echnologies. Consequenly, exracing empirical informaion is a

2 challenging exercise which requires he developmen of working rules and definiions (see Marinova and McAleer (003a) for a deailed analysis of he nanoechnology indusry). In he presen case, he definiion implies ha a regisered paen falls wihin he caegory of ani-polluion paens if is absrac, claims or specificaions include he word polluion. I was decided o analyse he ime series of paens according o he applicaion dae raher han he approval dae in order o avoid arificial disorions of he daa caused by organisaional delays in he process of graning paens. The daa were exraced on 6 January 003. Figure shows he rends in ani-polluion paening in he USA, based on monhly daa from January 1975 o December The graph exhibis wo rends, namely downward sloping from he mid-1970s o he mid-1980s, and upward sloping from he mid-1980s o he mid-1990s, followed by sabilisaion in he lae- 1990s. I would seem ha he ineres in ani-polluion paens diminished in he mid-1980s afer higher levels in he lae-1970s, bu has been resurreced slighly in he 1990s. 1 Figure 3 presens oal annual US paens and US anipolluion paens for he period Toal annual paens regisered in he USA have been increasing seadily, reaching a peak of close o 180,000 approved paens from he applicaions lodged in In addiion, he figure shows he annual rend in approved ani-polluion paens, which reached a global peak of,34 in 1995 and a slighly lower peak of,51 in A comparison of he wo rends shows very differen paerns beween oal annual US paens, which exhibi a clear upward rend, and annual US anipolluion paens, which have experienced wo separae rends, firs falling and hen rising. I is ineresing o noe ha ani-polluion paens are, in principle, a subse of ecological paens, which have ended o increase over he period (for furher deails, see Marinova and McAleer (003b)). Hence, polluion issues do no appear o have been of lasing imporance as far as he developmen of new echnologies have been concerned. Figure 3. Toal Paens and Ani-polluion Paens in he USA by Dae of Applicaion, Figure. Ani-polluion Paens in he USA by Dae of Applicaion, 1975(1) (1) ani-polluion paens Toal US paens ['00] Noe: The daa were exraced on 6 January 003. Noe: The daa were exraced on 6 January 003. During he sample period, he larges number of anipolluion paens awarded annually arose from applicaions lodged in 1995, namely,34 (see Figure 3), wih he monh of June 1995 being a period of exreme paening aciviy (see Figure ). Monhly paen daa show some seasonaliy for boh US anipolluion and oal US paens. The use of paen raios in he empirical analysis (see Secion 4 below) allows his problem o be largely eliminaed (see McAleer e al. (00) for furher deails). Such an inference is confirmed by an analysis of he annual raio of ani-polluion paens o oal US paens (see Figure 4). The highes share of.7% was recorded in 1975, and has been consisenly below % since 1981, reaching is lowes value of 1.% in This paern may be a warning of a lack of commimen from companies and individuals regarding he prevenion and abaemen of polluion. The long-erm implicaions of ignoring he effecs of polluion include adverse impacs on he naural environmen, a rise in sea levels, developmen of exreme weaher condiions, loss of biodiversiy, and greaer prevalence of infecious diseases, among oher. 1 The absence of a rend beween 1995 and 1999 may be arificial. As paens from more recen applicaions are approved, he numbers of approved paens would be expeced o increase. The remainder of he paper examines he volailiy in ani-polluion paens in he USA. Reasons for analysing his group of paens are very similar o ha

3 for any paen volailiy (for furher explanaions, see McAleer e al. (00)). In he absence of explici and acive markes for commodiies such as ecological and/or ani-polluion innovaions and inellecual propery, he esimaion of volailiies associaed wih such paens is fundamenal o risk managemen in financial and oher models ha describe he risk-reurn rade-off. The economeric modelling of volailiy would seem o be a useful firs sep in esablishing such markes, and will also lead o more informed business and policy decisions. Figure 4. Raio of Ani-polluion Paens o Toal Paens in he USA by Dae of Applicaion, Noe: The daa were exraced on 6 January MODEL SPECIFICATIONS The primary purpose of he following secions is o model he volailiy of he raio of he number of anipolluion paens regisered in he USA o he oal number of paens regisered in he USA (henceforh, he paen share ). This approach is based on Engle s (198) pah-breaking idea of capuring ime-varying volailiy (or uncerainy) using he auoregressive condiional heeroskedasiciy (ARCH) model, and subsequen developmens forming he ARCH family of models. Of hese models, he mos popular has been he generalised ARCH (GARCH) model of Bollerslev (1986), especially for he analysis of financial daa. In order o accommodae asymmeric behaviour beween negaive and posiive shocks (or movemens in he ime series), Glosen, Jagannahan and Runkle (199) proposed he GJR model, and Nelson (1991) proposed he Exponenial GARCH (EGARCH) model. Consider he saionary AR(1)-GARCH(1,1) model for he ani-polluion paen share, y : y = φ + φ + ε φ 1 (1) 1 y 1, < where he shocks, or movemens in he paen share, are given by: ε = η h, η ~ iid (0,1) () = + h + ω αε 1 βh 1, and ω > 0, α 0, β 0 are sufficien condiions o ensure ha he condiional variance h. The > 0 ARCH (or α ) effec indicaes he shor run persisence of shocks, while he GARCH (or β ) effec indicaes he conribuion of shocks o long run persisence (namely, α + β ). In equaions (1) and (), he parameers are ypically esimaed by he maximum likelihood mehod o obain Quasi- Maximum Likelihood Esimaors (QMLE) in he absence of normaliy of η. Ling and McAleer (00a) esablished he necessary and sufficien momen condiions for he univariae GARCH(p,q) model. Ling and McAleer (003) showed ha he QMLE for GARCH(p,q) is consisen if he second momen is finie, ha is,, Ling E( ε ) < and Li (1997) showed ha he local QMLE is asympoically normal if he fourh momen is finie, ha is, E( ε 4 ) <, and Ling and McAleer (003) showed ha he global QMLE is asympoically normal if he sixh momen is finie, ha is, 6 E( ε ) <. The necessary and sufficien condiion for he exisence of he second momen of ε for GARCH(1,1) is α + β < 1 and, under normaliy, he necessary and sufficien condiion for he exisence of he fourh momen is ( α + β ) + α < 1. Using a weaker condiion, Elie and Jeanheau (1995) and Jeanheau (1998) showed ha he log-momen condiion is sufficien for consisency of he QMLE for he univariae GARCH(p,q) model, and Boussama (000) showed ha i is also sufficien for asympoic normaliy. Based on hese heoreical developmens, he sufficien log-momen condiion for he QMLE of GARCH(1,1) o be consisen and asympoically normal is given by E (log( + β)) < 0. (3) αη However, his condiion is no sraighforward o check in pracice as i involves a funcion of an unknown random variable and unknown parameers. Alhough he sufficien momen condiions for consisency and asympoic normaliy of he QMLE for he GARCH(p,q) model are sronger han heir logmomen counerpar, he momen condiions are far more sraighforward o check in pracice. The effecs of posiive shocks (or upward movemens in he paens share) on he condiional variance,, h

4 are assumed o be he same as he negaive shocks (or downward movemens in he paen share) in he symmeric GARCH model. Asymmeric behaviour is accommodaed in he GJR model, for which GJR(1,1) is defined as follows: h ( ( )), (4) = ω + α + γ I η 1 ε 1 + β h 1 where ω > 0, α + γ 0, β 0 are sufficien condiions for h > 0, and I( η ) is an indicaor variable defined by: 1, ε < 0 I( η ) = 0, ε 0 as η has he same sign as. Such an indicaor variable differeniaes beween posiive and negaive shocks, so ha asymmeric effecs in he daa are capured by he coefficien γ, wih γ > 0. In he GJR model, he asymmeric effec, γ, measures he conribuion of shocks o boh shor run persisence, γ γ α +, and long run persisence, α + β +. Ling and McAleer (00b) showed ha he regulariy condiion for he exisence of he second momen of 1 GJR(1,1) under symmery of η is α + β + γ < 1, and he condiion for he exisence of he fourh momen under normaliy of η is 3 β + αβ + 3α + βγ + 3αγ + γ < 1. Using a weak novel condiion, McAleer e al. (00) showed ha he log-momen condiion for GJR(1,1), namely E(ln[( α + γi( η )) η + β ]) < 0, is sufficien for consisency and asympoic normaliy of he QMLE. 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 ε = + α η 1 + γη 1 + β log h 1 ω, β < 1. (5) The disinc differences beween EGARCH, on he one hand, and GARCH and GJR, on he oher, include he following: (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) noed 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. (00) observed ha β < 1 is likely o be a sufficien condiion for he exisence of momens; (v) in addiion o being a sufficien condiion for consisency, β < 1 is also likely o be sufficien for asympoic normaliy of he QMLE of EGARCH(1,1). As GARCH is nesed wihin GJR, based on he heoreical resuls in McAleer e al. (00), an asympoic -es of H 0 : γ = 0 can be used o es GARCH agains GJR. However, as EGARCH is nonnesed wih regard o boh GARCH and GJR, nonnesed procedures are required o es EGARCH versus GARCH and EGARCH versus GJR. A simple nonnesed procedure was proposed by Ling and McAleer (000) o es GARCH versus EGARCH, in which he es saisic is asympoically N(0,1) under he null hypohesis. Following a similar approach, McAleer e al. (00) derived a non-nesed procedure for esing EGARCH versus GJR, in which he es saisic is asympoically N(0,1) under he null hypohesis. In he nex secion, he volailiies of he ani-polluion paen raio will be modelled by esimaing he GARCH, GJR and EGARCH models hrough he use of rolling windows wih size 40. The dynamic pahs of he esimaes, second momens and log-momens will provide insighful informaion regarding he shor and long run persisence over ime, as well as he sabiliy and performance of each model. 4. EMPIRICAL RESULTS All he models in his paper were esimaed by EViews 4. The dynamic pahs of all esimaes for each model are available on reques. 4.1 GARCH(1,1) Varied movemens in he esimaes reflec he insabiliy of he shor run persisence in GARCH(1,1) for he ani-polluion paen share. For he oal of 6 rolling samples, he values of αˆ vary from 0.15 o wih a mean of and sandard deviaion of In paricular, αˆ increases from o 0.15 in December 1975, and hen decreases o in January Such wide flucuaions are no uncommon hroughou he remaining rolling samples. The oher wo samples for such behaviour occur in April 1977, when αˆ increases from o 0.096, hen declines o in he following monh, and in February 1979, when αˆ increases from o 0.10, hen declines o in March Furhermore, he α esimaes are negaive in 49 of 6 α

5 rolling samples, which violae he sufficien condiion for he condiional variance o be posiive. Movemens in βˆ correspond o he movemens in αˆ for GARCH(1,1): βˆ decreases from o in December 1975, hen increases slowly o in he following five monhs. In April 1977, βˆ decreases from o 0.84, and gradually increases o in Augus Moreover, βˆ decreases from o in March 1979 and hen increases immediaely o 1.00 in April I is ineresing o noe ha, while mos of he α esimaes are negaive, he β esimaes are greaer han one in 3 rolling samples. Overall, he mean β esimae is 0.980, varying beween and Ineresingly, all of he rolling samples saisfy he second momen condiion, excep in March 1979, when he second momen is Furhermore, he second momen, or long run persisence, seems unsable hroughou he rolling samples. This is no paricularly surprising since he second momen is he sum of he α and β esimaes in a GARCH(1,1) model. Overall, he mean second momen is 0.97 varying beween and All of he rolling samples saisfy he log-momen condiion. This is surprising since mos of he α esimaes are negaive, and i is possible for he negaiviy of he α esimae o lead o a failure in evaluaing he log-momen condiion for a paricular rolling sample. Furhermore, he dynamic pah of he log-momen condiion exhibis a downward rend, bu no such rend can be observed in eiher he α or β esimaes. Overall, he mean log-momen is 1.114, varying beween 0.93 and However, he saisfacion of he log-momen condiion in all rolling samples means ha he QMLE is consisen and asympoically normal. 4. GJR(1,1) Movemens in αˆ for GJR(1,1) show subsanial improvemen in sabiliy over heir GARCH counerpars. In paricular, he number of negaive αˆ values is now reduced o, wih mos of hem occurring oward he end of he rolling samples. In fac, he mean αˆ is for GJR(1,1) varying beween and For he firs 6 rolling samples, he α esimaes remain low and vary around 0, bu increase dramaically o in July I remains high around 0.08 before June 1977, and hen decreases dramaically o The α esimaes hen show greaer flucuaions in he remaining rolling samples. For GJR(1,1), he γ esimaes exhibi similar paerns o he α esimaes. However, he γˆ values are generally greaer han heir αˆ counerpars. This is refleced in he fac ha he mean γˆ is 0.078, varying beween and Ineresingly, γˆ is less han zero for he firs six rolling samples, bu increases dramaically o in July However, he esimae declines immediaely o and in Augus 1975 and Ocober 1975, respecively, says a around 0.15 unil June 1977, hen decreases o 0.03 and remains mosly negaive in he res of he rolling samples. This suggess he presence of asymmeric and ime-varying behaviour in he volailiies of he anipolluion paen shares. The β esimaes also exhibi a grea improvemen over heir GARCH counerpars. Firs, he number of βˆ values greaer han one is now reduced o 5, wih mean βˆ of 0.838, which is lower han is GARCH counerpar. Conrary o he movemens in αˆ and γˆ, βˆ is generally high and varies around 0.95 in he firs six rolling samples, hen decreases dramaically from 0.98 o in July 1975, bu bounces back almos immediaely o in Sepember I remains around unil June 1977, wih a dramaic increase from o 0.984, hen remains close o one in mos of he remaining samples. All of he rolling samples saisfy he second momen condiion, which suggess ha he QMLE are consisen and asympoically normal. The mean second momen is 0.910, varying beween and Furhermore, all he rolling samples saisfy he log-momen condiions wih he mean of.477 varying beween.37 and EGARCH(1,1) In general, he esimaes for EGARCH are disappoining. The dynamic pahs of αˆ, γˆ and βˆ exhibi periodic behaviour. These sugges ha he QMLE are exremely sensiive o he samples, so ha he esimaes are less likely o be reliable. Alhough βˆ is less han one in all he rolling samples, he mean βˆ is 0.49, varying beween and This is no paricularly saisfacory, considering he variabiliy in he esimaes. Similar conclusions can be found for boh αˆ and γˆ, where he mean αˆ is 0.318, varying beween and 0.079, and he mean γˆ is 0.056, varying beween and

6 5. CONCLUSION This paper analysed he rends and volailiy in anipolluion paens regisered in he USA using monhly US PTO daa for he period January 1975 o December The volailiies of ani-polluion paen shares, namely he raio of he number of ani-polluion paens regisered in he USA o he oal number of paens regisered in he USA, was analysed using hree ime-varying volailiy models, namely GARCH, GJR and EGARCH. In paricular, GJR and EGARCH are inended o capure he presence of asymmeric effecs in he volailiies of he ani-polluion paen shares. I was found hrough he use of rolling windows ha he asymmeric effecs were significan, bu varied over ime. The use of hreshold models for boh he condiional mean and condiional variance may provide a more accurae descripion of he dynamics in he daa. Overall, based on he variabiliy and robusness of he rolling esimaes, GJR was superior o boh GARCH and EGARCH in modelling he volailiies of ani-polluion paen shares. 6. ACKNOWLEDGMENTS The firs auhor wishes o acknowledge he financial suppor of an Ausralian Posgraduae Award and an Individual Research Gran, Faculies of Economics & Commerce, Educaion and Law, Universiy of Wesern Ausralia. The second auhor is mos graeful for he financial suppor of he Ausralian Research Council, Murdoch Universiy, and he Deparmen of Economics a he Universiy of Wesern Ausralia. The hird auhor wishes o acknowledge he financial suppor of he Ausralian Research Council and he Cener for Inernaional research on he Japanese Economy, Faculy of Economics, Universiy of Tokyo. 7. REFERENCES Boussama, F., Asympoic Normaliy for he Quasimaximum Likelihood Esimaor of a GARCH Model, Compes Rendus de l Académie des Sciences, Série I, 331, 81-84, 000 (in French). Bollerslev, T., Generalized auoregressive condiional heeroskedasiciy, Journal of Economerics, 31, , Dunn, S. and J.A. Peerson (eds.), Hydrogen Fuures: Towards a Susainable Energy Sysem, Worldwach Insiue, Washingon DC, 001. Elie, L. and T. Jeanheau, Consisency in Heeroskedasic Models, Compes Rendus de l Académie des Sciences, Série I, 30, , 1995 (in French). Engle, R.F., Auoregressive Condiional Heeroskedasiciy wih Esimaes of he Variance of Unied Kingdom Inflaion, Economerica, 50, , 198. Glosen, L., R. Jagannahan and D. Runkle, On he Relaion Beween he Expeced Value and Volailiy of Nominal Excess Reurns on Socks, Journal of Finance, 46, , 199. Jeanheau, T., Srong Consisency of Esimaors for Mulivariae ARCH Models, Economeric Theory, 14, 70-86, Ling, S. and W.K. Li, On fracionally inegraed auoregressive moving-average models wih condiional heeroskedasiciy, Journal of he American Saisical Associaion, 9, , Ling, S. and M. McAleer, 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. 6-4, 000. Ling, S. and M. McAleer, Necessary and Sufficien Momen Condiions for he GARCH(r,s) and Asymmeric Power GARCH(r,s) Models, Economeric Theory, 18, 7-79, 00a. Ling, S. and M. McAleer, Saionariy and he Exisence of Momens of a Family of GARCH Processes, Journal of Economerics, 106, , 00b. Ling, S. and M. McAleer, Asympoic Theory for a Vecor ARMA-GARCH Model, Economeric Theory, 19, , 003. Marinova, D. and M. McAleer, Nanoechnology Srengh Indicaors: Inernaional Rankings Based on US Paens, Nanoechnology, 14, R1-R7, 003a. Marinova, D. and M. McAleer, Modelling Trends and Volailiy in Ecological Paens in he USA, Environmenal Modelling and Sofware, 18(3), , 003b. McAleer, M., F. Chan and D. Marinova, An Economeric Analysis of Asymmeric Volailiy: Theory and Applicaion o Paens, paper presened o he Ausralasian Meeing of he Economeric Sociey, Brisbane, July 00, o appear in Journal of Economerics. Nelson, D.B., Condiional Heeroscedasiciy in Asse Reurns: A New Approach, Economerica, 59, , Shephard, N., 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. 1-67, World Bank, Making Susainable Commimens: An Environmen Sraegy for he World Bank, Washingon DC, 001.