Estimation of Asymmetric Garch Models: The Estimating Functions Approach

Transcription

1 Inernaional Journal of Applied Science and ecnology Vol. 4, No. 5; Ocober 4 simaion of Asymmeric Garc Models: e simaing Funcions Approac Mr. imoy Ndonye Muunga Prof. Ali Salim Islam Dr. Luke Akong o Orawo Deparmen of Maemaics geron Universiy geron Kenya Absrac is paper inroduces e meod of esimaing funcions (F) in e esimaion of e Asymmeric GARCH family of models. is approac uilises e ird and four momens wic are common in financial ime series daa analysis and does no rely on disribuional assumpions of e daa. Opimal esimaing funcions ave been consruced as a combinaion of linear and quadraic esimaing funcions. simaes from e esimaing funcions approac are beer an ose of e radiional esimaion meods suc as e maximum likeliood esimaion (ML) especially in cases were disribuional assumpions on e daa are igly violaed. We invesigae e presence of asymmeric (leverage) effecs in empirical ime series and fi wo of e mos popular Asymmeric GARCH models (GARCH and GJR-GARCH) under bo e ML and F approaces. An empirical example demonsraes e implemenaion of e F approac o Asymmeric GARCH models assuming a suden s disribuion for e innovaions. e efficiency benefis of e F approac relaive o e ML meod in parameer esimaion are subsanial for non-normal cases. Keywords: simaing funcion, Maximum likeliood esimaion, Asymmeric GARCH, Volailiy, Leverage effecs. Inroducion e financial sock marke as been an area of grea ineres o researcers for e las five decades. Saring wi e pioneering works on e random walk model, sock marke volailiy as been a key subec in mos subsequen researc works relaing o efficiency in e marke (Fama, 965; Fama, 97). In financial decision making, volailiy is an imporan facor in pricing of derivaives and porfolio risk managemen. is as warraned increased researc on modelling and forecasing an asse s pricereurns volailiy. Researc on canging volailiy using non-linear ime series models as been vibran since e inroducion of e Auoregressive Condiional Heeroscedasiciy (ARCH) model (ngle, 98). is model was e firs of is kind o akecondiional eeroscedasiciy ino consideraion. Bollerslev (986) generalised e ARCH model o include lagged condiional variances as well as lagged values of e squared innovaions. e GARCH family of models ave proved o be successful in capuring volailiy clusering and some amoun of e excess kurosis wic caracerize financial ime series daa. Since e works of ngle (98) and Bollerslev (986), various varians of e GARCH model ave been developed o model volailiy. Of grea imporance is e Asymmeric GARCH family of models wic address a maor limiaion of e Bollerslev s (986) basic GARCH model, relaing o e inabiliy of is model o capure e asymmeric impac of news on volailiy. News is undoubly a uge facor a affecs sock prices and erefore measuring is impac on sock marke volailiy is an imporan area of researc in financial eory (Neelab, 9). Differen volailiy models a capure is aspec ave been proposed and widely applied o real life problems in e las wo decades. Some of e mos popular models include e GARCH (Nelson, 99), GJR-GARCH (Glosene al., 993), NAGARCH (ngle and Ng, 993), APARCH (Ding e al., 993), GARCH (Zakoian, 994) and e QGARCH (Senana, 995). 98

2 Cener for Promoing Ideas, USA In e bulk of lieraure available for e Asymmeric GARCH models, e maximum likeliood esimaion meod as been e mos preferred in parameer esimaion due o is simpliciy and desirable properies. However, is meod is based on disribuional assumpions wic are ofen violaed in pracice and us alernaive parameer esimaion approaces are required. An alernaive meod of esimaion is based on e esimaing funcions (F) approac inroduced by Godambe (96). Under is approac, focus is usually on e esimaing funcion iself wic is a funcion of e observaions and e unknown parameers. is approac akes ino accoun iger order momens and does no rely on any disribuional assumpions on e daa for opimaliy. e focus of is paper is wofold. Firs we seek o inroduce e meod of esimaing funcions in e esimaion of e Asymmeric GARCH family of models based on Godambe and ompson s (989) opimal esimaing funcions for socasic processes. Secondly we will uilise e F meod in e esimaion of firs order GARCH and GJR-GARCH models based on empirical ime series from e USA and Japanese sock markes. An overview of ese wo Asymmeric GARCH models is presened in secion. e compuaion of opimal esimaing funcions for e firs order GARCH and GJR-GARCH models and e Asymmeric GARCH class of models in general, is presened in secion 3. An example involving e wo empirical financial ime series is presened in secion 4 o demonsrae e use of e F approac in esimaion of Asymmeric GARCH models paricularly in cases were ere are serious deparures from normaliy. Finally a conclusion of is paper is presened in secion 5.. Asymmeric GARCH Class of Models e convenional GARCH model besides is main virue of simpliciy imposes a number of sorcomings wi regard o volailiy modelling. However e primary limiaion of e GARCH model relaes o wa Black (976) firs documened. ere exiss a negaive correlaion beween sock reurns and volailiy implying a negaive reurns end o be followed by larger increases in volailiy wile posiive reurns of e same magniude end o be followed by lower volailiy. o model is penomenon, is paper will consider wo of e mos popular models a allow for asymmeric socks.. GARCH Model e GARCH model was inroduced by Nelson (99) o address some of e weaknesses of e convenional GARCH model inroduced by Bollerslev (986). is model capures asymmeric responses of e condiional variance o socks in e marke. e variance equaion in GARCH is specified as; were,,, g p ln () ln g z z z z z z. i i i i i i e lef and side is e log of e variance series. is makes e leverage effec exponenial and erefore e parameers,, and is e asymmery parameer. e value of g z i i are no resriced o be non-negaive. i mus be a funcion of bo magniude and sign of z in order o accommodae e asymmeric effec (Nelson, 99). e componens z and z z respecively and eac as a zero mean. is linear in z wi slope Over e range z, gz is linear in z wi slope. us g z allows e condiional variance q represen e sign effec and magniude effec wile over e range z, g z o respond asymmerically o canges in sock reurns. Consider e firs order GARCH model in (), z () ln ln z z z were, z is an idenically disribued sequence of random variables wi zero mean and a uni variance. 99

3 Inernaional Journal of Applied Science and ecnology Vol. 4, No. 5; Ocober 4 g z z z z gives e model capaciy o capure asymmery. If and, e innovaion (disurbance) in ln is now posiive (negaive) wen e magniude of z is larger (smaller) an is expeced value. On e oer and if and e innovaion in ln is now posiive (negaive) wen reurns innovaions are negaive (posiive). us e GARCH model is able o capure e leverage effecs under ese condiions. e erm. GJR-GARCH model is model was inroduced by Glosen e al., (993). I is an exension of e GARCH model o capure asymmeries beween posiive and negaive socks of e same magniude on e volailiy of reurns. e GJR- GARCH model is specified as; were, ~, z iid N. Consider e firs order GJR-GARCH model in (4); z i i s i i i i i i wen s oerwise p q (3) s (4) e model reduces o e radiional GARCH model wenever. e indicaor erm s ensures a e asymmeric effec is capured in e model. Wi, negaive socks increase volailiy more an posiive socks of equal magniude. e necessary and sufficien condiions o guaranee posiiviy of e condiional variance are,, and. is e asymmery parameer. 3. Opimal simaing Funcions In is secion we derive opimal esimaing funcions and sow eir applicaion o Asymmeric GARCH family of models. We draw exensively on e works of Godambe (96), Godambe (985) and Godambe and ompson (989).Wiou proof we firs presen an imporan eorem relaed o e eory of F's in socasic processes. Godambe and ompson (989) exended e concep of opimaliy of Godambe s (985) F ino a general seing using a more flexible condiioning meod wic is relaed o e concep of quasi-likeliood approac. aking as an arbirary sample space, ey considered e class of Fs M wic is a real funcion defined on suc a; were, is e parameer space and,,..., sample space. eorem: o esimae consider a class of Fs M y, y,..., yn; (5) k is e -field generaed by a specified pariion on e for wic; were, a is a real funcion on. k a M (6)

4 Cener for Promoing Ideas, USA e Fs M,,..., kare said o be orogonal if, An esimae ˆ defined as; were, i i i M M M M for i, i,,..., k. (7) of is obained by solving e equaion a k a y, y,..., y ;. e opimal F in is case is n M (8) M M y, y,..., yn; 3. Parameer simaion Using e simaing Funcions Approac o esimae parameers of e GARCH and GJR-GARCH models in a regression model se up using e Fs approac, opimal esimaing funcions approac o discree ime socasic processes by Godambe and ompson (989) was applied. Consider a general expression of e GARCH and GJR-GARCH models in a regression model se up wiou making any disribuional assumpions for e errors, y x () y ~ x, were, is e informaion se a ime, follows eier an GARCH or GJR-GARCH process and e componen x could be composed of exogenous variables and or lagged variables of e variable y wic is a discree ime series process. Consider e firs GARCH model in (). were, z ln ln z z z exp ln z z z, g z z z z and z is an independen and idenically disribued sequence of random variables. (,,, ). We seek o esimae e unknown parameer vecors and in e regression model () Le were is as defined in (). Similarly consider e firs order GJR-GARCH model in (), z s wen s oerwise were, z is an independen and idenically disribued sequence of random variables. (9) () ()

5 Inernaional Journal of Applied Science and ecnology Vol. 4, No. 5; Ocober 4 Le (,,, ). Similarly we seek o esimae e unknown parameer vecors and in e regression model () were is as defined in (). o evaluae e opimal esimaes of and i ( i, ) in eac case, Godambe and ompson s (989) eorem for socasic processes is applied. A good combinaion for basic unbiased and muually orogonal Fs is and suc a, y x y x (3) x and is no orogonal o e F. is implies a, e coice of ese wo esimaing funcions is based on e need o esimae e condiional mean condiional variance of y simulaneously. However e F (4) is erefore orogonalised using e Gram Scmid orognalisaion procedure (Hyde, 997) as follows, y x 3 = y x y x y x y x y x 3 y x y x From (5), consider e componen (6), Dividing and muliplying (6) by we ave, were, is e skewness of y condiional on. us, y 3 x y 3 x 3 y x erefore e wo elemenary Fs in (9) are now orogonal. (5) (6) (7) (8) y x y x o esimae e coefficien vecors and in e regression model (), opimal Fs are derived using e elemenary Fs in (9). e eorem by Godambe and ompson (989) is applied o form a linear combinaion of e elemenary Fs as, (9)

6 Cener for Promoing Ideas, USA g a a () g b b Le be e class of all Fs, a i a Were, i andb i i x b b gg given by (). e oinly opimal Fs g, g b for i, and,,3,..., are given by () wi, () y x y x () Solving e numeraor in () we ave, y x y x x y x x y x x Solving e denominaor in () we ave, y x y x y x 4 3 y x y x y x y x Muliplying and dividing (4) by leads o, 4 3 y x y x y x y x y x 3 y x y x 3 3 (5) (3) (4) 3

7 Inernaional Journal of Applied Science and ecnology Vol. 4, No. 5; Ocober 4 4 Were, 4 3 y x (6) quaion (6) represens e sandardized condiional kurosis (excess kurosis). Hence, x b (7) a (8) a y x y x y x (9) Subsiuing (), (7), (8) and (9) ino () gives e oinly opimal Fs as, g x g (3) Were, is given by equaions () and () and (,) (,) for GARCH for GJR GARCH (3) e resul in (3) is very general in a no disribuional assumpions on y ave been made. e esimaes for e unknown parameer vecors and are obained by solving e opimal F in (3). is means numerically minimizing g g.

8 Cener for Promoing Ideas, USA Were, g and g are as defined in (3). 4. simaion of Asymmeric GARCH Models on mpirical ime Series g, g g (3) is secion presens esimaion resuls. A brief descripion of e real daa a was used o fi e models is provided. Some preliminary and diagnosic ess for asymmery and normaliy are also conduced beforeand. 4. Daa In is sub-secion we model e volailiy of financial reurns of e Japanese and e USA markes for e period nd Jan 8 o 3 s May using Asymmeric GARCH models under bo e ML and F procedures. In eac marke we consider a compreensive and diversified sock index. For e New York Sock xcange we consider e Sandard and Poor s 5 index wile for e okyo Sock xcange we consider e Nikkei 5 index. In eac case daily reurns are compued as logarimic price ( P ) relaives. P R log (33) P were, R is e log reurn series (coninuously compounded reurn). ac empirical ime series comprises of daily observaions covering e period ( nd Jan 8 o 3 s May ).Sock markes in ese wo counries were among e mos volaile in e world during e 8 global financial crises and e early Japanese sunami disaser respecively and ence e impac of socks in e marke on volailiy of asse reurns was more pronounced during is period. 4. Preliminary ess able presens summary saisics (empirical properies) and preliminary ess of normaliy and asymmery for e daily sock reurns of e wo financial series. We noice a e daily volailiy for e Nikkei 5 index, represened by e sandard deviaion (.88%) is above e volailiy (.75%) for e S&P 5 index reurn series. able : Summary Saisics of e compounded reurns R Series Saisics SP5 INDX NIKKI 5 INDX Mean Sd Dev Skewness Kurosis Jarque Bera (Probabiliy) ADF es (Probabiliy) Cov, r r e skewness coefficien is negaive for bo series suggesing a ey a long lef ail. e kurosis coefficien on e oer and is very ig (7.44, 8.386) for e (S&P 5, Nikkei 5) a reflecion a e disribuions of e wo ses of real daa are igly lepokuric. e P-value corresponding o e Jarque Bera normaliy es is zero a 5% level suggesing a e es is significan for bo series. e es reurns a value of wic indicaes a e series R does no come from a normal disribuion in favour of wic indicaes a e series R comes from a normal disribuion wi unknown mean and variance. is implies a e wo series exibi non-normal beaviour. 5

9 Inernaional Journal of Applied Science and ecnology Vol. 4, No. 5; Ocober 4 e Augmened Dickey-Fuller (ADF) es reecs e uni roo null in bo daa ses. is is indicaed by e minimal p-values a 5% level and e values of. e es reurns a value of wic indicaes reecion of e uni roo in favour of e rend-saionary alernaive. indicaes failure o reec e uni roo null. us we conclude a e reurns of bo sock indices are saionary. Finally we es for presence of asymmeric effecs on condiional volailiy in bo empirical series. A simple diagnosic es for e leverage effecs involves compuing e sample correlaion beween squared reurns and e lagged reurns, Cov, r r (Zivo, 8). A negaive value for is coefficien provides evidence for poenial asymmeric effecs. Bo series ave a negaive value for is coefficien indicaing evidence of asymmery and ence Asymmeric GARCH family of models could perform well in explaining condiional volailiy in is case. Figure : Daily Logarimic Reurns (S&P 5, Nikkei 5).5 S&P 5. Nkkei Figure presens e plo of daily logarimic reurns for bo series over e considered ime period. We observe a volailiy clusering is presen in bo cases as e wo series sow periods of low volailiy wic end o be followed by periods of relaively low volailiy and oer periods of ig volailiy wic likewise end o be followed by ig volailiy. is aspec can be oug of as clusering of e variance of e error erm over ime, a is, e error erm exibis ime varying eeroscedasiciy. 4.3 Model simaes In is sub-secion, firs order GARCH and GJR-GARCH models are fied o e wo empirical series and esimaes obained using e maximum likeliood esimaion (ML) and esimaing funcion (F) approaces. In parameer esimaion under maximum likeliood meod, we assume a sandardized Gaussian or Suden s disribuion wi v degrees of freedom for e innovaions. Parameer esimaes, corresponding sandard errors (in pareneses), Akaike Informaion Crieria (AIC) and e log likeliood values are presened in ables -6. able : Parameer simaes of GARCH S&P 5 simaes Meod ML (.5) ML -.33 (.4578) F (.4379) (.8837) (.539).9765 (.4959) Sandardized Gaussian disribuion Suden s disribuion ( v ).76 (.96) (.3456).3898 (.3848) (.667) (.84) (.6946) 6

10 Cener for Promoing Ideas, USA able 3: Parameer esimaes of GJR GARCH - S&P5 simaes Meod ML ML F ( ) (.448-6) (.983-6).773 (.576).468 (.89).3883 (.985) Sandardized Gaussian disribuion Suden s disribuion ( v ).8899 (.3839).9695 (.55).973 (.9697) able 4: Parameer simaes of GARCH - Nikkei 5 simaes Meod ML (.455).9845 (.966) ML (.746) (.8745) F (.6957) (.837) Sandardized Gaussian disribuion Suden s disribuion ( v ).96 (.68537) (.3679).6956 (.34864) (.3999).473 (.3753) (.4877) (.543) (.3) able 5: Parameer esimaes of GJR GARCH Nikkei 5 simaes Meod ML ML F ( ) (3.3-6).5-5 (.96-6).584 (.6365).3799 (.569).689 (.3774) Sandardized Gaussian disribuion Suden s disribuion ( v ) able 6: AIC and Log Likeliood Values.866 (.368) (.759) (.66).68 (.739).889 (.484) (.3675) MODL GARCH GJR GARCH SRIS S&P 5 Nikkei 5 S&P 5 Nikkei 5 AIC Log - Likeliood Discussion From e resuls, i is seen a bo models ave almos similar AIC and Log likeliood values for e wo financial series. However GARCH (,) as relaively iger Log likeliood and lower AIC values an GJR GARCH (,) indicaing a i performs relaively beer in explaining condiional volailiy in bo empirical series over e considered ime period. e coefficiens and for e firs order GARCH and GJR GARCH models respecively reflecs e leverage effecs. e esimaes indicae e magniude and sign of e leverage effecs. e GARCH model sows a negaive parameer of asymmery in bo financial series suggesing a pas negaive socks (bad news) ave a greaer impac on subsequen volailiy of reurns an posiive socks (good news) do. e GJR GARCH model records posiive leverage effecs, aesing a bad news in e marke lead o a iger volailiy of asse reurns an good news. From our parameer esimaes i is clear a e F approac is more efficien an e ML meod in parameer esimaion of e firs order GARCH and GJR GARCH models in finie samples. 7

11 Inernaional Journal of Applied Science and ecnology Vol. 4, No. 5; Ocober 4 e sandard errors of e F approac esimaes are smaller an ose of e maximum likeliood esimaes assuming eier a Gaussian or a Suden s ( v ) error disribuion. e gain in efficiency follows from e fac a e F approac does no rely on disribuional specificaions for opimaliy and a i accouns for iger order momens presen in non normal daa suc as mos empirical financial ime series. However, i is eviden a e ML meod wen assuming a Suden s error disribuion compees reasonably well wi e F approac and provides a beer in-sample-fi an e ML meod wen assuming a Gaussian error disribuion across bo daa ses. is resul is expeced considering e Jarque Bera normaliy es in able wic implies a e empirical disribuions of e wo reurn series exibi eavier ails an e sandard normal disribuion. A Suden s disribuion exibis excess kurosis and fa ail beaviour. 5. Conclusion In is paper our main goal was o derive opimal esimaing funcions for e Asymmeric GARCH modes in general and demonsrae e applicaion of e F approac as an alernaive o e ML approac in parameer esimaion. We ave sown a e F approac compees reasonably well wi e ML meod especially in cases were ere are serious deparures from normaliy in finie samples. is approac erefore provides a useful alernaive meod of esimaion o e ML meod for e Asymmeric GARCH models especially in cases were e rue disribuion of e daa is unknown as i does no rely on disribuional assumpions for opimaliy. xending e F approac o e mulivariae GARCH model is a subec for fuure researc. Acknowledgemen is researc paper was prepared and made possible roug e elp and suppor of my academic supervisors; Prof. A.S Islam and Dr. L.A Orawo and e German Academic xcange Service (DAAD) wic sponsored my posgraduae sudies. Appendix (Proofs of quaions) Proof of equaion (4) y x y x Proof of equaion (9) 3 y x y x y x 3 Since y x, y x y x y x y x y x 3 3 8

12 Cener for Promoing Ideas, USA References Fama,. F. (965). e Beaviour of Sock-Marke Prices, Journal of Business. 38: Fama,.F. (97). fficien Capial Markes: A Review of eory and mpirical Work, Journal of Finance. 5: ngle, R. F. (98). Auoregressive condiional eeroscedasiciy wi esimaes of e variance of Unied Kingdom inflaion, conomerica. 5: Bollerslev,. (986). Generalized auoregressive condiional eeroscedasiciy, Journal of conomerics. 3: Neelab, R. (9). Condiional Heeroscedasic ime Series Models wi Asymmery and Srucural Breaks. Pd esis, Universiy of Pune, India. Nelson, D.B. (99). Condiional eeroskedasiciy in Asse reurns: A new Approac, conomerica. 59: Glosen, L. R., Jagannaan, R. and Runkle, D. (993). On e relaion beween e expeced value and e volailiy of nominal excess reurns on socks, Journal of Finance. 48: ngle, R.F. and Ng V.K. (993). Measuring and esing e impac of news on volailiy, Journal of Finance. 48 (5): Ding, Z., Granger, C. and ngle, R. (993). A long memory propery of sock reurns and a new model, Journal of mpirical Finance. : Zakoian, J.M. (994). resold Heeroskedasiciy Models, Journal of conomic Dynamics and Conrol. 9: Senana,. (995). Quadraic ARCH Models, Review of conomic Sudies.6: Godambe,V.P. (96). An opimum propery of regular maximum likeliood equaions, Annals of Maemaical Saisics. 3: 8-. Godambe, V.P. and ompson, M.. (989). An exension of e Quasi-likeliood simaion, Journal of Saisical Planning and Inference.: Black, F. (976). Sudies of sock price volailiy canges, Proceedings of e 976 Meeings of e Business and conomics Saisics Secion, American Saisical Associaion, pp Godambe, V.P. (985). e foundaions of finie sample esimaion in Socasic processes, Biomerika. 7: Hyde, C.C. (997). Quasi-Likeliood and Is applicaions, New York: Springer- Verlag Zivo.. (8). Pracical Issues in e Analysis of Univariae GARCH Models, Handbook of Financial ime Series, Springer, New York. 9