Influence Diagnostics in a Bivariate GARCH Process

Transcription

1 Influence Dagnoscs n a Bvarae GARCH Process an Qu Jonaan Dark Xbn Zang Deparmen of Economercs and Busness Sascs Monas Unversy Caulfeld Eas VIC 345 Ausrala Marc 6 Absrac: In s paper we examne nfluence dagnoscs for deecng oulers and nfluenal observaons n bvarae GARCH models. e paper exends e slope and curvaure based dagnoscs proposed by Zang and Kng 5 for e unvarae GARCH model. An nnovave perurbaon sceme s nroduced o e consan correlaon bvarae GARCH model of Bollerslev 99 and e varyng correlaon bvarae GARCH model of se and su. By examnng e nfluence grap we locae nfluenal observaons n a mulvarae conex. Some emprcal examples glg e usefulness of e approac. Keywords: consan correlaon curvaure-based dagnosc nfluenal observaons slope-based dagnosc me-varyng correlaon.

2 Inroducon e success of e auoregressve condonal eeroscedascy ARCH and generalsed ARCH GARCH class of processes n modellng volaly s well documened. A vas leraure exendng e unvarae GARCH class of processes o mulvarae GARCH MGARCH as also developed see Bera and Hggns 993 and Bollerslev e al 99 for revews. Despe e success of e GARCH class of processes e usual esmaon approaces are no very robus o exreme or nfluenal observaons Bera and Hggns 993; Zang and Kng 5. Oulers may sgnfcanly nfluence parameer esmaes and subsanally worsen e forecasng performance of GARCH models. Falng o allow for oulers may also serously affec e asympoc se and power of e Lagrange Mulpler es for ARCH effecs see Djk e al 999; Franses and Gjsels 999; Zang and Kng 5 for furer dscusson. e denfcaon of nfluenal observaons n unvarae and mulvarae GARCH models s erefore of grea mporance. e am of nfluence analyss s o examne e effec of relevan mnor perurbaons on some key resuls n a posulaed model. Cook 986 presened a geomerc meod for assessng s n a lnear regresson model. Here a lkelood dsplacemen funcon was defned for e purpose of assessng nfluence va a curvaure-based dagnosc. Bllor and Loynes 993 defned a modfed lkelood dsplacemen on wc nfluence s examned roug slope-based dagnoscs. Usng e modfed lkelood dsplacemen Zang and Kng 5 derved e curvaure-based dagnosc for e unvarae GARCH model. In s paper we exend e curvaure-based dagnosc proposed by Zang and Kng 5 o assess nfluenal observaons n bvarae GARCH models. Our nfluence analyss focuses on e consan-correlaon MGARCH model of Bollerslev 99 and e me-varyng correlaon MGARCH model of su and se. e res of s paper s organed as follows. Secon brefly revews e meods for ouler deecon and nfluence analyss n GARCH models. In secon 3 we derve e dervaves needed for e curvaure-based dagnoscs for e consan-correlaon bvarae GARCH model and e me-varyng correlaon GARCH model. Secon 4 apples e meodology o e compose prce ndex of e Sanga Excange and e Senen Excange; e Ausralan All Ordnares ndex and s SPI fuures; and e S&P 5 spo and s fuures. Concludng remarks follow n secon 5.

3 Influence Dagnoscs wo mporan sylsed facs of fnancal me seres are volaly cluserng and excessve kuross. e ARCH and GARCH models aemp o capure ese feaures owever many emprcal sudes observe a ese models are unable o capure all e excess kuross. One possble explanaon s a e me-seres are affeced by occasonal unpredcable evens wc make e condonal dsrbuon eavy-aled. Influence analyss s robus o e maskng effec and also assumes a e mng of oulers s unknown. In e conex of unvarae GARCH models Lug 4 employed e meod of forward searc wle Zang and Kng 5 conduced nfluence dagnoscs based on e second order approac. Bo procedures overcome e maskng effec owever ey do no gve a creron o correc e deeced oulers. o dae nfluence analyss as no been appled o mulvarae GARCH esmaon. In s paper we exend e second order approac of Zang and Kng 5 o a mulvarae seng. We focus on e correlaon esmaes obaned from e consan correlaon MGARCH model of Bollerslev 99 and e varyng correlaon MGARCH model of se and su. Assume a we ave observaons for a gven model le denoe e parameer vecor esmaed by maxmng e log-lkelood funcon L. Denoe L as e log-lkelood funcon obaned afer assgnng weg o eac observaon.e. nroducng a perurbaon vecor o e daa se. e followng perurbaon vecor K s assumed al for K were s e pon of null perurbaon. Defne e log-lkelood dsplacemen funcon as LD [ L L ] o nvesgae e nfluence of observaons Cook 986 suggesed evaluang e normal curvaure wc measures e local beavour of LD a e null pon. e e nervenon analyss Box and ao 975 and mssng value meods Ljung 989 are proposed o andle oulers w known mng. 3

4 mos mporan dagnosc of local nfluence analyss s e drecon vecor along e maxmum normal curvaure were e greaes local cange occurs. e frs order approac of Cook 986 uses e drecon vecor of e maxmum slope as nfluence dagnoscs. e second order approac of Wu and Luo 993 uses e drecon vecor of maxmum curvaure as nfluence dagnoscs. s paper examnes bo approaces bu focuses on e second order approac. Gven e perurbaon vecor of Zang and Kng 5 proved e normal curvaure of LD a e null pon s C l were l F l 3 / [ F F] l [ I F F ] l L F 4 L L L L F Denoe [ ] / A F B F F [ I F F ]. en e maxmum curvaures C l max s e larges egenvalue of e caracersc equaon A λ B 6 e egenvecor of e larges egenvalue s e drecon vecor l max of maxmum curvaure. e ndex plo of l max may reveal ose nfluenal observaons a ave a srong effec on LD because e MLE s mos sensve n e drecon of lmax under small perurbaons. e frs-order dagnoscs are equal o F n 4 wc s a by-produc of e second order approac. 5 e ouler searcng procedure appled n e presen paper s as follows. Frs we oban e quas maxmum lkelood esmaes QMLE of parameers from e unperurbed model. Second we use e analycal dervaves derved n Secon 4 o solve equaon 6. Las we use e ndex plo of l max o vsually denfy nfluenal observaons. Wu and Lou 993a argued a e second order dagnoscs can provde nformaon a e frs order dagnoscs fal o provde. 4

5 3 Curvaure-Based Dagnosc n Bvarae GARCH Models 3. Consan Correlaon Bvarae GARCH Consder e followng bvarae me seres{ e } w ero means and GARCH condonal varances. e H ~ γ α e for 7 / 8 were H s e condonal covarance marx and s e correlaon beween seres and. We assume a γ α and are nonnegave wα <. e log lkelood funcon may be expressed as L log log log - were s a K parameer vecor. e e e - e 9 e coce of perurbaon sceme maers because dfferen perurbaon scemes may resul n dfferen nfluence dagnoscsl max. Bllor and Loynes 993 dvded perurbaons no model and daa perurbaons. We employ a model perurbaon sceme because a daa perurbaon sceme would nroduce addonal complexy va a perurbaon vecor. Assume a e perurbaon vecor K s nroduced o e log-lkelood funcon L va equaon 8. e perurbed condonal covarance can be represened as. Under s perurbaon sceme e perurbed log-lkelood funcon s L log e e e e log log e pon of null perurbaon s an vecor of uny. o compue F and F we need e followng dervaves. 5

6 6 L s an vecor w e elemen gven by were are e sandarded resduals. L s a K marx wc can be paroned as 3 L L L L 3 Le. 3 α γ α γ en L s a 3 marx w e column gven by [ ] L s a 3 marx w e column beng [ ] L 3 s a row vecor w e elemen beng Gven e condonal volaly defned by 7 and are obaned va e followng recursve equaons e α α γ α γ 4 w e nal values beng. e α γ.

7 L s an dagonal marx w e dagonal elemen beng L s e second dervave esmae of e nformaon marx and s approxmaed numercally va e Hessan. 3 All of e dervaves n -5 are compued a and e MLE of. 3. Varyng Correlaon Bvarae GARCH e assumpon of consan correlaons can adversely affec e relably of sascal nference f volaed. se and su exended e consan-correlaon model o allow for me-varyng correlaon. For e bvarae varyng-correlaon GARCH model e correlaon parameer n 8 s replaced by ψ 6 M ψ w e /. 7 4 M M w and. 5 Assumng normaly e log-lkelood funcon can be expressed as 6 L log log log. 8 Assume a e perurbaon vecor s nroduced o e log-lkelood funcon va e condonal correlaon 6. Express e perurbaon sceme as 3 Engle and Bollerslev 986 Bollerslev Engle and elson 994 dscussed e analyc dervaves n e GARCH conex. Due o e complexy nvolved esmaon usually reles on e numercal esmaes. Forenn and Calolar 996 argue a e benef of usng analycal dervaves s small wo reasons. Frs a comparson of e Hessan marx obaned analycally and numercally for e unvarae GARCH conex found e dfference o be farly small. Second e dervaon of analycal dervaves for L n MGARCH s burdensome especally for e varyng correlaon srucure n secon In s paper M wc s a necessary condon forψ o be posve defne. 5 e CC-MGARCH model s nesed wn e VC-MGARCH model under e resrcon. 6 e SQPF algorm of Lawrence and s n OX verson 3. enables e condons and α and α o be mposed durng esmaon. 7

8 8 / 9 e perurbed me-varyng condonal correlaon s ψ en e log-lkelood funcon under s perurbaon sceme s. log log log L e pon of null perurbaon s an vecor of uny. e dervaves requred for compung F and F n equaons 4-5 are. L L 3 L L wc can be approxmaed numercally. e calculaons of a are used by -4 are gven as follows; k - k k k k. Denoe 3 α γ α γ.

9 9 3 s a 9 column vecor w 3 / for e α γ 5 [ ] j. and 3 j for j 6 e dervaves of w respec o n 5 and 6 can be obaned va 4. 3 A C A C B A 7 B C B C B A 8 3 B A C 9 w C B A.e nal values for 7-9 are and. - k k k k. { } k k dag s an dagonal marx w e k dagonal elemen gven by k k. All e dervaves are compued a and e MLE of.

10 4 Emprcal Examples 4. Consan Correlaon Bvarae GARCH 4.. Daa s secon examnes e frs daa se DS wc covers e compose prce ndex of Sanga Excange P and compose prce ndex of Senen Excange P. Afer adjusng for oldays eac seres represens 8 daly observaons from January roug May 3. Connuously compounded reurns R and R are calculaed as mes e log of e frs dfferences. Bo R and R are I and exb ARCH effecs. 7 able provdes a summary of e descrpve sascs. e me-seres plo of R and R are n Fgure and Fgure respecvely. I s clear R and R bo exb excess kuross. Inser able and Fgures and e coce of an adequae model s crcal because a poor model wll resul n an over-denfcaon of oulers. e ypoess of consan correlaons s erefore esed va e Lagrange Mulpler LM es of se. e es sascs of LMC.63 can no rejec e null ypoess of consan correlaon Resuls QMLE esmaes for all parameers n 7-8 are presened n able. e esmaes sugges a bo seres exb volaly perssence. ese esmaes are used o solve 6. e plo of curvaure dagnoscs l max s n Fgure 3. Inser able and Fgure 3 As argued by Zang and Kng 5 ere s lle gudance on ow o deermne weer e curvaure s sgnfcan. Gven a large curvaure value and s assocaed dreconal vecor we mg also ave dffculy n dervng a resold for locang nfluenal observaons. In emprcal sudes vsual nspecon mg be e mos plausble meod for locang nfluenal observaons accordng o e compued curvaure or slope and s assocaed dreconal vecor. e curvaure ndex plo 7 e resuls of e co-negraon es of Engle and Granger 987 sow P and P are co-negraed. Esmaon of e bvarae error correcon GARCH model owever resuled n lle cange n e varance parameer esmaes wen compared o e esmaes assumng a ero mean equaon. For smplcy s paper assumes reurns are of ero means. 8 All e parameers n e esmaed model also pass e yblom sably ess ndvdually and jonly.

11 suggess a observaons 55 and 583 are nfluenal. ese observaons and e prevous perod s reurn are presened n able 3. Inser able 3 For observaon 55 R remaned nearly uncanged wle R ncreases by 359%. For e oer nfluenal case bo markes ave large posve reurns. Fgure 4 plos e slope dagnoscs for DS were e mos nfluenal observaons are also 55 and 583. e wo markes are volale durng and e deeced oulers fall no s perod. However sould be noed a e oulers are no necessarly exreme reurns. Inser Fgure 4 We also derved nfluence dagnoscs n e unvarae GARCH model for R and R. er curvaure dagnoscs plos are n Fgures 5 and 6 w e nfluenal observaons presened n able 4. Inser able 4 and Fgures 5 & 6 ese oulers dffer from ose n e mulvarae conex w mos of em beng exreme reurns. e exreme negave reurns are more nfluenal an exreme posve reurns of e same magnude. s resul s conssen w e leverage effec. For e slope dagnoscs n Fgures 7 and 8 e mos nfluenal observaon s observaon for bo R and R. Imporanly none of e unvarae dagnoscs uncover observaon 55 as nfluenal. I sould be noed owever a no observaon can be regarded as nfluenal w % assurance. Inser Fgures 7 and 8 4. Varyng Correlaon Bvarae GARCH 4.. Example Daa e second daa se DS consss of e Ausralan All Ordnares Index and s SPI fuures. ere are 97 daly prces for eac seres coverng e perod from January 988 roug Sepember 99. Only ose days were ncluded were radng occurred n bo markes. We creae e daly reurns R3 and R4 e dfferenced logarmc prces presened n percenage for eac prce respecvely.

12 e summary sascs of R3 and R4 are oulned n able 5 w er me-seres plos n Fgure 9 and Fgure. Bo R3 and R4 are I exb ARCH effecs and excess kuross. e LMC es rejecs e null of consan correlaon a % level LMC Inser able 5 and Fgures 9 and Resuls e QMLE esmaes for all parameers are n able 6. Dagnoscs on e sandardsed resduals exb excess kuross ndcang e possbly of oulers. e curvaure dagnoscs are ploed n Fgure. e deeced nfluenal observaons are presened n able 7. Inser ables 6 and 7 and Fgure For e mos nfluenal observaon 453 bo markes exb er larges negave reurns e cange of R4 s muc larger an R3. e second nfluenal case observaon 96 also sees bo markes experence large negave reurns. Observaon 433 and 7 sow a dfferen paern e reurns of R4 cange grealy wle R3 reman nearly uncanged. e slope dagnoscs plo for DS s n fgure. e slope dagnoscs plo sn as nformave as a of curvaure dagnoscs were e four observaons are also among nfluenal observaons. Fgure We also examne e curvaure dagnoscs n e unvarae GARCH model for R3 and R4. e plos of curvaure dagnoscs are n Fgures 3 and 4 w plos of slope dagnoscs n Fgures 5 and 6. For R3 e curvaure dagnoscs are a lle nosy w observaons 63R3.9 and 97R3.9 e mos nfluenal. Is slope dagnoscs ndcae observaons 453 and 96 are mos nfluenal. For R4 observaon 453 and 96 are nfluenal of bo curvaure and slope dagnoscs. one of e dagnoscs uncover observaons 433 and 7 wc were deeced n e MGARCH model. Inser Fgures s s conssen w e rejecon of e lkelood rao es for e resrcon H. :

13 4.. Example Daa e rd daa se DS3 consss of e S&P5 Index and s fuures. R5 and R6 are e reurns for e spo and fuures prces respecvely. ere are observaons from December 999 o December 3. Summary sascs are n able 8. e deparures from normaly of bo R5 and R6 are no as serous as R3 and R5. e me-seres plos n Fgures 7 and 8 ndcae a ere are no exreme reurns. Inser able 8 and Fgures 7 and 8 e LMC of 4.6 s sgnfcan a e 5% level rejecng e ypoess of consan correlaon. s s conssen w MLE esmaon of 6 were e resrcon H s rejeced. : Resuls e QMLE esmaes for e VC-MGARCH models are n able 9. e sandarded resduals sll exb excess kuross. e curvaure dagnoscs are ploed n Fgure 9. e deeced nfluenal observaons are presened n able. Inser ables 9 and and Fgure 9 For observaons 48 R5 decreases from.96 o.76 wle R6 ncreases from.7 o Observaons 79 exb e smlar paern. For observaon 47 bo reurns ncrease bu e ncrease of R5 s muc greaer. e slope dagnoscs plo s n Fgure were e mos nfluenal observaon s observaon 79. e curvaure dagnoscs n e unvarae GARCH model for R5 and R6 are n Fgure - w e deeced oulers oulned n able. Inser able and Fgures e deeced oulers also end o be large reurns of eer sgn bu none of em were deeced n e VC-MGARCH model. For e slope dagnoscs of bo R5 and R6 Fgure 3-4 observaon 44R5-4.9 R6-5.9 s e mos nfluenal followed by observaon 8 R R In summary e resuls are conssen w our pror expecaon. e nfluenal observaons on e condonal varance were large posve and negave reurns. e nfluenal observaons on e condonal correlaon consan or me-varyng occur wen e relaonsp or correlaon beween e wo markes s dsurbed. Gven e g correlaon level n DS DS and DS3 nfluenal observaons were 3

14 denfed wen e percenage cange n reurns dffered grealy beween markes. s was denfed wen reurns remaned vrually uncanged n one marke bu moved grealy n e oer marke reurns ncreased n one marke wle decreased n e oer marke; and large absolue reurns moved n e same drecon. e resuls erefore glg e mporance of exendng e nfluence dagnoscs o e mulvarae GARCH seng. s s because caegores and may be dffcul o denfy wou nfluence dagnoscs gven a e magnudes of e reurns may be wn normal levels. 5 Concluson In s paper we appled nfluence dagnoscs for deecng oulers and nfluenal observaons n bvarae GARCH models usng bo e slope and curvaure-based dagnoscs dscussed n Zang and Kng 5. In parcular e paper examned e nfluence of mnor perurbaons n e consan-correlaon bvarae GARCH model of Bollerslev 99 and e varyng-correlaon bvarae GARCH model of se and su. Our aemp represens e frs emprcal work of assessng nfluence n a bvarae GARCH model. In addon o assessng nfluence n e conex of a bvarae GARCH model we examne nfluence n e conex of e unvarae GARCH model. Resuls sow a e condonal varance s affeced by large reurn observaons wle nfluenal observaons on e condonal correlaon occur wen e correlaon beween e wo markes s dsurbed. Gven e g correlaon levels beween e daa ses examned s occurred wen e relave cange n reurn dffered grealy beween e wo markes. ese observaons may no necessarly be nfluenal n a unvarae GARCH model. e resuls erefore llusrae e mporance of usng e dagnoscs developed n s paper wen esmang bvarae GARCH models. 4

15 REFERECES Bera A. and Hggns M. 993 ARCH Models: Properes Esmaon and esng Journal of Economc Surveys Bllor. and Loynes R.M. 993 Local Influence: A ew Approac Communcaons n Sascs eory and Meods Bollerslev. Cou R. and Kroner K. 99 ARCH Modellng n Fnance: A Revew of e eory and Emprcal Evdence Journal of Economercs Bollerslev. Engle R. and elson D. 994 ARCH Models Handbook of Economercs 4 Elsever Amserdam Bollerslev. 99 Modellng e Coerence n Sor Run omnal Excange Raes: A Mulvarae Generaled ARCH Model Revew of Economcs and Sascs Box G. E. P. and ao G. C. 975 Inervenon Analyss w Applcaon o Economc and Envronmenal Problems Journal of e Amercan Sascal Assocaon Cook R. D. 986 Assessmen of Local Influence w dscusson Journal of e Royal Sasc Socey Ser. B Djk D.V. Franses P.H. and Lucas A. 999 esng for GARCH n e Presence of Addve Oulers Journal of Appled Economercs Engle R. and Bollerslev. 986 Modellng e Perssence of Condonal Varances Economerc Revew 5-5. Engle R. and Granger C. 987 Co-Inegraon and Error Correcon: Represenaon Esmaon and esng Economerca Forenn G. and Calolar G. 996 Analyc Dervaves and e Compuaon of GARCH Esmaes. Journal of Appled Economercs Franses P.H. and Gjsels H. 999 Addve Oulers GARCH and Forecasng Volaly Inernaonal Journal of Forecasng 5-9. Lawrence C. and s A. A Compuaonal Effcen Feasble Sequenal Quadrac Programmng Algorm SIAM Journal on Opmsaon 9-8. Ljung G.M. 989 A oe on e Esmaon of Mssng Values n me Seres Communcaon sn Sascs. Smulaon and Compuaon se Y. A es for Consan Correlaons n a Mulvarae GRACH Model Journal of Economercs

16 se Y. and su A. A Mulvarae Generaled Auoregressve Condonal Heeroscedascy Model w me-varyng Correlaons Journal of Busness and Economc Sascs Wu X. and Luo Z. 993 Second-Order Approac o Local Influence Journal of e Royal Sasc Socey Seres B Zang X. and Kng M. L. 5 Influence Dagnoscs n Generaled Auoregressve Condonal Heeroscedascy Processes Journal of Busness and Economc Sascs

17 able Summary Sascs for DS R R Mean.5.8 Sd.Dev Maxmum Mnmum Skewness Kuross Jarque-Bera Sasc ** ** **Sgnfcan a e % level. Sascs for sandarded resduals are gven n pareneses able Esmaon Resuls for e Consan-Correlaon MGARCH for DS Equaons γ α e for Parameers γ α Esmaes.933**.53**.8393**.985** QMLE Sd. Error γ α Esmaes.894**.54**.85** QMLE Sd. Error raos are gven n pareneses **Sgnfcan a e % level 7

18 able 3 Second order nfluence dagnoscs for CCMGARCH-DS Observaon Reurn R -.6 R.64 R8.84 Prevous Reurn R able 4 Second order nfluence dagnoscs unvarae GARCH -DS R R Observaon Reurn Observaon Reurn

19 able 5 Summary Sascs for DS R3 R4 Mean.8. Sd.Dev Maxmum Mnmum Skewness Kuross Jarque-Bera Sasc 9.84** 596.3** **Sgnfcan a e % level. Sascs for sandarded resduals are gven n pareneses able 6 Esmaon Resuls for e me-varyng Correlaon MGARCH for DS Equaons γ α e for ψ Parameers Varance Equaon γ α Esmaes **.89** QMLE Sd. Error Varance Equaon γ α Esmaes.59.63**.946** QMLE Sd. Error Correlaon Equaon Esmaes.683** ** QMLE Sd. Error raos are gven n pareneses **Sgnfcan a e % level 9

20 able 7 Second order nfluence dagnoscs for VC-MGARCH DS Observaons R3 R4 R3 R4 R3 R4 R3 R4 Reurn Prevous Reurn

21 able 8 Summary Sascs for DS3 R5 R6 Mean Sd.Dev Maxmum Mnmum Skewness Kuross Jarque-Bera Sasc ** ** **Sgnfcan a e % level. Sascs for sandarded resduals are gven n pareneses able 9 Esmaon Resuls for e me-varyng Correlaon MGARCH for DS3 Equaons γ α e for ψ Parameers Varance Equaon γ α Esmaes **.933** QMLE Sd. Error Varance Equaon γ α Esmaes.433.5**.958** QMLE Sd. Error Correlaon Equaon Esmaes.3539** ** QMLE Sd. Error raos are gven n pareneses **Sgnfcan a e % level

22 able Second order nfluence dagnoscs for VC-MGARCH DS3 Observaon Reurn R5.76 R63.36 R5.96 R6.6 R5. Prevous Reurn R able Second order nfluence dagnoscs unvarae GARCH DS3 R5 R6 Observaon Reurn Observaon Reurn

23 Fgure me-seres Plo of R Sock Reurns of Sanga Excange Fgure me-seres Plo of R Sock Reurns of Senen Excange Fgure 3 Curvaure Dagnoscs for DS consss of R and R

24 Fgure 4 Slope Dagnoscs for DS consss of R and R Fgure 5 Curvaure Dagnoscs for R Sock Reurns of Sanga Excange Fgure 6 Curvaure Dagnoscs for R Sock Reurns of SenZen Excange

25 Fgure 7 Slope Dagnoscs for R Sock Reurns of Sanga Excange Fgure 8 Slope Dagnoscs for R Sock Reurns of SenZen Excange Fgure 9 me-seres Plo of R3 Ausralan All Ordnares Index

26 Fgure me-seres Plo of R4 SPI Fuure Fgure Curvaure Dagnoscs for DS consss of R3 and R Fgure Slope Dagnoscs for DS consss of R3 and R

27 Fgure 3 Curvaure Dagnoscs for R3 Ausralan All Ordnares Index Fgure 4 Curvaure Dagnoscs for R4 SPI Fuure Fgure 5 Slope Dagnoscs for R3 Ausralan All Ordnares Index

28 Fgure 6 Slope Dagnoscs for R4 SPI Fuure Fgure 7 me-seres Plo of R5 S&P5 Spo Fgure 8 me-seres Plo of R6 S&P5 Fuure

29 Fgure 9 Curvaure Dagnoscs for DS3 consss of R5 and R Fgure Slope Dagnoscs for DS3 consss of R5 and R Fgure Curvaure Dagnoscs for R5 spo

30 Fgure Curvaure Dagnoscs for R6 fuure Fgure 3 Slope Dagnoscs for R5 spo Fgure 4 Slope Dagnoscs for R6 fuure