EFFICIENCY EVALUATION IN MODELLING STOCK DATA USING ARCH AND BILINEAR MODELS ADOLPHUS WAGALA

Transcription

1 EFFICIENCY EVALUATION IN MODELLING STOCK DATA USING ARCH AND BILINEAR MODELS ADOLPHUS WAGALA A Thess Submed To The Graduae School In Paral Fulfllmen For The Requremens Of The Maser Of Scence Degree In Sascs Of Egeron Unversy EGERTON UNIVERSITY OCTOBER 008

2 DECLARATION AND RECOMMENDATION Declaraon Ths hess s my orgnal work and has no been presened o any oher nsuon for award of any degree. Sgnaure Adolphus Wagala SM/479/05 Dae Recommendaons Ths hess has been submed wh our approval as he supervsors. Sgnaure Prof. Dank K. Nassuma, Kabarak Unversy-Nakuru. Dae Sgnaure Dr. Al S. Islam, Egeron Unversy-Njoro. Dae

3 COPYRIGHT No par of hs hess may be reproduced, sored, n any rereval sysem or ransmed n any form or by any means, elecronc, mechancal, phoocopyng, and recordng whou pror wren permsson of he auhor or Egeron Unversy on ha behalf. 008 Wagala Adolphus All rghs Reserved

4 DEDICATION To My dad Wllam and lae mum Abgal. v

5 ACKNOWLEDGEMENT I wsh o hank he Lord God Almghy who has besowed upon me good healh, sane mnd, srengh, paence and suffcen grace o see me hrough hs sudy. All he honor and glory be uno you oh Lord! I am graeful o my lecurers and supervsors: Prof. Nassuma and Dr. Al for her commmen hrough ou my sudy and her paence o go hrough he earler manuscrps of hs work plus he many resourceful suggesons hereafer ha opened my eyes o see deeper sascally. Much hanks also o Dr. Mwang of he Mahemacs deparmen who nroduced me o me seres and has been of much suppor and encouragemen by provdng leraure maerals. To my famly and n parcular my ssers Nancy, Dora, Lllan and my cousn Isaya Jakawaya for moral and fnancal suppor, I say hank you. Specal apprecaon o my dad Wllam and sep-mum Isabella, I hank God for you and I am graeful for your encouragemens, nspraons, prayers and fnancal asssance you offered o me. You are smply he bes parens one could ever have. Fnally, I also wsh o acknowledge my frend Julana Chdumu, my classmaes: Mke, Anasaca, Mwangasha, Esha and Thga for he moral suppor and encouragemens whle n he urbulen waers of academcs. v

6 ABSTRACT Modellng of sock marke daa has wnessed a sgnfcan ncrease n leraure over he pas wo decades. Focus has been manly on he use of he ARCH model wh s varous exensons due o s ably o capure heeroscedascy prevalen n he fnancal and moneary varables. However, oher suable models lke he blnear models have no been exploed o model sock marke daa so as o deermne he mos effcen model beween he ARCH and blnear models. The underlyng problem s ha of denfyng he mos effcen model ha can be appled o sock exchange daa for forecasng and predcon. The purpose of hs sudy was o deermne he mos effcen model beween he wo models namely, ARCH and blnear models when appled o sock marke daa. The daa was obaned from he Narob Sock Exchange (NSE) for he perod beween 3 rd June 996 o 3 s December 007 for he company share prces whle for he NSE 0-share ndex daa was for perod beween nd March 998 o 3 s December 007.The share prces for hree companes; Bambur Cemen, Naonal Bank of Kenya and Kenya Arways whch were seleced a random from each of he hree man secors as caegorzed n he Narob Sock Exchange were used. Specfcally, he dfferen exensons of ARCH-ype models were ulzed wh ARMA and blnear models for modellng he weekly mean of he chosen daa se. The model effcency was deermned based on he mnmal mean squared error (MSE). The resuls show ha he Blnear-GARCH model wh he normal dsrbuon assumpon and he AR-Inegraed GARCH (IGARCH) model wh suden s -dsrbuon are he bes models for modellng volaly n he Narob Sock Marke daa. The resuls also ndcae ha he volaly n Narob Sock Exchange s sascally sgnfcan and perssen wh he posve reurn nnovaons havng a greaer mpac han he negave ones. Ths mples ha he leverage effec experenced n mos developed counres s no applcable o Narob Sock Marke. The resuls obaned are sgnfcan for plannng, predcon and managemen of nvesmens on shares n he Narob Sock Exchange. The chosen models are also helpful for decson makng especally by he nvesors, sockbrokers and fnancal advsors regardng he radng n shares a he Narob Sock Exchange. v

7 TABLE OF CONTENTS DECLARATION AND RECOMMENDATION... COPYRIGHT... DEDICATION... v ACKNOWLEDGEMENT... v ABSTRACT... v TABLE OF CONTENTS... v LIST OF TABLES... x LIST OF FIGURES... x ACRONYMS AND ABBREVIATIONS... x CHAPTER ONE... INTRODUCTION.... Background.... The Narob Sock Exchange....3 Saemen of he problem Objecves of he sudy Man objecve Specfc objecves Jusfcaon Defnon of erms used... 4 CHAPTER TWO... 5 LITERATURE REVIEW Fnancal Daa Models for Sock Marke Daa Lnear Tme Seres Models The Box and Jenkns Approach o ARIMA Modellng Non Lnear Tme Seres Models Auoregressve Condonal Heeroscedascy (ARCH) models Generalzed Auoregressve Condonal Heeroscedascy (GARCH)... 6 v

8 .5. GARCH-n-Mean (GARCH-M) model Exponenal GARCH (EGARCH) model Quadrac GARCH (QGARCH) model Threshold GARCH (TGARCH) model Glosen, Jagannahan and Runkle (GJR) model Parameer Esmaon of ARCH models Model denfcaon for he ARCH-ype models....6 Blnear Models (BL) Esmaon of parameers n blnear models Leas squares esmaon of model parameers for blnear models Order selecon for Blnear models... 6 CHAPTER THREE... 8 METHODOLOGY The scope of he sudy Daa collecon Daa analyss... 8 CHAPTER FOUR RESULTS AND DISCUSSIONS Prelmnary Analyss Emprcal Resuls and Dscussons Effcency Comparson beween he ARCH-ype Models Effcency Evaluaon n Blnear and Blnear-GARCH Models Comparson beween ARMA-GARCH, Blnear and Blnear-GARCH models CHAPTER FIVE SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary ARCH-ype models Blnear models Blnear-GARCH models Conclusons Recommendaons Furher Research REFERENCES v

9 LIST OF TABLES Table 4.: Basc sascal properes of reurns Table 4.: Maxmum lkelhood esmaes for he AR(p) Table 4.3: Maxmum lkelhood esmaes for he Varance Equaon (ARCH) Table 4.4: The goodness of f sascs for ARCH models Table 4.5: Dagnosc Tess for Sandardzed Resduals for ARCH models Table 4.6: Maxmum Lkelhood Parameer Esmaes for he AR(p) model Table 4.7: Maxmum lkelhood esmaes for he Varance Equaon (GARCH) Table 4.8: Goodness of f Sascs for GARCH(,)... 4 Table 4.9: The Dagnosc ess on he sandardzed resduals for he GARCH models... 4 Table 4.0: Maxmum Lkelhood Parameer Esmaes for he AR(p) model Table 4.: Maxmum lkelhood esmaes for he Varance Equaon (IGARCH) Table 4.: Goodness of f Sascs for IGARCH (,) Table 4.3: The Dagnosc ess on he sandardzed resduals for he IGARCH models Table 4.4: Maxmum Lkelhood Parameer Esmaes for he AR(p) model Table 4.5: Maxmum lkelhood esmaes for he Varance Equaon (EGARCH (, )) Table 4.6: The Dagnosc ess on he sandardzed resduals for he EGARCH models Table 4.7: Goodness of f Sascs for he EGARCH models Table 4.8 : Maxmum Lkelhood Esmaes for he AR(p) Table 4.9: Maxmum lkelhood esmaes for he Varance Equaon (TGARCH (, )) Table 4.0 :The goodness of f sascs for he TGARCH models Table 4.: The Dagnosc Tess n Sandardzed Resduals for he TGARCH models... 5 Table 4.: MSE for he fed ARCH-ype models... 5 Table 4.3: Maxmum Lkelhood Esmaes for he blnear models Table 4.4: Goodness of f sascs for he blnear models Table 4.5: Dagnosc Tess for he blnear models Table 4.6: Esmaed Blnear-GARCH models Table 4.7: Goodness of f sascs for he BL-GARCH models Table 4.8: Dagnosc ess for he Blnear-GARCH models Table 4.9: MSE for blnear and blnear-garch models x

10 LIST OF FIGURES Fgure 4.:Tme plo for weekly NSE- Index... 3 Fgure 4.: Tme plo for weekly average NBK prces... 3 Fgure 4.3: Tme plo for weekly Bambur prces... 3 Fgure 4.4: Tme plo for weekly Kenya Arways (KQ) prces... 3 Fgure 4.5: Tme plo for he Log dfferenced NSE-Index Fgure 4.6: Tme plo for he Log dfferenced NBK prces Fgure 4.7: Tme plo for he Log dfferenced Bambur prces Fgure 4.8: Tme plo for he Log dfferenced Kenya Arways (KQ) prces x

11 ACRONYMS AND ABBREVIATIONS AR ARMA ARIMA ARCH AIC ACF BL BIC CLSE EAA FPE FTA EXPAR FAR GARCH GED IID JB LSE LR MA MLE MSE NEAR NBK NSE NSSF NYSE PACF S & P Auoregressve Auoregressve Movng Average Auoregressve Inegraed Movng Average Auoregressve Condonal Heeroscedascy Akake Informaon Crera Auocorrelaon Funcon Blnear Bayesan Informaon Crera Condonal Leas Squares Esmae Eas Afrcan Arways Corporaon Fnal Predcon Error Fnancal Tmes Acuares Exponenal auoregressve Fraconal Auoregressve Generalzed Auoregressve Condonal Heeroscedascy General Error Dsrbuon Independen Idencally Dsrbued Jarque-Bera sascs for normaly Leas Squares Esmae Log lkelhood Rao es. Movng Average Maxmum Lkelhood Esmae Mean squared error Newer exponenal auoregressve Naonal Bank of Kenya Lmed Narob Sock Exchange Naonal Socal Secury Fund New York Sock Exchange Paral Auocorrelaon Funcon Sandard and Poor x

12 RCA SE WN Random coeffcen auoregressve Sandard Error Whe Nose x

13 CHAPTER ONE INTRODUCTION. Background Sock marke volaly s one of he mos mporan aspecs of fnancal marke developmens, provdng an mporan npu for porfolo managemen, opon prcng and marke regulaon (Poon and Granger, 003). An nvesor s choce of porfolo s nended o maxmze hs expeced reurn subjec o a rsk consran, or o mnmze hs rsk subjec o a reurn consran. An effcen model for forecasng of an asse s prce volaly provdes a sarng pon for he assessmen of nvesmen rsk. To prce an opon, one needs o know he volaly of he underlyng asse. Ths can only be acheved hrough modellng he volaly. Volaly also has a grea effec on he macro-economy. Hgh volaly beyond a ceran hreshold wll ncrease he rsk of nvesor losses and rase concerns abou he sably of he marke and he wder economy (Hongyu and Zhchao, 006). Fnancal me seres modellng has been a subjec of consderable research boh n heorecal and emprcal sascs and economercs. In recen leraure, numerous paramerc specfcaons of ARCH models have been consdered for he descrpon of he characerscs of fnancal markes. Engle (98) nroduced he Auoregressve Condonal Heeroscedascy (ARCH) for modellng fnancal me seres whle Bollerslev (986) came up wh he Generalzed ARCH (GARCH) o parsmonously represen he hgher order ARCH model. Owng o he emprcal success of he ARCH and GARCH models, researchers have concenraed on he wo models due o her ably o model heeroscedascy. There s a sgnfcan amoun of research on volaly of sock markes of developed counres. Gary and Mngyuon (004) appled he GARCH model o he Shangha Sock Exchange whle Berram (004) modelled Ausralan Sock Exchange usng ARCH models. Oher sudes nclude, Baudouha (004) who ulzed he GARCH model n analyzng he Nordc fnancal marke negraon, Waler (005) appled he srucural GARCH model o porfolo rsk managemen for he Souh Afrcan equy marke. Hongyu and Zhchao (006) forecased he volaly of he Chnese sock marke usng he GARCH-ype models.

14 The Sub-Saharan Afrca has been under-researched as far as volaly modellng s concerned. Sudes carred ou n he Afrcan sock markes nclude, Frmpong and Oeng-Abaye (006) who appled GARCH models o he Ghana Sock Exchange, Brooks e al., (997) examned he effec of polcal change n he Souh Afrcan Sock marke, Appah-Kus and Pasceo (998) nvesgaed he volaly and volaly spllovers n he emergng markes n Afrca. More recenly, Ogum e al., (006) appled he EGARCH model o he Kenyan and Ngeran Sock Marke reurns. From he avalable leraure, he NSE jus lke oher Sub Saharan Afrca Equy Markes has been clearly under-researched as far as marke volaly s concerned and herefore hs sudy conrbues o he lmed leraure avalable on he Narob sock marke. Mandelbro (963) ulzed he nfne varance dsrbuons when consderng he models for sock marke prce changes. Fama (965) when modellng sock marke prces arbued her dscrepances o he possbly of he process havng sable nnovaons and hus fed an adequae model on hs bass. These developmens n fnancal economercs sugges he use of nonlnear me seres models n analyzng he sock marke prces and he expeced reurns. The focus of fnancal me seres modellng has manly been on he ARCH model and s varous exensons hereby gnorng he oher suable nonlnear models lke he blnear class of models. As a maer of fac, he subjec of he effcency of he models for fnancal modellng has receved lle aenon as far as economerc modellng s concerned. Ths sudy herefore ams a fndng he mos effcen model from amongs he nonlnear models namely, blnear models and he auoregressve condonal heeroscedascy models.. The Narob Sock Exchange The Narob Sock Exchange (N.S.E) was formed n 954 as a volunary organzaon of Sock brokers. I s now one of he mos acve capal markes and a model for he emergng markes n Afrca n vew of s hgh reurns on nvesmens and a well developed marke srucure (Ogum e al., 005). The Narob Sock Exchange s a marke place where shares (also known as eques) and bonds (also known as deb nsrumens) are raded. The ordnary shares are also known as varable ncome secures snce hey have no fxed rae of dvdend payable, as he dvdend s dependen upon boh he profably of he company and wha he board of drecors decdes. The bonds are also known as he fxed ncome secures and nclude Treasury and Corporae

15 Bonds, preference shares, debenure socks; hese have a fxed rae of neres/dvdend, whch s no dependen on profably. As a capal marke nsuon, he Sock Exchange plays an mporan role n he process of economc developmen. The major role ha he Narob sock exchange has played, and connues o play s ha promoes a culure of hrf, or savng. The NSE also assss n he ransfer of savngs o nvesmen n producve enerprses hereby ulzng he money ha would oherwse le dle n savngs. Ths helps n avodng economc sagnaon. I also assss n he raonal and effcen allocaon of capal. An effcen sock marke secor wll have he experse, he nsuons and he means o prorze access o capal by compeng users so ha an economy manages o realze maxmum oupu a he leas cos. The NSE promoes hgher sandards of accounng, resource managemen and ransparency n he managemen of busness. In addon, he sock exchange mproves he access of fnance o dfferen ypes of users by provdng he flexbly for cusomzaon. Fnally, he sock exchange provdes nvesors wh an effcen mechansm o lqudae her nvesmens n secures. The nvesors are able o sell ou wha hey hold, as and when hey wan. Ths s a major ncenve for nvesmen as guaranees mobly of capal n he purchase of asses..3 Saemen of he problem The las wo decades have wnessed an ncrease n he modellng of sock marke daa usng he ARCH models and s varous exensons. However, effcences of compeng models such as he blnear models has so far no been deermned for modellng equy marke daa. Ths sudy herefore seeks o deermne he mos effcen model for applcaon o he Narob sock marke daa from he wo classes of models..4 Objecves of he sudy.4. Man objecve The overall objecve of hs sudy was o deermne he mos effcen model from he wo classes of models namely; ARCH, blnear and blnear-arch. 3

16 .4. Specfc objecves The weekly mean for he NSE 0-share ndex and he average weekly share prces for he followng companes; Naonal Bank of Kenya, Kenya Arways and Bambur Cemen Ld were used o acheve he followng specfc objecves; ) To model he Narob Sock Exchange sock daa usng ARCH-ype models. ) To apply Blnear models n fng he Narob Sock Exchange daa. ) To compare he effcency of he wo classes of models and make recommendaons regardng he bes model for modellng volaly..5 Jusfcaon The esablshmen of an effcen sock marke s ndspensable for an economy ha s keen on ulzng scarce capal resources o acheve s economc growh. I s herefore pruden o deermne he mos effcen model ha wll help n predcng volaly whch n urn s mporan n prcng fnancal dervaves, selecng porfolos, measurng and managng rsks more accuraely. The effcen model wll no only be useful n long erm forecasng and shor erm predcon bu also n helpng he nvesors on decsons regardng whch shares o sell, hold or buy..6 Defnon of erms used Bonds are fnancal nsrumens ha serve as an owe you ; an nvesor loans an ssuer, and reurns are fxed and guaraneed, no vong rghs and no benefs from exceponal performance by a company. Shares are fnancal nsrumens where one acqures ownershp sakes of a company. Reurns are neher fxed nor guaraneed. One acqures vong rghs and shares he company s profs and losses. Volaly s he varance or varaon of a gven me seres daa. Reurns are ransformaons gven by X = ln( P ) ln( P ), where X and P represens he reurn and weekly average value for each seres respecvely. 4

17 CHAPTER TWO LITERATURE REVIEW. Fnancal Daa Fnancal me seres daa ofen exhb some common characerscs. Fan and Yao (003) summarzes he mos mporan feaures of fnancal me seres as; he seres end o have lepokurc dsrbuon,.e hey have heavy aled dsrbuon wh hgh probably of exreme values. In addon, changes n sock prces end o be negavely correlaed wh changes n volaly, ha s; volaly s hgher afer negave shocks han afer posve shocks of he same magnude. Ths s referred o as he leverage effec. The sample auocorrelaons of he daa are small whereas he sample auocorrelaons of he absolue and squared values are sgnfcanly dfferen from zero even for large lags. Ths behavour suggess some knd of long range dependence n he daa. The dsrbuon of log reurns over large perods of me (such as a monh, a half a year, a year) s closer o a normal dsrbuon han for hourly or daly log-reurns. Fnally, he varances change over me and large (small) changes of eher sgn end o be followed by large (small) changes of eher sgn (Mandelbro, 963). Ths characersc s known as volaly cluserng. These are facs characerzng many economc and fnancal varables.. Models for Sock Marke Daa Researchers have appled dfferen models o he socks daa from me o me. Mandelbro (963) ulzed he nfne varance dsrbuons when consderng he models for sock marke prce changes. Fama (965) smlarly poned ou nally, her applcaon n cases of economcs parcularly n modellng sock marke prces. Fama e al., (969) used a random walk o model he prce changes. Andrew and Whney (986) esed he random walk hypohess for weekly sock marke reurns by comparng he varance esmaors. Here he random walk model was srongly rejeced. Omosa (989) appled he ARIMA model o he NSE daa and used he models for forecasng. Muhanj (000) suded he effcency of he Narob Sock Exchange and concluded ha he NSE had a weak form of effcency mplyng ha he marke s effcen a a parcular perod and becomes neffcen a anoher me. 5

18 In recen sudes, varous specfcaons of ARCH models have been consdered for he descrpon of he characerscs of fnancal markes. Some sudes n whch ARCH-ype models were ulzed nclude; Gary and Mngyuon (004) who appled he GARCH model o Shangha Sock Exchange, Berram (004) modelled Ausralan Sock Exchange usng ARCH models and Baudouha (004) used he GARCH model n analyzng he Nordc fnancal marke negraon. In addon, Curo (00) employed he GARCH model o explan he volaly of he Poruguese equy marke, Waler (005) appled he srucural GARCH model o porfolo rsk managemen whle Frmpong and Oeng-Abaye (006) modelled he Ghana Sock Exchange volaly usng he GARCH models. More recenly, Ogum e al., (006) appled EGARCH model o he Kenyan and Ngera daly sock marke daa. Smple regresson models have also been ulzed n modellng sock marke daa. Bodcha (003) appled regresson models o he NSE daa and found ou ha he regresson models are only approprae for shor erm predcon and no for long erm forecasng. The analyss of he general lnear regresson model forms he bass of every sandard economerc model. Mlls (999) appled he smple lnear relaonshp n modellng he expeced rsk and reurn n holdng a porfolo whle Gujara (003) appled economerc modellng o he NYSE daa..3 Lnear Tme Seres Models Le ς be a subse of he real numbers. For every on a probably space { Ω ω Ω} ς, le X (ω) be a random varable defned : ; hen he sochasc process { X ( ω) : ς} s called a me seres. Here,{ X, = 0, ±,... } s a realzaon a me for any gven ω. A me seres model for he observed daa {X } s specfcaon of he jon dsrbuons (or possbly only he means and covarances) of a sequence of random varables {X }. A me seres model accouns for paerns n he pas movemens of a varable daa and ha nformaon s used o conrol and predc s fuure movemens. The auoregressve movng average (ARMA) processes are he mos wdely known and appled se of lnear me seres models. For he ARMA(p,q) process, he observaon X s lnearly relaed o he p mos recen observaons (X -,, X -p ), q mos recen forecas errors ( q ε,..., ε ) and he curren dsurbance ε by he relaon: 6

19 p q + θε + = = X = φ X ε, ε ~ WN(0, σ ).3. where φ and θ are model parameers. The equaon n.3. represens an ARMA (p,q) process. Alernavely, usng he backshf operaor, an ARMA (p,q) process s represened as Φ ( B) w = Θ( B) ε,where B s a backshf operaor such ha BX =X -, ε } s a sequence of uncorrelaed random varables wh zero mean and varance σ.the polynomals: Φ... ( B) = + φ B + φ B + + φ B p p.3. Θ... ( B) = + θ B + θ B + + θ B q q.3.3 represen he auoregressve and he movng average operaors of order p and q respecvely. The coeffcens n Φ(B) and Θ (B) represen some of he model parameers. { A specal case of he ARMA(p,q) process s known as he auoregressve process whch dae back o he work of Yule (97) where he developed he frs order auoregressve process (AR()) whch s gven by he relaon X = φ + ε where φ s a model parameer and X ε ~ WN(0, σ ). In general, he AR(p) process s represened as X p = X = φ + ε.3.4 where φ are consans and ε ~ WN(0, σ ) Anoher ype of lnear me seres model s known as he movng average (MA) process whch was developed by Sluzky (937). The funconal form for he frs order movng average process (MA ()) process s gven by he equaon X = θε + ε, where θ s model parameer and ε ~ WN(0, σ ). The general represenaon of an MA(q) process s gven as X = q = θ ε ε where θ are model parameers and ε ~ WN(0, σ ). Here ε s no observable. When he saonary condon s assumed,.e. when he mean, varance and auocovarances of a process are nvaran under me ranslaons, hen he process s modelled usng he ARMA models. The ARMA models have been appled o modelng he UK neres raes, reurns on he 7

20 FTA All share ndex, S&P 500 sock ndex and he dollar/serlng exchange rae (Mlls, 999). Fan and Yao (003) also modelled he German Egg prces usng he Auoregressve Inegraed Movng Average (ARIMA) models. In addon Gujara (003) appled he Box Jenkns approach o model he money supply n he Uned Saes. The saonary condon resrcs he mean and he varance o be consan and requres he auocovarances o depend only on he me lag. However, hs s no rue n many fnancal me seres, hey are ceranly non saonary and have a endency o exhb me changng means and varances. Box and Jenkns (976), suggesed dfferencng as a means of ransformng a nonsaonary ARMA (p,q) process no a saonary ARMA(p,q) process known as he auoregressve negraed movng average (ARIMA) process. Ths s applcable o he fnancal me seres model buldng..3. The Box and Jenkns Approach o ARIMA Modellng Box and Jenkns (976) proposed hree major sages n ARIMA modellng, namely; denfcaon, esmaon, dagnosc checkng and forecasng. The approach s as follows; a) Idenfcaon Sage In hs sage he model selecon s done. The smples and mos basc ool for denfcaon s he me seres plo, whch s smply a graph n whch daa values are arranged sequenally n me. A plo s an effecve way of quckly percevng he evoluon of a sngle or a group of me seres. The plos are useful n deecng oulers, he seasonal, cyclc and he rend componens of a me seres daa. The nex creron for denfcaon s he Auocorrelaon Funcon (A.C.F). The auocorrelaon s gven by, Cov( x, x k ) ρ = / [ v( x ). v( x )] = k γ 0 γ k.3.6 The auocorrelaons consdered as a funcon of k s referred o as he auocorrelaon funcon ACF or somemes he correlogram. The ACF plays a major role n modellng he dependences among observaons. I ndcaes, by measurng he exen o whch one value of he process s correlaed wh he prevous, he lengh and srengh of he memory of he process. In general, 8

21 he correlaon beween wo random varables s ofen due o boh varables beng correlaed wh a hrd one. In he conex of me seres, a large poron of he correlaon beween X and X -k can be due o he correlaon beween X -,X -,,X -k+. To adjus for hs correlaon, he paral auocorrelaon funcon (PACF) may be calculaed. The PACF measures he addonal auocorrelaons beween X and X -k afer adjusmens have been made for nervenng lags. The ACF and PACF are useful n he denfcaon of orders n he ARMA processes. An AR(p) process has a declnng ACF, exponenally decayng o zero and he PACF s zero for lags greaer han p. An MA(q) process on he oher hand has an ACF ha s zero for lags greaer han q and PACF ha declnes exponenally. However, f he decay n he ACF sars afer a few lags lags hen he process could be an ARMA(p,q). If he seres s non-saonary, hen s ransformed by dfferencng o aan saonary. The decson abou dfferencng s based on he vsual examnaon of he correlogram. The adequacy of he fed model or an ndcaon of poenal mprovemens s deermned usng he followng dagnosc checks. These checks nclude he Fnal Predcon Error (FPE) creron whch was developed by Akake (969) for selecng he approprae order of an AR process. The dea here s o selec he model for {X } n such a way as o mnmze he one-sep mean squared error when he model fed o {X } s used o predc an ndependen realzaon {Y } of he same process ha generaed {X }. The FPE for order p s gven as, n + p FPEp = ˆ σ.3.7 n p To apply he FPE creron, p s chosen such ha mnmzes he value of FPEp. Anoher creron s he Akake nformaon creron (AIC) whch s more generally applcable for model selecon. The AIC was developed by Akake (974).The AIC s defned by: AIC (p,q) = log ˆ σ + ( p + q) T.3.8 where ˆ σ s he esmae of he error varance of an ARMA (p,q) and T s he oal number of observaons. For fng auoregressve models, he AIC has a endency o overesmae p,.e. for AIC an over parameerzed model s more lkely o be obaned. Schwarz (978) suggesed he Bayesan Informaon Crera (BIC) defned as 9

22 BIC (p, q) = log ˆ σ + ( p + q) T logt.3.9 The BIC aemps o solve he over parameerzaon of AIC and s hus srongly conssen, n ha deermnes he rue model asympocally. These procedures enal he comparson of he sample values wh he correspondng heorecal values. b) Esmaon of Parameers Once a enave formulaon of he me seres models has been accomplshed, esmaon of model parameers follows. The esmaon echnques used nclude he Yule-Walker esmaon creron. The Yule-Walker esmaes are obaned by machng paerns n he sample auocorrelaons wh heorecal paerns. For nsance, consder AR(P) process, X = φ ( B) + ε.3.0 X p Here E X ( X φ X ) = E( X ε ) for =0,,p = 0 γ ( 0) = σ φ γ p and γ p Γpφ = 0 where φ ( φ,..., φ ). Now, choose φ so ha γ = ˆ γ. The Yule Walker equaons can be wren = p n a marx form as ρ R p φ = where R p s a covarance marx. The Yule Walker equaons Γˆ ˆ pφ = ˆ γ p can also be presened as where Г ˆ σ = ˆ(0) γ ˆ p s he covarance marx φ ˆ γ p [ γ ( )] and p j = γ = [ γ (), γ (),..., γ ( p)]. Ths furher leads o p ˆ φ = Γ ˆ p γ where Γˆ p = R and φˆ s he Yule- p p Walker esmae. The nex esmaon mehod s he maxmum lkelhood procedure. Ths mehod s more approprae for small samples and especally when he parameer values approach he nverble boundares. In he maxmum lkelhood esmaon (MLE) of me seres models, wo ypes of MLEs are compued. The frs ype s based on maxmzng he condonal log-lkelhood funcon. These esmaes are he condonal MLEs defned by; 0

23 ˆ φc. m. l. e = arg max ln f ( X θ T = p+ / I, θ ). The second ype s based on maxmzng he exac loglkelhood funcon. These exac esmaes are called exac MLEs, and defned by; ˆ φ T = arg max ln f ( Y / I, θ ) + ln f ( y,..., y p; ).3. m. l. e θ θ = p+ For saonary models, ˆ φ c. m. l. e and m. l. e ˆ φ are conssen and have he same lmng normal dsrbuon. In fne samples however, ˆ φ c. m. l. e and ˆ φ m. l. e are generally no equal and may dffer by a subsanal amoun f he daa are close o beng non-saonary or non-nverble. Consder an AR () process X c +φ + ε = X where consan. The exac log-lkelhood funcon s hen; ε ~ WN (0, σ ) =,,3,,T and c a T T σ φ c ( T ) ln L( φ / y) = ln(π ) ln( ) ( x ) ln( σ ) ( x c φ φ σ φ σ = The exac log-lkelhood funcon s a non-lnear funcon of parameers φ and so here s no closed form soluon for he exac MLEs. x ) The nex esmaon creron s he condonal leas squares esmaon (CLSE) mehod. These esmaes are easer o compue compared o MLE. Ths procedure enals mnmzng he sum of squares Q = n ε. Thus for an AR (P) process, Q = = n p ( X φ X ). Now, for p=, = = n X X ˆ = φ =.3. n X = The opmal esmaon echnque s also useful n esmaon and was nally suded by Godambe (960) n comparson o he MLE. The opmal esmaes for sochasc process are obaned as follows: Consder an AR () process: X = φ + ε where ε ~ WN (0, σ ).3.3 X Le h = X φ X be a lnear funcon such ha E ( h / ) = 0 where I s he sgma algebra up o me -.e. σ-algebra on whch (x, x,,x n ) s defned. The opmal esmang funcon (Godambe, 985) s obaned as; I

24 g * = n = a h *.3.4 where a * = h E( / I φ E( h / I ) ) g * = X σ ( X φx ).3.5 and by seng g 0 leads o he opmal esmae of φ as * = n X X ˆ = φ =.3.6 n X = whch s he same as he CLSE. c) Forecasng When a me seres model has been denfed and parameer esmaes obaned, can be used for forecasng. The mos common forecasng creron s based on mnmzng he mean square error, ha s, for he process X, he am s o oban he forecas X, such ha E ˆ s [ X + X + ] mnmzed. ARIMA modellng s popular because of s success n forecasng. In many cases, he forecass obaned by hs mehod are more relable han hose obaned by economerc modellng (Gujara, 003). Mahemacally, lnear me seres models are he smples ype of dfference equaons and a complee heory of Gaussan sequences are readly undersood. The heory of sascal nference s also mos developed for lnear Gaussan models. The compuaon me requred for obanng a parsmonous ARMA model for he daa s well whn he reach of mos praconers. These models have been reasonably successful n analyss, forecasng and conrol of varous me seres daa. Lnear me seres models are easy o compue and manpulae. The assumpon of saonary makes modellng n hem smple. The esmaon procedures for he lnear models are also no complcaed. Some of he lmaons of lnear me seres models nclude he fac ha lnear dfference equaons do no perm sable perodc soluons ndependen of nal value. Havng symmerc jon dsrbuons, saonary lnear Gaussan models are no deally sued for daa exhbng

25 srong asymmery,.e. hey are dependen on he symmerc sysems. The ARMA models are no deally sued for daa exhbng sudden burss of very large amplude a rregular me epochs. Lnear me seres models do no capure seres ha exhb cyclcy. In pracce lnear me seres models are assumed o be Gaussan and hus have shor als. However, hs s no rue n mos me seres daa especally he fnancal daa whch usually follow non-normal dsrbuons. Lnear models are also unable o ulze hgher momens and assume only he frs wo momens whle n some cases, here exs a hrd or even a fourh momen. In seres ha are me rreversble, lnear models are of no use ( Tong,990; Kanz and Schreber, 005; Zvo and Wang, 005)..4 Non Lnear Tme Seres Models In ryng o address he lmaons of lnear me seres models, many non-lnear me seres models have been developed (Kanz and Schreber, 005; Zvo and Wang, 005). Non-lnear me seres models can be used o model seres ha show cyclcy. They are also useful n seres havng oulers. Non-lnear models are useful for hgher momen ulzaon snce hey are able o capure hgher momens. They are also useful n seres ha are me rreversble. Exreme nonsaonary especally n he varance makes non-lnear models more approprae. They are also useful n modellng daa ha are asymmerc (Tong, 990). In he recen years, a few non-lnear me seres models have been proposed. In many cases, he resuls are sll ncomplee and much research s gong on a presen. An example of a non-lnear me seres model s ha of non-lnear auoregresson models. Ths class of models s movaed drecly by dynamcal sysem (Tong, 990). The nex class of non-lnear models s he amplude-dependen exponenal auoregressve (EXPAR) models. These models were ndependenly nroduced by Jones (976) and Ozak and Oda (978). The EXPAR models are useful n modellng ecologcal/populaon daa (Ozak, 98), wolf s sunspo numbers (Haggan and Ozak, 98) and o a small exen, he economcs daa (Tong, 990). Anoher mporan class of non-lnear me seres models s he Fraconal Auoregressve (FAR) models. Ths class of models has no so far been exploed conclusvely o fnd her bes area of applcaon. Random coeffcen auoregressve (RCA) models have been appled o areas such as, ecology/populaon (Ncholls and Qunn 98) and Medcal daa (Robnson, 978).There 3

26 exs a subclass of RCA models wh he margnal dsrbuon ha s exponenal. They are known as he Newer exponenal auoregressve (NEAR) models. The NEAR models were appled o Geophyscs by Lawrence and Lews (985). The oher class of non-lnear me seres models s he Threshold models whch were nroduced by Tong (978).These models have a wde range of applcaons for nsance n fnance ( Peruccell and Daves, 986; Wecker, 98; Tyssedal and Tjøshem,988), populaon dynamcs (Senseh e al., 999), economcs (Tao and Tsay, 994). They are also applcable n ecology/populaon daa (L and Lu, 985). In addon, hey have been used n geodynamcs (Zheng and Chen, 98) and also n neural scence (Brllnger and Segundo, 979)..5 Auoregressve Condonal Heeroscedascy (ARCH) models An ARCH process s a mechansm ha ncludes pas varances n he explanaon of fuure varances (Engle, 004). The Auoregressve propery descrbes a feedback mechansm ha ncorporaes pas observaons no he presen whle Condonaly mples a dependence on he observaons of he mmedae pas and Heeroscedascy means me-varyng varance (volaly). These models were frs nroduced by Engle (98) when modellng he Uned Kngdom nflaon. In conras o he ARMA models whch focuses on modellng he frs momen. ARCH models specfcally ake he dependence of he condonal second momens n modellng consderaon. Ths accommodaes he ncreasngly mporan demand o explan and o model rsk and uncerany n fnancal me seres (Degannaks and Xekalak, 004; Engle, 004; Fan and Yao, 003). An ARCH process can be defned n erms of he dsrbuon of he errors of a dynamc lnear regresson model. The dependen varable y s assumed o be generaed by where y = ξ + ε =,,T.5. x x s a kx vecor of exogenous varables, whch may nclude lagged values of he dependen varable and ξ s a kx vecor of regresson parameers. The ARCH model characerzes he dsrbuon of he sochasc error ε condonal on he realzed values of he se of varables ψ y, x, y, x,...}. Specfcally, Engle s (98) model assumes = { ε / ψ ~ N ( 0, h ).5. 4

27 where h = α 0 + αε α qε q.5.3 wh α 0 >0 and α 0, =,..., q o ensure ha he condonal varance s posve. An explc generang equaon for an ARCH process s ε = η h.5.4 where η ~..d N (0,) and h s gven by equaon (.5.3). Snce h s a funcon of ψ and s herefore fxed when condonng on ψ, s clear ha ε as gven n (.5.4) wll be condonally normal wh E( ε / ψ ) = h E( η / ψ ) = 0 and Var ( ε / ψ ) = h, Var η = h. Hence he process (.5.4) s dencal o he ARCH process (.5.). ( / ψ ) Engle (98, 983) found ha a large lag q was requred n he condonal varance funcon when applyng he ARCH model o he relaonshp beween he level and volaly of nflaon. Ths would necessae esmang a large number of parameers, subjec o nequaly resrcon. To reduce he compuaonal burden, Engle (98, 983) parameerzed he condonal varance as; where he weghs q 0 + α ε = h = α w.5.5 q ( q + ) w = declne lnearly and are consruced so ha 0.5q( q + ) w = =. Wh hs parameerzaon, a large lag can be specfed and ye only wo parameers are requred n he condonal varance funcon. Despe he mporance of he ARCH model for many fnancal me seres, a relavely long lag lengh n he varance equaon wh he problem of esmaon of parameers subjec o nequaly resrcons s ofen called for o capure he long memory ypcal of fnancal daa. The ARCH model has been exended o varous generalzaons. Some of he generalzaons are gven n he followng secons. 5

28 .5. Generalzed Auoregressve Condonal Heeroscedascy (GARCH) The GARCH model was developed by Bollerslev (986) and s oday one of he mos wdely used ARCH-ype model (Engle, 004). He proposed an exenson of he condonal varance funcon (.5.3) whch he ermed as he generalzed ARCH (GARCH) and suggesed ha condonal varance be specfed as, h = α... β 0 + αε α qε q + βh + + ph p.5.6 wh he nequaly condons α 0 > 0, α 0 for =,,q, β 0 for =,,p o ensure ha he condonal varance s srcly posve. A GARCH process wh orders p and q s denoed as GARCH (p,q) and hs essenally generalzes he purely auoregressve ARCH o an auoregressve movng average model. The movaon for he GARCH process can be seen by expressng (.5.5) as h = α + + B) h α( B) ε β ( where α ( B) = α B α and B q q β ( B) = β B β are polynomals n he backshf B q q operaor B. Now, f he roos of β ( Z) le ousde he un crcle, equaon.5.7 be wren as h = α α( B) 0 * + ε = α 0 + β () β ( B) = δ ε.5.8 where α * 0 α 0 = and he co-effcen δ s he co-effcen of B n he expresson of [ β ()] α ( B)[ β ( B)]. The slope parameer β measures he combned margnal mpacs of he lagged nnovaons whleα, on he oher hand capures he margnal mpac of he mos recen nnovaon n he p condonal varance. When α + β j <, hen he process s weakly saonary and he = q j= condonal varance ( σ ) approaches he uncondonal varance ( σ ) as me goes o nfny.e E ( σ +s ) σ as p s. However, when α β > hen he process s non saonary. = q + There exss some suaons whereby parameer esmaes n GARCH (p,q) models are close o p he un roo bu no less han un,.e α + β j =, for he GARCH process. Here, he = mul-sep forecass of he condonal varance do no approach he uncondonal varance. j= q j= j 6

29 These processes exhb he perssence n varance/volaly whereby he curren nformaon remans mporan n forecasng he condonal varance. Engle and Bollerslev (986) refer o hese processes as he Inegraed GARCH or IGARCH. The IGARCH process does no possess a fne varance bu are saonary n he srong sense (Nelson, 990). From.5.8, s easy o see ha a GARCH (p,q) process s an nfne order ARCH wh a raonal lag srucure mposed on he co-effcen. The nenon s ha he GARCH process can parsmonously represen a hgh-order ARCH-process (Bera and Hggns, 993; Engle, 004; Degannaks and Xekalak, 004). The smples GARCH(,) s ofen found o be he benchmark of fnancal me seres modellng because such smplcy does no sgnfcanly affec he precseness of he oucome. A GARCH model can be appled wh he assumpon of normal, suden or general error dsrbuons. Besdes he emprcal success, GARCH models have wo major draw backs: Frs, hey are unable o model asymmery because n a GARCH model, posve and negave shocks of he same magnude produce he same amoun of volaly (.e only he magnude and no he sgn of he lagged resduals deermnes he condonal varance). However, volaly ends o rse n response o bad news and fall n response o good news (Nelson, 99). The second dsadvanage of GARCH models s he non-negavy consrans mposed on he parameers whch are ofen volaed by esmaed parameers (Curo, 00)..5. GARCH-n-Mean (GARCH-M) model The GARCH-M model was developed by Engle e al., (987) whose key posulae was ha me varyng prema on dfferen erm nsrumens can be modelled as rsk prema where he rsk s due o unancpaed neres raes and s measured by he condonal varance of he one perod holdng yeld. The GARCH (,)-M model s presened by, x = y β + h γ + ε.5.9 where x and h are defned as before whle y s a vecor of addonal explanaory varables. The resdual ε can be decomposed as n equaon.5.4. Jus lke he GARCH model, he GARCH-M s unable o capure asymmerc characerscs of fnancal daa. 7

30 .5.3 Exponenal GARCH (EGARCH) model EGARCH models were nroduced by Nelson (99) n an aemp o address he wo major lmaons of he GARCH models. Here he volaly depends no only on he magnude of he shock bu also on her correspondng sgns. The non-negavy resrcons are no mposed as n he case of GARCH snce he EGARCH model descrbes he logarhm of he condonal varance whch wll always be posve. The specfcaon for he condonal varance (Nelson, 99) s gven as, q p p ε ε Logσ = ϖ 0 + β log σ + α + γ.5.0 σ σ = Noe ha ε = η σ where η ~..d N(0,). = The parameer ( α ) n equaon (.5.0) measures he mpac of nnovaon on volaly a me whle parameer ( β ) s he auo-regressve erm on lagged condonal volaly, reflecng he wegh gven o prevous perod s condonal volaly. I measures he perssence of shocks o he condonal varance. The saonary requremen s ha he roos of he auo-regressve polynomal le ousde he un crcle. For EGARCH (,) hs ranslaes no β < (Ogum e al., 006). Unlke he lnear GARCH, n he EGARCH model a negave shock can have a dfferen mpac compared o a posve shock f he asymmery parameer γ s non-zero. =.5.4 Quadrac GARCH (QGARCH) model The QGARCH model was nroduced by Senana (995). The model can be nerpreed as a second order Taylor approxmaon o he unknown condonal varance funcon and hence s called a quadrac GARCH. The model of order p=q= s as follows, h = + ξε + αε + βh γ.5. where ε = η h, η ~..d N(0,) and γ, ξ, α and β are parameers o be esmaed. In hs model, f ε > 0, s mpac on h s greaer han n he case f ε < 0 (assumng γ, ξ, α and β are posve). Thus capures he asymmerc effecs from anoher pon of vew. The saonary n QGARCH s covarance based whenever he sum of α and β s less han one. Ths sum also provdes a measure of perssence of shocks o he varance process. 8

31 .5.5. Threshold GARCH (TGARCH) model Threshold GARCH models were nroduced by Zakoan (994). The generalzed specfcaon of he condonal varance equaon s gven by, h q p r = 0 + β jh j + αε + γ kε k I k j= = k= α.5. where ε = η h and I =, f ε < 0 and zero oherwse. In hs model, good news, ε > 0, and bad news ε < 0, have dfferenal effecs on he condonal varance. Good news has an mpac of α, whle bad news has an mpac of α + γ. If γ > 0, bad news ncreases volaly whle f γ 0, he news mpac s asymmerc. When he hreshold erm equaon.5. becomes a GARCH (p,q) model..5.6 Glosen, Jagannahan and Runkle (GJR) model I k s se o zero, hen Ths s a modfed GARCH-M model developed by Glosen e al., (993). The GJR model allows posve and negave nnovaons o reurns o have dfferen mpacs on he condonal varance. Ths s acheved by he nroducon of a dummy varable no he condonal varance equaon. The GJR-GARCH(,) model s gven by, h = 0 + αε + ϖs ε + βh where ε = η h, η ~..d (0,) and α.5.3 value one when ε 0 and zero oherwse. S denoes an ndcaor (dummy) funcon ha akes he.5.7 Parameer Esmaon of ARCH models In hs sudy, he focus s on he Maxmum lkelhood approach whch s he mos commonly used esmaon procedure for he ARCH models. Ths s based on he normaly assumpon of he condonal dsrbuon. Followng Bera and Hggns (993), consder he sandard ARCH- regresson model y ψ ~ N x ξ, h ) wh s log lkelhood funcon s gven by, T l = / ( ε l( θ ) = ( θ ),where l ( θ ) = C log( h ) and θ = ( ξ, γ ). Here ξ and γ denoe he T h condonal mean and condonal varance parameers respecvely. One aracve feaure of hs normal lkelhood funcon s ha he nformaon marx s block dagonal beween he 9

32 0 parameers ξ andγ. Now, he (,j) h elemen of he off-dagonal block of he nformaon can be wren as = = = T j T T j h h h E T l E T γ ξ γ ξ If h s an asymmerc funcon of he lagged errors n he sense of Engle (98), hen he las expresson n he square brackes s an-symmerc and herefore has expecaon zero. When he block s dagonal, under he lkelhood frame work, esmaon and esng for he mean and varance parameers can be carred ou separaely (Engle, 98; Bollerslev, 986; Bera and Hggns, 993; Fan and Yao, 003; Davdson, 008). Mos of he appled work on ARCH models uses he Bernd e al, (974) algorhm o maxmze he log lkelhood funcon [ ) (θ l ]. Sarng from esmaes of he r h eraon, he (r+) h sep of he algorhm can be wren as = = + + = T T r r l l l ) ( ) ( ξ ξ ξ ξ ξ and = = + + = T T r r l l l ) ( ) ( γ γ γ γ γ where dervaves are evaluaed a (r) ξ and (r) γ. When ε s Suden s -dsrbued wh v> degrees of freedom, he creron funcon maxmzed s gven by; = Γ + Γ = T h v v h v T v v T l ) ( )log ( log ) ( log ) / ( ) ) / (( log ) ( ε π θ.5.4 where > v conrols he al behavour. The suden s dsrbuon approaches normaly as v. To mprove numercal sably, he parameer esmaed s v (Davdson, 008). For he GED,.5.4 can be wren as / 3 ) (/ ) )( (3/ log ) / )( (3/ ) (/ log ) ( v v h X y v h v v v l Γ Γ Γ Γ = θ θ.5.5

33 where v > 0 conrols he al behavour. The GED corresponds o he Gaussan dsrbuon f v = and s lepokurc (fa aled) whenv <..5.8 Model denfcaon for he ARCH-ype models The selecon of he approprae model s one of he mos challengng areas n sascal modellng usng ARCH models. Ths area has had very lle developmen. The pormaneau Q-es sascs based on he squared resduals s used o es for he ndependence of he seres (McLeod and L, 983). Ths Q-sascs s used o es he ARCH effecs presen n he resduals. Snce s calculaed from he squared resduals, can be used o denfy he order of he ARCH process. The Lagrange Mulpler es proposed by Engle (98) s also used n a smlar manner as he Q-sasc. The Akake nformaon creron (Akake, 974) and he Schwarz Bayesan creron (Schwarz, 978) model selecon mehods have wdely been used n he ARCH leraure, despe he fac ha her sascal properes n he ARCH conex are unknown. These selecon mehods are based on he maxmzed value of he log-lkelhood funcon and evaluae he ably of he models o descrbe he daa as dscussed n secon.3..6 Blnear Models (BL) Followng Granger and Anderson (978), Subba (98), Subba and Gabr (984), a me seres { X } s sad o follow a blnear me seres denoed by BL(p,q,m,k) f sasfes he equaon X p q + a X = c ε + bj X jε k + ε j j = j= 0 = j= m k.6. where { ε } s a sequence of..d random varable, usually bu no always wh zero mean and varance σ ε and c 0 =, a,b j and c j are model parameers. I s easy o see ha blnear model s a specal case of ARMA (p,q) model. Usng lag (Backshf) operaor, equaon.6. can be specfed as φ ) ( B) w = ( λ( B) ε ) ψ ( B) w + θ ( B ε.6.

34 where = X, ε ~ WN(0, σ ε ).6.3 d w ( B) In equaon.6., ψ ( B ) = ψ + ψ B p ψ B and k p λ ( B) = + λ B λk B.Ths s equvalen o he Subba (98) BL(p,q,m,k) class of models (Davdson, 008). Blnear models have been appled n geophyscs daa (Subba, 988), Spansh economc daa (Maravall, 983) and n solar physcs daa by (Subba and Gabr, 984 ). These models are parcularly aracve n modellng processes wh sample pahs of occasonal sharp spkes (Subba and Gabr, 984).These phenomena are found n fnancal me seres daa..6. Esmaon of parameers n blnear models Consder he BL(p,0,m,k) model of he form X m k + a X a p X p = α + bj X ε j + ε.6.4 = j= where { ε } ~N (0, σ ε ). (Here he, MA erms have been dropped and a consan α has been added o he R.H.S o faclae he fng of such models o non mean correced daa). The lkelhood funcon of he unknown parameers s consruced, gven N observaons, X X N. Snce he model nvolves lagged values of he { X } X,...,, one canno evaluae he resduals for he nal srech of daa. The condonal lkelhood based on X X,..., X gven X,...,, X X N where = max( p, m, k) γ s hus consdered. γ +, γ + N, Le θ = θ, θ,..., θ ) denoe he complee se of parameers {a }, {b j },α.e. se ( n θ = a, =,,..., p, θ p + = b, θ p+ = b,..., θ p+ mk = bmk, θ p+ mk + = α and wre n=p+mk+ o denoe he oal number of parameers. The jon probably densy funcon of ε γ +, ε γ +,..., ε N s gven by ( N ( N γ ) / ) exp{ } ε πσ σ ε ε = γ +.6.5

35 and snce he Jacoban of he ransformaon from { X } o { ε } s uny, equaon.6.5 also represens he lkelhood funcon of θ, gven { X ; = γ +,..., N}. The (condonal) maxmum lkelhood esmaes of mnmzng θ,...,, θ θ n are hus gven by maxmzng (.6.5) or equvalenly by N Q (θ) = ε.6.6 = γ + The mnmzaon s performed numercally: for a gven se of values θ, θ,..., θ ) hen ε } s ( n evaluaed recursvely from equaon.6. and hen he Newon-Raphson mehod s used o mnmze Q(θ) ( Subba, 98). The Newon-Raphson erave equaons for mnmzaon of Q (θ) are gven by, θ (+) = θ () - H - (θ () )G(θ () ).6.7 where θ () s he vecor of parameer esmaes obaned a he h eraon, and graden vecor G and Hessan marx H are gven respecvely as, Q Q Q θ,,..., and H(θ)=. The paral dervaves of Q wh respec o θ θ θ n θ θ j N Q ε {θ } are shown o be = ε, =,,,n θ γ θ = + { Q θ θ j = ε ε ε N N + = γ + θ θ j = γ + θθ j,,j=,,,n.6.8 Subba and Gabr (984) developed a nea se of recursve equaons for hese dervaves as follows. Dfferenang equaon (.6.) wh respec o each of he parameers he followng are obaned ε a ε a ε a + φ( a ) = X j, =,,,p + φ( b ) = X ε, =,,,m j=,,,k + φ( α) =,where φ ( θ ) = m k = j= b X j ε θ 3

36 4 Assumng he nal condons = 0 = θ ε ε =,,,γ =,,,n he second order dervaves sasfy 0 = a a ε, 0 = α ε a,,j =,,p j j j a X b a b a = + ), ( ε ψ ε, j j j j j j j j b X b X b b b b = + ε ε ψ ε ), ( α ε α ψ α ε = + j j j j X b b ), (, 0 = α ε where l r m k j j l r X b θ θ ε θ θ ψ = = = ), (. For a gven se of parameer valuesα,{a }, {b j } he frs and he second dervaves of Q can be evaluaed from he above equaons and hence he vecor G and marx H evaluaed. The eraon equaon.6.7 s hen mplemened. When he fnal parameer esmae θˆ have been obaned, e σ s obaned as, + = = = N N Q N ˆ ) ( ˆ) ( ) ( ˆ γ ε ε γ θ γ σ Leas squares esmaon of model parameers for blnear models Followng he approach of Tong (990), consder he blnear model (equaon.6.). Rewrng n a Markovan represenaon wh monor changes n noaon. Se p=max(p,q,m,k). ξ =(A+Bε )ξ - +cε +d( ε σ ) X =Hξ - +ε.6.0

37 where ξ s a p-vecor and a a A= M a p a p 0 M M 0 0 L0 L 0 M L L0 b B= M b p 0 0 L L 0 M 0 a + c b c= M d= M H=[ 0 L 0] a p + c p b p By convenon, c j =0 f j>q, b jk =0 f j>m. Dagonaly of he model mples he relaon B=dH. The converse s no rue. Followng Guegan and Pham (989), ake θ=(a,,a p, c,,c p, b,,b p ) as he fundamenal parameer vecor and assume ha he represenaon n equaon (.6.6) s quasmnmal n he sense ha here s no oher Markovan represenaon wh he same nose srucure bu wh a sae vecor whch s a lnear ransformaon of he orgnal sae vecor and has a smaller dmenson. Furher, assume ha he model s nverble and saonary. Le (X,,X N ) denoe he observaons. I s plausble o esmae θ by mnmzng N ~ ε ( θ / ξ0 ) w.r. ~ ~ ~ θ Θ N where ε ( θ / ξ0 ) s he ε ( θ / ξ0 ) gven by ε ( θ / ξ0 ) = X -Hξ - (ξ 0 ) = wh ~ ~ θ as he parameer vecor. Inuvely, for hs o make sense he effec of ξ 0 on θ / ξ ) should dmnsh as ε ( 0 (see Tong,990). Le here exs a saonary me seres { ( ~ θ )} such ~ ~ ~ ha ε θ / ξ ) - ε (θ ) 0 as. Noe ha (θ ) s hen measurable w.r. σ -algebra ( 0 generaed by X s, s. Here, he model (.6.6) s sad o be nverble a ~ θ relave o he observaon {X }. A suffcen condon for hs s ~ ~ ~ ~ ~ [ E (ln A ch dhx )] < 0, where A ~, c ~, d ~ and H ~ are gven by θ. θ φ ε ε A reasonable choce of Θ N, δ Θ N s suggesed by he above suffcen condon and gven as ~ = { θ Θ 0 : = ~ ~ ~ ~ ~ A ch dhx φ < ( δ ) N }.6. 5