Factor-augmented GARCH with structural breaks for detecting breaks and forecasting VaR

Transcription

1 The 33 rd Inernaonal Symposum on Forecasng Facor-augmened GARCH wh srucural breas or deecng breas and orecasng VaR Jun-Seong Km Ch-Hyuc Jun Indusral and Managemen Engneerng POSTECH Jun-Seong Km

2 Fnanal daa are well nown o be subjec o srucural changes and have nernaonal comovemen acors among hemselves. For acor-augmened models exsng sudes usually x he number o acors and use me-varyng parameers o mae explanaon power hgher. However hey may no be able o accoun or comovemen and srucural changes. We propose a acor-augmened GARCH model wh srucural breas and presen an algorhm o model esmaon usng a Bayesan approach. To oban he number o acors rom daa and o esmae eecvely srucural breas or GARCH we employ a ully Bayesan probablsc PCA and Bayesan nerence usng a derenal evoluon MCMC mehod. Through he analyss o S&P 500 soc reurns usng soc daa o sx naons we evaluae he mpac o breas and comovemen. In addon we provde a reamen o accoun or he uncerany n he VaR predcons by garnerng emprcal poseror dsrbuon normaon o volaly n ou-o-sample. Keywords : Comovemen Brdge samplng Bayesan nerence Varaonal PC Derenal evoluon Change pon Regme swchng

3 Inroducon Bacground Proposed Mehod Expermenal Resul Concluson 3

4 GARCH Engle 98; Bollerslev 986. A smple and powerul way o model volalyvar. Too resrcve or long me seres due o breas n he volaly process. Srucural Brea One o he mos mporan challenges o modern emprcal macroeconomcs Evdence o parameer change n macroeconomc me seres Ang and Beaer 00; Soc and Wason 996; Andreou and Ghysels 009 Presence o srucural breas n predcve regresson models or asse reurns Pesaran and Tmmermann 00; Paye and Tmmermann 006; Rapach and Wohar 005; Leau and Van Neuwerburgh 008 Y = F β X + ε ε =σ z z ~.. d. N0 σ = c + αε + βσ 4

5 Mehodology or srucural brea /. Componen and smooh ranson Amado and Terasvra 008. Model wh me-varyng uncondonal varance Engle and Rangel Marov-swchng models Gray 996; Haas Mn and Paolella 004; Bauwens Premnger and Rombous 00; He and Maheu 00; Bauwens Duays De Bacer 0 5

6 Mehodology or srucural brea / A Bayesan approach would much beer su hs problem. Focus on change-pon model. <Inerence algorhm> The orward-bacward algorhm by Chb996 Canno be appled o models wh pah dependence Fruhwrh-Schnaer004. Sequenal Mone Carlo on-lne mehod He and Maheu00. O-lne MCMC mehods Lao008;Bauwens Premnger and Rombous 00; Bauwens Duays and Rombous 0; Bauwens Duays De Bacer 0 6

7 Inernaonal dynamcs/comovemen The exsence and mporance o common paerns n he nernaonal dynamcs o macroeconomc and soc ndces by common global dsurbances. Baglano and Morana 009 VaRValue a Rs A pon esmae o poenal nancal loss and a ceran degree o uncerany. Need o now dsrbuon o me-seres daa 7

8 Purpose o Research Marov-swchng GARCH models wh srucural breas Employ nernaonal dynamcs and comovemen usng acor Facor-augmened GARCH wh srucural breas or deecng breas and orecasng VaR more accuraely 8

9 DREAM Inerence algorhm The arcles ha nluence he presen paper are manly Bauwens Duays De Bacer 0 Braa and Vrug 008 Vrug er Braa Ds Robnson Hyman and Hgdon 009 When we sample poseror dsrbuon we can use he proposal o he Meropols algorhm auomacally generaed by DREAMDeRenal Adapave Evoluon Meropols algorhm Vrug er Braa Ds Robnson Hyman and Hgdon 009 I consders he brea daes as parameers o be esmaed and no anymore as hdden saes underlyng he daa generang process The proo o he convergence o hs algorhm o he saonary dsrbuon s n Bauwens Duays De Bacer 0 9

10 Varaonal PC/ Convenonally use PCAPrncpal Componen Analyss o nd acors n Mulvarae me seres daa. One o he cenral ssues n he use o PCA s ha o choosng he approprae number o reaned componens. Bshop 999 proposed probablsc modelng and Bayesan approach varaonal EM nerence algorhm. σ y α μ W T 0

11 Varaonal PC/ Fnd a soluon numercally by sarng wh a suable nal guess or he dsrbuons and hen cyclng hrough he groups o varables n run reesmang each dsrbuon based on varaonal EM algorhm. Each hyperparameer α conrols he nverse varance o he correpondng acor loadng W so ha a parcular α has a poseror dsrbuon concenraed a large values we do no use he correspondng acor. σ y α μ W T

12 Facor-augmened GARCH model/3 Selec he number o srucural breas : K Selec he lag o GARCH Assume ha a seres do no aec he ohers abou mean or vol. excep eec o acors. Assume ha a acor s ndependen o he oher acors. Assume ha all seres have same brea daes. For smplcy acor equaon o each regme has same lag o GARCH Each regme can have deren number o acors 0... ~ AR- GARCH regme acor or a a Facor Equaon o c N d wh '... ' ~ - Facor - AR- GARCH - Pon Change A Seres M an Equaon o T K K K F c c N d wh y y

13 Facor-augmened GARCH model/3 Noaon Θ : GARCH parameer Γ : brea daes F : acors Κ : acor-parameers '... ' ~ - Facor - AR - GARCH Change - Pon A Seres Man Equaon o T K K K F c c N d wh y y 0... ~ AR - GARCH regme acor or a a Facor Equaon o c N d wh

14 Facor-augmened GARCH model3/3 Modelng brea srucure or orecasng brea as AR duraon ξ = τ τ = K + τ 0 = τ K+ = T ξ = round[ν + ν ξ + ε ] ε ~N0 σ σ = T V = ν ν σ s xed snce usually we do no have a large sample o duraons 4

15 Samplng poseror dsrbuon Samplng Inal value Samplng GARCH parameer Θ rom PΘ Y T ΓFΚV DREAM Samplng brea daes Γ rom PΓ Y T ΘFΚV Dscree-DREAM Samplng acors F and acor-parameers Κ rom PFΚ Y T ΘΓV Poseror Mean o Varaonal PC Samplng brea-parameers V rom PV Y T ΘFΚΓ Poseror Mean Model selecon 5

16 Samplng Inal value GARCH Samplng Inal value N : The number o MCMC chans N Marov chans are launched Brea daes. Randomly generaed N vecors o brea daes. Ge GARCH parameer samples or each chan rom pror dsrbuon 3. Ge F rom poseror mean o Varaonal PC 4. Ge MLE GARCH parameers and acor-parameers 5. Repea 3-4 seps unl ermnave condon Facors and acor-parameers Brea-parameer 6

17 Samplng Inal value GARCH Brea daes Facors and acor-parameers Brea-parameer Samplng GARCH parameer DREAM/PΘ Y T Γ FΚV/. Propose a new Θ + = N chan = MCMC eraon denoed by Z Z = Θ + γ δd δ g= δ Θ r g h= Θ r h + ε ε~n0η Θ I Wh gh = δ r g r h r g r h g = h.. Replace each componen o Z one by one by her old value n Θ accordng o a cross-over probably CR. d s he number o cross-overs. γ δd =.38 Barra and Vrug 008 δd 3. Accep he proposed Z accordng o he accepance probably rao o he Meropols algorhm α Z Θ = mn{ P Z Y T ΓF Κ P Θ Y T Γ F Κ } by P Θ Y T ΓFΚV PY T ΘΓFΚ PΘ 7

18 Samplng Inal value Samplng GARCH parameer DREAM/PΘ Y T Γ FΚV/ Snce he suppor o he proposal s no consraned we map he consraned space o GARCH parameers no he real space beore applyng he DREAM algorhm. x R + ln x R GARCH x [0 ln x/ x R Brea daes Facors and acor-parameers Brea-parameer The Bayesan ramewor generally allows us o sp saonary condon bu we pu consrans on sampled parameers abou saonary condon n whole algorhm wh ny rcs. 8

19 Samplng Inal value GARCH Brea daes Facors and acor-parameers Brea-parameer Samplng brea daesdscree-dream/pγ Y T Θ F Κ V. Propose a new Γ + = N chan = MCMC eraon denoed by Z Z = Γ + round[γ δd δ g= δ Γ r g h= Γ r h + ε] ε~n0η Γ I Wh gh = δ r g r h r g r h g = h.. Replace each componen o Z one by one by her old value n Γ accordng o a crossover probably CR. d s he number o cross-overs. γ δd =.38 Barra and Vrug 008 δd 3. Accep he proposed Z accordng o he accepance probably rao o he Meropols algorhm α Z Γ = mn{ P Z Y T Θ F Κ P Γ Y T ΘFΚ } by P Γ Y T ΘFΚV PY T ΘΓFΚ PΓ V 9

20 Samplng Inal value GARCH Samplng acors and acor-parameers /4 Poseror mean o Varaonal PC/PF Κ Y T Θ ΓV PFΚ Y T ΘΓV=P FΚ Y T ΘΓ Brea daes P FΚ Y T ΘΓ = P Κ Y T ΘΓF P F Y T ΘΓ Facors and acor-parameers. Ge only F approxmaly rom he poseror mean o Varaonal PC P F Y T ΘΓ Assumpon o ge approxmae F: approxmaed dsrbuons can be acorzed do no consder heeroscedascy ge sample as he poseror mean.. Ge Κ rom P Κ Y T ΘΓF as he poseror meanmle by unorm pror. Brea-parameer 0

21 Samplng acors and acor-parameers /4 Poseror mean o Varaonal PC/PF Κ Y T Θ ΓV PFΚ Y T ΘΓV=P FΚ Y T ΘΓ. Ge only F rom he poseror mean o Varaonal PC P FΚ Y T ΘΓ Probablsc modelng Samplng Inal value GARCH Brea daes Facors and acor-parameers Brea-parameer σ y α W μ T z F v Y L D F Y 0 } 0.5 exp{ / where 0 / d c P b a P I N P w P W Y L D Y y I Wz y N z y P I z N z P m d m d m

22 Samplng Inal value GARCH Brea daes Facors and acor-parameers Brea-parameer Samplng acors and acor-parameers 3/4 Poseror mean o Varaonal PC/PF Κ Y T Θ ΓV PFΚ Y T ΘΓV=P FΚ Y T ΘΓ. Ge only F rom he poseror mean o Varaonal PC P FΚ Y T ΘΓ Varaonal EM nerence only assume q Z W q Z q W q q q by ln q j E j j [ln P Y ] cons. nally ge E[ z]: poseror mean σ y α μ W T

23 Samplng acors and acor-parameers 4/4 Varaonal EM nerence 3 Samplng Inal value GARCH Brea daes Facors and acor-parameers Brea-parameer EM varaonal b a q b a q m w N W q m N q m z N z q z F m d w w N n z n z n } ' ' ' ' ' { / / / / ' ' ' m W row o h : w n z n n n n n n N n n N n n n w N n n n w w N n n n z n z x z W x z W z z W W Tr x b b Nd a a w b b d a a z z dag x z m I N z W x m W W I x W where / / : ' ' : b a x x b a x b a x xx x x N use

24 Samplng Inal value GARCH Brea daes Facors and acor-parameers Brea-parameer Samplng brea-parameer Poseror mean /PV Y T Θ FΚΓ P V Y T ΘFΚΓ = PV Γ GeVrom P V Γ as he poseror meanmle by unorm pror 4

25 Model selecon Brdge samplng/ Ge approxmae margnal lelhoodmll o compare modelsk # o breas GARCH lag P Y T = PY T ΘΓFΚVPΘΓFΚVdΘdΓdFdΚdV I s hard o use all parameersespecally FΚV n Brdge samplng o ge MLL So assume ha samples o PΘΓFΚV Y T s consdered as samples o PΘΓ Y T Then ge samples o P ΘΓFΚV Y T = PΘΓ Y T PFΚ Y T ΘΓ P V Γ By samples poseror mean o varaonal PC poseror mean 5

26 Model selecon Brdge samplng/ Ge samples o P ΘΓFΚV Y T = PΘΓ Y T PFΚ Y T ΘΓ P V Γ By samples poseror mean o varaonal PC poseror mean q ΘΓFΚV s proposal dsrbuon p ΘΓFΚV s pror dsrbuon Funcon ΘΓFΚV = Meng and Wond 996 P ΘΓFΚV Y T +qθγfκv A = G G g = Θ g Γ g F g Κ g V g q Θ g Γ g F g Κ g V g A = G G g = Θ g Γ g F g Κ g V g P Y T Θ g Γ g F g Κ g V g p Θ g Γ g F g Κ g V g g s sampled rom PΘΓFΚV Y T g s sampled rom qθγfκv Then P Y T A A 6

27 Facor equaon Use MLE or acor poseror samples o each poseror mean regme Forecasng VaR The prce o he asse s p hen VaR φ = n p R F p φ where p depends on he dsrbuon o he reurn seres y. In each round o he MCMC mae a orecas o y T+ y T+h based on he random draws where T + h s nex expeced brea dae hen s emprcal poseror dsrbuon o he orecas. 7

28 Burn-n Replace ouler chans by he chan wh he hghes lelhood o ease convergence unl he specc number o burn-n MCMC sample. Ge he log-lelhoods o all chans a a gven one MCMC sample eraon. I some chan log-lelhood < Q3-Q3-Q Quanle deal wh as ouler. Gelman and Rubn 99 8

29 Pror dsrbuon and seng N # o chans: 0 MCMC samples : 500 use he acor hyperparameer α < 0 and x GARCH selec K # o breas and lag o AR. CR crossover probably : 0.5 δ = 3 η Θ = 0.00η Γ = n DREAM All hyperparameer pror o varaonal PC : 0.00 Broad pror Proposal dsrbuon o Brdge samplng : weghed mxure o ollowng dsrbuons same as Bauwens Duays De Bacer 0 AR/GARCH parameers Γ parameers Comp. # Wegh Dsrbuon Comp. # Wegh Dsrbuon 0.50 Nμ pm Σ 0.75 Nμ pm 0.0Σ 0.5 Nμ pm 0.5Σ 0.5 Nμ pm 4Σ Nμ pm 0Σ Nμ max 0Σ Nμ max Σ pm : poseror mean o samples max : a sample whch has maxmum lelhood value. 9

30 Smulaon seng Smulaon # o seres : 5 AR-GARCH K # o breas : 4 T # o obs. : 000 Smulaon parameer For seres M = Parameers Regme Regme Regme 3 Regme 4 Regme 5 μ c α β φar Brea daes For seres = 345 M = M

31 Smulaon seng For acor-parameer all regme are same Regme Regme Regme 3 Regme 4 Regme 5 # o acor κ κ For acor equaon M =M 3

32 Smulaon resul MLL # breas AR0- GARCH AR- GARCH For parameers o seres 0 smulaons; Focus on only seres Parameers Regme Regme Regme 3 Regme 4 Regme 5 μ c α β φar Brea daes [0.] [0.] [0.5] [0.7] [0.] 7774 [00] [0.9] [0.] [0.5] [0.7] [0.] 3983 [375] [0.9] 0.0. [0.] [0.5] [0.7] [0.7] 6568 [600] [0.9] [0.] [0.05] [0.7] [0.7] 8367 [850] [0.9] [0.8] [0.05] [0.9] [0.7] Average esmaed value o poseror meansandard devaon[rue values] - 3

33 Smulaon resul From he average esmaed value o poseror mean ge poseror mean o acor and acor-paramers by xng he ohers. Facor Regme Regme Regme 3 Regme 4 Regme 5 # o acor [] [] [] 3[] [] Duraon parameers ν 9. ν

34 Smulaon resul The proposed mehod selecs he rgh model. The esmaed values o parameers are close o he rue values. The sandard devaons can be smaller he number o smulaon become larger. The brea deecon s successul; The one sandard devaon nerval o a brea dae ncludes he rue brea dae. The proposed mehod canno nd exac acor value bu esmaed acor can ully explan acor pars. In addon he proposed mehod has good perormance o esmae he number o acor. Fnally usng duraon parameer acor equaon and esmaed values o GARCH parameers nex brea dae and nex predcon can be obaned. 34

35 Expermen on Soc reurn daa Monhly daalas spo value 307 observaons. S&P500 NASDAQ FTSE ALL SHARE NIKKEI 5 TOPIX S&P/TSX KOSPI EURO STOXX 8 seres daa o 6 naons From daasream. 35

36 Expermen on Soc reurn daa MLL # breas 3 AR0- GARCH AR- GARCH For parameers o S&P 500The oher resuls are n appendx Parameers Regme Regme Regme 3 μ c α β c α β uncondonal varance Brea daes

37 Expermen on Soc reurn daa Facor Regme Regme Regme 3 # o acor 3 3 Duraon parameers ν 8.50 ν

38 Expermen on Soc reurn daa The bes model s AR0-GARCH and he proposed mehod nds common srucural breas n and From uncondonal varance here exss a cycle relaed o volaly. The las brea corresponds o he begnnng o he recen subprme crss and he rs brea s relaed o he end o do-com bubble. Small number o observaon causes agaed resuls 38

39 Summary We nroduce a acor-augmened GARCH change-pon model wh srucural breas and nerence MCMC algorhm based on DREAM. We compare canddae models usng MLL obaned rom he Brdge samplng. We llusrae he ecency o he algorhm by means o smulaons n he vew o he number o acors and esmaon perormance o parameers and brea daes and apply o nancal me seres. We can apply poseror dsrbuon resul o seres o orecas VaR. Lmaon We can predc he dsrbuon o seresvar volaly and a nex brea dae usng acor-equaon and parameers o man-equaon. However he predcon o he dsrbuon and volaly s resrced o he days beore he nex brea dae. The nex expeced brea dae s based on a small se o observaons. 39

40 Fuure wor Compare o Regme Swchng modelrecurrence and o oher esmaon mehods o srucural brea GARCH. Equaon : Exenson o GARCH- EGARCH and o Facor Volaly modelapply acors o varance equaon 40

41 Andreou E. and Ghysels E. 008: Srucural breas n nancal me seres In T. G. Anderson R. A. Davs J. P. Kress T. Mosch Eds. Handboo o nancal me seres. Sprnger Berln. Amado C. and T. Terasvra 008: modellng Condonal and Uncondonal Heerosedascy wh Smoohly Tme-Varyng Srucure CREATES Research Papers Unversy o Aarhus. Ang A. and Beaer G. 00: Regme swches n neres raes Journal o Busness and Economc Sascs Baglano F.C. and C. Morana 009: Inernaonal macroeconomc dynamcs: a acor vecor auoregressve approach Economc Modellng Bauwens L. A. Premnger and J. Rombous 00: Theory and Inerence or a Marov-swchng GARCH Model Economercs Journal Bauwens L. A. Duays and J. V. K. Rombous 0: Margnal Lelhood or Marov-swchng and Change-Pon GARCH Models CORE dscusson paper 0/3. Bauwens L. B. De Bacer and A. Duays 0: Esmang and Forecasng Srucural Breas n Fnancal Tme Seres CORE dscusson paper 0/0. Bshop C. M. 999: Varaonal PCA In Proc. Nnh In. Con. on Arcal Neural Newors. ICANN Bollerslev T. 986: Generalzed Auoregressve Condonal Heerosedascy Journal o Economercs Engle R. 98: Auoregressve Condonal Heeroscedascy wh Esmaes o he Varance o Uned Kngdom Inlaon Economerca Engle R. and J. Rangel 008: The Splne-GARCH Model or Low-Frequency Volaly and s Global Macroeconomc Causes Rewewo Fnancal Sudes 87-. Gray S. 996: Modelng he Condonal Dsrbuon o Ineres Raes as a Regme-Swchng Process Journal o Fnancal Economcs Haas M. S. Mn and M. Paolella 004: Mxed Normal Condonal Heerosedascy Journal o Fnancal Economercs 50. He Z. and J. Maheu 00: Real Tme Deecon o Srucural Breas n GARCH Models Compuaonal Sascs and Daa Analyss Leau M. and S. Van Neuwerburgh 005: Reconclng he Reurn Predcably Evdence Worng Paper NYU. Pesaran M. H. and Tmmermann A. 00: Mare Tmng and Reurn Predcon under Model Insably Journal o Emprcal Fnance Rapach D. E. and Wohar M. E. 005: Srucural breas and predcve regresson models o aggregae U.S. soc reurns Manuscrp Unversy o Nebrasa-Omaha. Soc J. H. and Wason M. W. 996: Evdence on Srucural Insably n Macroeconomc Tme Seres Relaons Journal o Busness and Economc Sascs er Braa C. J. F. and J. A. Vrug 008: Derenal Evoluon Marov Chan Wh Snooer Updaer and Fewer Chans Sascs and Compung Tmmermann A. and Paye B. 006: "Insably o Reurn Predcon Models" Journal o Emprcal Fnance Vrug J. A. C. J. F. er Braa C. G. H. Ds B. A. Robnson J. M. Hyman and D. Hgdon 009: Accelerang Marov Chan Mone Carlo Smulaon by Derenal Evoluon wh Sel-Adapave Randomzed Subspace Samplng Inernaonal Journal o Nonlnear Scences and Numercal Smulaons