Bayesian Markov Regime-Switching Models for Cointegration

Transcription

1 Applied Mahemaics, 22, 3, hp://dxdoiorg/4236/am Published Online December 22 (hp://wwwscirporg/journal/am) Bayesian Markov Regime-Swiching Models for Coinegraion Kai Cui, Wenshan Cui 2 Deparmen of Saisical Science, Duke Universiy, Durham, USA 2 School of Science and Informaion, Qingdao Agriculural Universiy, Qingdao, China kc52@sadukeedu, wshcui@qaueducn Received Ocober, 22; revised November, 22; acceped November 8, 22 ABSTRACT This paper inroduces a Bayesian Markov regime-swiching model ha allows he coinegraion relaionship beween wo ime series o be swiched on and off over ime Unlike classical approaches for esing and modeling coinegraion, he Bayesian Markov swiching mehod allows for esimaion of he regime-specific model parameers via Markov Chain Mone Carlo and generaes more reliable esimaion Inference of regime swiching also provides imporan informaion for furher analysis and decision making Keywords: Coinegraion; Regime-Swiching; Bayesian; MCMC Inroducion Since he developmen of he concep of coinegraion [], here has been a rich lieraure on esing coinegraion and applying coinegraion approaches o real daa analysis One of he mos illusraive examples in pracice is he pair rading sraegy [2] The basic idea is ha: find wo securiies whose prices have been hisorically moving ogeher So when he spread beween hem widens, we shor he winner and buy he loser And if we believe ha he hisory would repea iself, prices will converge again and he arbirager will profi This moving-ogeher relaionship beween wo nonsaionary ime series is called coinegraion Mahemaically, if wo nonsaionary ime series U and V are coinegraed, hen here exiss a number called he coinegraion raio, such ha Y U V is saionary Alhough here have been many saisical sudies o find coinegraed ime series, here are sill many unsolved problems Firs of all, i is ofen hard simply o find coinegraion given a specific period of ime There are several saisical explanaions for failing o rejec he null of no coinegraion including he span of he daa se, srucural breaks [3] and he choice of es model [4] Secondly, here are few saisical decision-making rules afer idenifying candidae pairs Taking pair rading as an exmple, ypically, people simply use he decision rule ha hey open a long-shor posiion when he pair prices have diverged by a cerain amoun (eg wo sandard deviaions from he hisorical mean) and close he posiion when he prices have revered [5] This paper proposes he Bayesian Markov regimeswiching model ha allows he coinegraion relaionship beween wo ime series o be swiched on or off over ime via a discree-ime Markov process This is an improvemen o he radiional coinegraion ess considering ha he model flexibly allows local non-coinegraion raher han assuming global coinegraion over he whole period of ime By using a fully Bayesian models, uncerainy abou coinegraion raio is also incorporaed ino he model and inferred simulaneously wih all oher unknown quaniies Furhermore, inference of he hidden regime-swiching is also criical o decision making and furher generic analysis 2 Markov Regime-Swiching Models for Coinegraion Suppose we have wo nonsaionary ime series U and V wih inegraion order, and Y U V ( is known, ypically people propose a and hen es he saionary propery of Y ) If Y is saionary, hen we say ime series U and V are coinegraed To es for saionariy, he Engle-Grange mehod [6] ess he null hypohesis using he ADF uni roo es [5] based on he Error Correcion Model (EVM) wih lag order K (as compared o in which case i is saionary): i i k K Y Y Y () where is a consan, i s are auoregression coeffi- Copyrigh 22 SciRes

2 K CUI, W S CUI 893 N In comparison, he Markov regime-swiching model we proposed allows Y o swich beween coinegraed or non-coinegraed regimes in a Markovian manner, by inroducing he regime indicaor variable X, regime specific parameers and he Markov ransiion marix P X For he simpliciy of exposiion, we assume ha X,, wih X denoing ha ha Y is saionary (ie and U V are coinegraed) a ime and X meaning non-coinegraion Then he model can be wrien as: ciens and, X Y P X, N, p p I p p X Y K X k k X Y i where Y U V and hus Y U V is he Markov ransiion marix of X, wih pij Pr X j X i and iniial value X Clearly, when X he model reduces o model () wih negaive, while X specifies uni roo process for Y and hus no coinegraion exiss for ime series U and V By obaining inference of he underlining regimes X, regime-specific parameers and segmenaion of regime-specific daa, he model provides much informaion for furher generic analysis and decision making 3 Bayesian Compuaion We propose o use Bayesian analysis for he inference of parameers and laen regimes X, where poserior samples of all unknown quaniies are drawn using Markov Chain Mone Carlo (MCMC) Under his model (2), he likelihood funcion is:, L X Y P X P Y X PY XPX X PX PY X X : X : P Y X P X X P Y X P X X Conjugae prior disribuions are placed on model parameers [7] Specifically, conjugae Dirichle priors are assigned o each row of he ransiion marix and P X (2) P X q q, q, where qi Pr X i Conjugae Normal-Gamma priors are assigned for all he regression X coefficiens k and he corresponding precisions X q Dir i i 2 ij 2, ; p Dir, ; i, j, X N, X v SS k ;, X 2 2 To obained he poserior marginal disribuions of he unknown parameers and he hidden regimes X, Gibbs sampler is consruced o ierae he following seps: ) Sample q and PX from full condiional disribuions:, ;, X 2 X qdir I I p p Dir I I p, p,, 2 X X X, X i i i i Dir I I, X, 2 X X, X i i i i 2) Sample he regression coefficiens and variance from Normal-Inverse Gamma full condiional disribuions given he conjugacy of he priors 3) Sample he whole pah of X Since X s are highly correlaed, Gibbs sampler consruced via regular full condiional disribuion would be exremely inefficien [7] To overcome his, Forward Filering and Backward Sampling algorihm is applied o draw block samples of X To achieve his, define,,, n i P Y Yn Xn i, hen by recursion: i qp Y X i and i ; i jp PY X i n n ji n n j Wih his, he resuls follow ha:, P X i i P X j X i j p By using his algorihm, a sample of X T is firs drawn from a bernoulli (mulinomial if X akes more han wo values) disribuion, and T, samples of X are drawn sequenially and backward from he condiional bernoulli disribuion, wih unil he whole j j p ji PX j X i, jp i X ime series are sampled j ji Copyrigh 22 SciRes

3 894 K CUI, W S CUI 4 Simulaed Time Series Analysis 4 Model Assessmen To esify he performance of he proposed framework, we simulaed a Markov regime-swiching imes series of lengh T 5, which swiches beween one saionary AR(2) process (Sae ) and one non-saionary AR(2) process (Sae ) The wo AR(2) models and he corresponding Error Correcion Models (ECM) are shown as follows: (he (non-)saionary propery can be easily esed by he Uni Roo Tes) Sae : y 3 6y 28 y2, ECM : y 3 68y 28 y ; (3) Sae: y 4 7y 3 y2, ECM : y 43 y where N,and, N The ransiion marix is specified as p p 8 2 PX p p 8 2 A simulaed daa was shown in Figure The proposed model was applied o he ime series o find regime swiching, wih he priors specified as follows: X k q Dir,, p, p Dir,, p, p Dir,, X N, ;, X To infer he value of X based on poserior samples, we use poserior probabiliy 5 as he cu-off poin Shown in Figure 2, he inferred regimes are compared wih he rue values, which shows ha our model gives good recovery of he laen regimes (wih he firs 2 ime poins shown) Oher model parameers are also correcly inferred as shown in Table, where poserior disribuions cover he rue values well 42 Poserior Decision Making The imporance of inference of regimes when analyzing (non)saionary ime series lies in he fac ha commonly-used saionariy and coinegraion ess (eg ADF uni roo es and Engle-Granger coinegraion es []) may well give misleading resuls when regime swiching Value Ture Saes (: Saionary; : Non-Saionary) Inferrd Hidden Saes Time Figure 2 Inferred regimes X (in green) compared o he rue values (in blue) show good inference Table Poserior esimaes of model parameers compared o he rue values The parameers are defined as in model 2 and specified in (3) Parameer Mean STD Truh Time Figure Illuraion of a ime series simulaed by he markov swiching model Copyrigh 22 SciRes

4 K CUI, W S CUI 895 exiss in he process For illusraion, a quick ADF es of he previously simulaed daa concludes ha he null hypohesis wih uni roo is rejeced a 999% confidence level, indicaing he imes series is saionary If his ime series were generaed by he linear combinaion of wo nonsaionary ime series, hen he ADF es ells ha hese wo are co-inegraed, which is clearly wrong In he following par, we will use he conex of pair rading o illusrae how he Markov regime-swiching model can poenially help improve decision making in pracice Basically people do pair rading based on he radiional rule ha you open a long-shor posiion when he pair prices have diverged by more han wo hisorical sandard deviaions And you unwind he posiion when i reurns o hisorical mean Firs of all, he model clearly allows more reasonable esimaion of he hisorical mean and sandard deviaion, based soly on daa in he saionary (coinegraed) regimes, raher han including daa in he nonsaionary (non-coinegraed) regimes This difference can be observed in Figure 3, where he hisorical mean using daa in he saionary regime is differen from ha using all daa, and he sandard deviaion is also smaller Secondly, he idenificaion of saionary (coinegraed) and nonsaionary (non-coinegraed) regimes also help esablish more raional decision making rules, which should be: we open a posiion when i is boh in he saionary sae and has diverged from he hisorical saionary mean I is apparenly risky eiher o open a posiion when currenly we are in a non-saionary sae or he hisorical mean calculaion involves non-saionary daa Since people care much abou he ime poins where values are a leas 2 sandard deviaions away from he hisorical mean, he figure shows ha he we pick differen ime poins using our model and decision making rules from hose obained using all hisorical daa and radiional rules, which we believe are more reasonable choices For example, many spikes in Figure 3 are acually no good ime poins o open he posiion based on our Markov regime-swiching model simply because hose spikes are in he non-saionary (non-coinegraed) regime However in comparison, he radiional approach considers hem open posiions whenever he values are 2 sandard deviaions away from he mean, which is a very risky decision no considering he regimes 5 Coinegraed Price Series Analysis An possible example of a pair of coinegraed ime series is he gold ETF, GLD versus he gold miners ETF, GDX GLD reflecs he spo price of gold, and GDX is a baske of gold-mining socks I makes inuiive sense ha heir Figure 3 Resuls comparison beween our Bayesian Markov regime-swiching model and radiional coinegraion es and analysis using all hisorical daa Red lines indicae he mean and mean ±2SD using all hisorical daa, which is a radiional way afer you have done he ADF es o show he saionary propery; Green lines indicae hose using only hisorical daa in saionary regimes Red and green dos mark he ime poins where values a hose poins are a leas 2SD away from he hisorical mean based on radiional and our Markov regime swiching model respecively Copyrigh 22 SciRes

5 896 K CUI, W S CUI Figure 4 Disribuion of he probabiliy ha X is in he coinegraion regime ( =,,T) prices may move in andem Previous sudy via he wosep Engle-Granger mehod [] idenified ha a porfolio wih long share of GLD and shor 6766 share of GDX is likely a saionary ime series, wih lag bu he conclusion is laer quesioned by oher sudies [8] To es he possible co-inegraion, he wo-sae Markov regime swiching model is applied o he 5/23/6-/3/7 GLD and GDX ime series A hisogram shown he disribuion of he probabiliy of he ime poins being in he coinegraion sae is shown in Figure 4 According o he previous 5 cu-off poin, he Markov regime swiching model indicaes ha a mos of he ime, he wo ime series are no coinegraed wih he 6766 coinegraion raio This may serve as anoher counerexample (ogeher wih he simulaion resul) ha he widely-used ADF es migh provide misleading resuls when used o es co-inegraion regardless of possible regime swiching 6 Conclusions and Fuure Work In his sudy, we proposed o use he Bayesian Markov regime-swiching model as a flexible model for coinegraion and saionariy analysis, where he laen regimeswiching process is modeled via a Markov process A srong message of his sudy is ha, while idenifying coinegraion (or saionariy) is ofen hard globally, allowing local non-coinegraion (or non-saionariy) and inferring he regime swiching can provide much informaion for furher analysis and decision making Several exensions of he sudy are sill worh exploring, including relaxing he hidden Markov ransiion models and incorporaing uncerainy abou number of re- gimes in he model Hidden semi-markov models are naural exensions of hidden Markov models While he runlengh disribuion of he hidden Markov models impliciy follows a geomeric disribuion, hidden semi-markov models allow for more general runlengh disribuions, and hus are more flexible o describe he ime spend in a given regime As for he cases wih he number of regimes unknown, Bayesian inference hrough reversible jump MCMC mehods [9] could be a viable alernaive ha boh explores models wih differen number of regimes and esimaion of regime-specific parameers REFERENCES [] R F Engle and C W J Granger, Co-inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica, Vol 55, No 2, 987, pp doi:237/93236 [2] H Puspaningrum, Pairs Trading Using Coinegraion Approach, PhD Thesis, Universiy of Wollongong, Wollongong, 22 [3] J Campos and N R Ericsson and D F Hendry, Coinegraion Tess in he Presence of Srucural Breaks, Inernaional Finance Discussion Papers 44, Board of Governors of he Federal Reserve Sysem (US), 996 [4] E G Gaev and W Goezmann and K G Rouwenhors, Pairs Trading: Performance of a Relaive Value Arbirage Rule, Boson College, Boson, 26 [5] A W Gregory and B E Hansen, Residual-Based Tess for Coinegraion in Models wih Regime Shifs, Journal of Economerics, Vol 7, No, 996, pp doi:6/34-476(69) [6] D A Dickey and W A Fuller, Disribuion of he Esi- Copyrigh 22 SciRes

6 K CUI, W S CUI 897 maors for Auoregressive Time Series Wih a Uni Roo, Journal of he American Saisical Associaion, Vol 74, No 366, 979, pp doi:237/ [7] G E B Archer and D M Tieringon, Parameer Esimaion for Hidden Markov Chains, Journal of Saisical Planning and Inference, Vol 8, No, 22, pp doi:6/s (2)38-x [8] E P Chan, Quaniaive Trading, John Wiley and Sons, Hoboken, 28 [9] C P Rober, T Ryden and D M Tieringon, Bayesian Inference in Hidden Markov Models hrough he Reversible Jump Markov Chain Mone Carlo Mehod, Journal of The Royal Saisical Sociey Series B, Vol 62, No, 2, pp doi:/ Copyrigh 22 SciRes