An Investment Strategy Based on Stochastic Unit Root Models

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1 Inernaional Journal of Economics and Finance; Vol. 5, No. 3; 03 ISSN 96-97X E-ISSN Published by Canadian Cener of Science and Educaion An Invesmen Sraegy Based on Sochasic Uni Roo Models Mamadou A. Koné Universié Gason Berger, UFR Sciences Economiques e de Gesion, BP 34 Sain Louis, Sénégal Correspondence: Mamadou A. Koné, Universié Gason Berger, UFR Sciences Economiques e de Gesion, BP 34 Sain Louis, Sénégal. el: mamadou-abdoulaye.kone@ugb.edu.sn or konedoudou@yahoo.fr Received: Sepember 9, 0 Acceped: January 7, 03 Online Published: February 6, 03 doi:0.5539/ijef.v5n3p URL: hp://dx.doi.org/0.5539/ijef.v5n3p Absrac An algorihm is presened ha locally approximaes he nonlineariy of sochasic uni roo (SUR) models by n linear models. he previous ineger n is chosen so ha he Hadamard marix of order n can be defined. he sraegy SUR(n), hen consiss in creaing n linear models from his Hadamard marix and aking heir average forecas. A purchase (sell) signal is made if he obained average forecas is posiive (negaive). Subsequenly, a comparison is made wih respec o compeing models (Moving average sraegies) o assess heir abiliy o forecas he variaion of five inernaional indexes. I is found, afer aking accoun ransacion coss, ha SUR(n) generaes generally he highes profiabiliy in he ou-of-sample daa. Keywords: forecasing, rading rules, random coefficien auoregressive models, efficiency marke hypohesis. Inroducion he quesion of Efficiency Marke Hypohesis (EMH) has been sudied for many years by boh academics and marke paricipans. he aim is o see if he assumpions of marke fricionless and raders raionaliy are a good descripion of real markes where microsrucure (ransacion coss, informaion asymmery, ec.) and noise raders are presen. his is an ongoing debae and here has been no consensus. ha is why some auhors have ried o reconcile he EMH and Behavioral finance argumens hrough dynamic sysems, see for example Lo (005) and Koné (00). he empirical sudies of his hypohesis are based generally on hree classes. he firs is radiional regression models. heir aim is o es he validiy of he EMH in is weak form hrough radiional ime series forecas such as Auo Regressive Moving average models ARMA(p,q). If he marke is supposed o be a nonlinear dynamic sysem, one may consider nonlinear models such as he Random Coefficien Auoregressive RCA(p) or regime swiching models among ohers. he radiional regression models also conain analysis ools based on firms fundamenal (dividend, Book-o-Marke, ec.). In his case, he objecive is o es he EMH in is semi-srong form (fundamenal analysis). We refer o Ou and Penman (989) and references herein. he second class uses echnical Analysis ools such as Moving Average, Suppor and Resisance mehods. his approach is widely applied by raders o deec rends or reversal effecs by using informaion such ha prices, rading volume, ec. Here, he validiy of EMH is esed hrough is weak form, see for example Sullivan e al., (999). he las class, based on Machine Learning (Geneic Algorihms, Neural Neworks mehods) invesigaes he EMH in is weak and semi-srong form as for he class of radiional regression models. heir difference is ha Machine learning models are self-adapive mehods in ha here are few a priori assumpions abou he relaionship beween inpus while he radiional regression models make srong assumpions (parameric approach). he paper belongs o he firs class where he nonlineariy of financial asse prices is modeled by RCA(p). his economeric model generaes he main sylized facs of financial ime series, see Yoon (003). I may be also relaed o an Agen Based Model wih a swiching phenomenon beween fundamenaliss and noise raders, see for example Koné (0). here are many mehods proposed in he lieraure o esimae is parameers for rading or forecas purpose. For example Nicholls and Quinn (98) employed he radiional leas squares and he maximum likelihood mehods, see also Granger and Swanson (997). Wang and Ghosh (00) use Bayesian approach while Sollis e al. (000) work wih Kalman filer. We follow here anoher approach consising o approximae he RCA() model by n simple linear models where n is any ineger such ha he Hadamard marix

2 Inernaional Journal of Economics and Finance Vol. 5, No. 3; 03 H of order n can be defined. he laer is an n n marix wih all is elemens being eiher or, and such ha HH =n*i n where H is he ranspose of H, I n is he ideniy marix of order n. herefore, he Hadamard marix columns is an orhogonal binary basis of R n explaining why i is widely used in physics paricularly in he field of signal ransmission. he ineger n, in his sudy, mus saisfy he consrain n, n/ or n/0 is a power of. he predicion is hen made by aking he average forecas of hese n linear models exraced from he Hadamarix since many researchers agree ha combining muliple forecass leads o increased accuracy, see Granger and Ramanahan (984). he paper conribues in wo ways o he lieraure. Firs, conrarily o oher forecasing mehods, he esimaion procedure is made locally o capure raders feedback or ineracion since he variance and oher higher momens of SUR model do no exis. Only n daa are used in he linear regression models where 8 n 50. his consrain gives us exacly heigh (08) sraegies SUR(n) wih n {8,, 6, 0, 4, 3, 40, 48}. he second conribuion shows an applicaion of he Hadamard basis o reduce he complexiy of a problem (from exponenial o linear) for rading purposes. he paper is divided ino four addiional secions. Secion presens our mehodology and is compeing sraegies o forecas he variaion of asse prices of five inernaional indexes (CAC 40, DAX 30, FSE 00, Nikkei 5, S&P 500). Secion 3 describes he daa and he mehodology used in he empirical applicaion. Secion 4 presens he empirical resuls and he las secion concludes.. Some Forecasing Rules. Our Mehodology Consider he following sochasic uni roo SUR() model defined by: b ) ) 0, y b ( b ) y () ), ), where (ε ) is an i.i.d Gaussian process and y =logs (log of asse prices). cov( b, ) 0 he properies of eq. (), o replicae financial imes series, have been sudied by (Yoon, 003). he economeric model is also relaed o an agen based model wih ineracion beween fundamenalis and noise raders, see (Koné, 0). I is a special case of he Random Coefficien Auoregressive RCA() model which is defined by: y = (φ+b )y + ε E ( b ) ) 0, b ), ), cov( b, ). (Nicholls and Quinn, 98) have shown ha he RCA() process (y ) is a finie second-order saionary momen if he condiion φ +ω < is saisfied. Since φ= in our case, he saionary condiion is violaed (Noe ). ha means convenional mehods based on his assumpion such as Maximum likelihood mehod canno perform, see Yoon (006) (Noe ). Consequenly, we propose a mehodology ha locally approximaes he nonlineariy of asse prices by n linear models. For his purpose, b is supposed o ake only wo values α and α a any ime. herefore, i may be rewrien as b =αx where for any, X = or X =. he equaion () becomes r log S logs y y c X y () where a consan c is added, as usual, in he regression model. For he momen, he esimaion canno be proceed because he variable X is no known. o circumven his problem, regressions models are used condiional on he pah of (X ). For example, in he equaion (), if i is decided o use n daa for he esimaion process, we will have n pahs for (X ), =,,n since X akes or a any ime. Each rajecory generaes a linear model wih hree inpu variables X, y and he consan variable c. herefore, he nonlineariy is approximaed by n linear models (Noe 3). his feaure comes a a cos, as we need o sore a binary marix of size n n o make all linear regressions (Noe 4). Generally, we need he parameer n o be big for esimaion precisions bu no oo much o keep a local approximaion. In he applicaion, i is aken 8 n 50. o solve he dimensionaliy problem, echniques similar o Componen Principal

3 Inernaional Journal of Economics and Finance Vol. 5, No. 3; 03 Analysis (CPA) in exploraory daa analysis are used. Namely, we exrac "n orhogonal linear models". Here, he orhogonaliy of wo models i and j is defined by he orhogonaliy of heir corresponding pahs (X i k ) and (X j k ), k=,,n. If n is consrained o be an ineger such ha n/, n/ or n/0 is equal o k, k N, hen Hadamard marices exis. For example for n=, he Hadamard marix is H ha is a basis of R. Recursively, we can define he marix H 4, H 8,, by using he following formula H n H H n n H H n n his approach allows o pass from exponenial ( n ) o linear (n) complexiy since now for n daa used in he equaion (), n linear models are also employed where heir pahs correspond o he columns of he Hadamard marix of order n. For each model, deermined by he pah of (X ), he parameers α and c are esimaed by he Ordinary Leas Square mehod. hen a forecas is made a ime + hrough he equaion rˆ yˆ y cˆ X y (3) A recursive regression is applied. A any ime, he previous n daa are used in he regression model o deermine he new esimaed parameers. We denoe by SUR (n) he sraegy ha consiss o ake he average forecass of all n "orhogonal linear models". he procedure o creae he buy and sell signals is hen simple: a buy (sell) signal is produced if he average forecas, denoed by r ˆa, is posiive (negaive). o reduce he number of ransacion coss, we enhance he sraegy by allowing saic posiions in he case where he forecas signal is no significan. In oher words, he following sraegy is applied for SUR (n). sign( ra ˆ ) posiion(, n) posiion(, n) if ra ˆ ) c; ˆ c (max( R ) min( R )) / 30 oherwise. where R =logs logs represens he index reurn a ime (Noe 6). he sign funcion is defined by sign(x)= if x>0, sign(x)= if x<0 and sign(0)=0.. Compeing rading Rules If he marke is supposed o be efficien, an opimal sraegy is o buy and hold an index. he sraegy B&H consiss herefore o be long on he index a any ime and consequenly here are no ransacion coss. We consider also simple and exponenial moving average sraegies ha have been widely used by raders o cap momenums or reversal effecs. he idea is o consider wo moving average series M(n,) and M(m,) wih differen lenghs n and m. If we denoe by (S ) he asse price process, he simple and exponenial moving average are defined, for a given lengh k>0, by respecively he equaions (4) and (5). i0 k M ( k, ) S (4) i k M ( k, ) ( ) M ( k, ) S, wih M ( k,0) S (5) If m<n hen M(m,) (resp. M(n,)) is called he shor-erm moving average (resp. he long-erm moving average). he decision rule for aking posiions is specified as follows. If he shor-erm moving average M(m,) inersecs he long-erm moving average M(n,) from below, a long posiion is aken. Conversely, if he M(n,) is inerseced from above, a shor posiion is aken. he moving average sraegies are implemened by using he Malab funcion movavg. Noe ha in hese sraegies, ransacion coss appear only when an inersecion appears beween M(m,) and M(n,). In he decision making process of radiional regression models, if a hreshold is no used, he number of ransacions may be very high. 3

4 Inernaional Journal of Economics and Finance Vol. 5, No. 3; Daa and Mehodology In his paper, we consider he daily closing prices of five inernaional indexes CAC 40, Dax 30, FSE 00, Nikkei 5 and S&P 500 obained from Yahoo Finance websie. All ime series have he same lengh of daa as shown by able. able. Inerval of sudy of he five inernaional indexes Index CAC 40 DAX 30 FSE 00 NIKKEI 5 SP 500 in-sample 30 Jun Aug May Jan Apr Feb Feb Jan Dec Jan 009 ou-of-sample 0 Feb Feb Feb Dec Feb Dec 0 30 Dec 0 30 Dec 0 30 Dec 0 30 Dec 0 oal daa All he series end o 30 December 0, oaling N=943 rading days. heir difference appears only on he beginning period where he laer is chosen so ha o have he same lengh of daa han he Nikkei index. Each daa is afer divided ino wo periods: he firs period (in-sample daa) conains 07 (0.75*N) rading days. he remaining daa (736 or approximaively 0.5*N) is reained for he second period (ou-of-sample daa). he use of many geographic zones (Asia, Europa, Unied Saes) is o es he robusness of he differen algorihms. he mehodology is he following. For each class of rading rules, here SUR(n), Simple Moving Average SMA(m,n) and Exponenial Moving Average EMA(m,n), a raining period (in-sample daa) is used o find is bes model in erms of he Sharpe Raio which is an economic gain adjused for risk. If we le R =logs logs, he log reurn of he index a ime, hen he Sharpe raio (SR) is defined for any sraegy, say k, by where RM SR( k) RM ( k) ( k) ( k) z, z posiion(, k) R 0, 5 ) [ ( z RM ( ) ) ] ( k k Here posiion(,k) akes ( ) if he sraegy k is long (shor) a ime and represens he number of predicions. he second par consiss o compare he performance of he bes in-sample models wih respec o he ou-of-sample daa. he comparison is based on many crieria such as he Sharpe Raio, he winning up periods (W.U.P), he winning down periods (W.D.P), he correc direcional changes (C.D.C) and he Maximum Drawdown (M.D). Le R ~ and R be respecively he daily rading profi and acual reurn a ime, Q { R and 0} F { he indicaor funcions for rise and fall a ime, and finally U ~ and R 0} { R 0, R 0} D ~ he indicaor funcions for winning up and winning down periods, hen he above expressions can { R 0, R 0} be defined by ~ c ~ ~ c c R R R M D R max( ~ R C D C ~ { 0} i,. min( i ),),.. (8),, i,, where i U.. 00, W. U. D 00 (9) Q F W U P ~ ~ is he cumulaive rading reurns up o ime (Noe 5). c R i Ri D (6) (7) 4

5 Inernaional Journal of Economics and Finance Vol. 5, No. 3; 03 Finally, we inegrae he ransacion coss in he analysis. Namely, i is supposed ha any ransacion implies a consan cos of 0 basis poins. 4. Resuls We recall ha he in-sample daa conains approximaively 9 years of daa for each index. he SUR(n) class, wih he consrain 8 n 50 and n, n/ or n/0 is a power of, conains heigh (08) admissible sraegies characerized by he ineger n valued in {8,, 6, 0, 4, 3, 40, 48}. he simple and exponenial moving average classes are paramerized by wo inegers m and n, represening respecively he lead and lag parameer. In his sudy, sixeen (6) sraegies are proposed for each Moving Average class wih heir parameers given by m {, 5, 0, 5,} and n {50, 00, 50, 00}. All hese algorihms need some iniial daa o sar he forecasing procedure. For example, he SUR(n) sraegy needs n+ daa o make he firs forecas. For hese iniial daa, he agen decision is supposed o be always. he cos of one ransacion is aken o be 0 basis poin i.e 0.%. able shows he performance of he bes sraegies in each class hrough he differen indexes and hrough heir respecive in-sample daa given in able. For he SUR class, he bes sraegy is given by he parameer n=6 for he CAC, NIKKEI and S&P indexes and by n=0 and 4 for he DAX and FSE indexes, respecively. Overall, i is seen for he SUR class, he approximaion needs o be local or o have less daa (n 4) o generae good resuls. For he Exponenial Moving Average class, he lag parameer of he bes sraegy is always equal o n=50 for he differen indexes and he lead parameer lies o he se {0, 5}. For he Simple Moving Average class, he lag parameer varies hrough indexes where he parameer n=50 is more frequen. he same remark applies also for he lead parameer m where he mode is given by m=5. We also remark ha for boh moving average classes, a small lead (m= or 5) does no give saisfacory in-sample resuls. All bes compeing models (SUR, EMA, SMA), in he in-sample evaluaion, generae economic gains or a posiive Sharpe Raio. Furhermore, excep in he FSE index, he opimal sraegy of he EMA class ouperforms he oher bes models. able. Sharpe raio of he bes rading rules in each class (In-sample) Class SUR(n ) EMA (m,n) SMA(m,n) Buy and Hold CAC n=6 m=0, n=50 m=5, n=50 Sharpe Raio ,37 Dax n=0 m=5, n=50 m=5, n=50 Sharpe Raio FSE n=4 m=5, n=50 m=5, n=50 Sharpe Raio Nikkei n=6 m=0, n=50 m=0, n=50 Raio S&P n=6 m=5, n=50 m=5, n=00 Sharpe Raio On he oher hand, he Buy and Hold Sraegy has a negaive mean in he in-sample daa of all geographical zones showing consequenly a negaive Sharpe raio. his may be explained by he fac ha all five indexes are highly correlaed and herefore he probabiliy o have he same sign performance in he five indexes is very high. Afer geing he bes sraegy in each class, we make a comparison beween hem. Namely, hree rading rules are invesigaed for each index in heir ou-of-sample daa given in able. he aim is o see if i is possible o do beer han he benchmark sraegy afer aking ino accoun ransacion coss. o reduce he chance feaure, a long ime series of ou-of-sample is considered as conaining around hree years of daa. he resuls are shown in he able 3 and able 4. 5

6 Inernaional Journal of Economics and Finance Vol. 5, No. 3; 03 able 3. Ou-of-sample performance of he bes rading rules in each class (Par I) CAC SUR(6) EMA (0,50) SMA(5,50) Buy and Hold Sharpe Raio ransacions M.D DAX 30 SUR(0) EMA (5,50) SMA(5,50) Buy and Hold Sharpe Raio ransacions M.D FSE 00 SUR(4) EMA (5,50) SMA(5,50) Buy and Hold Sharpe Raio ransacions M.D Nikkei 5 SUR(6) EMA (0,50) SMA(0,50) Buy and Hold Sharpe Raio ransacions M.D S&P 500 SUR(6) EMA (5,50) SMA(5,00) Buy and Hold Sharpe Raio ransacions M.D Descripion: his able presens he ou-of-sample values of he Sharpe raio, he number of ransacions and he Maximum Drawdown (M.D) for each bes sraegy. able 4. Ou-of-sample performance of he bes rading rules in each class (Par II) CAC 40 SUR(6) EMA (0,50) SMA(5,50) C.D.C 50.4% 47.83% 50.8% W.U.P 34.3% 5.8% 57.9% W.D.P 66.94% 4.70% 43.53% DAX 30 SUR(0) EMA (5,50) SMA(5,50) C.D.C 5.58% 5.90% 5.77% W.U.P 73.6% 77.% 76.6% W.D.P 9.39% 3.63 % 3.9% FSE 00 SUR(4) EMA (5,50) SMA(5,50) C.D.C 5.90% 50.7% 50.8% W.U.P 38.60% 63.73% 60.0 % W.D.P 66.57% 35.43% 40.57% Nikkei 5 SUR(6) EMA (0,50) SMA(0,50) C.D.C 5.90% 5.58% 5.36 % W.U.P 43.6% 48.6 % 5.3% W.D.P 6.4% 57.30% 5.40% S&P 500 SUR(6) EMA (5,50) SMA(5,00) C.D.C 50.8% 5.63% 53.53% W.U.P % 66.67% 65.% W.D.P 55.69% 3.6% 38.77% Descripion: his able presens he ou-of-sample values of correc direcional change (C.D.C), he winning up periods (W.U.P) and he Winning down periods (W.D.P) for each bes sraegy. able 3 shows ha for he Sharpe Raio crierion, he SUR sraegy gives overall he bes resuls, namely hree over he five indexes. hen i is followed by he B&H sraegy which performs wo imes over he five cases. he resuls of SMA and EMA rading rules are no saisfacory in he ou-of-sample daa. For he Maximum Drawdown (M.D) measure, i is found over all ha he wo bes sraegies are also given by he SUR class and he Buy and Hold Sraegy ( over 5 indexes for each). Precisely, he SUR rading rule obains he good resuls from he CAC and Nikkei indexes while B&H does beer in he FSE and S&P indexes. 6

7 Inernaional Journal of Economics and Finance Vol. 5, No. 3; 03 he more M.D, he lile is he downside risk. he cumulaive reurns of Figure illusraes boh he concep of Maximumm downside and profiabiliy (Sharpe Raio) over he ou-of-sample daaa for he differen indexes. I is noed ha he riskies sraegies (downside risk) are usually reached by SMA and EMA in four amongg five indexes. For he profiabiliy measure, i is also seen ha he SUR and B&H cumulaive reurns end a he op of he oher sraegies for he CAC, DAX, Nikkei and S&P indexes. he excepion appears only for he FSE F index where SMA gives some ineresing resuls. Now, we are ineresed in he percenage of correc direcional changes (C.D.C), and he percenage of correc direcional changes in he rise and fall periods (W.U.P and W.D.P) ), see he eqs. (8) and (9) and able 4. For he C.D.C crierion, he wo bes sraegies are given by SMA and SUR. he simple moving average rading rule gives beer resuls for he CAC and S& &P indexes while he SUR sraegy performs for DAXX and FSE. he EMA, wih a C.D.C of 5. 58% only ouperforms he oher rading rules for he Nikkei index. Bu, for he W.U.P crierion, i is EMA he bes rading rules in hree over he five indices and hen i is followed by SMA. However, in financial markes, invesors are more concern o deec falling periods. Consequenly, he W.D.P W measure will play a major role for hem. I is noed ha, for all five indexes, he SUR sraegy gives hee bes resuls and someimes he difference is very significan as in he case of CAC and S&P indexes. Figure. he cumulaive reurn pah of he differen rading rules for each index We can resume he analysis, by saying overall, he SUR sraegy ouperforms he moving average rading rules since i belongs o he bes sraegies for many crieria such as he Sharpe raio, he Maximum Drawdown, he Correc direcional change and he Winning down periods. Is poor resuls in he ou-of-sample only come from Winning Up period crierion. 5. Conclusion In his paper, he sochasic uni roo model is used o forecas he direcion change of asse prices. he conribuion consiss in approximaing locally he nonlineariies of financial ime seriess by simple linear models. For his purpose, wo simplificaions are made o faciliae he esimaion of he parameers. hese simplificaions are also relevan o reduce he execuion ime and he memory cos involved by he algorihm. he firs is o consrain he sochasicc parameer o ake, a any ime, wo poss daa used in he regression, he nonlineariies are approximaed by n sible values. Consequenly, for n linear models. he second simplificaion is o solve he dimensionaliy problem by exracing only he mos significan linear models. We follow hee idea of he prin of R n ncipal componen analysis (PCA) by aking n orhogonal binary vecors (columns of Hadamard marix) and hen associae o each vecor, a simple linear model. hen, he sraegy SUR(n) is defined by aking he average forecas of hese n models and he decision making is simply o buy (sell) if his average forecas is posiive (negaive). o diminish he ransacion coss for profiabiliy reasons, bu also o capure significan 7

8 Inernaional Journal of Economics and Finance Vol. 5, No. 3; 03 informaions, an endogenous hreshold c is used o acivae a decision. Namely, he rader will ransac if he average forecas reurn is superior in absolue value o he hreshold, elsewhere he previous posiion is conserved. I is found ha he sraegies from SUR class dominae overall he moving average rading rules (simple and exponenial) and also he Buy and Hold sraegy for he Sharpe crierion. hese ineresing resuls may be explained by wo facs. Firs, i is known ha random coefficien auoregressive models are able o fi well financial asse prices. So i is expeced o have saisfacory resuls when his economeric model is used for forecasing. he second reason is due o our esimaion procedure which is local and allows o capure feedback or ineracion of raders raher using mehods based on saionariy assumpions in variance or higher momens ha are violaed in our case of sochasic uni roo model. References Dunis, C. L., Williams, M. (003). Applicaions of Advanced Regression Analysis for rading and Invesmen. C. In L. Dunis, J. Laws & P. Naïm (Eds.), Applied Quaniaive Mehods for rading and Invesmen. John Wiley & Sons. Granger, C. W. J., & Ramanahan, R. (984). Improved Mehods of Combining Forecass. Journal of Forecasing, 3, hp://dx.doi.org/0.00/for Granger, C. W., & Swanson, N. R. (997). An inroducion o sochasic uni-roo processes. Journal of Economerics, 80, hp://dx.doi.org/0.06/s (96) Koné, M. A. (0). A link beween Random Coefficien AuoRegressive Models and some kind of Agen Based Models. Journal of Economic Ineracion and Coordinaion, 6(), hp://dx.doi.org/0.007/s Koné, M. A. (00). Behavioural Finance and Efficen Markes: Is he join hypohesis really he problem? IUP Journal of Behavioral Finance, 7, 9-9. Lo, A. (005). Reconciling Efficien Markes wih Behavioral Finance: he Adapive Markes Hypohesis. Journal of Invesmen Consuling, 7(), -44. Nicholls, D. F., & Quinn, B. G. (98). Random Coefficien Auoregressive Models: An Inroducion. Lecure Noes in Saisics,. New York: Springer-Verlag. hp://dx.doi.org/0.007/ Nicholls, D., & Quinn, B. (98). he Esimaion of Random Coefficien Auoregressive Models. II. Journal of ime Series Analysis,, hp://dx.doi.org/0./j b003.x Ou, J., & Penman, S. (989). Accouning measuremen, Price-Earnings Raio, and he informaion conen of securiy prices. Journal of Accouning Research, 7, -5. hp://dx.doi.org/0.307/49068 Sollis, R., Leybourne, S., & Newbold, P. (000). Sochasic uni roos modelling of sock price indices. Applied financial economics, 0, hp://dx.doi.org/0.080/ Sullivan, R., immermann, A., & Whie, H. (999). Daa-snopping, echnical rading rule performance, and he boosrap. he journal of finance, 54, hp://dx.doi.org/0./ Wang, D., & Ghosh, S. K. (00). Bayesian Analysis of Random Coefficien Auoregressive Models. Model Assised Saisics and Applicaions, 3(), Yoon, G. (003). A simple model ha generaes sylized facs of reurns. Deparmen of Economics, UCSD. Paper Yoon, G. (006). A noe on some properies of SUR processes. Oxford Bullein of Economics and Saisics, 68, hp://dx.doi.org/0./j x 8