Asian Economic and Financial Review DEPENDENCE OF REAL ESTATE AND EQUITY MARKETS IN CHINA WITH THE APPLICATION OF COPULA

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1 Asian Economic and Financial Review, 205, 5(2): Asian Economic and Financial Review ISSN(e): /ISSN(p): RL: DEPENDENCE OF REAL ESTATE AND EQITY MARKETS IN CHINA WITH THE APPLICATION OF COPLA Iou-Ming Wang --- Ming Fang Chiu-Lan Chang 3,3 Assisan Professor, Deparmen of Inernaional Business, Providence niversiy, Taiwan 2 Deparmen of Inernaional Business Sudies, Naional Chi Nan niversiy, Taiwan ABSTRACT This paper examines he dynamic dependence and exreme co-movemens beween real esae and equiy markes in China. We illusrae hese ideas in simple empirical seings, implemening he relaively echniques from copulas. When comparing he real esae indices and equiy marke indices in China, our resuls show ha series in boh Shanghai Exchange and Shenzhen Exchange exhibi ail dependence wih heir respecive equiy indices. Time-varying SJC copula is he opimal dependence srucure while illusrae he exreme co-movemen beween real esae and equiy markes in China. 205 AESS Publicaions. All Righs Reserved. Keywords: Copula, Dependence srucure, Equiy, Exreme co-movemen, Real esae, Tail dependence. Conribuion/ Originaliy This sudy is one of very few sudies which have invesigaed he dependence srucure beween real esae and equiy markes in boh Shanghai Exchange and Shenzhen Exchange wih he applicaion of he copula. Moreover, we illusrae he exreme co-movemen effec beween China s real esae and equiy markes.. INTRODCTION China has made a huge progress forward since i opened is door o he ouside world in lae 970's and embraced a marke-oriened economy afer being ruled under he cenral conrolled sysem for almos 30 years. China's economic growh has a posiive impac on China's real esae marke. China s real esae marke has coninued o boom alarmingly. Invesors creae srong diversificaion gains in real esae marke. In his paper, we seek o examine he relaionship beween commercial real esae socks and he general equiy marke in he ails of heir respecive reurn disribuions. Hoesli and Reka (205) conduced an empirical invesigaion of he channels Corresponding auhor DOI: /journal.aefr/ / ISSN(e): /ISSN(p): AESS Publicaions. All Righs Reserved. 258

2 Asian Economic and Financial Review, 205, 5(2): underlying he risk of conagion beween real esae and socks in he nied Saes. Leung and Tang (205) sudied he sock marke and he real esae marke in Hong Kong and found ha improve he predicion of he China s real esae asse markes. Copulas are funcions ha connec mulivariae disribuion funcions o heir marginal disribuions of any dimension. They have all he relevan informaion abou he dependence srucure among he variables. This allows greaer flexibiliy in modeling he mulivariae disribuions and heir margins. Firsly, because here are more deails in he full se of copula specificaions han here are in he radiional mulivariae seing by iself. Secondly, he underlying mehodology allows join disribuions o be derived from heir marginal disribuions even when he laer are no normal disribuions. And finally, i is possible o separae he characerisics of each marginal from he dependence parameer. In he ess used in his paper, he univariae reurn series are assumed o be independen and idenically disribued. On he oher hand, i is well known ha financial ime series end o exhibi volailiy clusering. This problem can be solved by filering he daa using, for example, wih an auoregressive condiional model, i.e., a GARCH/EGARCH process. Poon e al. (2004) observed ha he correlaions for he reurns of equiies are jus lighly lower if filered and ha exernal dependence among volailiy-filered residuals is much weaker han for non-filered residuals. Thus, we can expec ha filering he series for heeroscedasiciy will reduce he ail dependence, such as in Paon (2006b). The res of he paper is comprised as follows. In Secion 2 we presen a summary of he heory of condiional copulas and heir esimaion via maximum likelihood also describing he parameric forms used in his work. In Secion 3 we describe our empirical mehodology. In Secion 4 we describe our daa, model he indexes margins and fi he parameric copulas along wih heir dependence parameers. Lasly, some closing commens are presened. 2. METHODOLOGY Copula funcions permi flexible modeling of he dependence beween random variables, by enabling he consrucion of mulivariae densiies ha are consisen wih he univariae marginal densiies. This separaion enables researchers o consruc mulivariae disribuion funcions, saring from given marginal disribuions which avoid he common assumpion of normaliy, for eiher he marginal disribuions or heir join disribuion funcion. In his paper, we employ singleparameer condiional copulas o represen he dependence beween wo index reurns, condiional upon he hisorical informaion provided by previous pairs of index reurns. The parameer of he condiional copula, like he marginal densiies of he separae index reurns, depends upon he condiioning informaion. The general heory of copulas is covered in he books by Joe (997) and Nelsen (2006). Imporan condiional heory has been developed and applied o financial marke daa by Paon (2006b). Paon (2006a) seems o be he firs o exend he heorem of Sklar (959) on copulas for a condiional version. Copulas provide a convenien way o express mulivariae disribuions. 205 AESS Publicaions. All Righs Reserved. 259

3 Asian Economic and Financial Review, 205, 5(2): Copulas are ofen defined in lieraure as disribuion funcions whose marginal disribuions are uniform in he inerval [0, ]. Tha is, for an n-dimensional vecor in he uni cube, a copula can be informally defined as Where i is a random variable wih uniform disribuion in [0, ] and ui is a realizaion of, i, 2,..., n. The bivariae dynamics of he reurns X and Y are deermined by he hree funcions f( x ), g ( ) x and c (, ) u v. Parameer esimaion is sraighforward when separae parameers are used in he funcions f, g and c, which we denoe respecively by he vecors x, and y c. The conribuion o he log-likelihood of all he daa made by he wo observaions a ime is hen i Wih x; y; c. To sum hese conribuions across a se of imes gives he log-likelihood of an observed ime series of n pairs of reurns { x, y, n}, which can be saed as Wih L k denoing he sum of he log-likelihood funcion values across observaions of he variable(s) k. Paon (2006a) proposes a wo-sage esimaion procedure ha is appropriae for large samples when he dependence vecor c does no have any impac upon he marginal disribuions. In he firs sage, he parameers of marginal disribuions are esimaed from univariae ime series as The second sage hen esimaes he dependence parameer(s) as The wo-sage ML esimaes x; y c are asympoically as efficien as one-sage ML esimaes. The variance covariance marix of has o be obained from numerical derivaives. We have only been able o obain saisfacory firs derivaives, from which he fully efficien wo-sage 205 AESS Publicaions. All Righs Reserved. 260

4 Asian Economic and Financial Review, 205, 5(2): esimaor ( ni) T of he variance covariance marix is given by I n ss, where he score n vecor log h s is evaluaed a. 2.. Esimaing Marginal Disribuions The condiional densiies of equiy index reurns are lepokuric and have variances ha are asymmeric funcions of previous reurns (Nelson, 99; Engle and Ng, 993; Glosen e al., 993). We sudy several models for he margins, boh for he condiional mean and for he condiional variance. For his univariae seing we assume suden- errors, wih he mean equaions following auoregressive processes, and variance equaions following TGARCH model (Glosen e al., 993) which allows he condiional variance o respond differenly o he pas negaive and posiive innovaions. Defining R ri, TGARCH(,) model can be wrien as : as he log-reurn of sock index i in r counry a ime, he AR () -- I Where denoes he indicaor funcion, : degree of freedom of -disribuion. When esimaing he TGARCH model wih equiy index reurns, γ is ypically found o be posiive, so ha he volailiy increases proporionally more following negaive han posiive shocks. This asymmery is someimes referred o in he lieraure as a leverage effec, alhough i is now widely agreed ha i has lile o do wih acual financial leverage (Bollerslev, 2008). ri 2.2. Copula Specificaions and Dependence Measures We employ various copula funcions ha are frequenly applied in finance, offering differen dependence srucures. Trivially speaking, ail dependence expresses he probabiliy of having a high (low) exreme value of Y given ha a high (low) exreme value of X has occurred. The analyic form for he coefficien of upper ail dependence ( ) is u y x lim P Y F u X F u provided ha he random variables X and Y are asympoically dependen while (0,], on he conrary, if 0, hen X and Y are asympoically independen. The coefficien of lower ail dependence can be defined in a similar way: lim Y X u0 y x P F u F u, L (0,]. Table shows he funcion form of L each copula and ail dependence. We perform a se of goodness-of-fi ess and use wo 205 AESS Publicaions. All Righs Reserved. 26

5 Asian Economic and Financial Review, 205, 5(2): informaion crieria for selecing he opimal copula model based on he maximized loglikelihood funcion(ln) are he Akaike informaion crierion(aic) and he Bayesian informaion crierion (BIC) where n is sample size and q is number of parameers. Table-. Copula funcion form and ail dependences Copula Family funcion form lower ail Gaussian Clayon Gumbel G C u v u v, exp log log Roaed RC C C u, v u v C u, v; Clayon SJC 2 2 u v x 2 xy y C u, v dx dy exp , C C u v u v Noe: are parameers of copula funcions. λ L andλ represens he lower ail and upper ail dependences, respecively. 0 CSJC u v CJC u v CJC u v u v 2,,,,,, CJC u, v u v upper ail L DATA AND EMPIRICAL RESLTS 3.. Daa The analysis of series pairs requires a highly deailed daase spanning a long ime period. Daily sock equiy reurns for he real esae and marke indices of Shanghai Exchange and Shenzhen Exchange from July 2, 200 o June 30, 205, including 3,393 rading days are obained from Taiwan Economic Journal Daabase (TEJ). Table II shows some descripive saisics for he log-reurns of such daa. The non-normaliy of he daa is apparen from he coefficiens of skewness and kurosis. We can see ha mos of he reurn series shows signals of posiive asymmery excep Shenzhen marke reurn. The kurosis excess which shows posiive asymmery and lepokurosis. Also, he Jarque-Bera es srongly rejecs normaliy. Meanwhile, esimaed mean and sandard deviaion are visually very similar. The sample period sars from July 2, 200 due o he sock indices of China being rebalanced on July, AESS Publicaions. All Righs Reserved. 262

6 Asian Economic and Financial Review, 205, 5(2): Table-2. Descripive and serial dependence es saisics Shanghai Shenzhen real esae sock real esae sock Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera *** *** *** *** Q 2 (30) *** *** 07.53*** 72.8*** Noe: Q(20) represen Ljung-Box saisics for serial correlaion up o he 20h order for he sandardized residuals. This empirical sudy is based on he daily equiy reurns. The aserisks 0** represens significance level a % Marginal Models We firs esimae he marginal models for each reurn series. We deermined he exensions in boh marginal models based on he model specificaion es resuls in Table 3. For all reurn series, he esimaed coefficiens β are saisically significan, indicaing ha volailiies have high persisence. In oher words, large (small) changes in he condiional variance are followed by oher large (small) changes. The coefficiens γ of sock reurns in Shenzhen Exchange is posiive and significan, indicaing ha he volailiy of he sock marke increases more afer a negaive shock compared o a posiive shock. Table-3. Marginal disribuions Shanghai Shenzhen real esae sock real esae sock a (0.030) (.95) (0.329) (0.027) a 0.03*.000***.00*** 0.082*** (0.08) (0.00) (0.00) (0.07) 0.04** 4.455*** *** (0.04) (.539) (0.56) (0.05) 0.059*** 0.072*** *** 0.070*** (0.00) (0.00) (0.009) (0.03) 0.930*** 0.93*** 0.945*** 0.90*** (0.009) (0.008) (0.006) (0.0) ** *** (0.02) (0.03) (0.0) (0.06) 6.8*** 4.74*** 6.43*** 6.600*** (0.835) (0.435) (0.773) (0.742) Noe: The able repors he marginal esimaion as equaion (6). The brackes repor sandard errors.this empirical sudy is based on he daily equiy reurns. The aserisks ***,** and * represens significance level a %, 5% and 0%, respecively. 205 AESS Publicaions. All Righs Reserved. 263

7 Asian Economic and Financial Review, 205, 5(2): Resuls for Copula Parameer and Dependence Esimaes We esimae he copula models discussed in secion 3 by applying he IFM esimaion procedure. Table 4 repors he esimaed copula wih he value of log-likelihood (lnl) and he diagnosic saisics AIC and BIC are repored. The opimal dependence srucure using hese informaion crieria is ime-varying SJC copula for boh Shanghai and Shenzhen indices pairs, SJC and Gaussian are he run-ups. In Figure we plo he esimaed condiional ail dependence from he consan and ime-varying SJC copula models. As in he condiional correlaions, we see subsanial ime variaion in ail dependence in Shangai, he upper dependence ranging from o 0.799, and he lower dependence ranging from o As in he condiional correlaions, we see subsanial ime variaion in ail dependence in Shenzhen, he upper dependence ranging from o 0.86, and he lower dependence ranging from o The ime-varying dependences of SJC copula enable risk managers o measure he risk effecively and avoid underesimaing he likelihood of a join crash. The sysemaically higher dependence beween real esae and sock markes by he lef ails in China implies he higher downurn comovemen of hese wo asses. Invesors who would like o inves in he real esae and sock markes in China shall ake his phenomenon ino accoun. Table-4. Dependence measures and comparing srucures using informaion crieria Shanghai lnl AIC BIC L Gaussian Clayon Gumbel Roaed Clayon SJC ime-varying SJC Shenzhen lnl AIC BIC L Gaussian Clayon Gumbel Roaed Clayon SJC ime-varying SJC Noe: and represen he righ and lef ail dependences respecively. lnl, AIC and BIC represen values of log-likelihood, Akaike informaion crierion respecively. The values of, of ime-varying SJC is he mean of he ime-varying dependences. 205 AESS Publicaions. All Righs Reserved. 264

8 Asian Economic and Financial Review, 205, 5(2): Figure-. Tail dependences of ime-varying SJC copula Source: All daa was obained from Auhors' calculaions 4. CONCLSION Dependence is quie imporan in modern financial economics. We naurally ry o undersand dependence in modern economic life. There has been a recen flurry of research seeking o undersand dependence in economic seings. We illusrae hese ideas in simple empirical seings, implemening he relaively echniques from copulas. This paper examines wheher he pairwise co-movemen beween real esae and equiy markes in China can be correcly quanified by copulas. Copulas allow us o consruc models ha go beyond he sandard ones a he level of dependence. When comparing he real esae indices and equiy marke indices in China, our resuls show ha series in boh Shanghai Exchange and Shenzhen Exchange exhibi srong ail dependence wih heir respecive equiy indices. The opimal dependence srucure using hese informaion crieria is ime-varying SJC copula for boh Shanghai and Shenzhen indices pairs, SJC and Gaussian copula are he run-ups. Our conribuion provides he abiliy o esimae he diversificaion effecs o inernaional invesors ha are aribued o he real esae marke in China. This has imporan implicaions for risk managemen and asse allocaions during exreme evens as he significan ail dependence implies higher han normal join risk. Furhermore, he finding is also imporan for inernaional asse pricing since he exposure o he join ail risk should be compensaed and hus should be included in pricing inernaional asses. 205 AESS Publicaions. All Righs Reserved. 265

9 REFERENCES Asian Economic and Financial Review, 205, 5(2): Bollerslev, T., Glossary o arch (Garch). Creaes Research Paper No Engle, R.F. and V. Ng, 993. Measuring and esing he impac of news on volailiy. Journal of Finance, 48(5): Glosen, L.R., R. Jagannahan and D.E. Runkle, 993. On he relaion beween he expeced value and he volailiy of he nominal excess reurns on socks. Journal of Finance, 48(5): Hoesli, M. and K. Reka, 205. Conagion channels beween real esae and financial markes. Real Esae Economics, 43(): Joe, H., 997. Mulivariae models and dependence conceps. Monographs in saisics and probabiliy. 73. London, England, K: Chapman and Hall. Leung, C.K.Y. and E.C.H. Tang, 205. Speculaing China economic growh hrough Hong Kong? Evidence from he sock marke IPO and real esae markes. Inernaional Real Esae Review, 8(): Nelsen, R.B., An inroducion o copulas Springer series in saisics. 2nd Edn., New York, New Jersey, SA: Springer. Nelson, D.B., 99. Condiional heeroscedasiciy in asse reurns: A new approach. Economerica, 59(2): Paon, A.J., 2006a. Esimaion of mulivariae models for ime series of possibly differen lenghs. Journal of Applied Economerics, 2(2): Paon, A.J., 2006b. Modelling asymmeric exchange rae dependence. Inernaional Economic Review, 42(2): Poon, S.H., M. Rockinger and J. Tawn, Exreme value dependence in financial markes: Diagnosics, models, and financial implicaions. Review of Financial Sudies, 7(2): Sklar, A., 959. Foncions de répariion e leurs marges. Paris: Publicaions de l Insiu de Saisique de l niversié de Paris, 8: Views and opinions expressed in his aricle are he views and opinions of he auhors, Asian Economic and Financial Review shall no be responsible or answerable for any loss, damage or liabiliy ec. caused in relaion o/arising ou of he use of he conen. 205 AESS Publicaions. All Righs Reserved. 266