Rolling ADF Tests: Detecting Rational Bubbles in Greater China Stock Markets

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1 Singapore Managemen Universiy Insiuional Knowledge a Singapore Managemen Universiy Disseraions and Theses Collecion (Open Access) Disseraions and Theses 2008 Rolling ADF Tess: Deecing Raional Bubbles in Greaer China Sock Markes Peng HUANG Singapore Managemen Universiy, Follow his and addiional works a: hp://ink.library.smu.edu.sg/ed_coll Par of he Asian Sudies Commons, Finance Commons, and he Porfolio and Securiy Analysis Commons Ciaion HUANG, Peng. Rolling ADF Tess: Deecing Raional Bubbles in Greaer China Sock Markes. (2008). Disseraions and Theses Collecion (Open Access). Available a: hp://ink.library.smu.edu.sg/ed_coll/36 This Maser Thesis is brough o you for free and open access by he Disseraions and Theses a Insiuional Knowledge a Singapore Managemen Universiy. I has been acceped for inclusion in Disseraions and Theses Collecion (Open Access) by an auhorized adminisraor of Insiuional Knowledge a Singapore Managemen Universiy. For more informaion, please

2 Rolling ADF Tess: Deecing Raional Bubbles in Greaer China Sock Markes Huang Peng MSc in Economics Submied in parial fulfillmen of he requiremens for he Degree of Maser of Science in Economics Singapore Managemen Universiy 2008

3 Absrac Following Phillips, Wu and Yu (2007), his paper exends heir bubble deecing work o several Greaer China sock markes. Two alernaive bubble deecing mehods, he forward recursive ADF ess raised by Phillips e al. (2007) and he modified version, forward rolling ADF ess, are implemened and compared. Mone Carlo simulaions are performed o deermine he criical values of he ADF saisic under differen sample size. Empirical resuls demonsrae ha only rolling ADF ess are successful in deecing raional bubbles by overcoming he problem of periodically collapsing bubble. As we have expeced, bubbles in China Mainland sock marke are deeced. Ou of our expecaion, significan and long sanding bubbles are also found in Hong Kong, Taiwan and Singapore sock markes. However, he syles of raional bubbles in differen sock markes are differen. Differences beween he ransiion sage of China Mainland and he maure sage of oher Greaer China economies should be one imporan reason ha leads o he differen sock marke speculaive behaviors during he same period. A las, he poenial ime when bubble begins o collapse is invesigaed. Keywords: Raional bubbles, forward recursive ADF ess, forward rolling ADF ess, bubble size, speculaive behavior, bubble collapse

4 Table of Conens 1 Inroducion Economic Background Lieraure Review Models and Mehods Theoreical Framework for Bubble Forward Recursive and Forward Rolling ADF Tess Daa Empirical Resuls Subsample Size Recursive and Rolling ADF Saisics Explosive Speed and Bubble Size When Bubble Begins o Collapse Conclusion...20 References...22 Appendices...25

5 Acknowledgemen I would like o express my graiude o all hose who gave me he possibiliy o complee his hesis. I wan o hank he School of Economics of Singapore Managemen Universiy for giving me permission o commence his hesis in he firs insance, o do he necessary research work and o use he relaed daa. I am deeply indebed o my supervisor Prof. Tse Yiu Kuen whose help, simulaing suggesions and encouragemen helped me in all he ime of research for and wriing of his hesis. I would also like o express my graiude o Prof. Yu Jun and Prof. Peer C.B. Phillips for heir enlighening advices o me on his research. My parens and friends suppored me in my research work. I wan o hank hem for all heir help, suppor, ineres and valuable hins.

6 1 Inroducion 1.1 Economic Background During he las season of 2007, sock marke price of China Mainland reached is highes level in hisory from is en-year lowes level in Dec In less han wo years, he index doubled wice or more. Many informal commenaors aribued his seep rise in sock prices o he presence of a bubble. However, price level and dividend level in he sock marke of China Mainland also changed grealy during his period. As a resul, wheher here was really a bubble or no is an issue ha should be subsaniaed using economic heories and economeric mehods. Wha s more ineresing, some oher Greaer China economies which are closely relaed wih China Mainland also experienced prosperiies in heir sock markes during he same period, alhough he sign of bubble in hese markes is no as obvious as in China Mainland. Do bubbles also exis in hese economies? If he answer is yes, wha are he differences among hese bubbles? Furher more, we should find and provide possible economic explanaions or mechanisms for he phenomenon. 1.2 Lieraure Review A large and growing number of papers in he lieraure are focusing on financial bubbles. Some of hem have similar saring poin of defining financial bubbles as raional bubbles based on raional expecaion models. In his paper, we also define bubble as raional bubble. By solving consumers opimizaion problem and assuming no-arbirage and no raional bubble, we can ge he sandard presen value model for asse price (Gurkaynak, 2005): 1 P E ( P 1 D 1) (1) 1 R 1 i 1 1 R i E ( D ) (2) i 1

7 where P is he real sock price a ime and D is he real dividend received from he equiy beween 1 and, and R is he discoun rae ( R 0 ). Equiy prices conain a raional bubble if invesors are willing o pay more for he sock han hey know is jusified by he value of he discouned dividend sream because hey expec o be able o sell i a an even higher price in he fuure, making he curren high price an equilibrium price. The pricing of he asse is sill raional (Gurkaynak, 2005). This can be explained by he following equaions: i 1 f i i 1 1 R P E ( D ) B P B (3) s. E ( B ) (1 g) B (4) 1 where he firs par in equaion (3) is he fundamenal par and he second is he bubble par. Equaion (4) reflecs he self-confirming belief. The prices are raional because hey are consisen wih invesors expecaion or belief. In accordance wih he idea of raional bubble, economiss were rying and are rying o deec bubbles using many differen mehods, bu ill now almos none of hem are enough saisfying (Gurkaynak, 2005). We will firs briefly review he bubble esing mehods ha have lile srucure on bubbles. Two of hem are famous. Variance bounds es for equiy prices are iniiaed by Shiller (1981) and LeRoy and Porer (1981). This mehod is firs iniiaed for esing random walk hypohesis in sock price bu hen used by some economiss as a crierion for he exisence of bubble. I proposes an upper bound on he variance of he observed price series based on dividend series. If he variance bound is violaed in daa, his may be an evidence of he presence of a bubble. Variance bounds es have sharp drawbacks: heoreically, i is no originally designed as a es for bubble bu as a es for he presen value model and he rejecion may be due o oher model failings bu no a bubble; pracically, Cochrane (1992) finds a sriking counerexample ha violaes he variance bound wihou requiring a bubble. 2

8 Anoher one is he duraion dependence es used by Chan e al. (1998), Harman and Zuehlke (2001, 2004), among many ohers. I is buil on he raional expecaion model and he saisical heory of duraion dependence. Similar o he heory of raional bubble, in an asse marke experiencing a bubble, invesors are aware ha prices of securiies exceed heir fundamenal values bu may sill wan o purchase securiies because hey believe ha prices will appreciae furher. Raional bubble exiss only if he probabiliy of a higher reurn rises o compensae for he increased probabiliy of crash. Thus according o he duraion dependence mehod, he probabiliy ha a run of posiive abnormal reurn ends should decrease wih he lengh of he run (sequence of he reurns wih he same sign) if bubble exiss in he securiy marke (Mokhar e al., 2006). This mehod is good a deecing bubble in some cases: Mokhar e al. deec bubble in Malaysia sock marke and Rangel e al. (2007) do so in Singapore sock marke. However, he definiion for he sign in a run is arbirary and differen from paper o paper, which can change he empirical conclusions for wheher bubble exiss (Rangel e al., 2007). According o he above wo mehods, i may be argued ha empirical ess for bubbles are unineresing because hey can be ruled ou by oher reasonable facors. Oher economiss, however, alleviae his problem by rying o add srucures on bubbles based on oher specific definiion or heory for bubble. The res of his paper looks ino he bubble model we choose. There is an insighful descripion for raional speculaive bubble in he sock marke made by Kousas and Serleis (2005) ha raional speculaive bubbles mus be coninually expanding since sock buyers mus pay a price higher han ha suggesed by he fundamenals if hey believe ha someone else will subsequenly pay an even higher price. As long as he belief is persisen, bubble will be growing explosively. This descripion implies ha bubble may appear in a form of explosive ime series. Differen from he above wo models, Diba and Grossman (1988) propose a es for bubble which allows for unobserved fundamenals, and imposes such srucure on 3

9 which deviaions from fundamenals in daa may be blamed on he presence of bubble. Two famous ess, Dickey-Fuller ess and coinegraion ess are used o deec bubble. However, boh of hem fail in finding evidence of bubble in a prosperous marke (Gurkaynak, 2005). Evans (1991) argues ha i is possible ha bubble will collapse o a small nonzero value and hen coninue increasing. This leads o he resul ha he uni roo based ess have difficuly in deecing collapsing bubbles because hey behave more like saionary processes han explosive processes as a resul of he periodic collapses involved. Evans criicism of uni roo ess for raional bubbles lead o a number of papers rying o overcome he difficuly of deecing collapsing bubbles, such as Hall, Psaradakis, and Sola (1999), Norden and Vigfusson (1998) and Driffill and Sola (1998). Recenly, Phillips, Wu and Yu (2007) develop a new mehod ha amelioraed he periodically collapsing bubble problem and obained saisfacory resuls using forward recursive ADF es. They no only succeed in deecing significan bubble during he expeced NASDAQ bubble period, bu also in daa-samping he sar and he end of he bubble. In his paper, we follow heir work and modify he recursive ADF es o rolling ADF es and apply he new mehod o four Greaer China sock markes. The resuls show some ineresing hings ha we have expeced and also somehing ha we have no expeced. Unlike previous sudies, his paper akes he problem of bubble collapse ino consideraion for he firs ime. However, due o he limi of sample size, no formal empirical work is done for bubble collapse. Noe ha we will only consider posiive bubble in his paper. Tha is, bubble exiss on he firs day of he marke and will no burs or become negaive. Argumens abou posiive bubble can be found in Diba and Grossman (1988) and Evans (1991). The conen of his paper is arranged as follows. In secion 2, he models which represen bubble are presened. Two differen bubble esing procedures, forward re- 4

10 cursive ADF ess and is modified version, forward rolling ADF ess are inroduced. Secion 3 discusses he daa we use in his paper. Secion 4 applies he wo models inroduced in secion 2. Imporan empirical resuls are shown and analyzed. The las secion gives several explanaions for he empirical findings and concludes. 2 Models and Mehods 2.1 Theoreical Framework for Bubble For he derivaion of he core of he bubble deecing model, his paper follows he seup in Phillips e al. (2007). In he raional bubble lieraure ha bubbles, if hey are presen, should manifes explosive characerisics in prices. This saisical propery moivaes an expression of bubble in erms of explosive auoregressive behavior propagaed by a process of he auoregressive (AR) form x x x 1 x, (5) where for cerain subperiods of he daa, 1. Figure 1 in he appendices shows ypical ime series plos for saionary ( 0.8), random walk ( 1) and explosive x ( 1.03) processes wih inercep 0 and error erm, ~ i. i. d. N(0,1). The differences in he rajecories are apparen and useful o help us undersand wha is differen beween a saionary process and a bubble process. x In Phillips e al. (2007), similar o many researches, he concep of raional bubble is illusraed using he presen value heory of finance whereby fundamenal asse prices are deermined by he sum of he presen discouned values of expeced fuure dividend sequence. We follow his idea and begin wih equaion (1), he sandard presen value model for asse price. Following Campbell and Shiller (1989), Phillips e al. obain a log-linear bubble 5

11 expression: p p b f (6) where p f i Ed 1 i 1 i 0 (7) b lim i E p i i 1 E ( b 1) b (1 exp( d p)) b (8) wih p log( P ), d log( D ), log(1 R), 1/ (1 exp( d p)), and d p being he average log dividend-price raio, and is a variable ha relaed o. Following convenion, f p, which is exclusively deermined by expeced dividends, is called he fundamenal componen of he sock price, and b which saisfies he difference equaion (9) below, is called he raional bubble componen. Boh componens are expressed in naural logarihms. 1 b b 1 b, (1 g) b 1 b, (9) E ( ) 0 1 b, 1 where g 1 exp( d p) 0 is he growh rae of he naural logarihm of he bubble and b, is a maringale difference. As eviden from (6), he sochasic properies of p are deermined by hose of f p and b. When bubble is no expanding i.e., g 0,, we will have p p b f 0, where 0 b is he consan iniial bubble which we do no know and p is deermined solely by f p and hence by d. In his case, from (7), we obain 6

12 d p E ( d ) b (10) i 1 i 0 1 i 0 However, when bubble is expanding, i.e., g 0, since equaion (9) implies explosive behavior in b, p will also be explosive by equaion (6). In his case here will be wo siuaions for dividend series. One is ha d is an inegraed process I(1) or a saionary process I(0). Anoher case is ha d is also explosive bu his is hard o be explained. In he empirical work, i will be easier o judge wheher bubble exiss when he firs siuaion is saisfied. Based on equaion (10), Phillips e al. look for explosive behavior in p and non-explosive behavior in d via righ-ailed uni roo ess. If p is explosive while d is non-explosive, here is evidence of bubble according o his special framework. Bubble may also exis in oher siuaions, bu in his paper we will only follow his idea o find evidence of bubble. 2.2 Forward Recursive and Forward Rolling ADF Tess Afer he model for bubble is deermined, he ess are implemened as follows. For ime series x (log sock price divided by log dividend), we apply he augmened Dickey-Fuller (ADF) es for a uni roo agains he alernaive of an explosive roo (he righ-ailed). Tha is, we esimae he following auoregressive specificaion by leas squares x x x x 1 j j x, j 1 2 x, ~ NID(0, x ) J (11) for some given value of he lag parameer J, where NID denoes independen and normal disribuion. The uni roo null hypohesis is H : 1 0 and he righ-ailed 7

13 alernaive hypohesis is H : 1. 1 The opimal lag order Jop in equaion (11) is deermined by using significance ess suggesed in Campbell and Perron (1991). The significance ess can be implemened like follows. Saring wih J max, he upper bound of he lag lengh, if he las included lag is significan (under he 5% significan level using he sandard normal asympoic disribuion), we selec Jop J max. If no, we reduce he order of he esimaed auoregression by one unil he coefficien on he las included lag is significan. If none is significan, we will suppose ha k 1. In he recursive and rolling procedures, such selecion procedure will be used for every subsample repeaedly. Oher lag lengh selecion crieria such as Bayesian informaion crierion (BIC) and Akaike informaion crierion (AIC) are more common and also applicable here. However, simulaion evidence presened in Hall (1990, 1994) suggess ha daa-based mehod induces lile size disorion in finie samples and he performance of he ADF es is considerably improved when he lag lengh is seleced from he daa. Thus, our seup in his paper is ha Jop should be deermined by he daa. Furher more, Said and Dickey (1984) argue ha o ge consisen esimaes of he coefficiens in equaion (11) i is necessary o le J max, he saring lag lengh in our significan es, be a funcion of sample size T. Schwer (1989) follows he inuiion 1/4 of Said and Dickey and suggesed ha Jmax in{12 ( T /100) }. This is also consisen wih he suggesion of Said and Dickey o use a high-order auoregressive process o approximae an unknown ARIMA process. In he empirical par, we will combine he significance es mehod and Schwer s seup for Jmax o find he opimal lag order J op. In forward recursive ADF ess, ADF saisic is compued for each recursive sub- 8

14 sample. The subsamples are all from he daa wih size T. The firs subsample includes he observaions from he firs observaion rec 1 o he jh observaion rec j and has a size of j. We exend each of he following subsample by adding one more observaion han he previous one. Thus he las subsample is equivalen o he full sample and he oal number of subsamples is T j. The size of he iniial subsample j can be seleced arbirarily as long as we only care abou he resul relaed wih he laer par of he daa. In Phillips (2007), j is chosen o be 10% of he NASDAQ sample. We will follow his seing in he empirical par. The rolling ADF procedure is a lile bi differen. The ADF saisic is compued for each rolling subsample. The subsamples are also from he daa wih size T. The firs subsample includes he observaions from he firs observaion rol 1 o he kh observaion rol k and has a size of k. The second subsample includes he observaions from he second observaion rol 1 o he ( k 1) h observaion rol k 1 and also has a size of k. Similarly, for each of he following subsample we jus move he subsample forward by one observaion bu keep he size fixed a k. Thus, every subsample has he same size and he oal number of subsamples is T In his paper, he conclusion of bubble is sensiive o he choice of k since ADF saisic changes as subsample size changes. Hence, how o deermine he bes subsample size is an imporan work before rolling ADF ess are implemened. k. 3 Daa We invesigae he sock markes from four Greaer China economies: China Mainland, Hong Kong, Taiwan and Singapore. Monhly observaions on he composie 9

15 sock price index and composie dividend yield are obained from DaaSream and compue he composie dividend series from hese wo series. Then we ake he monhly Consumer Price Index (CPI) from CEIC daase o conver he above nominal series o real series. Our sample covers he period from May 1994 o Jun 2008 and comprises 170 monhly observaions in each marke. Figure 2 in he appendices plos he ime series rajecories of real price and real dividend for he four markes. These ime series are all normalized o 100 a he beginning of he sample and hen ransformed o log form. The index series of China Mainland, Hong Kong and Singapore show some similariy ha hey climb up rapidly ogeher in he las several periods. However, heir dividend series look quie differen from each oher. The dividend series of China Mainland does no climb up wih index during he las periods while Hong Kong and Singapore dividend series do so. We may guess ha here mus be some bubbles in China Mainland sock marke bu we canno judge wheher bubbles exis in Hong Kong and Singapore markes a his sage. For he sock marke of Taiwan, here seems no sign of bubble according o he figure since he index series grows more slowly and seadily han he dividend series. However, i is oo early o make judgmen unil empirical ess are performed. 4 Empirical Resuls In his paper, R (ime discoun rae) is assumed o be ime invarian since he change of R is no significan relaive o he sock price and dividend during he whole sample. Anoher reason ha he change of R is ignored is: fewer variables lead o fewer measure and esimaion errors. 10

16 4.1 Subsample Size For recursive ADF es differen people may se differen ksar bu his has nohing o do wih oher subsamples excep for he earlier ones. Thus, here is no problem wih he selecion of subsample size for recursive ADF ess. Differen from recursive ADF ess, subsample size for rolling ADF ess is fixed in he rolling process. When sample size is oo small, saisical characerisics of he esimaors is poor, problem such as bias of coefficiens will become serious. However, when sample size is oo large, he problem of periodically collapsing bubbles demonsraed by Evan will become apparen. This is because some observaions ha do no belong o an explosive process may be included ino he es. Convenionally, a beer subsample size should be figured ou firs before furher analysis can be carried ou. In his paper, however, he bes choice is ha here is no single bes subsample size bu several good subsample size. We consider nine differen subsample size: 25, 30, 35, 40, 45, 50, 55, 60, and is a relaively small subsample size and 100 is a relaively large one. For each subsample size, forward rolling ADF ess are implemened. However, i is difficul and no rigorous o say which subsample size is beer while he empirical resuls are similar. In his paper, no benchmark is used o deermine he bes subsample size. Insead, here are many good subsample size ha can be used ogeher o do analysis. Furher more, good subsample size is differen among differen markes. Our judgmens can be improved by considering various condiions. Power analysis may be useful o figure ou an opimal subsample size, bu in his paper he imporance of his is small. This is because he main conclusion will no change when subsample size changes no oo much. We choose he subsample size under which he oal number of subsamples wih significan ADF saisic in he index series is large enough for us o find evidence of bubbles. This procedure seems o be inended bu is no. There are hree reasons. 11

17 Firsly, such crieria is in line wih he condiion under which evidence of bubble can be found by using equaion (10), as we have inroduced in secion 1 of par 2. Secondly, empirical ess find ou differen good sample size for differen sock markes, which is enlighening and can be inerpreed like follows. Differen markes have differen economic behaviors and he cycles of a bubble process may also be differen. If we use a fixed subsample size in every marke, we may make wrong judgmen due o incorrec selecion of subsample size or o periodically collapsing bubble. Thus, he hird reason is ha his procedure can furher reduce periodically collapsing bubble problem. Here we jump he deails of rolling ADF ess ha will be discussed in he nex secion and direcly obain our selecion for good subsample size. Table 2 in he appendices shows he number of subsamples wih significan ADF saisics under differen subsample size for each marke. As we have expeced, differen subsample size can lead o differen resuls in bubble deecing in he same marke. And for differen markes, here is also some difference in good subsample size. For China Mainland marke, he number of significan ADF saisics of he sock index series becomes very small when subsample size is 60 and diminishes when subsample size is 100, which ells us ha if here is bubble in China Mainland sock marke, he bubble cycle should be less han 60 monh. Similar o ha of China Mainland, he number of significan ADF saisics of Taiwan sock index series becomes very small when subsample size is 55 and diminishes when subsample size is larger han 60. Hong Kong and Singapore, however, may have longer bubble cycle as significan ADF saisics of heir sock index series can be found when subsample size is 60 or even 100. The above findings demonsrae ha, in Hong Kong and Singapore sock markes, bubble generaion cycle can be longer han in China Mainland and Taiwan. Shorer cycle may be resuled in more serious speculaive behavior in a sock marke. Wha s more, i ells us ha bubbles may be more likely o be formed and o collapse in China and Taiwan sock markes han ohers. 12

18 4.2 Recursive and Rolling ADF Saisics Fuller (1996) gives a able for ADF criical values under several sample size. However, more differen subsample size is involved in his paper and heir criical values are no available. Thus, Mone Carlo simulaions for criical values under he above sample size are implemened firs. Criical values for each recursive subsample size are obained approximaely by linear inerpolaion using he simulaed ADF saisics in Table 1. Boh he recursive and rolling ADF ess are applied o he daases. If he ADF saisic for index lies above he criical value (5% confidence level, see Table 1) and ADF saisic for dividend lies below i, here is evidence of bubble according o he argumen by Phillips e al. (2007). Boh he saisics of recursive ADF ess and of rolling ADF ess are shown in he appendices (See Figure 3 and Figure 4). Figure 3 plos he recursive ADF saisics of log real index and dividend in he four sock markes. The inerpolaed criical values are ploed in he figures oo. Forward recursive ADF ess are quie capable in deecing NASDAQ bubble in he paper by Phillips e al. (2007). Tha may be because of he relaively large sample size and smooh rack of NASDAQ ime series from he saring poin o he highes level. In his paper, however, he samples we choose are no smooh a all bu very flexuous. When forward recursive ADF ess are implemened he lines of ADF saisics are almos all below he lines of he criical value for all of hese four economies especially for he period of laes wo years. No maer wheher we follow he convenion and apply he ADF es o he full sample (from May 1990 o January 2008) or jus o he recursive subsamples, he empirical resuls can no rejec he null hypohesis H : 1 in favor of he righ-ailed alernaive hypohesis 0 H : 1 1 a he 5 percen significance level for he ime series, and herefore here is no evidence of bubble in hese sock markes. However, his is no consisen wih our inuiion a all. A leas for he sock marke of China Mainland, he resul should have been differen. 13

19 There are several reasons ha we do no believe in forward recursive ADF ess in his paper. The failure of forward recursive ADF ess o deec bubbles in his paper may be sill due o he problem of periodically collapsing bubbles. Since he subsample size is increasing in forward recursive ADF ess, as he subsample size approaches he oal sample size, he forward recursive ADF es approaches he convenional pure ADF es. Simulaneously, he problem wih periodically collapsing bubbles becomes apparen asympoically. Finally, he subsample equals he oal sample iself and he forward recursive ADF es becomes he convenional pure ADF es. Wha s more imporan, differen from NASDAQ sock marke, he sock markes of he four economies in our paper do no have saisfying explosive form during he whole sample period. Thus hey show more chance o suffer periodically collapsing bubbles han NASDAQ sock marke. Oher mehods are required o deec bubble for such more general condiions. Forward recursive ADF ess also suffer some exen of heoreical flaws. 1) They are self-conradiced inrinsically. In he recursive ess procedure, bubbles are probably no o be deeced for he firs fewer subsamples and we may conclude ha hese subsamples are no explosive. However, as long as we can deec bubbles laer for longer subsamples, we may conclude ha hese longer subsamples are explosive. Longer subsamples conain he firs fewer subsamples, so hese wo conclusions seem self-conradiced. 2) The subsample size is changing and differen in he recursive procedure especially when a daase is large. Due o difficuly in simulaing all criical values for differen sample size, however, hey are fixed for convenience. This may lead o incorrec judgmen on wheher bubble exis. When rolling ADF ess are applied, however, he resuls improve grealy. For each marke, many ADF saisics of sock index series where here may be bubbles are above he line of criical value as we have expeced. In he periods of he laes 14

20 wo years, here are successive significan ADF saisics under mos of he differen subsample size in each marke. The empirical resuls for rolling ADF ess show ha here is apparen evidence of long-sanding bubbles for he sock marke of China Mainland. Wha is ou of our previous expecaion, apparen and long-sanding bubbles are also deeced in Hong Kong, Taiwan and Singapore sock markes. Before we invesigae he exac periods of bubbles, we would like o firs define or o furher clarify he definiion of bubble. ADF ess are implemened for every subsample and he oucome is an ADF saisic for each subsample. The monhs where successive significan ADF saisics are found should no be considered as bubble period. Insead, each significan ADF saisic is relaed o a subsample which is experiencing bubble. Thus, when we observe a significan ADF saisic, is corresponding subsample is a bubble process. In oher words, a bubble is an explosive process, no poins wih significan ADF saisics. In he paper by Phillips e al. (2007), bubble is successfully daed bu he definiion is confusing ha bubble period is defined o be he poins where ADF saisics are significan. This is no proper. A poin where ADF saisic is significan is jus he las monh of is corresponding subsample where ADF es is implemened. Thus, he poins before he one where ADF es is significan should be included o analyze he bubble as a process. Tha is o say, a subsample should be considered as a whole. The firs observaion in a subsample is he sar of he bubble expanding process. Significan ADF saisics are found hrough he whole sample period, bu we will no consider all of hem since our focus is for he laes wo years. Table 4 liss all he periods from Jul 2006 o Jun 2008 which are he end of a subsample wih significan ADF saisics. We canno conclude which is beer han ohers bu we make a general analysis using all of he findings in he able. There are hree imporan findings. 1) Evidence of bubbles can be found in China Mainland marke mainly from Jan 2007 o Feb 2008, in Hong Kong mainly from early 2007 o Jun 2008, in Taiwan mainly from early 2007 o Apr 2008 and in Singapore mainly from Feb 2007 o Feb ) Suc- 15

21 cessive subsample size can lead o similar empirical resuls bu for subsamples wih large difference he conclusions are very differen. For China Mainland marke, resuls under subsample size 25 o 50 have much overlapping bu here is no evidence of bubbles for subsample size 100. Such kind of conclusions can also be found in oher hree markes. 3) No maer which subsample size is used, we canno find a bubble process ha can las as lae as Mar 2008 in China Mainland and Singapore. Tha is o say, when observaions afer Mar 2008 are included in an ADF es, he whole subsample canno form an explosive process. However, we find ha bubble process can las laer in Hong Kong and Taiwan. This may ell us ha bubbles collapse earlier in China Mainland and Singapore han in Hong Kong and Taiwan. 4.3 Explosive Speed and Bubble Size The definiion of bubble is he logarihm of he raio of index o is fundamenal. P f b log( ) p p (12) f P f where P, P are real prices. There are so many successive subsamples wih significan ADF saisics ha we canno consider hem all. In order o make he argumens more reliable, however, we inerpre he explosive speed and bubble size in a sock marke based on he subsample wih he mos significan ADF saisics only. This subsample has a form ha is closer o an explosive process as in Figure 1 han oher subsamples. We selec ha very subsample from all he subsamples under differen subsample size. Figure 5 plos hese four subsamples where price series and dividend series are separaed in order o make us clear abou he differen paerns. From he figures, we can find ha all he four price series are more likely o be in an explosive process han he dividend series. In Phillips e al. (2007), he iniial size of bubble is no compued by regressions bu is supposed o be 10 percen of he sock price. This kind of seup is very useful as a ool of analysis when we do no know wha he iniial bubble size is for he subsam- 16

22 ple we concern. In his paper, we do no assume iniial bubble size or compue he exac number of bubble size since ha is quie arbirary and can lead o various conclusions. Insead, we only look a how much he bubble grows in percenage from he firs period o he las period in he subsample we use. Here, we compue he coefficiens of AR (1) process for he subsamples we menioned above in he hope o capure some rough bu helpful informaion in he explosive process in he Greaer China sock markes (Since leas squares regression will produce downward biased coefficien esimaes in he firs order auoregression, i is useful o ake accoun of i and conduc inference on auoregressive coefficiens if we need exac informaion on bubble size). Table 5 in he appendices liss he auoregressive coefficiens for he subsample wih he mos significan ADF saisics in China Mainland, Hong Kong, Taiwan and Singapore. Firs order auoregressive coefficiens show he average growing speed of index in he above markes. The subsamples we choose have differen size of 25, 55, 40 and 50 for each marke. This is in line wih our previous finding ha China Mainland and Taiwan may suffer shorer bubble cycle han Hong Kong and Singapore. For he subsample mos likely o be explosive, China Mainland has he shores bubble expanding speed and fases price explosive speed of 0.92% per monh. For Hong Kong, Taiwan and Singapore, price explosive speeds are 0.38%, 0.46% and 0.44% per monh separaely. As shown in pas researches, leas squares regression will produce downward biased coefficien esimaes in he firs order auoregression especially when sample size is small. Among several saisical mehods iniiaed o deal wih his problem, we use Gouriéroux s indirec inference (1993) o adjus he simple leas square esimaes. Our indirec inference esimaes are obained via simulaion wih replicaions. The corresponding adjused AR(1) coefficien esimaes are 1.8%, 1.2%, 1.35% and 1.25%. f Suppose he iniial size of bubble in our subsample is b0 p0 p0, we can ge 17

23 he size of bubble a he end of he subsample by using equaion (12): b p p (1 c) p p (13) f T f T T T 0 T where bt is he bubble size of he subsample wih sample size T, c is he AR(1) coefficien and price growing speed, f pt is he fundamenal price a he end of he subsample. Thus, he percenage he bubble expands during he subsample is: b p p (1 c) p p b p p p p f T T T T 0 f f f T (14) Noice ha equaion (14) is difficul o implemen because we do no know wha f p0 is. In he four subsamples we concern, no evidence of explosive process can be deeced in corresponding dividend series, which lead o he conclusion ha fundamenal prices should no be explosive. However, dividend series seem o move in oher ype of form and hence f pt is also difficul o be deermined. To make hings easier, we suppose ha fundamenal price grows a he same speed wih acual price. Then he percenage he bubble expands during he subsample now become: b p p (1 c) ( p p ) T (1 c) (15) b p p p p f T f T T T 0 0 f f which can be figured ou by using our empirical resuls. If dividend series or he fundamenal price does no increase explosively, we can inerpre equaion (15) as a conservaive approximaion or a lower bound o he rue percenage ha he bubble expands from he iniial poin. China suffers a serious bubble expanding of 156% wihin only 25 monhs. The percenages ha bubble expand in Hong Kong, Taiwan and Singapore is separaely 192% afer 55 monhs, 171% afer 40 monhs and 186% afer 50 monhs. According o our derivaions above, he rue expanding percenages should be larger han hese figures, since here is no evidence of explosive process in heir dividend series. Since we do no know he iniial bubble in each marke, i is hard o conclude which marke 18

24 suffers he mos serious bubble. However, i is easy o see ha China suffers he fierces bubble expanding in much shorer period han ohers. For he above four subsamples wih he highes possibiliy of suffering explosive bubbles, subsample size are differen. China Mainland has he shores explosive process while he oher hree markes have much longer explosive process. Wha s more, he explosive speed in China Mainland is he larges. These may be because ha speculaive behavior in China Mainland is much sronger han in he oher hree markes, which leads o quick expanding in he bubble size and shorer explosive process. The peaks of bubbles for above four subsamples are differen. The larges bubble in China and Singapore are during Jun, 07 while in Hong Kong i s Jan, 08 and in Taiwan Nov, When Bubble Begins o Collapse I will be ineresing o invesigae he bubble collapse in a bubble sock marke. Previous researches always focus on bubble esing, which is only abou wheher bubbles exis or in anoher word, bubble generaion process. Scare research focuses on he opic of bubble collapsing process. Inuiively, bubble collapsing speed is always much faser ha bubble generaion. Sudden collapsing bubble can lead o very serious economic and social problems. And such process is relaively easy o ransmi from counry o counry. When bubble is collapsing, i has similar mechanism wih bubble generaion bu perform in he oher way round. Theoreically, bubble collapsing can also have an exponenial form because everyone believes ha nobody will buy he sock and hey mus sell heir sock in much lower price han i should be. In all of hese four bubble sock markes, here are obvious quick drops during he las periods. Back recursive or back rolling ADF ess which can be performed adversely o forward ADF ess seem quie capable if we wan o deec bubble collaps- 19

25 ing. Unforunaely, he sample size we can use is oo shor a he presen ime. We are no able o ge deeper findings unil we can find larger sample. Furher research on his opic is worhwhile o do so ha we can furher undersand wha bubble is. According o our empirical findings, we can guess when a bubble begins o collapse. When his happens, here is sill bubble in he marke bu he explosive process does no coninue anymore. Thus, rolling ADF saisic should be sill significan bu he saisic should become less significan han previous ones. Since we have already found he subsamples wih mos significan ADF saisics in he previous secion, i is very likely ha bubbles begin o collapse afer he end of hese subperiods. The poenial monh when he bubble begins o collapse in China Mainland and Singapore is afer Jun 07, in Hong Kong afer Jan 08 and in Taiwan afer Nov Conclusion In order o find evidence of raional bubbles in four Greaer China sock markes, wo differen mehods (forward recursive ADF ess and forward rolling ADF ess) are used and compared. Forward recursive ADF ess fail in deecing bubbles in our samples while rolling ADF ess do so. The same o he informal commens, we find srong evidence of long-sanding bubbles in China Mainland sock marke. We also deec bubbles in he res hree Greaer China sock markes (Hong Kong, Taiwan and Singapore). This is an evidence of srong ineracion in oday s inernaional financial markes. China Mainland sock marke may have suffered he fases bubble expanding, and wihin a very shor ime is bubble expands as much as oher markes whose bubble expanding duraion is much longer. Wha s more, he bubble in China Mainland sock marke collapses in he earlies ime. The empirical resuls for rolling ADF ess furher show ha he lengh of bubble cycles in differen markes may be differen. 20

26 I is a quie common economic phenomenon in China Mainland ha o inves in sock marke is much like a kind of gambling. Speculaive behavior in China sock marke is prevailing for various kinds of people. Wha s more, as an economy under a ransiion sage, financial regulaions in China Mainland are very weak. Due o he close economics relaionship beween hese hree economies (China Mainland, Hong Kong and Singapore), he conclusions of bubbles are similar. Speculaion may have spread among hem and China Mainland is very likely o be he saring marke. However, boh invesors undersanding abou invesmen and financial regulaions are more complee in Hong Kong, Taiwan and Singapore. This is why heir bubble processes are slower, longer and less serious. I is imporan for China Mainland ha hrough effecive financial policies like hose developed economies, speculaive behavior is possible o be conrolled or decreased. If bubbles exis in asse markes, marke prices of asses will differ from heir fundamenal values. Markes would no necessarily be allocaing he savings of individuals o he bes possible invesmen uses. As bubbles always lead o financial and social unres, public policies migh be designed o aemp o rid he markes of bubbles. I is more imporan ha afer coninuing educaion, invesor s in China Mainland can become really raional. There are sill some shorcomings in his paper ha may be improved. The model for bubble is only applicable for special cases when here is only explosive behavior in index bu no in dividend, so refinemen of model is needed for more general applicaions. The daases need o be exended o capure more hisorical informaion so ha deeper empirical research on he collapsing of bubble is available. A las, model comparison beween our mehod and previous mehods are also valuable and imporan. Finie sample performance should be examined among differen mehods so ha we can find ou he bes one. Fuure research will cover hese issues. 21

27 References Banerjee, Anindya, Robin L. Lumsdaine, and James H. Sock, 1992, Recursive and sequenial ess of he uni-roo and rend-break hypoheses: heory and inernaional evidence, Journal of Business and Economic Saisics 10, Campbell, John Y., and Pierre Perron, 1991, Pifalls and opporuniies: wha macroeconomiss should know abou uni roos, NBER Macroeconomics Annual 6, Campbell, John Y., and Rober J. Shiller, 1987, Coinegraion and ess of presen value models, Journal of Poliical Economy 95, Chan, K. McQueen, G.R and Thorley, S., 1998, Are here raional speculaive bubbles in Asian sock marke? Pacific-Basin Finance Journal 6, Cheung, Yin-Wong, and Kon S. Lai, 1995, Lag order and criical values of he augmened Dickey-Fuller es, Journal of Business & Economic Saisics 13, Cochrane, John H, 1992, Explaining he variance of price-dividend raios, Review of Financial Sudies 5, Diba, Behzad T., and Herschel I. Grossman, 1988, Explosive raional bubbles in sock prices, The American Economic Review 78, Dickey, David A., 1976, Esimaion and hypohesis esing in nonsaionary series, PhD Disseraion, Ames: Iowa Sae Universiy. Evans, George W., 1991, Pifalls in esing for explosive bubbles in asse prices, The American Economic Review 81,

28 Fuller, Wayne, 1996, Inroducion o Saisical Time Series, New York: Wiley. Gouriéroux, Chrisian, Alain Monfor, and Eric Renaul, 1993, Indirec inference, Journal of Applied Economerics 8, Gurkaynak, Refe S., 2005, Economeric ess of asse price bubbles: aking sock, Finance and Economics Discussion Series, Federal Reserve Board. Hall, Alasair, 1994, Tesing for a uni roo in ime series wih prees daa-based model selecion, Journal of Business & Economic Saisics 12, Hanim, Suraya, Annuar Md Nassir, and Taufiq Hassan, 2006, Deecing Raional Speculaive Bubbles in he Malaysian Sock Marke, Inernaional Research Journal of Finance and Economics 6, Harman, Y.S. and Zuehlke, T.W., 2001, Tesing for raional bubbles wih a generalized weibull hazard, Working Paper. Harman, Y.S. and Zuehlke, T.W., 2004, Duraion dependence esing for speculaive bubbles, Journal of Economics and Finance 28, Hassan, M., and Jung-Suk Yu, 2006, Raional speculaive bubbles in he fronier emerging sock markes, Presened a Financial Managemen Associaion. Kousas, Zisimos, and Aposolos Serleis, 2005, Raional bubbles or persisen deviaions from marke fundamenals? Journal of Banking and Finance 29, LeRoy, Sephen F., and Richard D. Porer, 1981, The presen-value relaion: ess based on implied variance bounds, Economerica 49, Liew, Venus Khim-Sen, 2004, Which lag lengh selecion crieria should we employ? Economics Bullein 3, 1-9. MacKinnon, James G., 1994, Approximae asympoic disribuion funcions for 23

29 uni-roo and coinegraion ess, Journal of Business & Economic Saisics 12, Phillips, Peer C. B., Yangru Wu, and Jun Yu, 2007, Explosive behavior in he 1990s NASDAQ: when did exuberance escalae asse values? Working Paper. Rangel, Gary, and Subramaniam S. Pillay, 2007, Evidence of bubbles in he Singaporean sock marke, Singapore Economic Review Conference Paper. Robers, George B., 1929, The price-earnings raio as an index of sock prices, Journal of he American Saisical Associaion 24, Supplemen: Proceedings of he American Saisical Associaion, Said, Said E., and David A. Dickey, 1984, Tesing for uni roos in auoregressive-moving average models of unknown order, Biomerika 71, Schwarz, Gideon, 1978, Esimaing he dimension of a model, Annals of Saisics 6, Schwarz, Gideon, 1989, Tess for uni roos: a Mone Carlo invesigaion, Journal of Business and Economic Saisics 7, Shiller, Rober J., 1981, Do sock prices move oo much o be jusified by subsequen changes in dividends? The American Economic Review 71, Taylor, Mark P., and David A. Peel, 1998, Periodically collapsing sock price bubbles: a robus es, Economics Leers 61, Waers, George A., and James E. Payne, 2007, REIT markes and raional speculaive bubbles: an empirical invesigaion, Applied Financial Economics,

30 Appendices Figure 1 Typical ime series 1. Simulaed AR (1) model wih 0.8 (sample size is 200) A saionary process wih mean 0 and variance less han infiniy 2. Simulaed AR (1) model wih 1 (sample size is 200) A random walk process 3. Simulaed AR (1) wih 1.03 (sample size is 200) An explosive process 25

31 Figure 2 Log normalized real index and dividend (May 1994-Jun 2008) China Mainland Hong Kong Taiwan Singapore 26

32 Figure 3 Recursive ADF ess China Hong Kong Taiwan Singapore 27

33 Figure 4 Rolling ADF ess China Mainland 28

34 29

35 Hong Kong 30

36 31

37 Taiwan 32

38 33

39 Singapore 34

40 35

41 Figure 5 Subsamples wih he mos significan ADF saisics in each marke China Hong Kong 36

42 Taiwan Singapore 37

43 Table 1 Mone Carlo simulaed criical values for ADF saisics This able repors he simulaed criical values for he ADF saisics in his paper. The firs column shows he sample size. The firs row shows he perceniles. The criical values are obained by Mone Carlo simulaion wih replicaions. The deailed procedure is like his: 1) for sample size of 25, we simulae AR (1) uni roo ime series each wih a fixed consan erm; 2) since ADF saisic has he same sampling disribuion as he DF saisic, we hen use he DF es o compue he corresponding ADF saisic for each simulaed sample, where he null hypohesis for he DF es is ha here is a consan erm bu no ime rend; 3) a las, we find he perceniles lised in he able for he ADF saisics; 4) procedures 1)-3) are repeaed for differen sample size. In he paper, we convenionally use he 95 percenile criical values o evaluae explosive evidence in recursive and rolling ADF ess. 38

44 Table 2 Number of significan ADF saisics under differen subsample size China Mainland Hong Kong Taiwan Singapore This able repors he number of significan ADF saisics under differen subsample size. The firs column shows he sample size. The lef column under each marke shows he number of significan ADF saisics of sock index series while he righ column shows ha of dividend series. 39

45 Table 3 Mos significan ADF saisics for sock price under differen subsample size China Mainland Hong Kong Taiwan Singapore This able repors he mos significan ADF saisics of sock index series under differen subsample size for each marke. The firs column shows he sample size. Significance of ADF saisic is he benchmark of his paper o selec he subsamples which are mos likely o suffer bubbles. The subsample size under which here exiss he mos significan ADF saisic (bold numbers) is differen from marke o marke. 40

46 Table 4 Periods in he laes wo years when significan ADF saisics are found under differen subsample size China Mainland Hong Kong Taiwan Singapore 25 Jan-Dec 07 Dec 06-Mar 07, May 07-Feb 08 none Feb, Mar, May-Sep Jan-Dec 07 Feb 07-Feb 08 May, July-Aug 07 Mar-Sep Feb-Dec 07, Feb 08 May 07-Jun 08 Oc 06-Feb 07, Jul 07-Feb 08 Jan 07-Jan Jan-Dec 07, Feb 08 Jan 07-Jun 08 Feb 07-Apr 08 Feb 07-Feb Jan-Dec 07, Feb 08 Apr 07-Jun 08 Jun 07-Apr 08 Feb 07-Feb Feb, May-Dec 07 Aug 07-Jun 08 Aug 07-Feb 08 Mar-Dec May-Dec 07 Dec 07-Jun 08 Jan, Feb 08 Mar-Jul, Feb May-Dec 07 Nov-Dec 07, Jun 08 none Feb-Jul, Sep, Nov none none none none This able repors he subsamples of which he ADF saisics of sock index series are significan. The firs column shows he sample size. The daes in he able are he las days of hese subsamples. We only repor he daes for he laes wo years (Jul 06-Jun 08) ha we concern. 41

47 Table 5 Auoregressive Coefficiens (Bubble Growing Speed) and Corresponding subsample ADF saisics period size ADF(price) ADF(dividend) cv AR(1) Adjused China Mainland Jun 05-Jun Hong Kong Oc 04-Jan Taiwan Jul 04-Nov Singapore May 03-Jun

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