Markov-Switching Unit Root Tests: A Study of Property Price

Transcription

1 1 Markov-Swiching Uni Roo Tess: A Sudy of Propery Price Bubbles in Hong Kong and Seoul Version 3(January 006) XIAO Qin i Division of Economics School of Humaniies and Social Sciences Nanyang Technological Universiy of Singapore S3-01C-10 Nanyang Ave. Singapore cqxiao@nu.edu.sg Tel. (65) Fax. (65) Gee Kwang, Randolph TAN Division of Economics School of Humaniies and Social Sciences Nanyang Technological Universiy of Singapore S3-BB-67 Nanyang Ave. Singapore arandolph@nu.edu.sg Tel. (65) Fax. (65)

2 Absrac: Evans (1991) demonsraes he failure of he uni roo ess recommended by Hamilon and Whieman (1985) and Diba and Grossman (1988) o deec periodically collapsing raional bubbles. Hall e al. (1999) however show ha he power of his es procedure can be improved by incorporaing a Markov-swiching sae variable. In his paper, we apply boh procedures o seleced daa from Hong Kong and Seoul. Boh poin o he possible exisence of a periodically-collapsing bubble in each price series invesigaed, wih he second procedure more precise on iming he bubble. Our Markov-swiching model is validaed using a symmery es and a Wald es. JEL classificaion: G1, C1, C5 Keywords: Markov-swiching; uni roo es; periodically-collapsing bubble; real-esae price.

3 3 1. Inroducion Hamilon and Whieman (1985) and Diba and Grossman (1988) sugges ha saionariy ess may be used o deec an explosive raional speculaive bubble. Use of such a es does no preclude he possible influences of unobservable marke fundamenals. The raionale of his procedure is as follows. If he firs-difference of a dividend and hose of unobservable fundamenals are saionary in mean, and if no raional bubble exiss, hen he firs difference of he associaed sock price mus be saionary. Differencing a sock price a finie number of imes would no render i saionary, however, if i conains a raional bubble. Due o he possible presence of unobserved variables, finding ha he firs-difference of a sock price is no saionary does no auomaically esablish he exisence of a raional bubble. However, he converse inference is possible. Tha is, evidence ha he firs-difference of a sock price has a saionary mean would be evidence agains he exisence of a raional bubble in ha price. Evans (1991) shows ha saionariy ess, suggesed by Diba and Grossman (1988) and Hamilon and Whieman (1985), are incapable of deecing periodically collapsing raional bubbles. Hall e al. (1999) show ha he power of hese ess can be improved significanly by incorporaing a sae variable ha follows a Markov process. They argue ha esing for a periodically-collapsing bubble essenially involves disinguishing he expanding phase from he collapsing phase of he bubble. The wo phases can be modeled by a wo-sae Markov chain. In his case, he daa generaing process (DGP) would ake differen parameers in differen saes. If he DGP is modeled wih a Markov-

4 4 swiching AR(p) process, hen a generalized ADF uni roo es would deec he bubble, if i exiss, quie effecively. In his sudy, we ake wo approaches. One is o break a long sample ino shorer ones, and apply he linear ADF and PP ess o hese samples (referred o as he shorer sample approach). The oher is o apply he Markov-swiching ADF es wihou breaking a sample (referred o as he Markov-swiching approach). The Markov-swiching approach is more appealing ex ane. This is because he swiching (breaking) poins are deermined endogenously inside a model, raher han pre-imposed by he researchers, as is he case in he firs approach. However, ex pos, he wo approaches give compaible resuls, wih he second more precise in iming a bubble, if i exiss. Our daa are drawn from he propery marke of Hong Kong and Seoul.. A Review of he Real-esae Price Bubble Lieraure In he language of economiss, a bubble exiss in a price if he price is oher han wha is warraned by is fundamenals. The issue of speculaive bubble arises because of uncerainies surrounding he fundamenals. In he real esae marke, for insance, a buyer of a propery is willing o pay a price which he perceives o be equal o he fundamenal value. In assessing his fundamenal value, he will make use of informaion on renal flows and fuure price changes. His assessmen mus ineviably rely on expecaions abou some relevan fuure evens. The expecaions in urn are based on subjecive, insead of he acual, probabiliies of such evens. Therefore, he assessmen of

5 5 marke fundamenals is inherenly subjecive (Shiller, 001). As a resul, he acual price will reflec he rue fundamenal value only by chance! Tha is, a price will conain a bubble elemen as a maer of course. The real-esae marke has he longes and he mos reliable hisory of boom and bus cycles sreching back o he early 1800s (Carrigan, 004). Researchers poin o speculaion as a prime force behind hese cycles (Malpezzi and Wacher, 00). Such speculaion could iniiae he formaion of speculaive bubbles. Evidence of speculaive bubbles in he real esae marke has been found in counries around he world. Abraham and Hendersho (1994) found ha, as of he end of 199, here was a 30% above marke premium in prices in he Norheas of he Unied Saes, a 15% 0% premium in prices on he Wes Coas of he counry, and probably a significanly negaive premium in Texas. In heir model, Abraham and Hendersho incorporae a proxy for he endency of a bubble o burs and a proxy for he endency of a bubble o swell. ii They found ha he proxy does indeed work and is especially useful in explaining he large cyclical swings in real house prices in he Wes. The lagged appreciaion erm ha represens speculaive pressures in he marke performs admirably in soaking up he volailiy. Bjorklund and Soderberg (1999) examine in heir paper he cycle in he Swedish propery marke. Their sudy shows ha he raio of he propery value o he ren increased oo much during his sample period, indicaing ha a bubble migh exis.

6 6 Sco (1990) and Brooks e al. (001) apply variance-bound ess in heir sudies o es he raionaliy of real-esae share prices. Sco analyzes price indices of 13 REITs. His sample sreches from he lae 1960s or early 1970s o Brooks e al. examine he prices of U.K. propery socks. Boh sudies indicae he exisence of irraional, speculaive bubbles. The large swings of propery prices in Japan in he lae 1980s and early 1990s have inrigued many researchers. Io and Iwaisako (1995) aemp o measure how much of he asse price variaion observed in Japan in he lae 1980s and he early 1990s can be aribued o changes in he fundamenals As represened by he sandard presen value model. They conclude ha i seems impossible o offer a raional explanaion of he asse price inflaion in he second half of he 1980s by changes in fundamenals (p. 10). This conclusion is reinforced by a variance decomposiion analysis (ref. heir p. 0). Basile and Joyce (001) use he mehod by Forune (1991) o measure he size of he asse price bubble, which is he difference beween he ex pos reurns of an asse and he required reurn. By his measure, hey find ha he land marke bubble grows evenly hrough mid-1990s before declining. There have been quie a number of sudies on he real-esae price bubbles in Souh Korea. Lee (1997) conduced a es of bubble in he land price of Korea beween 1964 and Using a srucural model wih GNP, ineres rae and money supply as fundamenals, he found evidence o rejec he hypohesis ha only marke fundamenal drove he land price in Korea. Kim and Lee (000) adaped he idea ha he exisence of

7 7 an equilibrium relaionship excludes he possibiliy of a price bubble. They found ha, in he long run, nominal and real land prices are coinegraed wih marke fundamenals (approximaed by nominal and real GDP respecively). However, in he shor run, such a coinegraion relaionship does no exis. This is consisen wih he noion ha speculaive bubbles are periodically collapsing (Blanchard and Wason, 198). In he shor run, speculaive forces could drive prices away from marke fundamenals. In he long run, fundamenal forces will evenual reasser hemselves. Lim (003) conduced wo ess for bubbles based on he presen value relaion on he housing price of Korea. One is a modified volailiy es (MRS es) suggesed by Mankiw e al (1985), while he oher combines he uni-roo es suggesed by Diba and Grossman (1988), and he coinegraion es of Campbell and Shiller (1987). Their MRS es show ha he null hypohesis of marke efficiency is rejeced, indicaing he exisence of an irraional bubble. Their uni-roo es and coinegraion es however sugges ha bubbles do no exis! This is in conras wih our findings in his paper, which employ a Markovswiching ADF approach. However, he daa series employed in he curren paper are no idenical o hose used by Lim (003). Hong Kong underwen exraordinary swings in he 1980s and 1990s. In size, he price swings have been as dramaic as hose happened elsewhere in he world. In frequency, hey have been more dramaic. This fac makes Hong Kong propery marke one of he mos ineresing for he sudying of speculaive bubbles. However, here have been relaively few papers devoed o his subjec. Chan e al. s (001) is one of hem. The sudy uses a sandard presen value model, bu allows for a specificaion error. The daa

8 8 hey use are monhly averaged renals and quarerly averaged prices of he privae domesic properies wihin he class A, which is defined as hose aparmens wih sizes less han 39.9 m. The sample period runs from he firs quarer of 1985 o he hird quarer of Their resuls show ha here exiss misspecificaion error in he model noises as well as a bubble. The pah of he bubble shows ha he bubble exploded mos sharply beween 1990 and 199, and beween 1995 and Xiao and Tan (006) use a similar approach, bu a differen echnique iii, o sudy he propery marke of Hong Kong. They use monhly observaions beween December 1980 and January 003 from four differen secors: he office, he domesic premises, he flaed facories and he reail premises secors. They conclude ha a periodically-collapsing raional speculaive bubble could be responsible for he observed volailiies in each secor. The peak of he bubble occurred in he mid-1994 and/or he mid-1997 in all cases. 3. Mehodology 3.1. Convenional Uni-roo Tess Dickey and Fuller (1979) suggesed a baery of ess based on a regression of he form y + = ρ y 1 u or y = + ρy 1 + u μ or y = μ + δ + ρy 1 + u, wih he rue process being y + = y 1 u or y = μ + y 1 + u. In hese ess, he disurbance erm is assumed o be i.i.d. and normal wih zero mean and consan variance (Hamilon, 1994a, p. 50).

9 9 Phillips (1987) and Phillips and Perron (1988) suggesed some modificaions o he DF es saisics o ake care of serially correlaed and heeroscedasic disurbance erms. The es suggesed by Phillips and Perron are referred o as he PP es. Dickey and Fuller (1979) provide an alernaive approach ha conrols for serial correlaion by including higher-order auoregressive erms in he regression. Tha is he model p ς i i 1 y = + + y μ δ ρ Δy i + u is o be esimaed wih possibly zero = coefficiens on he consan and he rend erms. This modified DF es is referred o as he ADF es. Various suggesions have been proposed regarding how o proceed when he process is deemed as AR( p) wih p unknown bu finie. Hamilon (1994a) suggess a simple approach ha akes p o be some pre-specified upper bound p (We se p = T in his paper. T is he sample size).the OLS -raio of ς p 1 can hen be compared wih he usual criical value for a saisic. If he null hypohesis is acceped, hen he OLS F es of he join null hypohesis (ha boh ς p 1 = 0 and ς p = 0 ) can be assessed using he usual F(,T-K) disribuion. The procedure coninues sequenially unil he join null hypohesis (ha ς p 1 = 0, ς p = 0,, ς p l = 0 ) is rejeced for some l. Greene (1997, p. 787) discusses he meris and flaws of his procedure. As our purpose is o remove serial correlaion among residuals, his procedure will suffice when combined wih he Durbin Wason es.

10 A Markov-swiching AR(p) Model and is Esimaion Procedure A MS model assumes ha ime series daa may display periodic changes in heir observed behavior, and i accouns for such changes hrough swiches in saes. The average duraion of each sae is allowed o differ. Furhermore, he saisical feaures and idenificaion of he saes are no imposed exogenously on he daa, bu deermined endogenously by he esimaion procedure. Consider Δy = where s { 1,,..., N} K s s s μ + φ y 1 + ψ k Δy k + ε ε k= 1 d, N(0, σ ) iid, ( 1), a sae variable following he firs order Markov chain: Pr = ( s+ 1 = j s = i, s 1 = i1,..., ζ ) Pr( s = j s = i) p ij + 1 ( ) where ζ = y, y,..., ) ( 1 y1, represening he informaion se available a ime, and pij is he probabiliy ha sae i will be followed by sae j given s = i, s i1,..., andζ. 1 = Equaion () says ha he probabiliy disribuion of s + 1 depends on pas evens only hrough he value of s iv. The sae variable s is no observable, bu is probabiliy for a given sample of size T, ( s = iζ ) Pr, can be inferred using he discree Kalman filer (refer o Hamilon, 1989, 1994a, 1994b for deail). T

11 In calculaing he smoohed inference of he sae variable, Pr ( s = iζ ) T 11, we assume ha s s s he DGP parameers, β = ( μ, φ, ψ, p, σ )', are known o us. In fac, hese parameers k ij need o be esimaed. We can esimae hem by maximizing he log likelihood funcion of he observed daa using he EM algorihm, since he EM algorihm is efficien, simple, and sable. v T The log-likelihood funcion o be maximized is LL = f ( y x, 1 ) ( ) = 1 log ζ, wih f y x, ζ 1 he densiy of y condiional on x and ζ 1. The esimaion seps are given below. s s s 1. Make an arbirary guess abou he values of μ, φ, ψ k, pij andσ.. Calculae he smoohed probabiliies of s using he discree Kalman filer. 3. OLS regress y Pr ( s = iζ ) on x ( s = iζ ) T Pr, i = 1, T, which gives he ML s ~ s esimaes μ, φ, ~ ψ ( k 1,,...K ) =. ~, ( k 1,,...K ) 4. Updae σ using he OLS residuals. s k =. Noice ha y Δ, y x 1, y, Δy,..., Δy )', ( 1 1 k ~ s σ 5. Updae p ij. = ~ ~ ( y x ' β )' ( y x ' β ) s ( T J ) s, ( 3) where J = he number of parameers esimaed in each sae. p ij T Pr = = T = ( s = j, s = iζ ) Pr ( s = iζ ) 1 1 T T ( 4)

12 1 6. Updae π. π = i = 1 ( s iζ ) Pr ( 5) T Repea seps hrough 6 unil he parameers and he likelihood converge Specificaion Tess of he Markov-swiching Model An imporan issue peraining o an MS model is he number of saes characerizing he daa. The sandard disribuional heory is no applicable for evaluaing he Markovswiching model agains some popular alernaives, such as a linear ime-series model. Hamilon (1989) shows ha convenional ess of a Markov-swiching model would render he Markov ransiion marix unidenified and he informaion marix singular, under he null hypohesis of a single sae. Several auhors have proposed alernaive esing procedures ha aemp o overcome hese problems. However, he applicaion of hese procedures can be problemaic. The problems arise because ha we have limied knowledge of he respecive powers of hese ess, and hese ess are, in general, compuaionally demanding (Raymond and Rich 1997). For hese reasons, perhaps, he previous sudies seldom validae heir Markovswiching specificaions vi. Breunig e al. (003) argue ha a Markov-swiching model should be pu o specificaion ess, like any oher model. Among he four ypes of ess suggesed, hey highly recommend a Wald-ype es, which hey call encompassing es. They show ha his

13 13 encompassing es is he mos powerful way of examining he abiliy of he model o mach he daa. We will pu our MS model o wo ypes of specificaion ess in his paper: a symmery es following Cecchei e al. (1990) and an encompassing es suggesed by Breunig e al (003). Under he null p 11 = p, he symmery es saisic has a sandard normal disribuion. The encompassing es procedure is described below. Le γˆ be a quaniy ha has been esimaed from he daa. This γˆ can be he mean, he variance, or somehing else. We denoe a comparable quaniy implied by he MS model by γ ( ˆ θ ) M, where θˆ is he MLE of he parameer θ associaed wih he MS model. In paricular, we simulae daa wih he esimaed MS model, and esimae γ ( ˆ θ ) M from he simulaed series. A scaling facor of T 1 + is applied o he variance of any es M saisic o make an allowance for he effec of he simulaion error upon he variance of an esimaor, where M is he number of replicaion and T he number of observaions in he sample. Consider he saisic ( ˆ θ ) ˆ τ = ˆ γ γ. Under he null M ( ) τ =, 0 γ 0 γ M θ0 where θ 0 is he rue value θ and γ 0 he rue value of γ, we have

14 14 T 1 d ( τ τ ) N( 0, V ) ˆ 0. Consider he es saisic 1 [ ( ˆ τ )] ˆ τ R * = ˆ` τ Var, τ which has a χ disribuion wih degree of freedom equals o he dimension of τ, wih Var ( ˆ τ ) Var( ˆ γ ) Var( γ ( ˆ θ ) =. This saisic can be replaced by 1 [ ( ˆ γ )] ˆ τ R = ˆ` τ Var. M As R < R*, if R exceeds he criical value, we would rejec he null even more srongly wih R*. Under he null, ha MS model is correc and characerized by parameer θˆ, we can simulae daa from he model and find ou Var ( γˆ ) from he simulaed series. Alernaively, we may use asympoic heory and compue a robus esimaor of Var ( γˆ ). In he curren sudy γˆ corresponds o SSE vii from he MS model. γ ( ˆ θ ) mean of SSE from he simulaion wih 10,000 replicaions, and var(γˆ ) variance of SSE. Under he null ˆ γ γ ( ˆ θ ) M M is he sample he sample =, he es saisic R has a χ () 1 disribuion. A scaling facor T + M 1 is applied o Var ( γˆ ), as discussed before Boosrapping

15 15 The null disribuions of he saisics for he linear uni-roo ess are abulaed in Hamilon (1994a). Those for he Markov-swiching ADF ess are unknown bu can be generaed by boosrapping. viii Boosrapping is a mehod for esimaing he disribuion of an esimaor or es saisic by resampling he daa. I amouns o reaing he daa as if hey were he populaion for he purpose of evaluaing he disribuion of ineres. Under mild regulariy condiions, Boosrapping yields an approximaion o he disribuion of an esimaor or es saisic ha is a leas as accurae as he approximaion obained from firs-order asympoic heory. Thus, boosrapping provides a way o subsiue compuaion for mahemaical analysis if calculaing he asympoic disribuion of an esimaor or saisic is difficul (Horowiz, 001). In fac, boosrapping is more accurae in finie samples han firs-order asympoic approximaions and does no enail he algebraic complexiy of higher-order expansions. The firs-order asympoic heory ofen gives poor approximaions o he disribuions of es saisics wih he sample sizes available in applicaions. As a resul, he nominal probabiliy ha a es based on an asympoic criical value rejecs a rue null hypohesis can be very differen from he rue rejecion probabiliy (RP). Boosrapping ofen provides a racable way o reduce or eliminae finie-sample errors in he RPs of saisical ess.

16 16 The mehod neverheless has is own limiaions and should no be used blindly, bu i works well in general. The readers are referred o he Handbook of Economerics, vol. 5, chaper 5 for deails on he sampling procedure and he consisency of boosrapping. The seps of boosrapping are described below. 1. Save he ML parameer esimaes θ ~ and residuals { } T ~ ε from he MS model. = 1. Consruc he random disurbance erm e (o be explained laer in his secion). 3. Take a random draw of e, denoe as (1) e 1, and se Δy Δy (1) 1 (1) K = ~ (1) μ + ~ ψ Δy e, k = 1 k k + = ~ (1) (1) μ + ~ ψ Δy + ~ ψ Δy e, 1 1 K k = 1 k k + 1 Δy K ( 1) ~ T k k = 1 = ~ (1) μ + ψ Δy e, (1) T k + 1 where Δ = simulaed values of Δ y, (1) y Δ y k = acual observed values of Δ y, and ~ μ ~ = ML esimaes., ψ k This gives a full sample { sample. (1) y } T = 1, where T is he number of observaions in he 4. Fi he arificial sample o equaion (1), producing esimaes of model parameers, ~ θ (1), and heir associaedτ and rho saisics.

17 17 5. Repea seps 3 and 4 10,000 imes, ix i) giving { } ~ ( θ and he 10,000 associaedτ and rho values. The 95% confidence inerval for he ML esimaes of θ ~, and he τ and ( rho saisics consruced under he null hypohesis include 95% of he values of θ i= 1 and he associaed values ofτ and rho, respecively. ~ i ) The random disurbance erm, e, is no consruced as an i. i. d. process. This is because ha our MS model residuals exhibi clusers when ploed agains ime (figures 5 hrough 8). We model he MS residuals, e, wih an ARCH(q) process of he form q = α 0 + α i i i = 1 1 ε u ε, where u d iid ( 0,1) and E [ ε ε ] Var Cov i = 0 [ ε ε ] i = α + 0 i= 1 i [ ε, ε ] = 0, for i 1 i q α ε i ( 6) where q is seleced by he usual F es. This model has he feaure ha disurbances are heeroscedasic bu serially uncorrelaed, as he covariance is zero beween ε and for all i 1 in his model. We can esimae his ARCH(q) model using he following procedure (Greene, 1997; Bollerslev, 1986) ε i

18 18 1. Regress he squared ML residuals on heir corresponding four lagged values o give he firs esimaes of α i, denoed by a i, i = 0,1,..., 4.. Compue he condiional variances using ˆ σ a + a ε + a ε + a ε a ε. = Run he regression e σ 1 1 e 1 e 1 e 1 e 1 d + d1 + d + d 3 + d 4. σ σ σ σ σ = 0 3. The asympoically efficien esimaor of α is given by αˆ = a + d, where ˆα, a, d are all 1 5 vecors and he Asy. Var( ) = ( Z' Z ) [ e e e e ] = z. α, where Z : T 5 and The esimaed parameers of he ARCH(q) model are hen used o generae he disurbance erm in he boosrapping procedure. 4. Daa The daa we use in his sudy are price and ren indices for building srucures, as opposed o raw land, of Hong Kong and Seoul. There is no convincing reason o believe ha a bubble is more likely o exis in he price of a building han in ha of a plo of land. Empirical sudies on speculaive bubbles use daa on boh. In fac, following he argumen of Homer Hoy (1933), if a bubble exiss in he land price, i is likely o be ransmied o he housing price, and vice versa. We do have a few reasons for choosing daa from Hong Kong and Seoul. During he 1980s and 1990s, he wo ciies experienced dramaic propery-price swings (figure 1).

19 19 These swings were suspeced by praciioners and academics o be he resuls of speculaive bubbles. Alhough qualiy daa on he propery secor are available, here have been relaively few research papers devoed o he sudy of he speculaive bubble in he propery marke of Hong Kong. More sudies are available on propery-price bubbles in Souh Korea, he mehodologies used in hese sudies are, however, crude in general. (inser figure 1 here) In Hong Kong, he daa available for building srucures are divided ino four caegories: domesic premises, office, reail premises, and flaed facories. Xiao and Tan (006) show ha each of hese four secors was plagued by a periodically collapsing speculaive bubble during he period of ineres, and he paerns of he bubble movemens in hese four secors are very similar. Thus, we selec arbirarily Hong Kong office price index and is associaed ren series in his sudy. We will use in his sudy he Seoul housingprice index and is associaed ren series, as i is he only daa available for building srucures in Seoul. All daa series come from he CEIC daabase, a comprehensive source of economic saisics for Asian economies x. The series are monhly daa deflaed by heir respecive CPI xi. Each series of Seoul has 10 observaions running beween January 1986 and June 003. Each series of Hong Kong makes use of wo daa ses of differen frequencies: he firs se is a quarerly daa beween January 1984 and Sepember 000, and he second a monhly daa from January 1993 o April 003. In order o combine hem, we have convered he firs se ino

20 0 monhly daa by means of cubic splining. Thus, he firs half of each series of Hong Kong, running from January 1984 o December 199, consiss of he splined oupu from he firs daa se; he remaining half of he series is drawn from he second one. The combined raw daa se has 3 observaions for boh price and ren series. A plo of he daa in figure 1 shows ha he Hong Kong office price is highly volaile. The price index more han doubled in a mere 15 monhs beween Dec and March Anoher sharp increase of he price occurred in he firs half of 1994, wih an average value of 6.1% per monh. The price crashed afer July 1997, following he Asian financial crisis. By April 003, he price index sood only a abou % of is peak value (occurred in May 1994). Xiao and Tan (006) show ha a periodically-collapsing speculaive bubble is responsible for his observed behavior, wih he bubble peaked in May 1994 and again in June The Seoul housing price is less volaile (compared wih he Hong Kong office price). Afer rising for four years beween 1988 and 199, i declined gradually hroughou he remaining par of he 1990s. The price sared climbing up again in 001. Theories sugges ha, in he absence of a bubble, a price and is associaed ren should move more or less hand in hand. Bu his seems no he case in our daa (figure ). The price-ren raio of Hong Kong office behaved very much like he price series. I increased coninuously beween 1990 and 1997, wih a few emporary reversals. This raio crashed o is hisorical low afer he lae On he oher hand, he price-ren raio of Seoul housing was on a declining rend hroughou he sample period, wih only a few brief

21 1 episodes of reversals. Does he behavior of he price-ren raio imply he exisence of a speculaive bubble in he prices of ineress? This is he quesion he curren paper is ineresed in. (inser figure here) 5. Empirical Resuls The empirical lieraure suggess ha bubbles are shor-lived phenomena, and esimaions using a long sample may fail o capure a bubble if i does no las long enough (Rappopor and Whie, 1994; Kim and Lee, 000). We ake wo approaches o resolve his problem in his sudy. One is o apply he linear uni-roo ess o shorer samples (referred o as he shorer-sample approach), he oher is o model he daa generaing process as Markov-swiching AR(p) (referred o as he Markov-swiching approach). The Markov-swiching approach has he advanage ha he swiching of a sae is deermined endogenously, raher han pre-imposed by he researchers, as is he case in he shorersample approach. In aking he shorer-sample approach, for purpose of comparison we invesigae he enire sample (he long sample), as well as sub-periods of each sample (he shor samples), of each daa series. The relevan sample periods used for Seoul are 86:1 03:6 (he long sample), 86:1 91:6, 91:7 98:6, and 98:7 03:6 (he shor samples); hose for Hong Kong are 84:1 03:4 (he long sample), 84:1 94:5, and 94:6 03:4 (he shor

22 samples). The selecion of sub-samples is based on graphical and empirical evidences. Empirical sudies show ha propery price bubbles may have occurred in Seoul during he lae 1980s and since he lae 1990s, and in Hong Kong during he early and mid- 1990s (Kim and Sub, 1993; Chung and Kim, 004; Chan e al., 001, Xiao and Tan, 006). A breaking poin is chosen a where a price shows a dramaic change in rend xii. An AR(p) model is fied for each sample for he purpose of linear and Markov-swiching ADF es. The AR(p) model we esimae for each sample has a consan erm bu no rend, ha is, p ς i i 1 y = + y μ ρ Δy i + u, as he samples show no signs of rending. The = number of lags, p, is seleced using mehod described in secion 3.1. The seleced AR(p) model for each sample is hen subjeced o a Durbin-Wason es. If a model fails he DW es a he 5% level, we add one more lag and, in all cases excep one, he auocorrelaion found among residuals is adequaely removed. In he excepional case, ha is Hong Kong office prices beween 84:1 and 94:5, adding one or more lags o he F-es seleced model does no allow he residuals o pass he DW es a he 5% level. Hence we sick o he F- es seleced model, as he null hypohesis of no auocorrelaion can be acceped a he 1% level. The number of lags seleced for each sample is lised in Table 1. (inser Table 1 here) In he shorer-sample approach, we conduc he linear ADF es as well as he linear PP es for each sample. When he long sample of Seoul (86:1 03:6) is invesigaed, all es saisics accep he null hypohesis of no bubble (a 5% significance level is used for all

23 3 ess unless oherwise specified). Recall ha a no bubble conclusion is reached if price and ren behave in a similar manner, ha is, boh are saionary, or have one uni roo, or are explosive (Table ). The alernaive hypohesis is, however, no unique, due o he unobserved marke fundamenals. Bu one likely alernaive is he exisence of a speculaive bubble (posiive or negaive). For his reason, when we rejec he null of no bubble, we will conclude ha a bubble migh or possibly exis. (inser Table here) The same conclusion of no bubble is drawn for Seoul using he ADF τ and rho saisics in he sample period 86:1 91:6. However, he ADF F saisic for he price series rejecs he join hypohesis of zero consan and uni roo in he lef ail, while ha for he ren series acceps he null hypohesis. Looking a he plo of he price and he ren in his sample period (Figure 1) we realize ha here was a wide gap beween he price and he ren a he beginning of he sample. Tha gap narrowed down in he firs half of his sample period, and sabilized in he remaining half. The PP es srongly suggess a posiive price bubble migh have occurred in his period, wih an explosive price pah accompanied by a uni-roo ren pah. We adop he conclusion drawn from he PP es, as his es akes care of he possible heeroscedasiciy (Table ). For he period beween 91:7 and 98:6, boh he ADF τ and rho saisics show a negaive bubble migh have occurred in he Seoul housing price, as boh saisics ell us price had one uni roo while ren was explosive. These resuls are reconfirmed by he PP es. Bu

24 4 he ADF F saisic acceps he join hypohesis of zero consan erm and uni roo for boh price and ren (Table ). For he 98:7-03:6 sub-sample, boh ADF τ and rho saisics show ha a posiive bubble possibly exised in he Seoul housing price, as price was explosive while ren had one uni roo. Again, he PP es confirms hese resuls. The ADF F saisic acceped he null in he price bu rejeced i in he ren (Table ). As he F saisic is difficul o inerpre, we again follow he conclusion drawn by he PP es. When he baery of linear ADF and PP ess is applied o Hong Kong office prices and ren indices beween 84:1 and 03:4 (he long sample), all saisics, excep PP-rho, acceped he null hypohesis of no bubble (Table 3). The PP-rho suggess a negaive bubble possibly occurred in his sample period. The PP-τ saisic only acceps he null marginally a 5% wih a P-value equal o All saisics (excep he ADF F saisic, which acceped he null for boh price and ren), rejeced he null of no bubble for he sample period 84:1 o 94:5. One possible alernaive is ha a posiive bubble exised in his period. For he sample period 94:6 o 03:4, boh τ and rho saisics acceped he null. The F saisic, while acceping he join hypohesis of zero consan and uni roo for he price series, rejeced i for he ren series in he righ region. A ploing of he series shows ha prices in his period moved from abou 0% above he ren o abou 10% below i (Figure 1). We suspec ha boh posiive and negaive bubble migh have occurred in his sample period, and he conclusion from τ and rho saisics could bear he effec of averaging. (inser Table 3 here)

25 5 In shor, he linear ADF and PP ess accep he null hypohesis of no bubble for he Seoul housing price, when he whole sample period 86:1 03:6 is invesigaed. PP saisics srongly sugges ha a posiive bubble migh exis in he period 86:1 91:6. Boh ADF and PP ess sugges ha here migh be a negaive bubble exising during 91:7 98:6 and a posiive bubble during 98:7 03:6 (Table ). For he Hong Kong office prices series, when he whole sample period 84:1 03:4 is invesigaed, he linear ADF es acceps he null of no bubble, and he PP saisics sugges he possible exisence of a negaive bubble. Boh ADF and PP ess rejeced he null of no bubble for he period 84:1 94:5, indicaing a posiive bubble migh exis hen, bu acceped he null for he period 94:6 03:4 (Table 3). The linear uni-roo ess sugges srucural changes in our daa ses (Seoul housing prices and ren 86:1 o 03:6, Hong Kong office prices and ren 84:1 o 03:4). These ess show ha he Seoul housing price series followed a uni-roo process in he period 91:7 98:6, an explosive process in he periods 86:1 91:6 and 98:7 03:6; and he ren series followed a uni-roo process in he periods 86:1 91:6 and 98:7 03:6, bu an explosive process in he period 91:7 98:6. The ess indicae ha Hong Kong office price series followed an explosive process beween 84:1 and 94:5, bu a uni-roo process beween 94:6 and 03:4. For he ren, hough boh rho and τ saisics of ADF and PP ess sugges he series has one uni roo in he firs (84:1-94:5) and he second (94:6-03:4) sub-sample periods, he F saisic acceped he join hypohesis of zero consan and uni roo for firs sub-sample,

26 6 bu rejeced he null in he second sub-sample. These conclusions imply ha a wo-sae Markov-swiching AR(p) model may beer represen he daa generaing processes. In fiing a wo-sae Markov-swiching model o he Seoul housing prices and ren, we scanned for a proper saring poin in he daa se, so ha he resuling smoohed probabiliies show reasonable flucuaions. xiii The saring poin finally chosen for Seoul housing was 90:7. Therefore, he seleced daa se (90:7 03:6) roughly correspond o he second and hird sub-sample period (91:7 98:6 and 98:7 03:6). The linear ADF and PP ess accep he null of no bubble for his new daa se (Table ). The long sample for Hong Kong office secor is adoped, for reasonable movemens in smoohed probabiliies are obained wih his sample xiv. The acual observed and he fied values using he MS model are shown in Figures 3 and 4 (all values are in firs differences). We lis he maximum likelihood esimaes of he MS model parameers in Tables 4 o 7, alongside heir Hessian and Whie s -raios xv. Noice ha he Whie s -raios are much less significan han heir Hessian counerpars. (inser Figures 3 and 4, and Tables 4-7 here) Boh symmery and Wald (encompassing) specificaion ess show ha he wo-sae Markov-swiching model is acceped for all daa ses under consideraion (Table 9). The alernaive is a linear model. Table 8 records he sae ransiion probabiliies. These ransiion probabiliies show ha whenever he Seoul housing price reaches he sae, i

27 7 will swich back o sae 1 wih cerainy, as p1=1; sae 1 is persisen in Seoul housing ren, because p11=0.99; boh saes are likely o appear in Hong Kong office price, given he values of he swiching probabiliies p1 and p1; he ren series of Hong Kong office is more likely o swich o and say in sae, as p1 is as high as 0.86, and p= These observaions are more or less confirmed by he uncondiional sae probabiliies π, i = 1, and he second eigenvalues of he sae ransiion marix, λ. λ shows ha he i saes of he wo prices have negaively serial correlaion. Tha is sae 1 is likely o be followed by sae, and vise versa, while he saes of he oher wo ime series are posiively serially correlaed. Wih posiive auocorrelaion, a sae is likely o persis once he daa generaing process eners ha sae. (inser Tables 8 and 9) A plo of he smoohed probabiliies of he saes in Figure 9 echoes he predicions of sae ransiion probabiliies and hose of he eigenvalues of he ransiion marix. There are frequen swiches of saes in Seoul housing price. On he oher hand, sae one of is corresponding ren series is highly persisen. There are also frequen swiches of saes in Hong Kong office price, bu sae wo is more persisen in is associaed ren series. Combine he above observaions wih he resul of he Markov-swiching ADF es (Table 14), we can conclude ha a bubble, if exiss in he price of ineress, is periodically collapsing. Furhermore, he Markov chain we esimaed is ergodic, because one eigenvalue of he sae ransiion marix is uniy and he oher is inside he uni circle.

28 8 Hence, he long erm forecas of he Markov chain is given by he uncondiional sae probabiliies (Hamilon, 1994a, p 68). The parameer esimaes of he ARCH model for he MS model residuals (along wih heir -raios) are summarized in Tables 10 and 11. No all -raios of he esimaed parameers are significan, bu he F es shows ha he seleced models are joinly significan a he 5% level (Table 1). The es saisics in Table 13 for ARCH effec, TR, are highly significan in all cases ( R is he goodness of fi measure of he regression, T he sample size). We plo he squared ML residuals and heir corresponding prediced values using he esimaes of ARCH model in Figures 5 hrough 8. These plos demonsrae ha he esimaed models capure quie well he paerns of he MS residuals. Thus, we incorporae ARCH disurbances ino he boosrapping procedure in generaing he disribuion of τ and rho saisics for he Markov-swiching ADF es. For each series, 10,000 replicaions are used. We hen compare heτ and he rho saisics of he MS models wih hese simulaed disribuions, and es he null hypohesis of uni roo (Table 14). (inser Tables 10-13, and Figures 5-8 here) A problem arises when we ry o draw conclusions. The wo saisics, τ and rho, sugges differen behavior of he series on hree occasions: Seoul housing prices and ren in sae wo, and Hong Kong Office price in sae wo (Table 14). Therefore, if τ is used, he

29 9 conclusion would be ha, for mos of he period beween July 1990 and he end of 199, Seoul housing prices migh conained a posiive speculaive bubble. This bubble disappeared hereafer bu reappeared in lae 001 (Table 15, Figure 9). This conclusion is consisen wih he observaions from he plos in Figure 1, and also parly compaible wih he resuls from he linear uni-roo ess (Table ). The τ saisic idenified four possible bubble episodes in Hong Kong office prices: lae 1987 o lae 1989, early 1994, lae 1997 o lae 1998, and early 001 (Table 15, Figure 9). These are largely consisen wih he facs shown in he plos of Figure 1, and wih he conclusions of Chan e al. (001) and Xiao and Tan (006). (inser Tables 14 and 15 and Figure 9 here) If rho is used, hen he conclusion is ha Seoul housing prices migh conained a posiive bubble hroughou mos of he period beween July 1990 and he end of 199. This possible-bubble disappeared hereafer unil lae 1997, when i reassered iself for a brief period (abou a year), and revived again in lae 001. These accouns are mosly compaible wih he conclusion ofτ, excep for he period involving lae 1997 and early Bu he rho side of he sory abou Hong Kong is less ineresing. I says ha he ciy migh have experienced a posiive bubble hroughou he enire sample period (Table 15). 6. Conclusions

30 30 A Speculaive bubble is likely o occur in an asse, such as a real esae, whose fundamenal value is difficul o assess. The real-esae marke is more prone o speculaive bubbles han oher ypes of asse marke for reasons such as resricions in supply and insiuional arrangemens (Xiao and Tan, 006). Idenifying a speculaive bubble is neverheless a horny issue, mainly because of he unobservable marke fundamenals. Hamilon and Whieman (1985) and Diba and Grossman (1988) sugges ha saionariy ess may be used o deec an explosive raional speculaive bubble. These ess need no preclude he influences of unobserved marke fundamenals. If he firs-difference of a dividend and hose of unobservable fundamenals are saionary in mean, and if no raional bubble exiss, hen he firs difference of he associaed sock price mus be saionary. Differencing a sock price a finie number of imes would no render i saionary, however, if i conains a raional bubble. Due o he possible presence of unobserved variables, finding ha he firsdifference of a sock price is no saionary does no auomaically esablish he exisence of a raional bubble. However, he converse inference is possible. Tha is, evidence ha he firs-difference of a sock price has a saionary mean would be evidence agains he exisence of a raional bubble in ha price. The empirical lieraure suggess ha bubbles are shor-lived phenomena, and saisical ess using a long sample may fail o capure a bubble if i does no las long enough (Evans, 1991; Rappopor and Whie, 1994; Kim and Lee, 000). In his paper, we ake wo approached o deal wih his problem. One is o divide a long sample ino shorer

31 31 samples, and apply he linear Phillips-Perron and he augmened Dickey-Fuller ess o hese samples (referred o as he shorer sample approach); he oher is o deploy a Markov-swiching AR(p) model in he ADF es (referred o as he Markov-swiching approach). The Markov-swiching approach is more appealing han he shorer sample approach ex ane. This is because ha he swiching (breaking) poins are deermined endogenously, raher han pre-imposed by he researchers. Neverheless, he wo approaches in our sudy give compaible resuls ex pos, wih he Markov-swiching approach more precise in iming a bubble, if i exiss. By breaking Seoul daa se (1986:1-003:6) ino hree sub-samples, we idenified wih he linear ADF and PP ess ha a posiive bubble possibly exised beween 1986:1 and 1991:6, and beween 1998:7 and 003:6, and a negaive bubble possibly occurred beween 1991:7 and 1998:6. The Markov-swiching ADF es does no poin o he exisence of a negaive bubble in Seoul housing price. I neverheless idenified hree episodes in which a posiive bubble migh be acive: from July 1990 o he end of 199, beween he lae 1997 and he lae 1998, and beween he lae 001 and he early 003. For he purpose of linear uni-roo ess, we break he Hong Kong daa se ino wo subsamples. The linear ADF and PP ess rejeced he null of no bubble. Therefore, a posiive bubble migh have occurred for he firs sub-sample (1984:1-1994:5). We acceped he null in he second sub-sample (1994:6-003:4). The Markov-swiching ADF es

32 3 idenified four possible bubble incidences beween 1984:1 and 003:4. They occurred beween he lae 1987 and he lae 1989, in he early 1994, beween he lae 1997 and he lae 1998, and again in he early 001. We suspec, however, he las one migh be a false alarm, given he behavior of he price and he ren in his period (Figure 1). Our findings are consisen wih hose in Chan e al. (001) and in Xiao and Tan (006).

33 33 Table 1. Number of Lags Seleced in an AR(p) Model The AR(p) model we esimae for each sample has a consan erm bu no rend, ha is, p ς i i 1 y = + y μ ρ Δy i + u, as he samples show no signs of rending. The number of = lags, p, is seleced using mehod described in secion 3.1. Seoul Housing Secor Hong Kong Office Secor Sample periods 86:1 o 03:6 90:7 o 03:6 86:1 o 91:6 91:7 o 98:6 98:7 o 03:6 84:1 o 03:4 84:1 o 94:5 Price Ren :6 o 03:4

34 34 Table. Linear ADF and PP Uni Roo Tess for Seoul Housing Price and Ren 1. Phillips Perron Tes is adoped in drawing conclusions when conflicing resuls arise beween PP and ADF ess and PP srongly sugges H0 or H1, as PP akes care of he Heeroscedasiciy exising in our daa.. * means significan in he lef ail. 3. ** means significan in he righ ail.4: in all cases, he Durbin-Wason saisic accep H0: no serial correlaion among residuals. ADF Tess H0: uni roo H1:no uni roo (5% level) PP Tess H0: uni roo H1:no uni roo (5% level) ADF Tess H0: uni roo H1: no uni roo (5% level) PP Tess H0: uni roo H1:no uni roo (5% level) 1986:1 o 003:6 1990:7 o 003:6 Price Ren price ren DW Rho (Pr<Rho) τ (Pr<τ ) F (Pr>F) Rho (Pr<Rho) τ (Pr<τ ) (0.614) (0.633) 0.86 (0.851) -1.5 (0.861 ) (0.765) -.05 (0.078) -.35 (0.157).91 (0.330) -4.8 (0.450) (0.414) (0.331) -.83 (0.057) 4.31 (0.070) -.35 (0.734) -.4 (0.137) (0.080) -.16 (0.0).38 (0.465) -.31 (0.740) (0.77) 1986:1 o1991:6 1991:7 o 1998:6 1998:7 o 003:6 price ren price ren price ren DW Rho (Pr<Rho) τ (Pr<τ ) F (Pr>F) Rho (Pr<Rho) τ (Pr<τ ) (0.918) -0.9 (0.90) 0.47* (0.957) 0.3** (0.966) 0.14** (0.967) (0.587) (0.549) 1.5 (0.687) -.34 (0.73) -1.5 (0.647) (0.777) (0.565) 4. (0.079) (0.805) -.13 (0.34) 7.7** (0.999) 1.66** (0.999).69 (0.395) 8.87** (0.999) 3.86 ** (0.999) 0.68** (0.980) 0.41** (0.98) 1.57 (0.675) 1.30** (0.99) 1.37** (0.999) (0.847) (0.45) 4.84* (0.046) -1. (0.861) -1.6 (0.468)

35 35 Table 3. Linear ADF and PP Uni Roo Tess for Hong Kong Office Price and Ren 1. Phillips Perron Tes is adoped in drawing conclusions when conflicing resuls arise beween PP and ADF ess and PP srongly sugges H0 or H1, as PP akes care of he Heeroscedasiciy exising in our daa.. * means significan in he lef ail. 3. ** means significan in he righ ail.4: in all cases, he Durbin-Wason saisic accep H0: no serial correlaion among residuals. ADF Tess H0: uni roo H1:no uni roo (5% level) PP Tess H0: uni roo H1:no uni roo (5% level) 1984:1 o 003:4 1984:1 o1994:5 1994:6 o 003:4 Price Ren price ren price ren DW Rho ** (Pr<Rho) (0.371) (0.85) (0.993) (0.380) (0.75) (0.616) τ ** (Pr<τ ) (0.480) (0.443) (0.990) (0.336) (0.430) (0.115) F ** (Pr>F) (0.743) (0.711) (0.669) (0.513) (0.061) (0.038) Rho **.49 ** (Pr<Rho) τ (Pr<τ ) (0.83) (0.797) (0.951) (0.948) (0.999).06 ** (0.999) (0.885) (0.791) (0.768) (0.408) (0.771) (0.015)

36 36 Table 4. Markov Swiching Model Parameer Esimaes for Seoul Housing Price 1. iner j: ML esimae of he consan erm in sae j;. lpj: ML esimae of coefficien on y 1 for sae j; 3. ldpij: ML esimae of coefficien on Δ in sae j. 4. (H): raio obained from Hessian; y i 5. (W): raio obained from Whie covariance; 6. sig j: variance esimae in sae j; 7. Pi j: uncondiional sae probabiliy in sae j. These noaions are applicable hroughou his chaper. Sae one Sae wo Parameer (H) (W) Parameer (H) (W) iner iner lp lp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp Sig Sig Pi Pi Table 5. Markov Swiching Model Parameer Esimaes for Seoul Housing Ren Sae one Sae wo Parameer (H) (W) Parameer (H) (W) iner iner lp lp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp Sig Sig Pi Pi

37 37 Table 6. Markov Swiching Model Parameer Esimaes for Hong Kong Office Price Sae one Sae wo PARM (H) (W) PARM (H) (W) iner iner lp lp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp ldp Sig Sig Pi Pi Table 7. Markov Swiching Model Parameer Esimaes for Hong Kong Office Ren Sae one Sae wo Parameer (H) (W) Parameer (H) (W) iner iner lp lp ldp ldp ldp ldp ldp ldp sig sig pi pi

38 38 Table 8. Sae Transiion Probabiliies 1. pij is he probabiliy of sae j in ime +1, given he ime sae is i.. π i, i = 1, is he uncondiional probabiliy of sae i. 3. λ i, i = 1, is he eigenvalue of he ransiion marix. 4.he average duraion of sae i (i=1,) is given by ( 1 p ) 1 (Raymond and Rich, 1997, page 0). Seoul Housing Secor Hong Kong Office Secor Price Ren Price Ren Parameer (H) Parameer (H) Parameer (H) Parameer (H) p p p p π π λ λ ave. duraion sae 1 (monhs) ave. duraion sae (monhs) ii

39 39 Tesing Significance of Markov Swiching Model Table 9. Tesing for he Significance of he Markov Swiching Model (i) The symmery es saisic has a sandard normal disribuion. (ii) The Wald saisic has a () 1 χ disribuion. The null disribuion of he saisic is generaed using boosrapping wih 10,000 replicaions. Symmery es: H0:p11=p (linear model) H1:p11<p (MS model) Wald Tes H0: MS model H1: linear model Saisic (Prob(Z z)) Conclusion Wald saisic (Prob ( χ n (1) ) c) ) Conclusion Seoul Housing Secor Hong Kong Office Secor Price Ren Price Ren (0.996) (1.000) (0.000) (0.000) rejec H0 rejec H0 rejec H0 rejec H0 in favor of in favor of in favor of in favor of H1: MS H1: MS H1: MS H1: MS model model model model (0.750) accep H0 MS model (0.500) accep H0 MS model (0.050) accep H0 MS model (0.750) accep H0 MS model

40 40 Table 10. Parameer Esimaes of ARCH Model for MS Residuals of Seoul Housing Price and Ren Sae one Sae wo Parameer raio Parameer raio Price a a a a a a Ren a a a a a Table 11. Parameer Esimaes of ARCH Model for MS Residuals of Hong Kong Office Price and Ren Sae one Sae wo Parameer raio Parameer raio Price a a a a a Ren a a a a a Table 1. F Tes for Join Significance of ARCH Model of MS Residuals All es saisics are significan a 5% level, showing he models are acceped. Sae one Sae wo Seoul Housing Secor Hong Kong Office Secor Price Ren Price Ren F df (5, 150) (4, 151) (4, 18) (4, 3) F df (5, 150) (4, 151) (4, 18) (4, 3)

41 41 Table 13. TR Tes For ARCH Effec in MS Residuals The es saisic TR has χ ( ) disribuion. All saisics are significan a 5% level, indicaing q he ARCH effec is indeed presen in he MS residuals. Seoul Housing Secor Price ( q = 5) Ren ( q = 4) Hong Kong Office Secor Price ( q = 4) Ren ( q = 4) Sae one Sae wo