Asian Economic and Financial Review

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1 Asan Economc and Fnancal Revew ISSN(e): /ISSN(p): URL: THE POWER OF A LEADING INDICATOR S FLUCTUATION TREND FOR FORECASTING TAIWAN'S REAL ESTATE BUSINESS CYCLE: AN APPLICATION OF A HIDDEN MARKOV MODEL Yun-Lng Wu --- Cheng-Huang Tung Chun-Chang Lee 3 Lecturer, Department of Real Estate Management, Natonal Pngtung Unversty, Tawan 2 Assocate Professor, Department of Computer Scence and Informaton Engneerng Natonal Pngtung Unversty, Tawan 3 Professor, Department of Real Estate Management, Natonal Pngtung Unversty, Tawan ABSTRACT Ths paper employ the dscrete hdden Markov model (HMM) n order to capture nformaton about the Markov swtchng model s nner states that s not drectly observable, and to pre-detect the real estate busness cycle s volatlty trend. The emprcal results show that ths HMM can capture the asymmetry n the duraton of states. Compared wth the real estate leadng ndcator announced by the Tawan Real Estate Research Center, ths HMM yelds the same results n terms of forecastng the trends of cycle fluctuatons. The explanatory power of the HMM n 4-steps out-of-sample forecastng s supported both conceptually and methodologcally. Keywords: Real estate cycle, Real estate cycle leadng ndcator, Asymmetry, Duraton, Hdden Markov model, Tawan real estate research center. Receved: 9 July 206/ Revsed: 2 September 206/ Accepted: 9 October 206/ Publshed: 5 November 206 Contrbuton/ Orgnalty Ths paper uses the HMM to capture the optmal path of state transton to observe the trends of fluctuatons of out-of-sample data. The results confrm that trends of the real estate busness cycle fluctuatons are asymmetrc and that the average duraton of recesson perods s longer than that of expanson perods.. INTRODUCTION The long-term trends of the busness cycle n the real estate market and how to predct fluctuatons n that busness cycle are subjects of great topcal concern n macroeconomc analyss and decson makng. Tvede et al. (2008) found that, accordng to a conservatve estmate, constructon actvtes account for 0% of global GDP, of whch one ffth comes from stable market fluctuatons n the publc sector and the rest results from sgnfcant perodc busness cycle fluctuatons. In other words, when actvty n the real estate market decreases by 33%, there wll be a 3% decrease n GDP, wth that decrease not ncludng the wealth spllover effects caused by the plunge n real estate prces. In addton, market observatons over the past several decades and even century suggest that there are repeated busness cycle fluctuatons n the real estate markets around the world and that real estate market Correspondng author DOI: /journal.aefr/207.7./ ISSN(e): /ISSN(p):

2 fluctuatons often lead to consderable fluctuatons n raw materal prces. Hence, montorng the trend of real estate market cycle fluctuatons s an mportant actvty n the feld of macroeconomc research. Prevous studes on busness cycles can be categorzed n two ways. Frst, n prevous studes amed at analyzng the real estate busness cycle and ts causes, researchers have appled proxy varables such as vacancy rates (Grenader, 995; Gordon et al., 996) and real estate rental growth rates (Mueller, 999) n ther analyses. Second, the relatonshp between resdental housng market prce and quantty (Edelsten and Tsang, 2007) mbalances n real estate market supply and demand (Roulac, 996) nvestment expectatons (Huang, 203) and the mortgage loan supply and asset prce relatonshp (Arsenault et al., 203) have been used to descrbe the causes of and changes to the real estate busness cycle. However, those studes only descrbe the busness cycle fluctuatons or causes of fluctuatons wthout comprehensvely explanng the mplcatons of the real estate busness cycle. Moreover, some studes have dscussed the real estate busness cycle and macroeconomc relatonshps, or the mutual nfluences of relevant ndustres (Prtchett, 984; Clayton, 996; Goetzmann and Wachter, 2000; Kan et al., 2004; Leung, 2007; Huang, 203). However, research on the composte ndcators of the real estate busness cycle s stll relatvely rare. In order to drectly quantfy real estate busness cycle fluctuatons, recent studes have adopted the composte real estate ndex. Krystalogann et al. (2004) used the leadng composte ndex to predct the features and performance of the Brtsh commercal real estate cycle. Lee et al. (2009) appled the real estate leadng composte ndex and Markov-swtchng model to explore the change n the real estate busness cycle turnng ponts. They found that the performance of the leadng composte ndex for dentfyng real estate busness cycle turnng ponts s good. Although state transtons were excessvely frequent and there were some dspartes n terms of the duratons of busness recesson and expanson perods, the model ftness performance and applcaton were both found to be good. It was concluded that the composte ndex s superor to the sngle seres for understandng busness cycle fluctuatons. Many recent studes on macroeconomc cycles or real estate busness cycles have focused on the observaton of busness turnng ponts (Scott and Judge, 2000; Baum, 200; Barras, 2005) and busness duraton perods. For the measurement model, the Markov swtchng model proposed by Hamlton (989) has been the most wdely appled, havng been appled n varous studes such as those of Hamlton (989); Pelagatt (200); Krolzg (997) and Cruz (2005). The man advantage of the Markov-swtchng model s that t can capture the random changes of the busness state over tme through a group of unobserved state factors n the model settng mechansm. Moreover, the model can slghtly modfy the model pattern wth dfferent data patterns, such as the Markov model of ntercept changng wth state (Hamlton and Susmel, 994) the Markov model of varance changng wth state (Ca, 994; Hamlton and Susmel, 994; Gray, 996) or a varety MSVAR (Markov-swtchng vector auto-regresson models) (Krolzg, 997). Hence, the applcaton and analyss process of the model s more flexble and practcal. The Markov-swtchng model s manly domnated by a group of unobservable states and the random transton jumpng mechansms between those states. The state settng s the unobservable varable and s able to descrbe the features of the state random change. However, n the preset vewpont, f the state s set as unobservable, t s not consstent wth the observable data. As such, durng the analyss process, t may be able to capture the state of random change but not able to observe state features, thereby losng a more accurate estmaton of state path swtchng probablty. As a result, t may lead to estmaton error n estmatng the trend of busness cycle fluctuatons. The state random swtchng theory of the Markov-swtchng model n the framework of probablty theory, namely the hdden Markov model (HMM), can be regarded as a double-embedded stochastc process (Huang et al., 200). A complete HMM conssts of two random processes: one layer s the hdden unobserved state swtchng seres correspondng to a pure frst-order Markov process, whle the other layer s the observable random seres n the hdden state. In the Markov - swtchng model, although random state swtchng seres cannot be observed, through the observable seres of state, a hdden Markov model can predct the orgnally unobservable state swtchng seres 82

3 probablty. As the hdden state seres s transformed nto the observable state feature seres, ths model s known as the HMM. The HMM was frst appled n the late 970s for the dentfcaton of sound sgnal fluctuatons (Baum and Petre, 966) and s now wdely used n engneerng, genetcs and other felds, such as communcatons audo classfcaton and speech recognton, wth consderable research achevements. Accordng to Hassan and Nath (2005) the HMM model has the followng advantages: () a strong statstcal bass; (2) the ablty to handle new nformaton robustly; (3) the capacty to calculate and forecast smlarty modes more effcently. In recent years, the HMM model has been appled n economc, fnancal, and management felds, ncludng economcs n general (Leng and Wang, 204) and the study of fnancal seres fluctuatons more specfcally (Gregor and Lenglart, 2000; Hassan and Nath, 2005; Korolkewcz and Ellott, 2008; Oguz and Gurgen, 2008; Lu, 200; Roamn et al., 200; Zhu and Cheng, 203). In the management feld, t has been used to study customer relatons management (Bouchaffra and Tan, 2004; Shen and Zhao, 2006; Netzer et al., 2008; Sepdeh and Aaghae, 20) and onlne purchasng behavors (Wu et al., 2005). By followng the current lterature on the forecastng of real estate busness cycle fluctuatons and measurement models, based on the developed Markov-swtchng model, ths study uses the real estate busness cycle leadng ndcator composte ndex (footnotes ) released by the Tawan Real Estate Research Center to establsh a dscrete hdden Markov-swtchng model (dscrete HMM; HMM wth dscrete output observatons), n order to capture unobservable state mplcatons. It s expected that the model wll be able to predct the trends and changes n the real estate busness cycle, as well as detect busness turnng ponts and state duraton perods. The remander of ths paper s organzed as follows. Secton 2 ntroduces the HMM theory and model parameter estmaton; Secton 3 explans the data descrpton and analyss, as well as the features extracted from the economc cycle leadng ndcators seres to comply wth HMM model settng prncples and deas; Secton 4 dscusses the emprcal results analyss; Secton 5 offers conclusons. 2. HIDDEN MARKOV MODEL AND MODEL PARAMETER ESTIMATION In prevous studes, economsts have developed a seres of non-lnear econometrc models by descrbng the changng process of lnear model parameters. The state transton model (regme swtchng model) can process multple jumpng processes of the parameters (footnotes 2). In order to observe the real estate busness cycle leadng ndcator s trend characterstcs and the predctablty of seres fluctuatons, ths paper uses a non-lnear HMM model as the measurement research tool. The model and parameter estmaton results are as explaned n the sectons that follow. 2.. HMM In terms of the overall parameters and random concepts, the HMM can be dvded nto two parts. The frst part can be descrbed as a Markov chan to generate hdden state random seres; the second part of the random process s descrbed by the dstrbuton of the observaton varable probablty n the state. The basc elements are as shown below (Huang et al., 200; Kosknen and Öller, 2004): ).Hdden state set: s,s,..., S 2 s N, where N s the state number and t 2). In-state output observaton seres set: o,o,..., (footnotes 3) 3). State transton probablty dstrbuton:, 2 om a j q s the state at tme pont t., where M s the number of observatons n a state. 83

4 where a q S q S j swtchng from t j t,, j N, and a j 0, a j represent the probablty of S to S j and from t to t. 4). Under the condtons of state where, b m N S, the output observaton varable probablty dstrbuton s: b, s the output observaton spatal sample n the state, and N, m O f O q S m t t m ;, O t s the output observaton random varable at tme t, whch can be a unvarate varable or multple varable. Ths paper sets the output observaton value as a unvarate dscrete number. 5). Model ntal state probablty dstrbuton:, N, where q S To sum up, the parameters needed to descrbe a complete HMM model are general lterature n the form of, S,,, t, whch s smplfed n,. In other words, the HMM model can be descrbed by three settng parameters ncludng the ntal state probablty dstrbuton, the hdden state transton probablty dstrbuton and the n-state output observable seres probablty dstrbuton. The model makes two major assumptons. The frst s the frst-order Markov assumpton, whch assumes that the nter-state swtchng probablty s related to the ntal probablty and the last term probablty. The current term probablty s subject to the nfluence of the prevous term probablty. The nter-state swtchng probablty s nontme-varyng as shown n Eqs. () and (2): T Q Pq,...,q t,...,q T Pq Pq t q t, t 2 a Pq jq,, j N P j () t t (2) The second assumpton s the n-state observatons observatons and state s as shown n Eq. (3). P o m (output-ndependent) assumpton, whch assumes that qt are dependent but that the observatons are mutually ndependent. The condtonal equaton T OQ,λ Po,..., o,..., o q,..., q,..., q, λ b (3) m M t T Π q o t m t In parameter settng, the HMM model s flexble and varyng n pattern (footnotes 4). The proposed model pattern settngs are as shown n Fgure. There are three states ncludng S, S 2, and S 3, and the n-state observatons are dvded nto seven types by feature extracton method nto...7. The nter-state swtchng 84

5 probablty j can only delay the current term or swtch forward over tme to be set as, 2, 22, 23, 33 and 3. The HMM pattern s a left-to-rght pattern. By swtchng probablty, the correspondng dfferent busness state duraton perods can be nferred. Wth the three-state settngs as an example, the swtchng probablty s a 3 3 matrx and the dagonal estmate s the probablty of the current term state remanng as the prevous term state (, 2, 3 ), respectvely, beng, 22 and 33. For ths reason, t s nferred that the state duraton perods are ( ) ˆ, ( ˆ ) and ( ˆ ) The left-to-rght pattern s a type of generalzed HMM pattern. The generalzed HMM model pattern means that the nter-state swtchng probablty can swtch randomly wthout lmtaton. The man dfference between the left-torght pattern and the generalzed pattern s that the nter-state swtchng probablty can delay the current term or swtch forward only (.e., t cannot swtch backward). The concept of such a settng manly comes from the noton of movng forward n tme and the dea that the state swtchng path wll naturally jump forward. As shown n the above HMM model, the observaton output sample space and probablty dstrbuton n hdden space may have dfferent settng patterns accordng to the lterature. Ths paper emphaszes the capablty of the HMM model to capture the nter-state swtchng. Hence, t s set as a unvarate dscrete observaton data pattern. Fgure-. The HMM model pattern proposed n ths study Source: Compled by ths study HMM Model Parameter Estmaton Method There are three basc problems to solve by usng the HMM model to obtan the optmal parameters, namely, model tranng, hdden state optmal path estmaton, and the computaton of the maxmum lkelhood estmate (Baum and Petre, 966). The estmaton method s brefly descrbed as follows: ). Computaton of maxmum lkelhood estmate ˆ ML Wth a gven model optmzaton parameter,,, the computaton of the maxmum lkelhood value of the observaton seres n the state O o,..., s used to compute the lkelhood functon O or logarthm lkelhood functon f O o m ln value; ndcatng the model parameter,, f value ˆ ML smulates the 85

6 observaton seres o,..., o m Asan Economc and Fnancal Revew, 207, 7(): 8-98 O accuracy. Ths study uses the forward-backward procedure/algorthm to conduct the optmzaton of the HMM model to obtan the optmal soluton. 2). Hdden state optmal path estmaton Wth a gven model parameter,, most possble hdden state optmal path Q q,..., ˆ and observaton seres O o,..., ML q t o m, the estmaton of the s used to estmate the optmal possble state path of the observaton seres. Ths paper uses a Vterb algorthm for estmatons. 3). Model learnng/tranng Model tranng s requred for the soluton of the model parameter estmaton. After settng the HMM ntal model and the observable sample seres for the computaton state O o,...,, o m ^, smulates the n-state observable sample seres to obtan ML arg max f O ˆ ML the optmzaton model parameter:,,, computng the model parameter whch determnes. The EM algorthm or the Baum-Welch algorthm can be used for the estmatons. The Baum-Welch algorthm s used heren. In the evaluaton of the model predctablty, ths paper uses expected loss functon prncples of turnng pont error (TPE) and mean squared error (MSE) to evaluate the forecastng errors between the actual observaton values of the leadng ndcator and the estmated predcted values by usng the HMM model to llustrate the performance of applyng the HMM model n predctng the trends of the real estate busness cycle leadng ndcator fluctuatons. The two nspectons can smply and drectly compute the expected loss or expected cost error between the predcted values and the actual values. 3. DATA SOURCE AND ANALYSIS The data used n ths study were sourced from the real estate busness cycle leadng ndcator composte ndex values from 97 Q to 2009 Q4 released by the Tawan Real Estate Research Center, a perod ncludng 56 samples of seasonal data. The Center releases data regardng the trends of the real estate busness cycle va quarterly reports and elaborates on the complaton of economc cycle leadng ndcator seres n the appendces of those quarterly reports. As the seres data contaned n the economc cycle leadng ndcators have dfferent characterstcs, after the verfcaton of the quarterly and trend statstcs method, all the contaned seres have been de-trend adjusted, whle some data are seasonally adjusted by X2 software. The constructon stock prce ndex representng nvestments, the constructon loan credt balance change representng producton, and the CPI representng transactons have been adjusted accordng to the trend. In addton, the GDP and monetary supply values n the nvestment dmenson have also been revsed accordng to the de-trend adjustment and seasonal adjustment. Accordng to the prevous lterature, when analyzng a relevant tme seres by usng an emprcal model, the data should consst of a statonary seres (footnotes 5) for parameter estmaton and statstcal nference. Ths paper conducts the ADF (augmented Dckey-Fuller test) unt root test of economc cycle leadng ndcators n advance to determne whether the seres data are statonary. On the assumpton of null hypothess wth a unt root, the ADF test confrms that the real estate busness cycle leadng ndcator has a unt root, suggestng that seres data are nonstatonary. After the frst dfferentaton of the data, ths study conducts the ADF unt root, fndng that the economc cycle leadng ndcator seres at the % sgnfcance level rejects the null hypothess after the frst dfferentaton. In 86

7 other words, after the frst dfferentaton, the economc cycle leadng ndcator seres s statonary. Therefore, ths study adopts the frst dfferentaton seres of leadng ndcator for analyss. Regardng the settng of the HMM model, the number of states should be set n the begnnng. Accordng to the purpose and pre-concept, the number of states can be ncreased or decreased ndependently. Most of the prevous studes on the overall busness cycle or real estate busness cycles have descrbed busness cycle fluctuatons n terms of two states (Lee et al., 2009). Some research has used three states. For example, Cruz (2005) ponted out that the expanson and shrnkage stages of the three-state busness cycle model descrbed n prevous studes was relatvely smlar to the turnng ponts released by the NBER (Natonal Bureau of Economc Research). The proposed model sets three busness states, namely, market recesson, market of no sgnfcant change, and market expanson. The settng of two states may result n excessve range to capture the feature nformaton n the HMM model. In the estmaton of the state swtchng probablty, nter-state swtchng wll be too slow to lead to an excessvely lengthy duraton perod for each state. As a result, t s dffcult for the HMM model to capture n-state features. Second, wth regard to the observable nformaton value of each n-state type and the correspondng unvarate busness leadng composte ndex, the unvarate varable s set as dscrete. Ths study uses the feature extracton method to process by dvdng the nformaton of the economc cycle leadng ndcators seres by change nto {sgnfcant ncrease, general ncrease, modest ncrease, small change, modest decrease, general decrease, sgnfcant decrease) output sgnals. Therefore, from 97 Q2 to 2009 Q4, the leadng ndcators dscrete observable data were extracted. Ths paper apples the feature extracton method n a manner smlar to the concept of complng busness countermeasure sgnals (footnotes 6) to categorze seres nto a number of features by change. In addton to the emphass on the applcaton of the HMM model n the forecastng of the trends of busness cycle fluctuatons, t s expected to hghlght the market state fluctuatons degree and data swtchng features ncludng busness data expanson, recesson, and others. Huang et al. (200) extracted Dow Jones ndustral average ndex features to represent the bull market, bear market, and fluctuatons market of no change. Unlke the fve types of busness montorng ndcators, that study extracted seven types of trend sgnals of features fluctuatons, expectng to further categorze the fluctuaton degree of the leadng ndcator to observe more characterstcs of the trends of the busness cycle fluctuatons. As mentoned above, Huang et al. (200) argued that the contnuous data contan a consderable degree of nformaton despte the feature extracton of dscrete data. However, f they are further dvded nto nne types or more, the value of each change s too small and the dstngushng of features wll be too nsgnfcant. As a result, t may result n excessvely frequent jumpng of state swtchng path probablty. Therefore, ths paper sets seven types of features. To measure the forecastng capabltes of the HMM model aganst the trends of fluctuatons, ths paper dvdes the busness leadng composte ndex seres (97 Q2~2009 Q4) nto two groups. One group conssts of the n-sample (97 Q2~2008 Q4) samples; ths group, wth a total number of 52 samples, s used as the tranng data for parameter estmaton. Another group s the out-of-sample (2009 Q~2009 Q4) observaton data for 4-step-ahead forecastng. In addton, by usng the rollng wndow samplng method (footnotes 7), wth 4 seasons as a unt tme length, the sample data are sorted out by tme. The one-term lag data start wth 97 Q2 and develop at the nterval of one season to 2008 Q4 to result n a total of 48 pattern tranng samples. The rollng wndow for the preprocessng of the observaton samples can ncrease the number of samples to mprove the accuracy of the parameter estmaton. Meanwhle, t s expected to smulate the smlar fluctuaton path pattern seres by usng the n-sample samples as the four-term pattern samples n the forecastng of out-of-sample 4-step-ahead forecasts to mprove forecastng accuracy. 87

8 4. ANALYSIS OF EMPIRICAL RESULTS Ths secton contans two parts. The frst part s the descrpton and analyss of the HMM model estmaton and 4-step-ahead forecasts. The second part further elaborates on the forecastng capablty of the applcaton of the model n the trends of the economc cycle leadng ndcators out-of-sample fluctuatons. 4.. Analyss of HMM Model Estmaton Results The settngs of the model ntal parameters are shown n Table. The ntal settngs are the same as the model pattern shown n Fgure : left-to-rght pattern and state swtches wth tme n a forward and rreversble away. Hence, regardng the nter-state swtchng probablty of a 3, a 2 and a 32, before parameter estmaton, the probablty s set as 0, the rest (, a a, a, a,, and a a3 33 ) are preset as 0.5. Regardng ntal probablty, before tranng, the bult-n software program uses an automatc settng that s equal to the aj probablty. The settngs of the overall state swtchng probablty a are smlar to the assumpton condtons n Eqs. () and (2), that j s, the frst-order Markov assumpton and the assumpton of non-tme-varyng nter-state swtchng probablty. As shown n Table, the ntal settng of each n-state observaton value b m probablty s 7, that s, the probablty of each n-state observaton value before parameter estmaton s the same, whch s consstent wth the concept of the random appearance of samples. For model tranng and program computaton, ths paper uses the bult-n HMM program of MATLAB 7.0 software to obtan the parameter optmal solutons by teraton computaton of the logarthm maxmum lkelhood value. Fgure 2 llustrates the teraton tme s curve of the logarthm maxmum lkelhood durng the optmal parameter estmaton process. When the tolerance rate s , the model parameter estmaton converges when the teraton tmes are 432 durng the model parameter estmaton, and the logarthm maxmum lkelhood values are Table-. Intal parameter settngs of the model before tranng ( ) Parameter Intal settng (before tranng) State State 2 State State a j State State Sgnfcant General Modest Small Modest b m General ncrease ncrease ncrease change decrease decrease State State State Sgnfcant decrease Source: Compled by ths study. Note: () s the model ntal probablty; a j s the nter-state swtchng probablty; b m s the probablty of observaton value m at state., j 3, m 7. (2) State : recesson market. State 2: unchanged market. State 3: expanson market. (3) n-state observaton value m : sgnfcant ncrease, general ncrease, modest ncrease, small change, modest decrease, general decrease, and sgnfcant decrease. 88

9 Fgure-2. The teraton tmes curve of the logarthm maxmum lkelhood of the dscrete HMM model Source: Compled by ths study. After tranng, the optmzaton model parameters are as shown n Table 2. The state ntal probablty suggests that the probablty of the model ntal state n state (recesson market state) s the hghest at Ths can be verfed by the trends of the real estate leadng composte ndex as shown n Fgure 3. As shown there, the duraton of the market recesson s not lengther than the duraton of the market expanson or market of no sgnfcant fluctuatons. Obvously, the estmaton of the probablty of the ntal state n state s reasonable. Moreover, the ˆ Table 2 nter-state swtchng probablty value â j suggests that the probablty of the prevous term state of 3 and the current term state of s the hghest, followed by the probablty of the prevous term state of and the current term state of. The probablty of the prevous term state of 2 and the current term state of state 3 s the thrd rank. Based on the swtchng probablty of the current term of state and the followng term of state and the current term of state 2 and the followng term of state 2, the degree of state contnuaton s hgh. However, the probablty of the number of teratons ˆ current term of state 3 and the followng term of state 3 ( 33 ) s almost 0, suggestng that the contnuaton rate for state 3 s very low. Ths can be verfed by the long-term trends of the leadng ndcator as shown n Fgure 3. When the market s n the expanson state of state 3, n most cases, t wll swtch to the recesson market state of state. In other words, there are more downward turnng ponts than upward turnng ponts, ndcatng that the probablty of ˆ remanng n state 3 s low. All of the above verfes the reasonablty of 33 estmaton. â By usng the swtchng probablty estmate j, ths paper estmates the busness state s average duraton perod (footnotes 8). The probablty of the current term busness beng n the market recesson state (state ) and the â followng term busness beng n the recesson state s = The probablty of the current term busness state beng n the unchanged market state (state 2) and the followng term busness beng n the unchanged market state s â 22 =0.4099, whle the probablty of the current term busness beng n the market expanson state (state 3) and the â followng term beng n the expanson state 33 s almost 0. The average duraton perod of busness n the state of 89

10 recesson (state ) s 5.23 seasons, the average duraton perod of busness n the state of market of no change (state 2) s.69 seasons, and the average duraton perod of busness n the expanson state (state 3) s very short. From the perspectve of state and state 2 swtchng probablty, the state s contnuous, whch confrms the characterstc of Tawan s real estate market busness state beng dffcult to change and supports the vewpont of contnuous busness state. The state 3 (expanson market state) probablty estmaton suggests that t s dffcult to reman n the same state. Ths paper nfers that the settng of the HMM model shfts from left to rght. As a result, the probablty shfts from state 3 to state. In addton, the leadng ndcator trends n Fgure 3 suggest that the trends n the expanson stage are steep and that the downward fluctuaton sectons are frequent. Hence, the trend deepness n the expanson state s relatvely less steep as compared wth the deepness n the state recesson perod. Therefore, t s relatvely easer to shft to the recesson perod. The probablty value of state 3 suggests that t s unlkely to reman n state 3. The estmaton verfes that the perod of busness recesson s longer than the expanson perod. The above estmaton results of the busness average duraton perod suggest that the real estate busness cycle s recesson and expanson perods are asymmetrc (footnotes 9). In ths respect, the fndngs of ths study are consstent wth most of the avalable emprcal results for the overall busness cycle or the real estate busness cycle, ndcatng that expanson perods and recesson perods have asymmetrc trends. The real busness cycle model s consdered to be subject to the nfluence of the same random varables n busness expanson perod and recesson perod states, and thus they have the same degree of fluctuatons and the same dynamc characterstcs. However, Fgure 3 suggests that the trend of ncreasng real estate busness cycle leadng ndcator data durng the perod from 97 to 2009 s steeper and shorter n duraton. On the contrary, the trends of the decreases n most ndcators are less steep and last longer. The long-term trends of tme seres data are consstent wth the emprcal results. Accordng to the bˆ m estmaton results, when the market s n the recesson state, the total probablty of observatons of decreases n leadng ndcators (.e., modest decreases, general decreases, and sgnfcant decreases) s When the market s n a state of no sgnfcant change, the probablty of the trend change observaton (small change) s close to 50%, and up to When the market s n a state of expanson, the feature value of the n-state observaton value general ncrease s Roughly speakng, the n-state feature value probablty dstrbuton meets our expectatons, as the features n the expandng market state (state 3) (sgnfcant ncrease, general ncrease and modest ncrease) may concentrate on the observaton feature of the modest ncrease. The probablty of the occurrence of the two observaton features s almost 0. Ths may be caused by the steep trends of the real estate busness cycle leadng ndcator n the expanson perod, and the n-state feature closer to the boundares of other states can be more easly captured by the small change features of state 2 n probablty. As a result, t s dffcult for the model to capture n the expandng market state the probablty of two features ncludng sgnfcant ncrease and general ncrease but concentrate on the boundary secton of state 2 features. Another possble reason s that the samples are not categorzed by smaller features and thus the abnormaltes of each n-state feature can easly affect the bas of the probablty estmaton. In 4-step out-of-sample forecastng, through the known n-sample observaton data, the dscrete HMM model s establshed to allow the known out-of-sample forecastng observatons and known n-sample observaton data to have a smlar pattern and smlar statstcal characterstcs. The forecastng results error rate s as shown n Table 3. Accordng to the absolute error and mean absolute error rato of the smple state estmates, the forecastng accuracy of the out-of-sample data s not satsfactory, whch s as expected. Accordng to real estate busness cycle leadng ndcator trends as shown n Fgure 3, the ndcator has been consderably decreasng snce There s a possblty of seres structural change. 90

11 Parameter ˆ â j Table-2. After HMM tranng, model ( ˆ ) parameter Optmal parameter (after tranng) State State 2 State State State State e-057 bˆ m Sgnfcant ncrease General ncrease Modest ncrease Small change Modest decrease General decrease Sgnfcant decrease State State State Source: Compled by ths study. Note: () s the model ntal probablty; a j s the nter-state swtchng probablty; b m represents the probablty of observaton value m at state., j 3, m 7. (2) State : recesson market. State 2: unchanged market. State 3: expanson market. (3) In-state observaton value m : sgnfcant ncrease, general ncrease, modest ncrease, small change, modest decrease, general decrease, and sgnfcant decrease. Source: Compled by ths study. Fgure-3. Trends of the real estate busness cycle leadng ndcator (97Q~2009Q4) Table-3. HMM model 4-step out-of-sample forecastng results - mean absolute error rate Forecastn Actual state Predcted state Absolute error Mean absolute error g perod y value ŷ value (%) 2009 Q Q % 2009 Q Q4 3 Source: Compled by ths study. Note: () In ths study, the busness state s dvded nto three types: state : recesson market. State 2: unchanged market. State 3: expanson market. (2) Forecastng perod: 4-step-ahead forecasts. (3) The absolute error n the table s set as y yˆ, then 0, and the rest s. (4) Mean absolute error t n t n y t ŷ Ths paper further examnes the model-estmated data of each year, and deletes the samples of suspected of structural changes. The total sample HMM model s then appled n the estmaton of parameters, and the 9

12 means absolute error rate s appled n the detecton of total sample of each year n four-step forecastng as shown n Table 4. The results are consstent wth those of Wu (2009). The mean estmaton error rate of the total sample s 44.90%; the accuracy of forecastng samples out of the four-step forecastng (2007Q-2007Q4) s up to 75%. It s concluded that the HMM model has good accuracy n forecastng fluctuatons of the busness cycle leadng ndcator. The results valdate the nference of ths study. In the perod of 2009 Q to Q4, there are structural changes n Tawan s real estate busness cycle leadng ndcator. Prevous lterature (Rau et al., 200; Chen, 2006) on the overall busness cycle of Tawan ndcates that structural change occurred durng the 990s. In the feld of real estate busness cycle research, many artcles on the measurement of the real estate busness cycle by housng prce, such as those by Chen (2003) and by Peng et al. (2004) argue that housng prces dd undergo structural changes. However, to avod dvergence of research topcs and to smplfy the model to hghlght the characterstcs of the HMM model for capturng the jumpng state, the topcs of structural change are not consdered and explored n ths study. Table-4. HMM model forecastng results of total sample n each year-mean absolute error rate detecton table Perod of Mean absolute Perod of sample Mean absolute Perod of Mean sample forecastng error rate (%) forecastng error rate (%) sample forecastng absolute error rate(%) 60Q2-Q4 0 73Q-Q Q-Q4 0 6Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q4 0 89Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q4 50 9Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q Q-Q4 50 8Q-Q Q-Q Q-Q Q-Q4 0 95Q-Q Q-Q Q-Q Q-Q4 25 7Q-Q Q-Q4 0 72Q-Q Q-Q4 25 Source: Compled by ths study Ths paper further uses the MSE crteron to estmate the accuracy of estmaton by determnng the degree of error as ndcated by the dfference between the actual value and the forecast value of the busness ndcator. The total MSE result s.048, as shown n Table 5, where ŷ s the leadng ndcator value deduced from the optmal path by the estmaton of the HMM model. Before the applcaton of the HMM model, ths study has extracted the features of the economc cycle leadng ndcators for ntermttent classfcaton. In the process, to extract features, the category average number s estmated. Hence, ŷ s the average feature category deduced from the optmal path of forecastng. Table-5. HMM model 4-step-ahead forecasts-mse crteron test Forecastng Actual leadng ndcator Estmated leadng ndcator perod value ( y ) value ( ŷ ) 2009 Q Q Q Q Source: Compled by ths study. MSE.048 y Note: () s the real leadng ndcator value; (2) MSE crteron testng value ŷ s the estmated leadng ndcator value. 2 ŷ n MSE y t n 92

13 4.2. Analytc Comparson of HMM Model Forecastng Results and Condtons Released by Tawan Real Estate Research Center For the busness duraton perod, the estmated average duraton perod of busness n the state of recesson (state ) s 5.23 seasons, the estmated average duraton perod of busness n the state of market of no change (state 2) s.69 seasons, and the estmated average duraton perod of the busness n the state of expanson (state 3) s very short. These estmatons are n lne wth the current condton that busness recesson duratons are longer than the duratons of the expanson perods. Accordng to the trends of economc cycle leadng ndcators released by the Real Estate Research Center as shown n Fgure 4, the state of recesson s more frequent and longer than expanson n the busness cycle. However, the results on the average of the long and short duratons are dfferent from those released by the Tawan Real Estate Research Center (footnotes 0). Ths s caused by the dfference n the number of states. As some state paths of the model are affected by state 2 of market of no change, thus the probablty of state and state 3 estmated paths s decreased. The possble reason s due to HMM model left-to-rght pattern and ntal state swtchng probablty settng. The regresson computaton of dfferent ntal settngs has a consderable mpact on the parameter estmaton of the optmal HMM model. In addton, the overall parameter of ˆ 3 s consstent wth the settngs of the HMM model. The cause of the low duraton of state 3 may lead to the dfference between the estmaton of busness duraton and asymmetry and the results released by the Real Estate Research Center. Accordng to the leadng ndcator trends released by the Real Estate Research Center (Fgure 4), compared to recesson market states, the probablty of the current term beng n the expanson state and the followng term beng n the expanson state s not very hgh. As seen n the fgure, the market often swtches to the recesson market state after t enters nto the expanson state. Ths s consstent wth the estmated swtchng probablty of ˆ 3. Furthermore, the HMM model estmaton results of swtchng probablty suggest that there s an asymmetrc relatonshp between the recesson and expanson perods of the busness cycle. Prevous emprcal research nto the overall busness cycle and real estate busness cycles suggests that there are asymmetrc trends for the expanson perods and recesson perods. As shown n Fgure 4, from 97 to 2009, Tawan s real estate busness cycle leadng ndcator data had steep ncreases of shorter duratons. On the contrary, the trends of decreases for most ndcators tended to be less steep and last for longer perods of tme. The long-term trends of the tme seres data can verfy the asymmetrc estmaton results. Fgure-4. Real estate busness cycle leadng and concdent ndcator composte ndex trends Source: Archtecture and Buldng Research Insttute, Mnstry of the Interor, Tawan Real Estate Research Center, Natonal Chengch Unversty (Quarterly report of Tawan s real estate busness cycle trends, December 2009) 93

14 In order to observe the changes of the real estate busness cycle leadng ndcator and HMM model forecastng capabltes by followng the practce of the 4-step out-of-sample forecastng method, ths study uses the average change rate of the features of the state optmal path to estmate the real estate busness cycle leadng ndcator. The model estmated trend forecastng s as shown n Fgure 5. Between 972 Q3 and 974 Q4, the model estmaton has apparent laggng trends. The trends of the upward and downward fluctuatons of the model forecastng are n the same drecton wth those released by the Tawan Real Estate Research Center. Wheren, there are fve decreasng forecastng values, whch are consstent wth the real estate leadng ndcators released by Tawan Real Estate Research Center, n partcular, those for the perod of 97 Q2~Q4 and the perod of 2008 Q~Q4. As far as the busness cycle fluctuatons deepness s concerned, the HMM-estmated leadng ndcator s deeper than the real ndcator n the state of recesson. Changes n forecastng average fluctuatons suggest sgnfcant dfferences. These dfferences are caused by the dfferent number of states. Source: Compled by ths study. Fgure-5. predcted trends of real estate busness cycle leadng ndcator 5. CONCLUSION Based on the tme seres data of the real estate busness cycle composte leadng ndcator jontly compled and regularly released by the Archtecture and Buldng Research Insttute, as well as the Tawan Real Estate Research Center of Natonal Chengch Unversty, ths paper uses the HMM to capture the optmal path of state transton to observe the trends of fluctuatons of out-of-sample data. Regardng the dentfcaton and applcaton of the economc cycle leadng ndcators and HMM model n the trends of busness cycle fluctuatons, the results confrm that trends of the real estate busness cycle fluctuatons are asymmetrc and that the average duraton of recesson perods s longer than that of expanson perods. By overcomng the unobservable lmtaton of usng the Markov-swtchng model to capture state seres settng, ths study apples the unvarate dscrete HMM model and feature extracton method to extract sgnals from the unobservable state to ncrease data smulaton. Overall, the n-state observaton value probablty s consstent wth the market fluctuatons that the HMM model extracts for the n-state observaton values to observe the unobservable state of the Markov-swtchng model. In terms of the out-of-sample 4-step-ahead forecasts, except for the sectons of the fluctuatons of structural change, the results of the mean absolute error rate of the estmaton accuracy suggest that the average accuracy of usng the HMM model to forecast 4-step out-of-sample state s up to 55.4%. Ths ndcates that the proposed model s advantageous n model estmaton. However, the proposed model has not consdered the dfferences caused by the fluctuaton trend path estmaton due to structural change. Therefore, t cannot effectvely master the fluctuaton path of total samples n the current 94

15 term. Prevous studes have suggested that the leadng ndcator s not only useful for settng and selecton but can also serve as the leadng remnder of macroeconomc phenomena. Therefore, future studes can use the mproved dual-layer bult-n HMM measurement models based on the leadng ndcator seres compled by the Tawan Real Estate Research Center, such as the multvarate Markov vector auto-regresson model for descrbng the common fluctuaton characterstcs between seres (Krolzg, 997) the swtchng probablty tme-varyng Markov-swtchng model (Peersman and Smets, 200) the duraton dependent Markov-swtchng model of swtchng probablty wth duraton dependent characterstcs (Pelagatt, 200) or the contnuous hdden Markov-swtchng model, to further reduce forecastng error. Note Note : The leadng ndcator composte ndex s compled by the Tawan Real Estate Research Center, College of Socal Scences, Natonal Chengch Unversty, at the delegaton of Constructon and Plannng Agency, Mnstry of the Interor. Startng wth the frst quarter of 97, t ntally conssted of relevant macroeconomcs ndcators ncludng GDP, monetary supply, and CPI. In 98 and 989, ndcators relatng to the real estate ndustry and the constructon stock ndex as well as the constructon credt balance change were used to form the current real estate busness cycle leadng composte ndex. Note 2: The state transton model s a non-lnear model. It can be dvded nto two types by the state transton observaton. If the model s state transton process s determned by the observable varables, t s known as the threshold model, n whch case state change s determned by whether the observable varable s above the threshold value. If t s above threshold value, the state transton occurs. For another type of the model, the state transton process s determned by the unobservable varables. Therefore, the model should defne the state transton process. The Markov swtchng model s such a model. The Markov swtchng model consders that data come from dfferent parent matrces. By settng a group of selfregresson equatons and usng a Markov chan to understand the nter-state swtchng process, the current term state wll be subject to the nfluence of the prevous state, so that the data of varous perods wll be contnuous and relevant. The probablty theory can be appled to estmate the non-lnear shftng of state transtons. Note 3: In the HMM model, there are a number of patterns of hdden n-state observaton output sample spaces and probablty dstrbutons such as unvarate dscrete observatons, contnuous observatons, and mxed contnuous observatons. Ths paper sets the output observatons as type of unvarate dscrete data dstrbuton. Note 4: There are a number of general patterns for the HMM model. The man change s that nter-state swtchng probablty can swtch n between states wthout lmtaton. Second, the pattern of the n-state observable value can be unvarate dscrete data, unvarate contnuous data, multvarate dscrete data, or multvarate contnuous data. To summarze, the two assumptons of ths study should be satsfed, that s, n-state observaton value o m and state qtare mutually dependent and mutually ndependent wth o m. The frst-order Markov assumpton and the current term swtchng probablty are subject to the nfluence of the current term and prevous term swtchng probablty only. Note 5: The term statonary seres means that the statstcal characterstcs of the tme seres data generaton process, ncludng average, varance, and covarance, should be lmted constants. It expects the tme seres varable s mportant statstcal characterstcs ncludng average, varance to be non-tme-varyng 95

16 to facltate statstcal nference and parameter estmaton. On the other hand, f t s msused, t can easly lead to estmaton bas. The most well-known example of such bas s proposed by Granger and Newbold (974) when they note that spurous regresson often occurs n between non-statonary varables. Note 6: In Tawan s real estate busness cycle trend quarterly reports, the scores are gven n terms of changes n the ndcators. The real estate busness cycle trend sgnals are categorzed nto fve lghts, specfcally, red, yellow-red, green, yellow-blue, and blue lghts, to represent busness ndcators of rangng from a sgnfcant ncrease to a sgnfcant decrease ncludng overheated busness, busness boom, busness stablty, poor busness, and busness recesson. Note 7: The term rollng wndow refers to the samplng method of a gven sample length and samplng perod range shftng by term. Note 8: The estmaton of the busness duraton perod s determned by the computaton of nter-state swtchng probablty, that s,, and ˆ ˆ22 represent the average duraton perods of the market recesson, market of no change, and market expanson. Note 9: The term busness cycle asymmetry refers to the nconsstency n the duratons of expanson and recesson perods n busness cycles. Schel (993) proposed two features relatng to busness cycle asymmetrc fluctuatons: deepness and steepness. In that study, deepness s descrbed as the nconsstency n the dstance from the trend values n the case of valley and peak of the busness cycle n fluctuatons. In addton, steepness s manly used to descrbe the nconsstency n the slope of the valley and peak movements n the busness cycle, that s, the speeds at whch rebounds from the peak and the valley occur are not the same. Note 0: Accordng to the data released by the Tawan Real Estate Research Center, the current average expanson perod lasts 9 seasons and the average recesson perod lasts 24 seasons, ndcatng an apparent asymmetry of longer busness recessons than expansons. Fundng: Ths study receved no specfc fnancal support. Competng Interests: The authors declare that they have no competng nterests. Contrbutors/Acknowledgement: All authors contrbuted equally to the concepton and desgn of the study. REFERENCES Arsenault, M., J. Clayton and L. Peng, 203. Mortgage fund flows, captal apprecaton, and real estate cycles. Journal of Real Estate Fnance and Economcs, 47(2): Barras, R., A buldng cycle model for an mperfect world. Journal of Property Research, 22(2): Baum, A., 200. Evdence of cycles n European commercal real estate markets and some hypotheses. A global perspectve on real estate cycles. US: Sprnger. pp: Baum, L.E. and T. Petre, 966. Statstcal nference for probablstc functons of fnte state Markov chans. Annals of Mathematcal Statstcs, 37(6): Bouchaffra, D. and J. Tan, Introducton to the concept of structural HMM: Applcaton to mnng customer's preferences n automatve desgn. 7th Internatonal Conference on Pattern Recognton. IEEE. pp: 493. Ca, J., 994. A Markov model of swtchng-regme ARCH. Journal of Busness & Economc Statstcs, 2(3): Chen, M.C., Tme-seres propertes and modellng of house prces n Tape area: An applcaton of the structural tme-seres model. Journal of Housng Studes, 2(2): Chen, S.W., An analyss of the effect of output volatlty on turnng ponts dentfcaton: Internatonal evdence. Journal of Socal Scences and Phlosophy, 8(): ˆ33 96

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