Modeling House Price Volatility States in the UK by Switching ARCH Models

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1 Modeling House Price Volailiy Saes in he UK by Swiching ARCH Models I-Chun Tsai* Dearmen of Finance Souhern Taiwan Universiy of Technology, Taiwan Ming-Chi Chen Dearmen of Finance Naional Sun Ya-sen Universiy, Taiwan Absrac This aer analyzes volailiy saes of he UK house rices. We use boh ARCH and GARCH models o esimae rice condiional heeroscedasiciy and find he evidence of ime-varying roery in rice series. We coninue o use he SWARCH model and found here are a leas wo volailiy saes in he rice series. Our esimaions sugges he UK housing markes are relaively sable and differen saes do no swich very ofen. The magniude of high rice volailiy is as high as 4.89 imes of low volailiy for all housing marke and.87 imes of low volailiy for new housing markes. However, low volailiy is he normal condiion in hese wo markes. Keyword: Markov-swiching, ARCH, SWARCH, House rices *Corresonding auhor: I-Chun Tsai, Dearmen of Finance, Souhern Taiwan Universiy of Technology, Tainan, Taiwan; risa@mail.su.edu.w; Tel: ex5335

2 I. Inroducion Over he las few decades, susained long run growh in house rice and is recurren flucuaions around he growh ah seems o be common henomena in many counries in he world. The risk consideraion is now widely recognized by boh heoreical and alied economiss and he modeling of house rice behavior has araced much research aenions. Emirical work on he dynamics of he housing marke has ried o caure he shor-erm adjusmen rocess (for examle, Meen, 99; Drake, 993; Malezzi, 999). Mos of hese sudies ry o model rice behavior based on he ineracion of demand and suly side facors using radiional regression analysis on condiional mean. However, mos of hese economic models may no come u wih a saisfying erformance cauring he boom and bus behavior of he housing markes. Alhough highly volaile behavior in house rice series has been recognized in revious sudies, some more effors sill can be done. Tyically, we can see ha house rices are always convered o naural logarihms in emirical es in revious house rice sudies and his is because he heeroscedasiciy roblem usually occurred in house rice regression model. Therefore, volailiy in house rices has o be reduced so ha he model will no violae he assumion of OLS. Some sudies have ried o caure he volailiy. Hendry (984) and Giussani and Hadjimaheou (99) have ried o model he volailiy by using non-linear secificaion o caure exreme movemens in house rices. Hendry (984) uses a cubic aroximaion funcion, calculaed as he cubic erm of house rice changes. Giussani and Hadjimaheou (99) use boh square and cubic erms o caure raid adjusmen of house rice. In addiion, alhough many revious sudies used he error correcion aroach, he volailiy exceeding equilibrium adjusmen sill canno be correced.

3 Because high volailiy is very common in financial daa, he family of ARCH and GARCH (Engle, 98 and Bollerslev, 986) models are develoed and widely alied o modeling he variance of financial variables. These yes of models allow he condiional variance of a series o deend on he as realizaions of he error rocess and simulaneously model he ime-deenden mean and variance. Because of heir excellence in cauring volailiy, hese yes of models are alied o oher areas including housing sudies (Dold and Tirioglu, 997, ). However, imeseries behavior is always comlex and someimes srucural breaks occurr. Lamoureux and Lasraes (99) argue ha he near inegraed behavior of he condiional variances migh be due o he resence of srucural break, which are no accouned for by sandard ARCH models. In he housing marke, some evens such as marke crashes, financial liberalizaion and changes in governmen olicy always have aaren effecs. Cyclical volailiies in he house rice series are also clearly observed. During hese abru evens or cycles, house rice behavior changes subsanially. Following he work of Hamilon (989) on swiching regimes, Hamilon and Susmel (994) roose a new ARCH model, he Swich ARCH (SWARCH) model ha is ime-varian and allows for condiional volailiy rocess o swich sochasically among a finie number of regimes. In addiion, Bloomfield and Hales () sugges regime-shifing model seems o be a reasonable framework in which o inerre some marke anomalies. This SWARCH model can incororae he ossibiliy of volailiy regime swich in he condiional variance in exlaining volailiy ersisence, a henomenon ha is commonly observed in he housing marke. Hence, he SWARCH model may be a useful ool o exlain more realisically he ime-series roery of house rice series.

4 The main urose of his aer is o sudy volailiy roeries in he UK house rice series. We examine wheher volailiy of house rice changed over ime by ARCH and GARCH model. We also analyze wheher here are differen saes of volailiy in he rice series. We alied he SWARCH model because his model can hel us o analyze differen saes of volailiy and esimae robabiliy of changes in hese saes. The aer is srucured as follows. The nex secion describes he alied mehodologies. Secion 4 reviews our daa and ess is ime-series roeries. Esimaion resuls are reored and discussed in Secion 5, and he las secion rovides a summary of he main finding and draws some conclusions. II. The Swiching ARCH Model for House Prices For cauring volailiy of house rice, we emloyed he ARCH-ye model o model he volailiy of house rice changing over ime, and use he SWARCH model o see wheher here is regime-shifing in he variance rocess of house rice. II.A. Modeling Volailiy of House Prices Over Time: ARCH and GARCH Models Many economic ime-series do no have consan mean and volailiy. Engle (98) shows ha i is ossible o simulaneously model he ime-deenden mean and variance. I is he widely known ARCH (Auoregressive Condiional Heeroskedasic) model. I allows he condiional variance of a series o deend on he as realizaions of he error rocess. Bollerslev (986) exended Engle s original work by develoing he GARCH (Generalized Auoregressive Condiional Heeroscedasiciy) model ha allows for boh auoregressive and moving average comonens in he heeroskedasic variance. Now we briefly illusrae he feaures of hese wo models in he following. () ARCH model 3

5 Le y denoe he series of an asse reurn, he error rocess obained from a firsorder auoregression for y follows ARCH(q) model and i can be secified as: y = a + a y + ε ε Ω ~ N(, h ) h = ω q + α iε i i= Where q is he number of ARCH erms, and h is he heeroskedasic condiional variance, which is correlaed wih he lagged error erms. () GARCH model If he error rocess obained from a firs-order auoregression for y follows GARCH(,q) model hen i can be secified as: ε Ω ~ N(, h ) q + β i i α iε i i= i= h = ω h Where h is he heeroskedasic condiional variance, which is correlaed wih he lagged error erms and condiional variance. II.B. Deecing Volailiy Saes in he Housing Marke: SWARCH While he ARCH and GARCH models has been widely alied o modeling variance of financial series, Lamoureux and Lasraes (99) show ha hese models 4

6 may no be aroriae if srucural changes are exised in daa. Hamilon and Susmel (994) roose a regime swiching ARCH model (SWARCH model) allows for he condiional volailiy rocess o swich sochasically among a finie number of regimes. The feaures of he SWARCH model are briefly described as followed. Le y denoe he series of an asse reurn, he error rocess obained from a firsorder auoregression for y follows SWARCH(K, q) model and i can be secified as: y = a + a y + ε ε Ω ~ N(, h ) ε = g s u u = h v h = ω q + α iε i i= Where q is he number of ARCH erms, K is he number of regime saes, v follows Gaussian disribuion, s denoes an unobserved random variable ha can ake values,, K, and g s are scale arameers ha caure he size of volailiy in differen regimes. From he discussed of las secion, we roosed ha he volailiy of house rice can be large, median and small, hence we use SWARCH(3,q) model in emirical es of his research, i.e. K=3. The scale arameer for he firs sae g is normalized a uniy wih g s for s =, 3, and we also assume g < g 3, hence he sae ( s =) can denoe he low-volailiy regime, he sae denoes he median- 5

7 volailiy regime, and he sae 3 denoes he high-volailiy regime. The 3-sae regime swiching is assumed o follow a Markov rocess. The swiching robabiliies beween hree saes follow he ransiion robabiliies Marix P: P = Where ij denoe he robabiliy of sae i swich o sae j, assume ij < for all i and all j, and i ij = for all j; j ij = for all i. III. Daa Descriion Since he SWARCH model requires relaively longer daa o have beer esimaion, we use he UK naion-wide house rice daa saring from 955Q4 o 5Q4. All and new house rice daa are comared. The daa se used in our analysis consiss of quarerly observaions on all house rices (Allh), and new house rices (Newh). Table resens a summary of he descriive saisics for wo house rice variables. Table also reors he oucome of ess for saionariy. Augmened Dickey-Fuller (Said and Dickey, 984) es and also Phillis-Perron (988) es boh confirm ha wo house rice variables are I(). Evidenly, he uni-roo hyohesis canno be rejeced a he 5% significance level for wo series in levels. In addiion, ess alied o differenced daa favor he saionary alernaive for wo series. To avoid he roblem of surious regression, hroughou he aer, we use he firsdifference daa o esimae he emirical models. 6

8 Table Descriive Saisics Variable All Ph New Ph No. of observaions 3 3 Mean Sd. Dev Skewness.4.7 Kurosis Variables in Level ADF es.79 (.99).3 (.98) PP es.83 (.99).39 (.98) Variables in Differenced ADF es (.) (.) PP es (.) -8.3 (.) * Mackinnon (996) one-sided -values are reored in arenheses. To observe he volailiy of house rices, Figure los he quarerly ime series of wo house rices for he samle eriod. We can observe ha wo rice series have increased during he samle eriod in non-monoonic ways. Two series seems o have changes in volailiy saes or bubbles in four eriods: , , , -5, he series are highly volaile in hese four eriods. Unlike some researches ha are devoed o discuss he bubble-like behavior of house rices (ex. Hall e al., 997), his aer emhasizes on he volailiy roeries of house rices. 7

9 A L L P H N E W P H Figure House Price in he all and new housing Markes IV. Emirical Resuls IV.A. Does he Volailiy of House Prices Change Over Time? Before esimaing he house rice volailiy, we need o deerminae he mean equaion. We use he lagged daa of house rice o be he indeenden variables, and choose he model which can minimize he value of AIC (Akaike Informaion Crierion) and SBC (Schwarz Bayesian Crierion) o decide he number of lag erms. Due o he consideraion abou he degrees of freedom, only lags of lengh o 4 are esed. The emirical resuls of differen AR model are shown in Table. 8

10 Table Esimaes of AR models All house rice AR() AR() AR(3) AR(4) AIC SBC New house rice AR() AR() AR(3) AR(4) AIC SBC From he Table, we found AR() model for new house rices is he mos aroriae because hese wo model selecion crieria derived from he firs-order auoregression model erform beer han oher AR models. For all house rices, he SBC crieria sugges similar resuls alhough AIC suggess ha AR(4) migh be beer. Because he SBC is asymoically consisen, whereas he AIC is biased oward selecing an over arameerized model, hence we choose AR() model for all house markes. Before we use he ARCH and GARCH models, i is necessary o es wheher here are ARCH effecs exising in daa. We use he formal Lagrange mulilier es for ARCH disurbances roosed by Engle (98) and he resuls of LM es are shown in Table 3. Table 3 Resuls of ARCH effec es lags of lengh 3 4 All house rice TR value New house rice TR value.8... * Ho: There are no ARCH effecs. From he resuls in Table 3, we found ha he disurbances obained from he 9

11 firs-order auoregression of wo series are auocorrelaed. I means he variance (risk) in wo house markes are ime deenden. Now we use he ARCH and GARCH models o esimae he variance of wo series a a aricular oin in ime. We esimae hree differen ARCH and GARCH models o deermine he mos aroriae model for volailiies of wo housing markes. Tha are AR()-ARCH(), AR()-ARCH(), and AR()-GARCH(,) model. The resuls of hese hree models esimaions are shown in Table 4.

12 Table 4 Emirical Resuls of ARCH-Family Model: Le y denoe he series of he differenced house rice y = a + a y + ε ε ~ N(, h ) Ω q + β i i α iε i i= i= h = ω h Model ARCH() ARCH() GARCH(,) Mean equaion All house New house All house New house All house New house a.44.48*.76*.* (.93) (.37) (.89) (.43) (.47) (.48) a.66*.53*.7*.59*.6*.46* (.5) (.) (.5) (.4) (.7) (.8) Variance equaion ω 4.4* 56.38* *.6.58 (8.67) (.38) (7.5) (5.7) (.48) (.53) α.33*.6*.33*.6*.5*.7* (.) (.3) (.) (.8) (.4) (.3) α *.44* - - (.) (.4) β *.87* (.3) (.) Residuals es Q().79* 46.39* 9.8* 4.49* 3.8* 4.* Q () 58.79* * Noes: Numbers in arenheses are sandard errors. * indicaes significance a he 5% level. Q() and Q () are he Ljung-Box saisic based on he sandardized residuals and he squared sandardized residuals resecively u o he h order. Three ARCH-family models for wo series are esimaed and he resuls are resened in Table 4. The esimaed coefficiens of ARCH and GARCH effecs are highly significan in each series. The sum of ARCH and GARCH coefficiens in every esimaion is close o or larger han one, suggesing ha shocks o he condiional variance are highly ersisen. Diebold (986) and Lamoureux and Lasraes (99)

13 argued ha he high ersisence migh reflec regime swich in he variance rocess. Therefore, differen volailiy saes migh also occur in hese wo housing markes. We can verify his henomenon from figure 3 and 4 of condiional variances for hese models. These wo figures aear o show differen saes of volailiy. We can see several high volailiy saes. Volailiies are smaller before 97 bu are geing larger hereafer. Consequenly, we coninue o model hese volailiy saes by using SWARCH model. Figure 3 Esimaed condiional volailiies for he all house marke c o n d i i o n a l v a r i a n c e o f A R C H ( ) c o n d i i o n a l v a r i a n c e o f A R C H ( ) c o n d iio n a l v a r ia n c e o f G A R C H (, ) Figure 4 Esimaed condiional volailiies for he new house marke

14 5 c o n d i i o n a l v a r i a n c e o f A R C H ( ) c o n d i i o n a l v a r i a n c e o f A R C H ( ) c o n d i i o n a l v a r i a n c e o f G A R C H (, ) IV.A. Does he Housing Marke Exhibi Differen Volailiy Saes? To see wheher he condiional volailiies of house rices swich sochasically and caure he swich oins endogenously, we use AR()-SWARCH(3,) o esimae he variance of he housing markes. The resuls of SWARCH model esimaions are shown in Table 5. 3

15 Table 5 Emirical Resuls of SWARCH Model Model: Le y denoe he series of he differenced house rice y = a + a y + ε, ε Ω ~ N(, h ) ε = s u, u = h v g, s =,, 3 h = ω + αε Model SWARCH(3,) Mean equaion All house New house a.4(.55)*.68(.6)* a.6(.6)*.48(.6)* Variance equaion ω 7.53(4.7)* 3.49(7.4)* α.(.7).4(.7) Transiion rob..93(.6)*.86(.8)*.6(.6).(.)* 3.5(.).5(.47).7(.3).(.).86(.3)*.78(.5)* 3.7(.6).4(.68) Sae variable g 8.8(.74)* 8.7(.46)* g 3.88(7.73)* 5.(844.) 3 Noes: Numbers in arenheses are sandard errors. * indicaes significance a he 5% level. From able 5, all housing marke model has significan sae variable g and g 3, suggesing housing exhibis hree differen volailiy saes. The coefficien g sugges ha variance in medium-volailiy sae ( s =) is more han.97 ( g ) imes ha in he low-volailiy saes ( s =). The coefficien esae of g3 sugges ha variance in high-volailiy sae ( s =3) is more han 4.89 ( g 3 ) imes ha in he low-volailiy saes. Consequenly, he differences of volailiy saes are very obvious. 4

16 Because s is assumed o by governed by a firs order Markov chain wih ransiion robabiliy ij, which means differen saes are swich according o he ransiion robabiliy. The all housing marke has significan ransiion robabiliy and wih value of.93 and.86 resecively. These large values sugges ransiion beween saes is no ofen and every sae mainains a relaively long ime. We can furher esimae he average duraion of every sae by ii and have 4.9, 7.4 and 8.33 quarers for sae,, 3 resecively. These esimaions sugges he housing marke is relaively sable and differen saes will no swich very ofen. This is consisen wih our execaion of he housing marke because of inefficiency characerisics. The housing marke is no like he sock marke ha volailiy saes changes frequenly. The cycles in he housing markes usually las several years. The resuls of he new housing marke is differen because only one sae variable g is significan, suggesing here are only wo volailiy (high and low) saes in he new housing markes. The variance in high-volailiy sae is more han.87 imes ( g ) ha in he low-volailiy saes. The ransiion robabiliies, and, which are all significan, sugges ransiions beween saes are more easily when some evens occur. We can same esimae he average duraion of every sae by ii and have 7.4 and 4.55 quarers for sae and resecively. These esimaions sugges he saes of new housing marke change more easily, which means he marke is more efficien. Therefore, he low-volailiy saes are longer in he new housing marke as he all housing marke. However, when some evens occur, he magniude of volailiy will be double and high-volailiy sae will las one year 5

17 more before he marke reurns o low-volailiy sae. We can furher observe differences of he volailiy saes by using smoohed robabiliy shown in figure 5 and 6. I can be seen ha before 97, he marke is in low-volailiy sae because he robabiliy of sae is close o. Figures show ha 97 and 977 aear o swich o volailiy sae bu hese volailiy are medium. However, here are wo srong regime-swiches during 988 and because he robabiliy of volailiy sae changed o high is close o. However, new housing marke does no have as high volailiy as in all housing markes in hese wo ime oins because is no significan in he new housing marke. 6

18 . s m o o h e d r o b a b i l i i e s o f s a e s m o o h e d r o b a b i l i i e s o f s a e s m o o h e d r o b a b i l i i e s o f s a e Figure 5. Smoohed robabiliies of he all house marke 7

19 s m o o h e d r o b a b il i i e s o f s a e s m o o h e d r o b a b il i i e s o f s a e s m o o h e d r o b a b il i i e s o f s a e Figure 6. Smoohed Probabiliies of he new house marke VI. Conclusion House rice recurren flucuaions around he growh ah seem o be common henomena in he world. This aer ried o analyze he UK house rice volailiy when changes of volailiy sae are allowed in he housing marke. We use boh ARCH and GARCH models o esimae rice condiional heeroscedasiciy in order o verify ime-varying roery in he rice series. Then we coninue o use he SWARCH model o esimae heir volailiy saes. Our findings indicae ha he esimaed coefficiens of ARCH and GARCH 8

20 effecs are highly significan in our samle series, suggesing ha he volailiy of house rice changes over ime. Because he shock o he condiional variance are highly ersisen in he ARCH esimaion and his high ersisence migh reflec regime swich in he variance rocess, we furher es wheher here are differen saes of volailiy by using he SWATCH model. Two volailiy saes for new housing marke and hree volailiy saes for all housing markes are found. The esimaions also sugges he housing markes are relaively sable and differen saes do no swich very ofen. The magniude of high volailiy is as high as 4.89 imes of low volailiy for all housing markes and.87 imes of low volailiy for new housing markes. However, low volailiy is a normal condiion in hese wo markes. Reference. Bloomfield, R. and Hales, J. () Predicing he Nex Se of a Random Walk: Exerimenal Evidence of Regime-shifing Beliefs, Journal of Financial Economcis, 65: Bollerslev,T., (986), Generalized auoregressive condiional heeroskedasiciy, Journal of Economerics, Vol. 3, Dolde, W. and Tirioglu, D (), Housing Price Volailiy Changes and Their Effecs, Real Esae Economics, 3(): Drake, L. (993). Modelling UK house rices using coinegraion: an alicaion of he Johansen echnique, Alied Economics, 5: Engle, R. F. (98). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion, Economerica, 5: Giussani, B. and Hadjimaheou, G. (99). Modelling regional house rice in Unied Kingdom, The Journal of he Regional Science Associaion Inernaional, 7 (): -9. 9

21 7. Hall. S., Psaradakis, Z. and Sola, M. (997). Swiching error-correcion models of house rices in he Unied Kingdom, Economic Modelling, 4 (4): Diebold (986) 8. Hamilon, J. D. and R. Susmel (994), Auoregressive condiional heeroskedasiciy and change in regime, Journal of Economerics, 64, Hendry, D. F. (984). Economeric modelling of house rices in he UK, Economerics and Quaniaive Economics, Hendry, D. F. and Wallis, K. F. (eds), Basil Blackwell, Oxford.. Holly, S. and Jones, N. (997). House rices since he 94s: coinegraion, demograhy and asymmeries, Economic Modelling, 4 (4): Lamoureux, C. G. and W. D. Lasraes (99), Persisence in variance, srucural change and he GARCH model, Journal of Business and Economic Saisics, 8, Meen, G. P. (99). The removal of morgage marke consrains and he imlicaions for economeric modelling of UK house rices, Oxford Bullein Economics and Saisics, 5 (): Malezzi, S. (999), A Simle Error Correcion Model of House Prices, Journal of Housing Economics 8, Hamilon, J. D. (989). A new aroach o he economic analysis of nonsaionary ime series and he business cycle, Economerica, 57: Phillis, P. and Perron, P. (988). Tesing for a uni roo in ime series regression, Biomerica, 75: Said, S. and D. Dickey (984). Tesing for Uni Roos in Auoregressive-Moving Average Models wih Unknown Order. Biomerica. 7: