Table 3. Yearly Timeline of Release Dates Last Quarter Included Release Date Fourth Quarter of T-1 First full week of April of T First Quarter of T

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1 3 Mehodological Approach 3.1 Timing of Releases The inernaional house price daabase is updaed quarerly, bu we face grea heerogeneiy in he iming of each counry s daa releases. We have found a significan number of counries repor he daa needed o include a new quarerly observaion of he (nominal and real) house price and PDI index series hree monhs afer he end of each quarer. 17 As a pracical compromise beween he imeliness of he release and he counry coverage, we schedule he posing of he inernaional house price daabase during four fixed periods each year incorporaing already a hree monh lag. The schedule for a given year T, herefore, is: Table 3. Yearly Timeline of Release Daes Las Quarer Included Release Dae Fourh Quarer of T-1 Firs full week of April of T Firs Quarer of T Firs full week of July of T Second Quarer of T Firs full week of Ocober of T Third Quarer of T Firs full week of January of T+1 We aim o have he daase posed during he week indicaed in Table 3. We will apply he same schedule for daa releases every year, unless he release lag of he counry daa on which he panel depends were o change, requiring us o adjus his schedule o preserve boh is wide coverage and represenaiveness. The schedule of daa releases for he curren and subsequen years can be found on he web. Whenever observaions are missing for a counry s house price or PDI index, we complee he series for he curren release wih nowcass derived wih he basic srucural ime series (BSTS) model ha we describe in Secion 3.7 below. The daase will be revised in subsequen quarers o incorporae he counry daa, as he corresponding quarerly observaions become available. 17 Counry coverage usually represens more han 80 percen of oal oupu from all counries included in he sample, measured by heir 005 GDP in purchasing power pariy adjused erms according o he IMF. Prices from around he world, all under one roof.

2 3. Approach o Consruc he Counry Daa: Sep-by-Sep The FHFA house price index series (formerly called OFHEO house price index) serves as our benchmark measure when selecing a house price index for he oher counries in he daabase. The main selecion crieria and preference are given o: geographic coverage (naionwide); vinage of dwellings (exising); ype of dwellings (single-family); priced uni (per dwelling); availabiliy of daa ( presen); and frequency (quarerly). The disposable personal income series by he Bureau of Economic Analysis (BEA) serves as our benchmark measure when selecing or reconsrucing he PDI index for he oher counries, alhough no all accouning differences across counries can be fully reconciled wih he BEA concep. The PDI measure for some counries is derived from naional accoun aggregaes using he accouning relaionships discussed in Secion. above. The mehodological decision ree found in Figure 3 below describes he seps we follow o derive consisen series by combining he available sources for each counry in he panel. The same approach is applied o all series in he inernaional house price daabase. The moivaion behind each sep is o mainain consisency across counries, while limiing daa disorions as much as possible. The reference series for each counry is is mos curren one, since i is seleced for being he mos consisen wih he benchmark series (FHFA house price index and disposable personal income) and for having daa available up o he presen. In mos cases, he reference series does no exend all he way back o he firs quarer of 1975, and i has o be compleed wih oher series which we refer o as he hisorical series. We label he combined series ha goes back o he firs quarer of 1975 as he exended or he full-lengh series. Old vinages of he inernaional house price daabase are kep for researchers ineresed in realime daa analysis.

3 Figure 3. Mehodological Decision Tree for he Consrucion of Time Series Daa Does primary series exend back o 1975Q1? Is hisorical daa available since 1975? Backcas (if 1- years missing) Are daa frequencies consisen across series? Conver lower frequency o mach higher frequency daa. Are irregular componens presen in any series? Smooh he series wih excess volailiy Splice wih growh raes Is he primary series curren? Is i quarerly? wcas (if curren observaion is no available) Frequency Conversion (if he primary series is no quarerly) Is seasonaliy or volailiy sill presen in he daa? Smooh and Seasonally-Adjus Consruc Index Prices from around he world, all under one roof.

4 3.3 Frequency Conversion Mehods Frequency conversion is ofen required in order o repor he house price and PDI indexes a a quarerly frequency. We are confroned wih he problem of convering high-frequency daa (monhly) o quarerly daa as well as he problem of convering low-frequency daa (semiannual, annual, bi-annual) o quarerly daa. If he series has a higher frequency, hen we use a simple average of he monhly observaions in order o repor he ime series a quarerly frequency. This makes use of all available daa and produces observaions ha summarize he quarerly paerns of he daa. 18 Inerpolaion is applied o he laer problem, bu i is worh poining ou ha inerpolaing procedures can only provide esimaes beween known daa poins, since quarerly daa are no observed. In oher words, i ries o complee he series by fiing a curve o he known daa poins which can hen be used o infer plausible oucomes for he unknown quarerly observaions. The mehod of inerpolaion will be deermined by he feaures ha are mos desirable for he inerpolaed daa. We discuss he implemenaion of he frequency conversion mehods in greaer deail in Secion below Quarerly Inerpolaion Inerpolaion mehods are used for he conversion of low frequency o high frequency daa ha is, for emporal disaggregaion whenever no addiional source of high frequency daa is available o faciliae he conversion. 19 Inerpolaion in he conex of he inernaional house price daabase can be defined as fiing a curve over measuremens made a he sampled periods o infer unsampled quarers wih which hey mus conform. Sandard inerpolaing mehods include consan piecewise, linear, polynomial (quadraic, cubic, ), and spline, among ohers. If he observed ime series has a lower frequency (semi-annual, annual, or bi-annual) han he daabase (quarerly), our preferred choice for inerpolaion is o use he quadraic-mach average (or he quadraic-mach sum) mehod o inerpolae he daa o a quarerly frequency. 0 The quadraic-mach mehods do no guaranee ha he curve fied would saisfy non-negaiviy, even if he observed daa poins are all posiive. This is a problem especially in counries ha experienced periods of hyperinflaion or severe inflaion wihin he sample covered by he inernaional house price daabase (such as Croaia). We resolve he issue inerpolaing he 18 A noe of cauion, averaging may induce a spurious firs-order serial correlaion effec in he differenced series as shown by Working (1960) among ohers. 19 Differen economeric disaggregaion echniques, such as he Denon (1971) and Chow and Lin (1971) approaches o cie jus wo of he mos popular mehods, can also be used for quarerizaion. These echniques inerpolae he low frequency daa a quarerly frequency using relaed indicaor variables ha are repored hemselves a quarerly frequency. We generally do no have access o quarerly indicaors ha can be used wih he available daa for emporal disaggregaion of he series in he inernaional house price daabase. 0 A noe of cauion, inerpolaing wih a quadraic funcion inroduces a sysemaic source of serial correlaion in he regressors because daa poins are relaed o each oher by a quadraic polynomial. This mus be aken ino accoun when using ime series wih inerpolaed daa for he purpose of saisical inference ha is, sandard errors should be made robus o auocorrelaion in hypohesis esing.

5 logged series, and hen use he growh rae in log-differences o splice and exend he daa backwards. In all oher cases, inerpolaion is conduced wih he series expressed in levels. Inerpolaion is performed in wo sages. In he firs sage of inerpolaion, he objecive is o repor he curren and hisorical series a he same frequency, so hey can be spliced ogeher. 1 If he hisorical series is repored a a lower frequency han he curren series, bu he frequency of he curren series is lower han quarerly, hen he hisorical series is inerpolaed o he same frequency as he curren series raher han a quarerly frequency. For example, in he case of Ialian house prices, he hisorical series is repored bi-annually and he curren series is repored semi-annually. In he firs sage, he hisorical series is inerpolaed o a semi-annual frequency and hen spliced wih he curren series. In he second sage of inerpolaion (if his sep were necessary), he objecive is o repor he exended series a a quarerly frequency. Again, using he Ialian house price series as an example, once he bi-annual series is inerpolaed o a semiannual frequency and spliced wih he curren series, he exended series is inerpolaed o quarerly frequency. 3.4 Seasonal-Adjusmen Mehod In order for quarerly daa o be useful for researchers or for policy analysis, he series mus be repored o reflec he rue underlying paerns of he daa. Seasonal-adjusmen aims o pinpoin economically relevan feaures of he daa (rend and cycle), bu irregular componens (ouliers, breaks) may also be presen. In which case, seasonal-adjusmen migh no be sufficien in removing all he irrelevan noise in an observed series. Hence, we develop and implemen a smoohing algorihm based on he basic srucural ime series (BSTS) model, fied o he daa in order o remove boh seasonaliy and he effec of unrelaed irregular noise. We give preference o daa no seasonally-adjused by he source because hen we can apply he BSTS model in a manner ha consisenly reas he effec of seasonaliy across counries and series. The seasonal adjusmen procedures used by he sources ofen vary by counry he U.S. Census Bureau family of seasonal adjusmen procedures (such as X-1) or he Bank of Spain/Eurosa TRAMO-SEATS package which inroduce heerogeneiy in he daa. In cases when he original daa is only repored seasonally-adjused (such as Souh Africa s PDI), we use i wihou furher correcion unless he seasonal adjusmen by he source appears insufficien o remove all seasonaliy and he irregular noise in he ime series. We would hen apply a varian of he BSTS model o his daa in order o obain he final seasonally-adjused, smoohed series for he inernaional house price daabase. The specificaion of he BSTS model in sae-space form is boh simple o esimae and very flexible capuring he relevan componens of he daa. I produces esimaes of he seasonal facors ha are comparable o hose obained wih oher convenional mehods (such as X-1 and TRAMO/SEATS), while providing an inegraed framework for he removal of irregular componens, and can also be used for backcasing and nowcasing. We furher discuss he uses of he BSTS model in Secion 3.7 below. For a more formal evaluaion of he BSTS model for 1 We provide deails on he implemenaion of he splicing mehod in Secion 3.5. We discuss he backcasing and nowcasing uses of he BSTS model in Secion 3.7. Prices from around he world, all under one roof.

6 seasonal adjusmen and he reamen of irregular componens, he ineresed reader is referred o Marínez-García (014). If seasonaliy or large irregular noise componens are presen in he curren series bu no in he hisorical series (especially when he hisorical series is derived from inerpolaed daa), we remove he seasonaliy and irregular componens from he curren series before splicing using he BSTS model. In general, we seasonally-adjus he individual daa series conained in an exended series separaely whenever hey display very differen seasonaliy or irregular paerns, hen splice he smoohed series. If he seasonal paerns of he individual series appear similar, hey are firs spliced, hen seasonal adjusmen is applied o he exended series. The seasonal adjusmen wih he BSTS model is performed on daa wih a frequency higher han annual, bu i can be implemened wih inerpolaed daa as i removes some of he irregular paerns in he daa ha may follow from he implemenaion of he quadraic-mach procedure. 3.5 Splicing Mehod All series are spliced ogeher using he growh raes of he longer hisorical series in order o exend he level of he shorer curren ime series backward in ime. Splicing wih exac growh raes is preferred, bu we use log-differences in he case of counries ha have undergone a period of severe inflaion wihin he sample (such as Croaia). The hisorical ime series are seleced by availabiliy in order o reasonably rack he changes of he reference series. Splicing he curren series wih he growh raes of he hisorical series may no exend he series as far back as In which case, he hisorical series is complemened wih backcased daa, where addiional observaions no represening more han wo years are obained using he bes fied BSTS model for he available hisorical ime series. Time series backcasing is used o exend he house price indexes of Spain and he Neherlands from he firs quarer of 1976 back o he firs quarer of Before splicing he series, we perform he minimal adjusmens needed o obain consisen frequency across series and remove any irregular componens or seasonaliy. wcasing is applied primarily in order o ensure he imeliness and compleeness of he inernaional house price daabase release when he curren series of reference for a counry is missing some recen observaions due o lags in reporing from he naional sources or simply because he repored daa is produced a a lower-han-quarerly frequency. wcasing is used, for example, wih he annual consumpion of fixed capial series of Belgium and he U.K. The daa a a lower-han-quarerly frequency is exended wih a hisorical series and augmened wih he nowcased value for he unobserved year ha includes he curren quarer. Then, he series is inerpolaed o a quarerly frequency. If he curren series is repored a quarerly frequency, bu he reporing lag resuls in missing observaions for he mos recen quarers, we use nowcased values for hose missing observaions. The nowcased values will subsequenly be replaced wih acual daa from heir respecive naional sources, as hose 3 Backcasing esimaes are obained by re-ordering he original daa x backwards from he end of he sample ( = T) o he beginning of he sample ( = 1), and hen running a regression model o forecas he daa of he series x prior o = 1.

7 observaions become available. We finally seasonally-adjus he exended quarerly series, excep when he seasonal paerns or oher irregular componens of he daa appear no o be saisically-significan. All nowcass and backcass are derived from he bes fied specificaion of he BSTS model. This is an esimaed regression model ha is used for seasonal-adjusmen and smoohing as well, where he underlying componens of each ime series decomposed in rend, cycle, seasonal facors, and irregular noise are fied a each poin by Maximum Likelihood and updaed ieraively wih he help of he Kalman filer. In Secion 3.7 below, we give furher inerpreaion o he BSTS model and a shor overview of he mehodology as i is implemened in he inernaional house price daabase. We leave he more echnical deails for he ineresed reader o be found in Marínez-García (014). 3.6 Approach o Aggregae he Counry Daa All counry series a quarerly frequency and exended going back o he firs quarer of 1975 are hen indexed o 005=100. The counry series are aggregaed o produce global indicaors of he housing marke, using weighs ha accoun for he size of he economy of each counry incorporaed in he daabase. We use he 005 purchasing power-pariy adjused (annual) GDP shares of all counries in he sample as consan weighs o derive our aggregae nominal house price index and our aggregae real house price index. Similarly, we use hese consan GDP weighs o aggregae he nominal and real PDI series. All GDP shares are obained from he IMF World Economic Oulook daabase. Prices from around he world, all under one roof.

8 3.7 More in Deph: An Overview of he Basic Srucural Time Series (BSTS) Model We use he BSTS approach as follows, - The esimaion of he BSTS model correcs for seasonaliy a quarerly frequency and also accouns for oher exraneous irregular noise ha may be presen in he observed ime series (hisorical or curren). isy series and/or series wih a saisically-significan quarerly seasonal paern are repored as he combinaion of he smoohed rend and cycle componens, herefore excluding boh he noise and seasonaliy from he series. - The BSTS model is used for backcasing and nowcasing in order o exend some ime series a he beginning and end of heir samples. The esimaed model produces accurae backcass/nowcass wih oherwise minimal effor in idenificaion, bu i is mos appropriae eiher in shor- o medium-horizons where observed daa paerns are likely o coninue or whenever changes in he ime series occur slowly over ime. Backcass/nowcass are used in pracice wih a mos 1- years back/ahead for ha reason All-Encompassing BSTS Model Specificaion Given a ime series y, = {1,,T}, he sandard BSTS model of Harvey (1989) decomposes he daa addiively as follows,, 0, e, 1,,...,, y c s e e NID T (1) where is he ime-varying rend, which capures he long-run evoluion of he series as a funcion of ime; c is he cycle, which denoes ransiory movemens in he series around is rend; s represens he seasonal facor, which capures a recurren movemen in he series ha repeas iself a fairly regular inervals; and e is an irregular (and random) componen called noise, error erm, or disurbance erm, which includes oher non-modeled, exogenous, and inrinsically irrelevan facors affecing he observed series (he signal). These four componens can also be combined muliplicaively in he ime series o produce he log-addiive model specificaion, y exp exp c exp s exp e, e NID 0,, 1,,..., T, () e which has many applicaions in he inernaional house price daabase. This log-addiive model specificaion reduces o he addiive specificaion in (1) whenever we work wih logged values. Hence, he BSTS model can be discussed solely in is addiive form even hough we acually consider he log-addiive case as well in our implemenaion.

9 All four consiuen componens, c, s, e are sochasic, and he disurbances driving hem are muually uncorrelaed in general. The exraneous noise componen e is modeled as whie noise, while he rend, cycle c and seasonal facor s are all assumed o follow a paricular model. As saed earlier, we consider a sochasic polynomial model of order wo or lower for he rend componen. We assume a covariance-saionary AR(p) process for he ransiory componen of he ime series described by he cycle c. We also assume a sochasic version of he sandard seasonal facors model adding up o a random shock o capure ime-variaion in he seasonaliy. For he reasons saed before, we consider boh he addiive as well as he logaddiive specificaions of he BSTS model under consideraion bu describe he modeling choices for he unobserved componens, c, s on he addiive case only. For more deails on he exac sae-space form of each varian of he BSTS model ha we acually explore and a furher discussion of he mehodology and is advanages, he ineresed reader is referred o Marínez- García (014). The addiive, univariae specificaion of he all-encompassing BSTS model is given by, {1,..., T} y c s e e NID 1b1, NID 0,, sae equaion (local polynominal rend), b b 11, NID 0,,, NID 0,,, 0, e, signal equaion (decomposiion equaion), 1 1 c 1c 1c1, NID0,, sae equaion (AR() cycle), 1 c c 1, 1 s s 1s1s1, NID0,, sae equaion (sochasic seasonal), 1 s 1 s, s 1 s, (3) where he componen represens he unobserved level, b deermines he unobserved linear erm on he ime-rend, and is he unobserved quadraic erm on he ime rend. For specificaions wih a cyclical componen c, an AR() (or AR(1)) process is used for reference. For specificaions wih seasonaliy, he quarerly seasonal componen s is recurren and is ime- Prices from around he world, all under one roof.

10 invarian coefficiens mus sum up o a random disurbance. The random disurbances e,,, and are normally disribued, and muually independen a all leads and lags. j,, The parameers of he model include up o six variances for he random disurbances, 0, 0, 0, 0, 0, and 0. The lis of parameers also includes p (p in our specificaion of reference) auoregressive coefficiens used o describe he AR(p) cyclical componen of he ime series. The model descripion is incomplee wihou a specificaion of 1 1 priors for 0, b0, 0, c0, c0, s0, s0, s0, which we generally ake o be diffuse priors in all our esimaions. e

11 3.7. BSTS Model Esimaion, Backcasing/wcasing and Forecasing We represen all varians of he BSTS model in sae-space form and esimae heir ime-varying, unobserved componens by Maximum Likelihood. The advanage of assuming he normaliy of he disurbances is ha i simplifies he esimaion of he model in sae-space form by Maximum Likelihood, and also allows us o use he Gaussian Kalman (Bucy) filer. The Kalman filer algorihm is applied o recursively updae he model wih new observed daa ha conains noise (random variaion) o produce a saisically opimal predicion and forecas of he unobserved saes (cycle, rend, and seasonal) ha characerizes he ime series along wih is corresponding uncerainies. The main assumpions of he Kalman filer are ha he model be a linear dynamic sysem and ha he error erm (irregular noise on he observed series) and all sae disurbances have a Gaussian disribuion. wcasing and forecasing are fairly sandard in his case. Backcasing is accomplished by reordering he original daa y from =T o =1 (in reverse order) and hen running he esimaion and forecasing algorihm under he same sae-space model specificaion chosen o fi he daa. Maximum Likelihood esimaion and he recursive Kalman filer are exploied o obain esimaes of he iniial saes and forecass of he series in reverse order (backcass) BSTS Model Selecion Selecing an appropriae model specificaion depends on: (a) wheher he daa is rending or no (in oher words, he series is non-saionary)? (b) wheher he series has a ransiory componen or cycle ha is saionary? (c) wheher here is seasonaliy in he daa? and (d) wheher he noise and irregular componens are well-approximaed as independen and idenically disribued Gaussian random disurbances? Model selecion would consider alernaive specificaions of he cyclical AR(p) componen and he seasonal model including varians wihou hem based on sandard likelihood-based selecion crieria (primarily he Akaike Informaion Crierion), srongly favoring he less-parameerized and less-complex specificaion of he model. For furher discussion on he goodness-of-fi in-sample and he forecas accuracy ou-of-sample as well as he acual implemenaion of he model selecion procedures, he ineresed reader is referred o Marínez-García (014). Prices from around he world, all under one roof.