A method for combining transaction- and valuation-based data in a property price index

Transcription

1 A mehod for combining ransacion- and valuaion-based daa in a propery price index Olof Nezell Building and Real Esae Economics Royal Insiue of Technology Sockholm 202

2 A mehod for combining ransacion- and valuaion-based daa in a propery price index Absrac: This paper presens a mehod for combining ransacion- and valuaion-based daa in a propery price index. The mehodology is devised for a world where observable ransacion prices can be used o consruc a price index ha consiues a noisy, unbiased signal of he rue price index. I is furhermore assumed ha valuaions can be used o consruc a marke value index which does no conain noise bu ha suffers from so called appraisal smoohing. The valuaion-based index is hus assumed o lag he rue value index and exhibi lower volailiy. The model of he valuaion-based index follows Gelner (993). By regressing he ransacion-based index on he valuaion-based index (conemporaneous and lagged one period) i is possible o filer ou he noise in he observable price index hus esimaing he rue price index. The mehod may be seen as a way of desmoohing a valuaion-based index wihou knowing he smoohing parameer beforehand. The mehodology may also be used as a way of esimaing he smoohing parameer. 2

3 Inroducion Price (or marke value) indices for propery markes are imporan for several reasons. Price indices are for example used as benchmarks by propery owners and by invesors as a means o compare average reurns on propery and alernaive asses such as socks and bonds. High qualiy price indices are also imporan in porfolio allocaion decisions (indices can for example be used o calculae correlaions beween asse classes). Price indices are furhermore imporan in research on propery markes. Research opics where price indices are used include propery cycles and he relaionship beween propery markes and oher financial markes. Unforunaely i is no a simple ask o consruc propery price indices of high qualiy. Two imporan reasons for his are ha properies are heerogeneous - differen properies have differen characerisics (size, age, echnical ameniies ec) - and ha properies are ransaced seldom. This means ha here exiss relaively few observable propery prices during a given ime period on a given marke and ha hose prices are no direcly comparable. The difficuly of consrucing price indices is less severe for cerain ypes of propery. Single-family homes is an example of a propery ype wih relaively many sales where hose properies ha are sold also are relaively comparable. For his propery ype i is herefore comparaively easy o design a reliable index. For commercial properies, on he oher hand, here may exis only a few ransacions in a given year and marke. In hese condiions i may be impossible o consruc a reliable index. The difficuly of consrucing an index is relaed o he level of aggregaion. If he index is inended o capure he price level for properies in Europe we will mos likely have enough ransacion prices. This is likely also he case if we wan o consruc an index for Swedish offices. If we however wan o consruc an index for Sockholm CBD offices or single family homes in a paricular parish of Sockholm here may no be enough daa o consruc a reliable index based on ransacions. One way of circumvening he problem of low liquidiy is o make use of valuaions insead of ransacion prices. This approach depends heavily on he qualiy of valuaions. If valuaions are inaccurae his may no be a reliable way of obaining a price index. As an index is an aggregae of many observaions, inaccuracy of individual valuaions is no necessarily problemaic. Errors may cancel ou. There is however research ha suggess ha valuaions of properies lag behind and underesimae he volailiy of acual value movemens 3

4 (Gelner e al. 2003). This valuaion bias, popularly ermed appraisal smoohing, does no cancel ou when valuaions are aggregaed (Gelner e al. 2003). This paper presens a mehod for combining ransacion- and valuaion-based daa in a price index. The poin of he mehod is o a leas parly provide a remedy for inheren problems in he wo ypes of daa: noise in ransacion daa and smoohing in valuaion daa. The mehodology is devised for a world where here are a leas some observable ransacion prices ha can be used o consruc a price index ha consiues a noisy signal of he rue price index (an index free of bias and noise). Furhermore, i is assumed ha valuaions from he populaion can be used o consruc a noiseless bu smoohed valuaion index. This valuaion index is a lagged, smoohed-ou version of he rue index. By regressing he price index on he valuaion index (conemporaneous and lagged one period) i is possible o filer ou he noise in he observable price index and hence esimae he rue price index. The naure of price indicaion daa in propery markes An asse price index is an index ha measures price movemens in a populaion of asses. For some asses he consrucion of he index is fairly sraighforward. For common socks for example, we may simply collec price observaions of every sock in he populaion for every ime period, add hem and divide by he price level in he chosen base period. Price daa in propery markes is generally no as easily ransformed o a reliable index. For some markes here simply are oo few ransacions for his procedure o be feasible and when ransacion daa acually is available, heerogeneiy of properies ofen makes i difficul o consruc a reliable index. Unless we conrol for differences in propery characerisics, ransacion prices are no comparable. Transacion price A may differ from ransacion price B because he wo ransacions occur a differen poins in ime and prices have changed or because propery A and B are of differen qualiy (propery B may have a nice view for insance). Unless we can conrol for differing qualiy, heerogeneiy will inroduce noise in observed ransacion prices. Hence, an index consruced by aking he average of ransacion prices will be noisy. Noise will pose less of a problem he more ransacion daa ha is available. Heerogeneiy may also inroduce bias in an index. There are wo reasons for his. Firs, he characerisics of properies may change sysemaically over ime. If properies echnical ameniies are improved across a whole marke for insance, we should observe price increases 4

5 due o qualiy improvemen. For a given level of qualiy however, prices may have been consan. Second, properies of differen characerisics may ransac a differen poins in ime. If high qualiy properies ypically ransac in cerain ime periods, failing o conrol for his may lead us o believe ha prices have increased more han hey acually have during hese periods. Noe ha if we had coninuous price daa for every propery, his would no be a problem. Heerogeneiy and low liquidiy hus ogeher make i difficul o creae indices. I should be noed ha wha we mean by bias may depend on wha use he index is inended for. For some applicaions i may no be necessary or even desirable o conrol for all ypes of differences in characerisics. One may for insance wan o consruc an index for which depreciaion and improvemens are no conrolled for. This, along wih oher basic issues regarding index consrucion, is discussed in more deail by Wang and Zorn (997). The fac ha propery markes are search markes is anoher source of noise in propery ransacion daa (Fisher e al. 2007). Transacion prices are he oucomes of negoiaions beween buyers and sellers. For any ransacion he oucome of he ransacion process is jus one realizaion of many possible oucomes. The acual selling price can be viewed as a random variable disribued around he marke value (where I hink of he marke value as he expecaion of he selling price in a normal ransacion, i.e. no forced sales for example). To exemplify, he price may end up below marke value if he buyer has an excepionally skilled negoiaor a he negoiaion. A subsanial lieraure has addressed index consrucion mehodology and has suggesed soluions o he inheren problems. The repea sales regression (firs developed by Bailey e al., 963) is a mehod for producing an index ha compares prices of houses ha have ransaced a leas wice during he period for which he index is consruced. The regression model is consruced so as o compare he ransacion price for he same propery a wo (or more) ransacions. The mehodology hus a leas in par avoids he problem of heerogeneiy. There are hree main problems wih his ype of index. Firs, he mehod requires pleny of ransacion daa and is herefore no feasible for many propery markes. Only properies ha have ransaced a leas wice during he index period can be used. Second, in is simples form, he mehod does no adjus for he fac ha he properies in he index may change over ime (depreciaion, renovaions ec). Laer lieraure has suggesed ways of dealing wih his problem (Case and Quigley, 99, is one example). Third, he mehod necessarily means ha we build he index on properies ha ransac ofen (properies ha have ransaced only once 5

6 during he index period will no ener he regression). These properies may no be represenaive of he populaion. One sudy ha invesigaes his problem is Englund e al. (999). Their sudy shows ha Swedish single family homes ha are ransaced ofen ypically are of lower qualiy (small los ec). Anoher way o design ransacion-based indices ha conrols for differences in characerisics is o use a hedonic regression model. In he hedonic approach, a propery is viewed as a composie good: When buying a propery one is really buying a se of goods. The hedonic approach aims o find he marginal conribuion of each of hese goods or characerisics on he value of he composie good (in our case a propery). This is achieved by regressing he ransacion price of a propery on a number of is characerisics (locaion, area, age ec). By inroducing ime dummies in he regression, i is possible o capure he price level in differen ime periods while he included propery characerisics conrol for heerogeneiy. An alernaive approach is o esimae a hedonic regression for each ime period and revalue a represenaive propery each ime period using each respecive period s characerisics prices. Miles e al. (990) and Webb e al. (992) are examples of sudies where a hedonic mehodology is used. Clapp and Giacoo (992) suggesed an efficien way of conrolling for heerogeneiy among properies. They argue ha valuaions of each respecive propery provide an excellen heerogeneiy conrol. Using valuaions as a conrol for differing characerisics is an aracive idea for wo reasons: They are likely o capure very much of he heerogeneiy and hey are fairly easy o obain unlike oher conrols ha may require collecion of an exensive array of propery aribues. Fisher e al. (2007) presen a new quarerly index for commercial propery ha uses his echnique. As wih repea-sales mehods he hedonic mehod is only feasible when here is pleny of daa. For he hedonic approach no only ransacion daa is needed bu also daa on he characerisics of he properies in he index. A compleely differen approach o consrucing price (or value) indices for propery is o use valuaions insead of observed ransacion prices. A valuaion-based index is consruced by revaluing he same sample of properies each ime period. Valuaion-based indices hus in par avoid he problem of heerogeneiy. However, assuming ha he properies in he sample change in qualiy over ime, his should be aken ino accoun. Using valuaions as a means of racking price (or value) movemens hinges criically on he naure and qualiy of valuaions. There is a fairly subsanial lieraure ha shows ha valuaions are prone o a cerain ype of bias. More specifically, a number of aricles sugges ha valuaions end o lag acual prices and also end o smooh ou acual price movemens, 6

7 so called appraisal smoohing (Gelner e al. 2003, Diaz and Wolveron 998, Fisher e al. 999 and Fisher and Gelner 2000). This phenomenon can be shown o be he resul of opimal valuer behaviour (Quan and Quigley, 989 and 99, Childs e al. 2002) bu is no opimal from an index-consrucion poin of view as smoohing in individual valuaions is likely o spread o an aggregae index. If smoohing is presen in valuaions, an index based on valuaions will simply no show acual price movemens bu movemens in valuaions. The phenomenon will be deal wih in some deail in he following. A valuer ha ries o esimae he marke value of a propery uses ransacion prices from properies ha are as comparable as possible o he propery ha is being valued. Ideally hese comparable sales (comps) should (), come from properies ha are idenical o he propery being valued, (2), he ransacions should have occurred very recenly (ideally a he same momen ha we are making he valuaion) and (3), we should have access o pleny of hem. This will ypically no be he case as properies are heerogeneous and ransac seldom. The valuer will have o make do wih less perfec daa. The daa ha is available o he valuer will conain noise (due o heerogeneiy) and i will no be compleely up o dae (old ransacions). We can hink of he value esimae as a simple average of he ransacion prices from comparable sales. If we use only very recen comps he value esimae will be up o dae bu noisy due o he fac ha we have very few comps in he average, perhaps only one comparable sale. As we include older and older comps he number of comps in he average will be larger reducing he effec of noise. The value esimae will however be less up o dae he farher back in ime we go. Thus here will be a rade-off beween noise and bias depending on how far back he valuer decides o go. Using only recen comparable sales will give an esimae ha conains a lo of noise bu very lile bias. Using comparable sales from a longer ime period will resul in less noise bu more bias. How far back i is opimal o go depends on wha use he valuaion is inended for. If we aim for as small error as possible in he individual value esimae i may be opimal o go quie far back as his will reduce he noise. If we wan o have an esimae wih as lile bias as possible i may be opimal o use only very recen comps. If we wan o value an enire porfolio of properies for example i is arguably beer o have unbiased bu noisy esimaes of he individual properies as noise will filer ou in he aggregae. This descripion of he valuers problem is simplified (valuaions are usually no simple averages of comps) and is mean o give an inuiive explanaion for appraisal smoohing. The general idea is ha valuers use old informaion and ha his behaviour is jusified. Quan and Quigley (99) have sudied he valuers problem more formally. They find ha, given a 7

8 number of assumpions, i is opimal for he valuer o behave according o he following model: ( ) vi = αi pi + αi vi () Where v i is he valuaion of propery i in ime, p i are (noisy) conemporaneous comparable sales and v i- is he valuaion in he previous period. α i is a parameer ha ells us how much weigh is given o curren informaion relaive o how much weigh is given o old informaion. A large α i corresponds o much weigh being given o conemporaneous informaion and vice versa. Noe ha previous valuaions (v i-, v i-2 ec) will follow he same model. I can easily be shown ha formula () is simply a weighed average of he curren and all previous comparable sales (i.e. p is where s=, -, -2,..) wih lower and lower weighs he farher back we go (see formulas (6) and (7) below). α i is usually wrien wihou ime or individual subscrips bu hey are included here in order o emphasize ha alpha may differ over ime as well as for differen properies. An index based on valuaions ha follow he paern in formula () will be smoohed. For many applicaions his is problemaic and research has herefore been devoed o he quesion of how o derive unbiased price indices from valuaion-based indices. Two groups of soluions are he zero-auocorrelaion mehod and he reverse engineering mehod. The zero-auocorrelaion mehod builds on he idea ha reurns in propery markes should be unpredicable. Using his assumpion i is possible o back ou rue (non-auocorrelaed) reurns hrough a regression where auocorrelaed reurn is filered ou. Once rue reurns have been calculaed hese can be used o calculae rue price levels. Blundell and Ward (987) proposed his echnique and a number of aricles have used and/or developed he mehod (Fisher e al. 994, Cho e al and Brown and Maysiak, 998). The main cavea of he mehod is he problemaic assumpion of zero-auocorrelaion in reurns, which may no hold. The reverse engineering mehod is relaed o model () and was proposed by Gelner (993). Gelner (993) argues ha if individual valuaions follow he paern in formula (), a valuaion-based index will be well described by he following model: I is opimal in he sense ha if v i is chosen in accordance wih his formula, i will converge o he rue marke value of he propery faser han any oher simple linear valuaion rule. 8

9 ( ) V = αp + α V (2) where V and V - are he valuaion-based index levels a ime and - respecively and P is a price (or marke value) index level a ime (he ilde is merely here o disinguish P from a differen price index P below). Noe ha V and V - are observable. P on he oher hand is here regarded as a non-observable componen of V. If (2) holds and if we know α i is possible o consruc a price index by backing ou ( reverse engineering ) formula (2): P from ( α ) V V P =. (3) α α (3) is jus a simple manipulaion of formula (2). Gelner furhermore argues ha he noise in p i will largely diversify away in heir aggregae counerpar P so ha P may be viewed as a rue (unbiased, noiseless) price index. We may also assume ha P conains noise and employ some noise-reducion echnique. The i subscrips have been dropped in formula (2) and (3) in order o emphasize ha P, V and V - are measured a he index level in hese formulas and ha α when used in his way usually is assumed o be consan (an assumpion ha may no hold). One of he main problems wih reverse engineering is ha we mus esimae α, which is inherenly difficul as we do no have access o he rue price index and probably no he valuers comps (p i in formula ()) eiher. One of he few sudies on he subjec is Clayon e al. (200). The difficuly of obaining α is aggravaed by he fac ha α may vary over ime and over differen properies (empirical suppor for his can be found in Brown and Maysiak (998)) and ha α on he individual propery level no is necessarily immediaely ransferable o he aggregae (index) level (Bond and Hwang, 2007). So far we have discussed how boh valuaions and ransacion prices are imperfec measures of price movemens in propery markes. There are, however, oher more indirec indicaors of propery prices. One prominen example is prices on socks of lised propery companies (or REITS). These prices refer o indirecly owned propery which means ha hey canno be used as price indicaors for he direc propery marke wihou adjusmen (or a 9

10 leas no wihou cauion). Propery socks are for example usually leveraged asses. This has o be aken ino accoun as we usually creae propery indices for properies as such, no leveraged propery holdings (which does no preclude ha he properies in indices are owned by leveraged owners). Empirical research has also found ha propery sock prices move parly independenly from he direcly owned propery marke (Chau e al. 200). Ling e al. (2000) and Fu (2003) are wo examples of aricles ha presen mehods of using indirec indicaors for compuing price indices. Boh aricles make use of laen variable models. Wih his ype of model i is possible o calculae an unobservable laen variable wih he help of a number of observable indicaor variables. Applied o propery price indices, he laen variable is he rue value index while valuaions and propery sock prices may be used as indicaor variables. Proposed index consrucion mehod In shor, he seing is as follows. I is assumed ha indicaions of curren marke value can be obained from wo sources; ransacion prices and valuaion daa. The ransacion prices are assumed o be unbiased esimaes of marke value bu conain a lo of noise. The valuaion daa on he oher hand is assumed o suffer from he effecs of appraisal smoohing (lag, lower volailiy). Assume ha here are hree indices in he marke, wo of hem observable and one unobservable. Firs we have he unobservable rue price index, I, ha we wan o esimae. There is also a ransacion-based index, P, which is buil on noisy ransacion price daa. I is assumed ha he price index P is dispersed around he rue marke value index: P = I + u (4) where u is a random error disribued around he marke value index and E( u ) = 0. u is assumed o be uncorrelaed wih I s where s =, +2, +,, -, -2.. The variance of u may differ in differen ime periods. In words, P is a noisy measure of I. Assume furhermore ha we have a valuaion-based index, V. This index is buil on individual appraisals. The individual appraisals are assumed o follow he paern discussed above (formula ()). I is furhermore assumed ha his paern carries hrough o he index so ha we have 0

11 V ( α ) V = α I +, (5) where α is he smoohing parameer. In words, he valuaion series V provides a smoohed bu noiseless signal of I. Regarding he behaviour of I and V, he presened seing is he same as Gelner (993). One could use reverse engineering on he valuaionbased index V presuming ha we have an idea of he value of α. Afer considering he se-up, he following quesion may arise: Why does he price index P conain noise while he signal of rue value in he valuaion-based index does no? In he presened se-up, he individual valuaion is buil on noisy price informaion and he previous valuaion, bu when we combine valuaions in an index, he noise in he price informaion filers ou. Why can we no simply collec he price informaion ha valuers use and creae a ransacion-based index free of noise? The noise filers ou in he valuaion-based index why no in he ransacion-based index? The se-up implicily assumes ha he price informaion ha valuers have access o is richer han he price informaion available o he person consrucing he index. This requires some moivaion. Firs of all, he informaion available o valuers may be cosly or impracical for he index-consrucor o acquire. I may for example be he case ha he daa are no colleced in one place or ha he raw daa needs exensive processing before use. Secondly, valuers may have access o informaion ha simply is no available o he index-consrucor. Some ransacion prices may no be disclosed publicly bu leak o valuers. Some ransacions are par of a larger deal ha includes oher asses as well. In his ype of deal he implici ransacion price of he propery may no be known o he public bu o valuers. Furhermore, he noisy price informaion ha valuers use may no be acual ransacion prices. Knowledge of deals ha did no happen, rumours ec may be seen as par of he noisy price informaion used by valuers. Despie his argumen one may argue ha he rue price index componen in (5) should include an error erm. The effecs of allowing for his are discussed in a subsequen secion (equaions (25) and (26)). Simulaion (A) in Figure shows visually how I, P and V relae o each oher. In his simulaion I is assumed o follow a random walk: I = I + v (6) ( 0,) v ~ N (7)

12 I was consruced by generaing 25 random numbers (v ) and hen using formula (6). P was generaed using formula (4) where ~ N( 0,4) u. V was consruced using formula (5). α was se equal o 0.4. The figure illusraes ha P is a noisy (more volaile) version of I and ha V is a smoohed (less volaile, lagging) version of I I P V Figure. Simulaion (A) of a rue value index (I ), an index consruced wih observable ransacions (P ) and an index consruced wih valuaions (V ). Equaion (5) is equivalen o: ( α ) V V I = (8) α α Equaion (8) is a descripion of how I is relaed o V and V - where I is expressed as a linear funcion of V and V -. Of course, I is no lierally driven by V and V -. (8) merely shows how variaion in I can be capured wih V and V - if we assume ha equaion (5) holds. Assuming ha we can observe he hree variables we could esimae (8) by OLS. If we were o regress I on V and V - we would be able o capure all variaion in I since I only 2

13 depends on V and V -. The coefficien for V would equal /α and he coefficien for V - would equal ( α) / α. If we included an inercep in he regression i would equal zero. I use he word depend here in he sense ha he variaion in I can be capured by V and V -. Now, we can observe V and V - bu no I. We can however observe P which is jus a noisy measure of I : V = α ( α ) V P + α u (9) I have simply insered he righ-hand side of equaion (8) insead of I in equaion (4) in order o arrive a (9). Model (9) is possible o esimae since we have assumed ha P and V are observable. We would hen run he following regression model: P = β0 + βv + β2v + e (0) where we know from (9) ha he rue parameers are β 0 = 0, β = / α, ( ) β2 = α / α and ha e = u. Assuming ha u is uncorrelaed wih V and V - he coefficiens for he explanaory variables will be unbiased. In oher words, heir expeced values are heir respecive rue populaion counerpars: ( 0 ) E ˆ β = 0, () ( ) E ˆ β = / α, (2) ( 2 ) ( ) E ˆ β = α / α (3) We can obain prediced P : P ˆ ˆ ˆ ˆ = β0 + βv + β2v (4) The expeced value of P ˆ given V and V - is: 3

14 ( ˆ ) ( ˆ ˆ ˆ, β0 β β2, ) E P V V = E + V + V V V (5) β β V β V = V α) V α α ( = = I In words, prediced P is an unbiased esimae of I. As he number of observaions increases, he coefficiens are beer and beer esimaed and he prediced P will come closer and closer o I. Figure 2 shows simulaion (B) which is similar o simulaion (A) in Figure bu in which I have also included P ˆ which is prediced P from a regression where P is regressed on V and V - (regression model (0)). As is eviden from he figure, he prediced P comes close o I I P V pred. P Figure 2. Simulaion (B) of a rue value index (I ), and an esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. 4

15 We do no acually have o assume ha u is uncorrelaed wih V and V -. I follows from previously made assumpions: (i) he assumpion ha u is uncorrelaed wih I in all ime periods and (ii) he assumed model of he appraisal-based index, equaion (5). To see his, noe ha equaion (5) implies ha V can be expressed as a funcion of he curren and lagged values of I. We have (equaion (5) resaed): V ( α ) V = α I + (6) on yields: Inserion of I + ( α ) V 2 α insead of V, I 2 + ( α ) V 3 α insead of V 2 and so V 2 3 ( α ) αi + ( α ) αi + ( α ) αi... = I α (7) Equaion (7) shows ha V is a funcion of I s where s =, -, -2. which are all uncorrelaed wih u by assumpion. Hence, u is uncorrelaed wih V. The same argumen holds for V -. The reader may objec ha esimaing P on V and V - resuls in biased coefficien esimaes due o simulaneiy (he argumen migh be ha prices drive valuaions, no he oher way round). Then we have o remember wha we are rying o achieve wih regression equaion (0). The poin of he regression is no o es a causal relaionship. The poin is insead o reduce he noise in he P observaions (or o ge rid of he lagging/smoohing behaviour in V if you will). β and β 2 should no be hough of as measuring causal effecs bu raher he linear relaionship beween P, V and V -. We know from he assumpions ha we have made ha his relaionship follows formula (9). How can valuaions compleely capure rue price movemens in his seing? In order o give an inuiive explanaion why his may be he case le us sar wih he basic model of how he valuaion-based index relaes o he rue price index: V ( α ) V = α I + (8) The formula shows ha V conains boh he rue price I scaled down by a facor α and he previous valuaion V -. Thus, by scaling up he rue price componen and geing rid 5

16 of he V - componen we have he rue price. This is exacly wha happens when we regress P on V and V -. From (5) we have ha: ( ˆ V ( α) V, ) E PV V = (9) α α The firs erm in his expression may be hough of as he erm ha scales up he I componen of V. To see his noe he following: V = ( αi + ( α) V ) = α α ( α ) V = I + α (20) ( α) V Subracing he previous-valuaion-componen, V, from we ge: α α V ( ) ( ( α) V α V α) V = I + = I (2) α α α α Relaxing assumpions The proposed mehod relies on a number of assumpions. If hese assumpions are fulfilled, he index consrucion mehod works well in he sense ha i produces an unbiased esimae ha converges o he rue index series. Of course, he assumpions may no be fulfilled or a leas may no be compleely fulfilled. The res of he paper discusses how he resuls are affeced if he assumpions are no fulfilled. Price process In he presenaion of he mehodology, he process of he rue price index was no discussed and no assumpions were made abou wha i looks like. In oher words, he index consrucion mehod is no dependen on a paricular process of he rue price index. Simulaion (C) was made o illusrae his. I is assumed o follow an ARMA(,) process: 6

17 I I v v (22) = ( 0,) v ~ N (23) V and P are consruced in he same way as in simulaion (A) and (B) bu α is assumed o be 0.3 in his simulaion I P V pred. P -6-8 Figure 3. Simulaion (C) of a rue value index (I ), and an esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. I is assumed o follow an ARMA(,) process. As in simulaion (B), prediced P follows I closely: he mehodology is no sensiive o he process of he rue price index. The simulaion serves a second purpose. In his simulaion, 200 observaions were generaed insead of 25 observaions as in simulaion (B). This means ha when regressing P on V and V - in his simulaion, coefficiens are esimaed wih more accuracy. Consequenly, prediced P follows I more closely han in simulaion (B) illusraing he fac ha he more observaions, he beer he proposed mehodology works. Valuer model The assumpion of how he valuaion index behaves, equaion (5), is explicily used in he derivaion of he index consrucion mehod. In general, herefore, he mehod does no 7

18 work unless his assumpion holds. The mehod may however sill work as an approximaion even if equaion (5) does no hold in a sric sense. Wheher he approximaion is reasonable or no depends on exacly how realiy deviaes from equaion (5). As he rue behaviour of V may deviae from equaion (5) in counless ways i is impossible o give an exhausive discussion of wha happens when model (5) is invalid. This secion will discuss some possible deviaions. Firs, one may hink of several models ha share imporan rais wih model (5) bu deviae in some sense. Model (24) is one such example: V ( α α 2 ) I 2 α I α I (24) = This model will lag he rue index and will smooh ou is movemens jus like model (5). The difference beween he models is he weighs and he fac ha model (5) goes furher back in ime. Model (24) is moivaed for example if we hink ha valuers do no go as far back in ime as suggesed by model (5). A simulaion was run where he rue price index is assumed o follow a random walk as in simulaion (B), P is generaed as in simulaion (B) and V is now assumed o follow model (24) wih weighs chosen o be α = α = α α 3. The resuls of simulaion 2 2 = (D) are shown in figure 4. As expeced, he resuls are no as good as in he previous simulaions. The mehodology does however no collapse compleely. There is lile lagging and much of he noise is eliminaed. If we have more observaions he resuls are even beer. Simulaion (D) was made wih 25 observaions. Appendix A shows he resuls when he simulaion is made wih 000 observaions. While he resuls for model (24) are encouraging, hey canno be generalized. Simulaion (D) does however show ha he mehodology does no necessarily collapse if model (5) is no rue. 8

19 I P V pred. P Figure 4. Simulaion (D) of a rue value index (I ), and an esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. V is assumed o follow model (24). An aleraion o model (5) ha makes sense inuiively is o assume ha insead of I in model (5) we have I* which is I plus random noise n : V ( α ) V = I * + α (25) I * = I + n (26) The raionale for his model is ha maybe no all of he noise from he individual valuaions is filered ou when valuaions are aggregaed ino an index. If (25) holds he rue populaion model of P is: V ( α ) V P = n + α α u (27) 9

20 If we regress P on V and V - when he rue populaion model is equaion (27) he coefficien esimaes will be biased as V is correlaed wih he error erm in equaion (27). This can be seen from equaion (25) and (26): V is a funcion of n. In general herefore, his ype of deviaion from he assumpions is problemaic. Three simulaions were made in order o see how problemaic. The simulaions are all similar o simulaion (B) excep ha V is consruced using formula (25) and (26). They differ beween each oher in how large he variance of n is. Simulaion (E) has he lowes variance of n, , which can be compared wih each ime periods innovaion in I which has a variance of. When he variance of n is his low he problem associaed wih his ype of deviaion is relaively small (see figure 5) I P V pred. P Figure 5. Simulaion (E) of a rue value index (I ), and an esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. V is assumed o follow model (25) and he variance of n is If he variance of n is as in simulaion (F) here are bigger problems as can be seen from figure 6. Appendix B shows he resuls when he variance of n is When he variance is his high, he prediced P follows V raher han I. This simulaion is however no included as a pracical example bu raher o show ha he esimae of P is biased owards V. 20

21 The resuls show ha he effec of his ype of noise depends criically on he variance of he noise I P V pred. P Figure 6. Simulaion (F) of a rue value index (I ), and an esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. V is assumed o follow model (25) and he variance of n is Consan alpha The proposed model implicily assumes ha he smoohing parameer α does no change over ime. Quan and Quigley (99) showed in a heoreical model ha α can be expeced o be differen in differen marke condiions. This is inuiively appealing since differen periods exhibi differences in ransacion volume and hence he number of comps ha valuers can use. Brown and Maysiak (998) show empirical evidence ha α differs over ime and circumsances. A simulaion (G) was made in order o see wha happens when α changes over ime. In he simulaion, α follows a simple process: for he firs 3 ime periods, 2

22 α is 0.4, for he laer 2 ime periods α is 0.2. Excep for he changing α he simulaion is similar o simulaion (B) I P V pred. P Figure 7. Simulaion (G) of a rue value index (I ), and an esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. The smoohing parameer α shifs over ime in his simulaion. The simulaion, shown in figure 7, shows ha he mehod is sensiive o changing α: For he firs par of he index, rue marke movemens are exaggeraed while he opposie is rue for he laer par. This sems from he fac ha α is esimaed a 0.3 or he average α over he ime period. Consequenly α is underesimaed for he firs half of he period and overesimaed for he second half. This in urn has he effec ha movemens in I is exaggeraed in he firs half and he oher way round in he second half. Simulaion (G) has shown bu one way in which α may change bu has demonsraed ha he mehod is sensiive o his assumpion. A feasible remedy o his problem is o use a rolling regression echnique. 22

23 Conclusion This paper presens a mehod for combining ransacion- and valuaion-based daa in a price index. The poin of he mehod is o a leas parly provide a remedy for inheren problems in he wo ypes of daa: noise in ransacion daa and smoohing in valuaion daa. The mehodology is devised for a world where he observable ransacion prices can be used o consruc a price index ha consiues a noisy signal of he rue price index. Furhermore, i is assumed ha valuaions can be used o consruc a marke value index which is a noiseless bu smoohed version of he rue index. By regressing he observable price index on he valuaion index (conemporaneous and lagged one period) i is possible o filer ou he noise in he observable price index. If here are many observaions, he prediced observable price index comes very close o he rue price index. The mehod may be seen as a way of de-smoohing a valuaion-based index. The advanage ha his mehod gives compared o earlier de-smoohing echniques is ha i does no require us o know he smoohing parameer beforehand. On he conrary, he mehodology may be seen as a way of esimaing he smoohing parameer. The paper discusses some of he assumpions made. I is shown ha he mehod is insensiive o he rue price process. The model of he valuaion index is a more crucial assumpion bu i is demonsraed ha deviaion from he model assumed is no necessarily criical. I is furhermore poined ou ha over ime varying smoohing of he valuaion index is problemaic. This may however be remedied by a rolling regression echnique. 23

24 References Bailey, M.J., Muh, R.F. & Nourse, H.O. (963). A Regression Mehod for Real Esae price index consrucion, Journal of he American Saisical Associaion, 58, Blundell, G.F. & Ward, C.W.R (987). Propery porfolio allocaion: A mulifacor model, Land Developmen Sudies, 4(2), Bond, S.A., & Hwang, S. (2007). Smoohing, nonsynchronous appraisal and cross-secional aggregaion in real esae price indices, Real Esae Economics, 35(3), Brown, G.R. & Maysiak, G.A. (998). Valuaion smoohing wihou emporal aggregaion, Journal of Propery Research, 5(2), Case, B. & Quigley, J.M. (99). The dynamics of real esae prices, The Review of Economics and Saisics, 73(), Chau, K.W., Macgregor, B. D. & Schwann, G. M. (200). Price discovery in he Hong Kong real esae marke, Journal of Propery Research, 8(3), Childs, P., O, S. & Riddiough, T. (2002). Opimal valuaion of noisy real asses, Real Esae Economics, 30(3), Cho, H., Kawaguchi, Y. & Shilling, J.D. (2003). Unsmoohing commercial propery reurns: A revision o Fisher-Gelner-Webb s unsmoohing mehodology, Journal of Real Esae Finance and Economics, 27(3), Clapp, J. & Giacoo, C. (992). Esimaing price indices for residenial propery: A comparison of repea sales and assessed value mehods, Journal of he American Saisical Associaion, 87, Clayon, J., Gelner, D. & Hamilon, S. (200). Smoohing in commercial propery valuaions: Evidence from individual appraisals, Real Esae Economics, 29(3), Diaz, J. & Wolveron, M. (998). A longiudinal examinaion of he appraisal smoohing hypohesis, Real Esae Economics, 26(2),

25 Englund, P., J. M. Quigley & Redfearn, C.L. (999). The choice of mehodology for compuing housing price indexes: comparisons of emporal aggregaion and sample definiion, Journal of Real Esae Finance and Economics, 9(2), 9-2. Fisher, J. & Gelner, D. (2000). De-lagging he NCREIF index: ransacion prices and reverse-engineering, Real Esae Finance, 7(), Fisher, J., Gelner, D. & Webb, B. (994). Value indices of commercial real esae: A comparison of index consrucion mehods, Journal of Real Esae Finance and Economics, 9(2), Fisher, J., Gelner, D. & Pollakowski, H. (2007). A quarerly ransacions-based index of insiuional real esae invesmen performance and movemens in supply and demand, Journal of Real Esae Finance and Economics, 34(), Fisher, J., Miles, M. & Webb, R. (999). How reliable are commercial propery appraisals? Anoher look, Real Esae Finance, 6(3), 9-5. Fu, Y. (2003). Esimaing he lagging error in real esae price indices, Real Esae Economics, 3(), Gelner, D. (993). Esimaing marke values from appraised values wihou assuming an efficien marke, Journal of Real Esae Research, 8(3), Gelner, D., Macgregor, B.D. & Schwann, G.M. (2003). Appraisal smoohing and price discovery in real esae markes, Urban Sudies, 40(5-6), Ling, D.C., Naranjo, A. & Nimalendran, M. (2000). Esimaing reurns on commercial real esae: A new mehodology using laen-variable models, Real Esae Economics, 28(2), Miles, M., Cole, R. & Guilkey, D. (990). A differen look a commercial real esae reurns, Journal of he American Real Esae and Urban Economics Associaion, 8(4), Quan, D.C. & Quigley, J.M. (989). Inferring an invesmen reurn series for real esae from observaions on sales, Journal of he American Real Esae and Urban Economics Associaion, 7(2),

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27 Appendix A I P V pred. P Simulaion of rue value index (I ), and esimaion of I (prediced P ) using ransacions-based and valuaion-based indices. V is assumed o follow model (24). The simulaion is based on 000 observaions. 27

28 Appendix B I P V pred. P Simulaion of a rue value index (I ), and an esimaion of I (prediced P ) using ransacionsbased and valuaion-based indices. V is assumed o follow model (25) and he variance in n is