The Housig Bubble ad a New Approach o Accouig for Housig i a CPI W. Erwi iewer (Uiversiy of Briish Columbia), Alice O. Nakamura (Uiversiy of Albera) ad Leoard I. Nakamura (Philadelphia Federal Reserve Bak) (July 7, 29) Absrac Over he course of he rece house price bubble i he Uied Saes, he price of homes rose rapidly from 999 Q4 o 25 Q4 (.3 % aually as measured by he Case-Shiller idex, ad 8.4 perce aually as measured by he Federal Housig Fiacig Agecy) bu slowly as measured by ower equivale res (3.4 perce), so measured core iflaio remaied relaively docile durig his period sice oly res are used o measure iflaio for housig services i he Uied Saes. Over he las several decades, he U.S. Bureau of Labor Saisics (BLS) has experimeed wih boh real equivalece ad user cos approaches for accouig for ower occupied housig (OOH) services i he CPI. We explai he basics of hese approaches, ad oulie he BLS experieces wih usig hem. This assessme leads us o coclude ha he ime has come o ry a ew approach: he opporuiy cos approach. We argue his approach has advaages over boh he coveioal real equivalece ad user cos approaches, hough i embeds compoes of he measures for boh hose approaches ad builds solidly o he research of Verbrugge ad ohers a he BLS. Also, we ake up empirical issues ha mus be faced regardless of which of he approaches discussed is adoped. We explai how he repea-sales ad various hedoic regressio mehods ca be placed i a commo framework, hereby faciliaig udersadig of he properies of ad he radeoffs bewee he mehods. We also cosider measureme complicaios ha arise because he lad ad srucure compoes of properies depreciae a differe raes.. Iroducio How is he cos of housig services chagig over ime for hose livig i heir ow homes? Good measures are eeded by ecoomic policy makers maagig everyhig from he moey supply o beevole icome rasfer programs, bu are hard o come by sice homeowers do o acually pay hemselves for he services of heir owed homes. Over he course of he rece house price bubble i he Uied Saes, he price of homes rose rapidly from 999 Q4 o 25 Q4 (.3 % aually as measured by he Case-Shiller idex, ad 8.4 perce aually as measured by he Federal Housig Fiacig Agecy) bu slowly as measured by ower equivale res (3.4 perce). Oe cosequece was ha measured core iflaio remaied relaively docile durig his period sice oly res are used o measure iflaio for housig services i he Uied Saes. Ye, as Galli (29) ad Croe e al. (29) Fiacial assisace from he Caadia SSHRC is graefully ackowledged, as is he hospialiy of he Cere for Applied Ecoomic Research a he Uiversiy of New Souh Wales. This paper reflecs he opiios of he auhors ad does o represe he views of he Federal Reserve Bak of Philadelphia or of he U.S. Federal Reserve Sysem. The auhors hak he paricipas i a semiar ad a workshop held a he Bak of Caada, a semiar a he eparme of Ecoomics a Oawa Uiversiy, meeigs a he Philadelphia Federal Reserve Bak, ad a sessio of he 29 aual meeig of he Caadia Ecoomics Associaio for helpful commes. Special haks are due o Radal Verbrugge, Thesia Garer, Pierre uguay, Adrew Baldwi, Marc Prud homme, Emi Nakamura, Kam Yu, Mario Seele, Maso Gaffey, Ae Laferrère, eis Capozza, eis Fixler, Johaes Hoffma, ad Paul Schreyer for helpful commes o various versios of his paper ad relaed discussios, bu all remaiig fauls of he paper are our sole resposibiliy. - -
have argued, res ad home prices are coiegraed over he log ru. Moreover, Galli has preseed evidece ha home prices ed o overshoo while res ed o lag. If he housig bubble ook he form of ususaiable icreases i home prices, he expecaio ha hese icreases could be susaied i he shor ru perhaps lead o a shor ru divergece bewee res ad home prices. If here is iformaio abou iflaio reds i boh res ad home prices, i migh be useful o combie boh ypes of measures. The opporuiy cos approach which we develop combies iformaio o home price chage, as par of a fiacial user cos compoe, ad iformaio o res as a real equivale compoe. Over he las several decades, he U.S. Bureau of Labor Saisics (BLS) has experimeed wih boh real equivalece ad user cos approaches for accouig for ower occupied housig (OOH) services i a Cosumer Price Idex (CPI). We explai he basics of hese approaches i secios 2 ad 3, respecively, ad oulie he BLS experieces wih usig hem i secio 4. This assessme leads us o coclude ha he ime has come o ry a ew approach o accouig for OOH services coss i measures of iflaio: a ew approach ha everheless builds o BLS experise ad research fidigs, especially icludig he work of Radall Verbrugge ad his collaboraors. 2 The opporuiy cos approach, iroduced i secio 5, was firs suggesed a a 26 OEC Workshop by Erwi iewer. 3 We argue his approach has advaages over boh he coveioal real equivalece ad user cos approaches, hough i embeds compoes of he measures for boh hose approaches. Also, i secios 6 ad 7, we ake up empirical issues ha mus be faced regardless of which of he approaches discussed is adoped. We explai how he repea-sales ad various hedoic regressio mehods ca be placed i a commo framework, hereby faciliaig udersadig of he properies of ad he radeoffs bewee he mehods. We also cosider measureme complicaios ha arise because he lad ad srucure compoes of properies depreciae a differe raes. Secio 8 cocludes. 2. The Real Equivalece Approach The real equivalece approach values he services yielded by a owed dwellig a he correspodig marke real value for he same sor of dwellig for he same period of ime. This is he approach used by he BLS a prese for he CPI. 4 The price daa eeded for he CPI real equivalece compoe for OOH services are observaios o res paid by reers: he same price daa also used by he BLS o compile he real compoe of he CPI. The locaio of each real ui for which re ad oher daa are colleced is uique. Empirical sudies have show locaio o be a key deermia o oly of boh res ad resideial real esae price levels, bu also of he raes of chage over ime i he levels. Hece, afer choosig a sample of dwellig uis o use for he collecio of re daa, he BLS repeaedly samples hose uis. I is assumed ha he chages i owers equivale res wihi small geographic areas (areas of 3-4 ciy blocks, someimes called segmes) will move similarly o chages i acual res. (The aure of his re daa, ad some of he mai daa ses for 2 See Verbrugge (28) ad Garer ad Verbrugge (29), ad also Poole, Pacek ad Verbrugge (25). 3 See iewer (26/29). 4 This secio draws o he U.S. Bureau of Labor Saisics (BLS) (27). - 2 -
housig price daa oo, are why, i secio 6, we explore he relaioship bewee he repea sale ad hedoic esimaio mehods.) Each real ui ha is priced does double duy: i represes he res for reers wihi he segme, ad i also separaely represes he re equivales implicily paid by owers wihi he segme. 3. The User Cos Approach The oly aios ha use he user cos approach o accou for he cos of OOH services i heir official measures of iflaio omi he propery appreciaio erm. 5 However, repors o he reame of OOH by official saisics agecies, icludig he BLS, make freque referece o he shared heoreical uderpiigs for he user cos ad he real equivalecy approaches, ad i is he user cos, icludig he propery appreciaio erm, ha is releva i his regard. The propery appreciaio erm of he user cos formula also plays a impora role i he research of Verbrugge ad his collaboraors. Hece, i his secio, we describe he user cos approach ad show why ad how he propery appreciaio erm eers io he user cos formula. The user cos approach is rouiely used i a variey of oher measureme ad accouig coexs oo, such as i he capial asse pricig lieraure, i producio fucio sudies, i he measureme of oal facor produciviy growh, ad i he aalysis of ax depreciaio rules. The uderlyig heoreical framework is provided by he fudameal equaio of capial heory. Accordig o his equaio, i equilibrium, he price of a durable asse equals he prese discoued value of he fuure e icome ha is expeced o be derived from owig i. Thus, if he fuure icome flow ha a asse such as a machie ca geerae is kow or ca be readily forecas, he his iformaio ca be used o ifer wha he asse would be worh o a buyer. O he oher had, i he lieraure o iflaio measureme for OOH services, wha is direcly observed are he purchase prices for houses ad here are o observable rasacios for he re ha ower occupiers implicily charge hemselves for use of heir homes. Isead, he fudameal equaio of capial heory is used o ry o back ou he period by period coss o he ower occupier of he OOH services hey are usig. iewer (974, p. 54) ses ou he user cos priciples for cosumer durables: 6 To form he real price (or user cos) for he services of oe ui of he h good durig period, we imagie ha he cosumer purchases he good durig period ad he sells i durig he followig period (possibly o himself). The he discoued expeced real price for he h cosumer good durig period is give by he discoued cos of he purchase of he h good durig period mius he discoued resale value of he depreciaed good durig period +. Ad, Bajari, Bekard, ad Kraier (23, p. 3) oulie he coecio i he housig ecoomics lieraure of user coss o he real equivalece approach: 5 See iewer ad Nakamura (29) for summary iformaio ad refereces regardig he use of he user cos approach by Saisics Icelad ad Saisics Caada. 6 iewer (974, 98) followed Fisher (897) ad Hicks (939) i derivig he user cos usig a discree ime approach raher ha he coiuous ime approaches used by Jorgeso (963, 967), Griliches (963), Jorgeso ad Griliches (967, 972) ad Chrisese ad Jorgeso (969, 973). See also Schreyer (29a, 29b). - 3 -
oughery ad Va Order (982) were amog he firs o recogize ha he user cos... should be equal o he real price of a sigle ui of housig services charged by a profi-maximizig ladlord. Thus, he iherely difficul ask of measurig a uobservable margial rae of subsiuio is replaced by he much easier ask of measurig res. 3. The Fudameal Equaio of Capial Theory A clear udersadig of he basics of he fiacial heory jusificaio coveioally cied as he basis for boh he real equivalece ad he user cos approaches is helpful o have before we discuss he opporuiy cos approach. We review his fiacial heory basis here. Aeio o imig maers for udersadig user coss. Realized prices are deermied a pois i ime. Raes of ieres are regarded as fixed a pois i ime. I coras, raes of iflaio are defied for ime iervals. If here is iflaio, moey is less valuable whe received a he ed versus he begiig of a period. A ed of period value ca be covered o is equivale a he begiig of ha same (o he ex) period by discouig by he erm + r, where r is he period omial ieres rae. As Arold Kaz (29) of he U.S. Bureau of Ecoomic Aalysis (BEA) 7 explais, he user cos of capial is based o he fudameal equaio of capial heory. I box, he derivaio of he user cos by Kaz (29, appedix A) is show, recas usig he oaio for our paper. We deoe he value of a home ha is v periods old a he sar of period by V v. Give oly he iformaio available a he sar of, he expeced price a home could be sold for a he ed of period, which is he sar of period +, is deoed by V v + +. Ov deoes he aicipaed operaig coss icludig maieace expeses. Kaz explais ha he user cos measure is ypically derived by assumig ha flow rasacios wihi a period occur a he ed of he period. 8 Followig his coveio leads o he ed of period user cos, show i box as equaio (3-4): 9 u v r V v + O v (V + v + v V ). The symbol m i box (above expressio (3-)) deoes he remaiig service life of he home. The expeced marke value of a home a he ed of period ( havig a remaiig service life of m periods. V v + + ) is codiioal o he home 7 The BEA makes use of he OOH compoe of he CPI o suppleme he oher iformaio hey use for accouig for OOH iflaio i compilig he Naioal Icome ad Produc Accous for he Uied Saes. 8 iewer (25a, 25bb) also carefully disiguishes bewee begiig ad ed of period user coss ad recommeds he use of ed of period user coss sice hey are more cosise wih fiacial accouig coveios. 9 Ulike he home value variable where we eed o refer o boh he begiig ad he ed of period values, we oly eed o refer o he ed of period values for he oher aicipaed variables ad deoe hem simply usig as he superscrip, as Kaz does. We also forego usig a special desigaio for expeced values. - 4 -
Box. erivaio of he User Cos Measure from Kaz (28, Appedix A) The user cos of capial measure provides a esimae of he marke real price based o coss of owers. I is direcly derived from he priciple ha, i equilibrium, he purchase price of a durable good will equal he discoued prese value of is expeced e beefis; i.e., i will equal he discoued prese value of is expeced fuure services less he discoued prese value of is expeced fuure operaig coss. To see his, le purchase price of a v periods old durable a he begiig of period ; begiig of period + whe he durable is oe period older; period services of his durable; O v u v V + v+ V v deoe he deoe is expeced purchase price a he deoe he expeced ed of period value of he deoe he expeced period operaig expeses o be paid a he ed of period, for his v periods old durable i period ; ad r deoe he expeced omial discou rae (i.e., he rae of reur o he bes aleraive ivesme) i period. Expeced variables are measured as of he begiig of period. Assume ha he eire value of he durable s services i ay period will be received a he ed of he period, ad ha he durable is expeced o have a service life of m periods. From he defiiio of discoued prese value, + + m v + + m v u v u v um Ov Ov Om (3-) Vv = + + + K + + K + + m v i + + m v i + r (+ r )(+ r ) Π (+ r ) + r (+ r )(+ r ) Π (+ r ) i= Whe he durable is oe period older, he services i reders i period will have bee received ad he operaig expeses of period already icurred. Thus, he expeced price of he durable a he begiig of period + is: (3-2) V + + 2 + m v + + u v u v 2 um Ov v = + + + + + + + K K + + + 2 + m v i + + r (+ r )(+ r ) Πi = + (+ r ) + r ividig boh sides of (3-2) by (+ r (3-3) Vv + Vv + u v O = v. + + r + r + r Π i= + m v Om + m v i = + i (+ r ) ), subracig he resul from equaio (3-), ad rearragig erms yields Muliplyig hrough equaio (3-3) by (+ r ) ad rearragig erms, he yields he ed of period user cos: v v + v+ (3-4) u = r V + O (V V ). v v 3.2 The Verbrugge Varia (VV) of he User Cos Approach Radall Verbrugge, i research of his ow ad wih various colleagues, has sough o deermie wheher res ad user coss move ogeher, as he fiacial ecoomics heory oulied i box seems o imply should be he case. We refer o he specificaio of he user cos implemeed i Poole, Pacek ad Verbrugge (25) (PPV hereafer), ad ha is explaied ad ivesigaed more fully i Verbrugge (28) ad Garer ad Verbrugge (28 ad 29), as he Verbrugge varia user cos: he VV user cos for shor. The VV user cos ca be saed as: H (3-5) u = [r + γ + Eπ]V, where V is he begiig of period home value; r is a omial ieres rae; γ H collecs ogoig housekeepig operaig expeses; ad Eπ is a esimae of he expeced home price appreciaio. Noe ha (3-5) is esseially he same as formula (3-4) i box. - 5 -
Wha disiguishes he VV user cos from he geeric oe is ha Verbrugge uses aleraive forecasig equaios for he E π erm i (3-5). I his preferred equaio, documeed mos fully i Verbrugge (28) ad Garer ad Verbrugge (29), he erm Eπ is specified o be he expeced value of he 4-quarer home price appreciaio. Wih his seup, chages i home prices have a immediae wihi-year impac o he user cos. Verbrugge shows empirically ha, sice 998, he VV user cos racks eiher res or house prices whe evaluaed usig his preferred forecasig equaio. 4. The Real Equivalece versus he User Cos Approach The BLS has experimeed over he years wih boh he user cos ad real equivalece approaches. Uil he early 95s, homeowers coss o re were impued by he BLS: a real equivalece approach. issaisfacio wih his approach developed due o widespread re corols, ad his led he BLS o swich o wha came o be called he Asse Price approach. Wih his simplified user cos approach, which Greelees (23) erms a ad-hoc user cos approach, OOH services coss wihi he CPI were buil up from five elemes: () home purchase prices, (2) morgage ieres coss, (3) propery axes, (4) homeower isurace charges, ad (5) maieace ad repair coss. Over ime, problems arose wih his BLS approach oo. By he early 98s, he qualiy of he daa available o he BLS o house prices ad morgage ieres raes was deerioraig. The source of he house price ad morgage daa uilized by he BLS he was he Federal Housig Admiisraio (FHA) admiisraive daabase for FHA-isured houses: a small, aypical ad shrikig segme of he housig marke. Also, he iflueial Sigler Repor (Sigler 96, p. 53) had come ou srogly wo decades earlier i favor of real equivalecy. 2 Thus, i 983 he BLS swiched back o a real equivalece procedure for dealig wih OOH services i he CPI. 3 Res, of course, were beig colleced all alog ayway for he real compoe of he CPI. The real equivalece approach is sill beig used. However, PPV oe ha, by 25, he rapid rise i housig prices i he pos 999 years coupled wih slow icreases i he OOH compoe of he CPI had led o cocer amog may ecoomic aalyss abou he use of real equivalece for accouig for OOH services coss. Youger people, may of whom faced deerioraig employme ad earigs oucomes alog wih risig home prices, were amog hose who argued ha risig housig prices were pushig up heir cos of livig sharply. Ye, he The Poole, Pacek ad Verbrugge (25) paper was prepared for preseaio o he U.S. Federal Ecoomic Saisics Advisory Commiee (FESAC) o ecember 9, 25. The FESAC is a Federal Advisory Commiee sposored joily by he Bureau of Labor Saisics of he U.S. eparme of Labor ad by he Bureau of Ecoomic Aalysis ad he Bureau of he Cesus of he eparme of Commerce. See Poole, Pacek ad Verbrugge (25). Also, Kaz (982, 983) a he BEA explored he sesiiviy of user cos esimaes o aleraive assumpios abou expeced raes of iflaio ad paers of depreciaio. 2 The Sigler Repor (Sigler 96, p. 53) saes ha: The welfare of cosumers depeds o he flow of services from houses ad o upo he socks acquired i ay give period. The repor cocluded ha (p. 48) If a saisfacory re idex for uis comparable o hose ha are ower-occupied ca be developed, his commiee recommeds is subsiuio i he CP for he prese series for he prices of ew houses ad relaed expeses. 3 See Gilligham ad Lae (982). The real equivalece approach was implemeed for he CPI-U i Jauary 983 ad for he CPI for Urba Wage Earers ad Clerical Workers (CPI-W) i Jauary 985. - 6 -
real equivale compoe of he CPI rose lile for he aio ad ideed fell for some urba areas. Bauer, Halom, ad Peerma (24) wih he Federal Reserve Bak of Alaa argue ha, i he Uied Saes, he geeral sofeig of res i 22-23 is causally relaed o icreases over he same period i he demad for owed homes. They argue ha may of hose who had plaed o purchase homes over he comig years isead rushed o buy ou of fear hey would be shu ou of he owed housig marke by he price icreases. May home buyers eered io morgage coracs wih excepioally lile equiy dow based o expecaios, o heir par ad held as well by leders, of risig home values. Ye, cocerig possible aleraives o real equivalece ha he BLS migh cosider, PPV wrie ha he user cos approach is he oly serious aleraive o real equivalecy. We agree, a leas, ha eiher of he oher mai approaches currely i use by major official saisics agecies -- amely, he acquisiios ad he paymes approaches -- would be a suiable aleraive. 4 We agree oo wih some of he problems ha PPV raise cocerig a user cos approach. They oe, for example, ha he capial heory used o derive he user cos approach oly holds uder equilibrium codiios. Ye, a housig bubble is a sig ha he housig marke is o i equilibrium. Furhermore, as coveioally specified, i a period of rapid house price iflaio, he appreciaio compoe of he user cos expressio 5 ca grow large eough so ha he user cos urs egaive. Ye i makes o sese o have a egaive figure for he value ower occupiers place o livig i heir homes! We coclude ha he ime has come o fid a ew approach o accouig for OOH services i a CPI. This is he ask we ur our aeio o i he followig secio. The ew opporuiy cos approach ha we recommed builds direcly o suggesios made by PPV i heir semial 25 paper. O a operaioal level, he proposed approach also builds o he experise ad daa collecio isrumes amassed over he previous decades a he BLS. 5. The Opporuiy Cos Approach As oed i he iroducio, i a 26 address, iewer recommeded a opporuiy cos approach for dealig wih OOH i a CPI: 6 [P]erhaps he correc opporuiy cos of housig for a ower occupier is o his or her ieral user cos bu he maximum of he ieral user cos, which is he fiacial opporuiy cos of housig, ad wha he propery could re for o he real marke. Afer all, he cocep of opporuiy cos is supposed o represe he maximum sacrifice ha oe makes i order o cosume or use some objec. iewer ad Nakamura (29, p. 2) followed up wih a wo-par suggesio for how a opporuiy cos approach migh be used o compile a price idex for OOH services: 4 See iewer (23/2) for more o his issue. 5 The appreciaio compoe is he fial erm i pareheses o he righ had side of (3-4) i box. 6 See p. 3 i iewer (26/29), which is he published versio of iewer s preseaio a he OEC-IMF Workshop o Real Esae Price Idexes held i Paris, November 6-7, 26.. - 7 -
For each household livig i ower occupied housig (OOH), he opporuiy cos is he maximum of wha he dwellig could have bee reed ou for, which is he real equivale, ad he fiacial user cos of he fuds ied up by owig he propery. The OOHOC idex for a aio ca be defied as a expediure share weighed sum of a real equivalecy idex ad a fiacial user cos idex, wih he expediure share weighs depedig o he proporio of ower occupiers for whom he fiacial user cos is esimaed o exceed he real equivale cos. Our purpose i his secio is o furher develop ad explore he properies of wo compoes of he proposed OOHOC idex. The erm opporuiy cos refers o he e value of he bes of he aleraives give up i akig he opio chose. Before proceedig furher wih he derivaio of he household level compoes of a OOHOC idex, i he followig secio, we briefly discuss he cocep of a opporuiy cos. 5. Opporuiy Cos Basics Cosider he example provided i he Wikipedia aricle o opporuiy coss. 7 The example give is for a ciy ha has decided o build a hospial o vaca lad i ows. I opig o build a hospial, he ciy passed up wo compeig developme proposals: (A) a spors area, ad (B) aoher reveue geeraig commercial proposal. The Wikipedia wrie-up saes ha: If he ciy decides o build a hospial o he vaca lad i ows, he opporuiy cos is he value of he beefis forgoe of he ex bes hig ha migh have bee doe wih he lad. Thus, he opporuiy cos is he prese value of he greaer of he wo developme opporuiies ha were passed up i choosig o use he vaca lad for he hospial. I he Wikipedia aricle, i is explaied ha he reaso ha he opporuiy cos is he greaer -- ha is, he maximum -- raher ha he sum of he expeced oucomes for he wo proposals passed up is because hose oher opporuiies ad he oe chose were muually exclusive. Noe oo ha he opporuiy cos is beig evaluaed afer he choice o use he lad for he hospial has bee made. Equivalely, he opporuiy cos could have bee assessed prior o he decisio o go forward wih he hospial, bu from he perspecive ha his decisio should be adoped. To compue he opporuiy cos of some course of acio, he aalysis mus be carried ou from he perspecive ha he desigaed acio was or will be chose ad ha he oher aleraives were rejeced. I he Wikipedia example, he cosrucio cos of he hospial ad a esimae of he marke value of he vaca lad could be calculaed, jus as he full purchase price of a home ca be observed or esimaed. However, aoher feaure of he Wikipedia example ha is releva is ha he wo opios ha he ciy passed up i choosig o build a hospial would boh have geeraed e reveue flows ha could be forecased, ad ha furhermore should be cosidered i deermiig he full ecoomic cos of he decisio o use he lad for a hospial. This is a 7 hp://e.wikipedia.org/wiki/opporuiy_cos. - 8 -
poeial reaso for assessig he opporuiy cos eve if a appropriae direc cos measure could be developed. Noe hough ha i would make o sese o rea he acquisiio coss of eiher a hospial or a home as he coss of usig he asses for a period of ime such as a year; raher, some way mus be foud o allocae he acquisiio coss for hese asses over heir useful lives. The user cos approach is oe possible meas of achievig his cos allocaio. However, housig markes are ofe o i equilibrium: a requireme for he derivaio of he real equivale measure of OOH services usig a user cos approach as i secio 3.. Thus, ieres i deermiig he opporuiy cos of OOH services, arises, i par a leas, because of he lack of some oher valid measure of he rue ecoomic cos ivolved. 5.2 The PPV Opporuiy Cos Jusificaio of he Real Equivalece Approach PPV sae ha he measureme quesio ha mus be addressed is: How much richer would he homeower be if he or she did o cosume he housig services provided by a dwellig? They go o o provide he followig guidace o his opporuiy cos calculaio. They explai ha a homeower always has he opio of movig ou of his/her house i order o re i o a ea for re r, ad of he movig io aoher real ui whose re is r2, wih r2 < r. oig so would free up (r r2) icome for oher uses. This demosraes, hey oe, ha a homeower is, i effec, givig up icome equal o r (o r2, ad o r-r2) if he/she occupies his/her ow house. They go o o oe ha: The fac ha sheler services are cosidered esseial for survival ( oe has o live somewhere ) is irreleva o he argume. PPV argue ha every ower occupier passed up he opporuiy o re ou heir home by choosig o occupy he home hemselves. Owig ad occupyig, ad owig ad reig, a dwellig ou are muually exclusive aleraives. Treaig his pair of opios as he oly releva aleraives a homeower faced provides a opporuiy cos jusificaio for he coveioal real equivalece approach: a jusificaio wih he advaage ha i perais o he use of he services of he durable asse -- he home -- for some period. Also, his jusificaio does o res o a assumpio ha real esae markes are i equilibrium, ulike he usual heoreical jusificaio of he user cos approach (oulied i secio 3. above). The developme of a opporuiy cos jusificaio for he real equivalece approach by PPV is, i our view, a impora sep forward. The argumes hey level agais he user cos approach would oherwise apply as well o he real equivalece approach if boh are viewed as based o he same heoreical argumes. A heoreical jusificaio of he real equivale ha does o eail imposig a housig marke equilibrium codiio meas oo ha he real equivale is o ivalidaed by empirical evidece ha res ad housig prices ofe seem o move quie differely. This is helpful sice here are may reasos why house prices ad res may have differig developmes over ime. Trasacios coss ca be subsaial for real esae. Verbrugge (28) suggess ha he large coss associaed wih real esae rasacios would have preveed risk eural ivesors from earig expeced profis by usig he rasacio sequece buy, ear re o propery, sell, ad would have preveed risk eural homeowers from earig expeced profis by usig he rasacio sequece sell, re for oe year, repurchase. - 9 -
Owers ad reers are subjec o differig sors of uceraiy regardig chages over ime i housig relaed expeses. 8 Ad, because of agecy problems, a ladlord may o wish o cusomize a real ui o he same exe as a homeower. Also real uis may be subjec o greaer depreciaio (Croe e al., 29). Ladlords seem o be boud oo by real marke coveios o chage res ifrequely. Real rae sickiess has bee show empirically o be paricularly impora for coiuig eas. 9 I addiio, he ax reame of ower occupiers ad reers differs i may couries icludig he Uied Saes. 2 Also, he real marke for luxury homes is hi. Someimes he owers of luxury homes wa or eed o re ou heir homes. Luxury homes ed o be offered for re mosly uder codiios ha limi he opios of a reer. To fid reers, he owers of luxury homes ofe mus compee o price for eas who would ormally re lower qualiy housig uis ad cao afford o pay much more ha wha hey ormally would pay. 2 Ye aoher reaso may be ha a ladlord s horizo m may be loger ha for a ower occupier. Whe he ladlord builds or buys a real propery, he ladlord will wa o se res a leas equal o he user cos. However, oce he propery is buil or bough, he cos is suk ad supply ad demad facors for real properies could cause he marke res o diverge from he expeced user coss. Evidece of his facor a work icludes cycles i he cosrucio of real uis; whe he ladlord s user cos exceeds marke re, buildig of ew real properies slumps ad vice versa whe he ladlord s user cos is below curre marke re. 5.3 The Aleraive Opios Passed Up by Ower Occupiers I he Wikipedia example, he se of aleraive opios he ciy had o cosider whe choosig he hospial was kow: he opios were he proposals submied before a fixed deadlie (which icluded he opio chose). Now cosider he problem of ryig o measure he opporuiy cos for a ower occupier. We ca observe whe a home ower has decided o coiue o ow ad occupy heir dwellig for he period (ad perhaps for may periods o come). Bu wha are he opios ha he ower occupier had, bu passed up i choosig o coiue o ow ad occupy heir home? PPV argue ha each ower occupier i each period gave up he aleraive of reig ou heir dwellig for ha period. Wha we add o PPV s coribuio is he isigh ha mos ower occupiers also gave up aleraive fiacial ivesme opporuiies. The opporuiy cos of a choice ake is he value of he ex bes available aleraive. Thus, o properly deermie he fiacial ivesme compoe of he opporuiy cos of a choice o ow ad occupy i period, 8 See Siai ad Souleles (25). 9 See, for example, Gordo ad Goehem (24) ad he fidigs of Geesove (23). Also, Hoffma ad Kurz- Kim (26, p. 5) repor he followig: I our sample, prices las o average more ha wo years... bu he chage by early %. The loges price duraios are foud for housig res, which, o average, are for more ha four years. Also, Hoffma ad Kurz-Kim (26, p.5) repor ha Germa res chage oly every 4 years o average. 2 See, for example, Poerba ad Siai (28). 2 I also seems likely o us ha, movig up he value scale, a icreasig perceage of homes offered for re are, i fac, offered wih he erms of payme icludig house siig duies alog wih he moeary re obligaios. Siuaios like his should, of course, be caugh by he quesios asked as par of he collecio of he re daa, bu i seems likely ha o all he cases like his are properly ideified. - -
i is o ecessary o kow he eire aleraive fiacial ivesme choice se ha he ower occupier faced. Raher, i is oly ecessary o cosider he highes valued of hose aleraives. I addiio, he aleraives cosidered as foregoe opporuiies mus be muually exclusive wih each oher ad wih he opio chose of owig ad occupyig i period. 22 5.4 The Fiacial User Cos of a Ower Occupier The user cos approach is a appropriae way of allocaig over ime iiial fiacial ivesmes made i asses ha yield earigs over muliple subseque ime periods. Ulike real esae markes, fiacial markes are geerally believed o saisfy he assumpios o which he user cos derivaio is based. Thus, we derive he fiacial user cos for a ower occupier wih o-egaive home equiy a he sar of period i sub-secio 5.4.. We explore he aure of he fiacial user cos compoe of he OOH opporuiy cos for a homeower wih oegaive home equiy i various special circumsaces i sub-secio 5.4.2. We he ake up he egaive equiy case i sub-secio 5.4.3. Throughou secio 5.4, we absrac from axes, we assume here are o defauls, ad we do o explicily deal wih home real or purchase or fiacig or refiacig choices. Raher, we ake he perspecive of devisig ways of accouig for OOH services i a CPI codiioal o he produc choices people have made, icludig heir housig services cosumpio choices. This is he usual pracice for price idex cosrucio. 5.4. Homeowers wih No-Negaive Equiy I wha follows, we proceed uder he cojecure ha he highes value of he forgoe fiacial ivesme aleraives for a ower occupier wih o-egaive home equiy is he expeced earigs ha would have resuled from ivesme of he home equiy fuds ha he sale of he home a he sar of period would have freed up: a muually exclusive aleraive o he choice acually made of coiuig o ow ad occupy ad o he oher aleraive cosidered ad rejeced of owig ad reig ou he dwellig. A homeower is viewed as havig made he choice o coiue owig heir home wih some level of deb ( ) ad equiy ( V ) ad wih a kow, required morgage payme (I ) due a he ed of each ime period, 23 which (I ca be expressed as I = (I / ) = r wih beig defied by / ) r. The fiacial user cos for owig he home i period ad livig i i, discoued o he sar of he period, is: r 22 There are muliple ways i which a homeower migh have wihdraw equiy from heir home. For example, refiacig les a homeower sell (or buy back) a fracio of a owed home. However, for he purposes of deermiig he opporuiy cos of a choice ake, aleraives foregoe ha could have bee joily seleced, such as reig he home ou for period ad wihdrawig some home equiy durig ha ime period oo, mus be cosidered as sigle, combied iems i he se of muually exclusive choices. 23 Morgage coracs ofe coai fixed mohly payme requiremes ad also a fixed rae (or raes) of ieres o he ousadig pricipal of he morgage. The mohly payme ypically represes a combiaio of ieres ad pricipal, bu all ha maers i our aalysis is ha he amou is fixed a he sar of each period for he duraio of m subseque periods. - -
(5-) u + r [V r ] O + (V + r + ), V + where is defied as he value of he home a he begiig of he period plus he expeced per period average appreciaio of he home over he m subseque ime periods ha he dwellig is assumed o be able o provide housig services. We recommed ha m should be se a a value a leas as large as he media umber of years ha he releva populaio of homeowers repor havig lived i heir prese homes. 24 This implies ha a very log erm rae should be used for he expeced home value appreciaio. Now, if we muliply expressio (5-) hrough by he discou facor, + r, we obai he followig equivale expressios for he ed of period user cos: + (5-2) u = r + r (V ) + O (V V ) (5-3) = O + (r r ) [V ( + r )V ]. + 5.4.2 Properies of he Fiacial User Cos for Homeowers wih No-Negaive Equiy The user cos expressio give i (5-3) ca be beer udersood by cosiderig some specific ypes of siuaios. Cosider firs of all a homeower wih o morgage deb. For hem, expressio (5-3) reduces o + (5-4) u = O [V ( + r ) V ]. This expressio is esseially he same as he cusomary user cos derived by Kaz (29) ad show i (3-4) i box above, ad specified by Verbrugge ad show i (3-5). We ex cosider he exreme case of ower occupiers wih posiive equiy whose morgage payme rae equals heir expeced rae of reur o fiacial holdigs (i.e., r = r ). I his case oo, (5-3) reduces o (5-4) above. Thus, he coveioal user cos expressio implicily assumes ha homeowers wih morgages make paymes such ha equals heir expeced rae of reur o aleraive fiacial ivesmes. Amog ower occupiers wih posiive home equiy, well off households ofe ca ge morgages wih ieres raes ha are less ha prime ad moreover so ha r < r. I his case, he user cos expressio (5-3) ca be wrie as: + (5-6) u = O (r r ) [V ( + r )V ], r 24 Mos people ow heir homes for 7 years or more. I 24, for example, here were 72 millio ower-occupied homes ad exisig home sales were 6.8 millio. Ideed, ower occupiers ypically roll forward he equiy accumulaed i oe owed home io aoher owed home whe heir housig eeds chage. Few reur o beig reers, eve for brief periods. May people move io heir ow owed homes as soo as hey ca afford o afer reachig adulhood ad die sill owig heir ow homes. - 2 -
where he erm (r r ) is ow posiive. Thus, all else equal, for hese homeowers, higher morgage deb reduces he fiacial user cos of OOH services. occupiers wih O he oher had, mos subprime loas are high cos. For posiive equiy ower r > r, he user cos expressio (5-3) reduces o: + (5-7) u = O + (r r ) [V ( + r )V ], where (r r ) is posiive. All else equal, for hese homeowers, higher morgage deb meas a higher fiacial user cos for OOH services. 5.4.3 Homeowers wih Negaive Equiy We ow ur our aeio o he egaive equiy case: a case of special relevace i he wake of he rece burs of a housig marke bubble. Righ ow, may households are livig i homes ha could o be sold for eough o cover heir debs. Ashcraf ad Schuerma (28) cie daa showig ha secod morgages were quie rare i he 99s, bu ha by 26, may firs morgages were accompaied by secod morgages. I a comprehesive survey repor of he Bak for Ieraioal Selemes (a ceral bak forum), for he pos-2 period i he Uied Saes, Ellis (28) calls aeio o a icreased use of secod morgages, ad observes ha may U.S. households were able o obai perce fiacig i his way. Ellis oes ha, i addiio o iiial deb-o-value raios beig higher ha before, he deb-o-value raios for a growig proporio of home owers failed o declie over ime as expeced based o previous paers, or eve icreased because of rece declies i home values, movig icreasig umbers of home owers io egaive home equiy siuaios. Caga (27) esimaes ha aroud 5 perce of loas made i he early 2s were already i egaive equiy a he ed of 26, hough he figure for older loas was lower. Caga esimaes ha aroud 8 perce of morgages origiaed i 26 were i egaive equiy by he ed of ha year. A ower occupier who has borrowed usig heir home as collaeral basically is reig par of heir home from hemselves ad par from heir credior, wih he re ha he credior is chargig (i.e., he ieres ad he required parial repayme of pricipal ha mus be made a he ed of each ime period for he ower o be able o eiher occupy or re ou he home) beig specified i he morgage corac. As before, he morgage deb i omial erms is deoed by, he required payme o he morgage ha is due a he ed of period is r, he begiig of he period marke value of he home is, ad he expeced ed of period marke value is V +. Morgage deb i he amou of ca cosis of some combiaio of a coveioal firs morgage ad subprime or secod morgage fuds. Negaive equiy a he begiig of period meas ha deb is larger ha begiig of period equiy V so ha (5-8) V <, ad hece V >. If he homeower had le go of heir home a he begiig of period by eiher sellig i or givig i back o he bak for he curre marke value of V, ad if hey eiher could o or chose o o cosider simply walkig away from heir residual deb, he household e worh V - 3 -
would have had o decrease a he sar of period by he amou of he homeower s egaive equiy posiio a he sar of, which is V >. Of course, here are differeces i price levels a he begiig ad ed of he period ad so he ed of he period coss mus be discoued relaive o he begiig of he period coss. Bu wha is he appropriae discou facor i his case? r If he homeower has ivesmes ha are earig more ha he morgage payme rae, i would o make sese o liquidae hese ivesmes o pay off he egaive equiy. Nor would i make sese for he homeower o borrow a a rae higher ha o pay off he egaive equiy. 25 Raher, igorig possible isiuioal or moral cosrais, i would oly make sese for he homeower o borrow or o liquidae ivesmes wih a associaed rae of less ha r (say he rae of ρ r ). We le ρ deoe a suiable discou rae for he homeower. The cheapes fuds for payig off egaive equiy a he sar of, ad hece he bes fiacial aleraive, are fuds i a coveioal savigs or checkig accou ha are earig o ieres, or household savigs achieved by cuig back o oher begiig of period household expeses, i which case he appropriae discou rae migh be he aicipaed Cosumer Price Idex Iflaio rae over period, sice discouig by oe plus his rae will make asse values a he ed of he period comparable o asse values a he begiig of he period i erms of cosumpio equivales. To kow he ed of period value of he begiig of period egaive equiy, we eed o kow where hose fuds would have come from. I compuig he e value of he mos aracive way i which he homeower could have le go of he home a he sar of period, we mus also allow for he fac ha he ed of period egaive equiy posiio, give coiued owership ad occupaio of he dwellig, is expeced o be as well of r ad O due a he ed of he period. V I wha follows, we will assume ha he appropriae period discou rae, r +, wih he coss, is a rae a leas as high as he aicipaed CPI iflaio rae ad less ha he morgage payme rae of r. Thus assumig ha morgage ieres ad operaig coss are paid a he ed of he period, we fid ha he begiig of he period user cos u /( + ρ ) is defied as follows: (5-9) u + ρ + r O ) [ V ] ( V + ρ + r O [ V ] + + ( V ) = +. + ρ Muliplyig boh sides of he above expressio hrough by ( + ρ ) leads o he followig expressio for he ed of period user cos for ower occupied housig if begiig of he period equiy is egaive: ρ 25 Some egaive equiy homeowers will have low ieres morgage opios. However, i is ypically he case ha egaive equiy homeowers have relaively high ieres morgage deb. - 4 -
+ (5-) u = ( + ρ )( V ) [ r O ( V )] + (5-) = O + (r ρ ) [V ( + ρ )V ]. The expressio for he user cos i (5-) says ha user cos is equal o operaig expeses plus real morgage ieres ( r ρ ) O less he aicipaed real capial gai i he value of he + house [V ( + ρ )V ]. This expressio makes sese iuiively. 26 We will o develop a user cos formula for a homeower wih egaive equiy who ca walk away from payig off heir deb wih o cosequeces. The problem of rasacios ad adjusme coss becomes criical i his case. If here were o rasacios ad o oher adjusme coss of buyig ad sellig ad buildig homes ad here were o oher cosequeces of defaulig, he i he egaive equiy case, i would make sese for he homeower o defaul ad immediaely repurchase heir home a he ew lower price (or buy aoher home of equivale value). However, wih rasacios ad adjusme coss, i ca make sese for he egaive equiy homeower o say i he home wihou defaulig. Thus he size of he rasacios ad adjusme coss becomes criical. 6. Iflaio Measureme Give wellig Uiqueess A decisio by he BLS ad oher official saisics agecies o move o a opporuiy cos approach for accouig for OOH i a CPI would be a sep forward, i our view. However, regardless of wheher a opporuiy cos or a user cos or a real equivale approach is used, saisical agecies will sill face empirical mehodology problems arisig from dwellig uiqueess. Each dwellig has a uique locaio ad dwelligs coiually evolve via dwelligspecific depreciaio ad reovaio. wellig uiqueess preses similar empirical challeges regardless of wheher he price daa are sale prices or real prices. Thus, i his secio, we use he erm price o refer o eiher dwellig sale or re observaios, uless oherwise specified. The depreciaio rae for a dwellig ca be defied ad ca hypoheically be measured by he raio of he same period prices for ideical dwelligs ha have bee used for differe leghs of ime. The iflaio rae for a dwellig ca be defied ad ca hypoheically be measured by he raio of he prices a differe pois i ime for ideical dwelligs ha were used he same legh of ime. However, o wo dwelligs are ideical. Amog oher differeces, hey all have differe physical locaios. Cross secioal iformaio o used asse sale prices or res a ay oe poi i ime will o allow us o separae ou he separae effecs of depreciaio ad iflaio for durable asses ha mus be viewed as uique for price measureme purposes. 27 26 We have o modeled various ax cosequeces associaed wih home owership. This is lef o furher research. 27 Special cases of his fudameal ideificaio problem have bee oed i he coex of various ecoomeric housig models: For some purposes oe migh wa o adjus he price idex for depreciaio. Uforuaely, a depreciaio adjusme cao be readily esimaed alog wih he price idex usig our regressio mehod. I applyig our mehod, herefore, addiioal iformaio would be eeded i order o adjus he price idex for depreciaio, Bailey, Muh ad Nourse (963, p. 936). The price idex ad depreciaio are perfecly colliear, so if oe cares abou he price idex, i is ecessary o use exeral iformaio o he geomeric depreciaio rae of houses, Palmquis (23, p. 43). - 5 -
However, his separaio ca be based o empirical evidece if some way ca be foud for decidig whe dwelligs ca be viewed as comparable for price measureme purposes. 28 Achievig his separaio is especially impora for ay caegory of durables where maieace ad reovaio expediures ad cosequeces are subsaial, ad also where iflaio ad deflaio movemes i prices ca be subsaial. The prospecs for separaig ou depreciaio ad iflaio effecs are much improved if dwelligs ca be viewed as he same for price measureme purposes provided hey have cerai shared characerisics. This is rue for he hedoic mehods, as used for measurig iflaio for boh dwellig sale prices ad dwellig res, bu o for he repea sales mehod, which oly compares he same ui over ime.. The repea sales ad hedoic mehods are wo seemigly very differe ypes of empirical mehods which we explai here ca be placed wihi a commo mahemaical framework, hereby allowig users o ake advaage of isighs from research io he properies of boh hese mehods. Wih a pure hedoic mehod, daa colleced over muliple periods are classified by value deermiig characerisics ha are o uique such as eighborhood or disace from he dow ow core, ype of dwellig ui (e.g., sigle deached or a ui i a muliple ui buildig), some meric for size such as floor space, ad he age of he propery. To apply a hedoic mehod, daa are eeded o he seleced lis of value deermiig characerisics, ad here mus be agreeme ha his lis is appropriae ad adequae o corol for differeces i he value of he housig services provided by he differe purchased or reed dwelligs. Thus he hedoic mehod daa requiremes are usually very exesive. The iflaio rae is he esimaed usig observaios over ime while corollig ecoomerically for chages i he value deermiig characerisics. The repea sales mehod compares he price observaios for housig properies ha were sold, or ha were reed, muliple imes over he ime ierval spaed by he available daa. 29 This mehod is popular wih U.S. real esae researchers ad praciioers because i uses oly he iformaio readily available i all localiies of he Uied Saes: sale or real prices ad he uique legal propery descripios. A key uderlyig assumpio is ha, wih similar maieace expediures, owers of resideial properies ha hey occupy or re ou usually maage o maiai heir properies i uchaged codiio over he dwellig service lives. This mehod does, however, corol for he qualiy deermiig aribues of each dwellig ha do o chage over ime, ad i does so wihou he eed for havig daa o (or eve kowig abou) all of hose value deermiig aribues. We ow ur our aeio o he specificaios of he repea sales ad hedoic mehods ad how hese mehods are relaed. We begi wih he repea sales mehod which is due o Bailey, Muh ad Nourse (963). Sice hedoic regressio models, as usually used i he price measureme lieraure, have price levels (ofe i logarihmic form) as depede variables, raher ha price raios as is he case for he repea sales mehod, i is helpful o iroduce he repea sales mehod as i arose hisorically: as a geeralizaio of he chaied mached model mehodology. 3 This is he moivaio for how he repea sales mehod is preseed i box 2. 28 For CPI re measureme, BLS age adjuss iflaio raes usig he hedoic mehod of Radolph (988a, 988b). 29 The repea sales procedure, ow i widespread use, daes back o Bailey, Muh, ad Nourse (963). See also reima ad Peigo-Cross (24) for he uses of his mehod ad see Gree ad Malpezzi (23, pp. 32-6) for a review of he repea sales idex lieraure. 3 See Wygarde (927) ad Wezlick (952). - 6 -
Box 2. A Exposiio of he Repea Sales Mehod Le S(,) deoe he se of housig uis ha are i scope for he idex ad were sold i boh periods ad. eoe he price for propery sold i period by periods ad, so S(,). Le, P V. Here aeio is cofied o jus wo ime be he real esae price idex goig from period o. For housig uis i S(,), suppose he sochasic model relaig he propery sales price raio, V / V, o P is:,, (4-) V / V = P expε,, ε where is assumed o be a idepedely disribued error erm wih mea ad cosa variace. Takig logarihms of boh sides of (4-) leads o he followig liear regressio model: ],, (4-2) l[ V / V =π + ε,,, where π l P. The leas squares esimaor for π is he arihmeic average of he logarihms of he sales price raios. Expoeiaig his esimaor yields a prelimiary mached model propery price idex goig from period o :,* S(,) / N(,) (4-3) P [V / V ], where N(,) is he umber of houses i he se S(,). This idex is see o be he equally weighed geomeric mea of sales price raios V / V for all he properies ha chaged hads i boh periods ad : a ypical mached model esimaor for a elemeary price idex. Nex le N(,2) deoe he umber of sales of houses i se S(,2) ad cosider he se S(,2) of houses ha sold i boh periods ad 2. Now he prelimiary mached model price idex goig from period o 2 ca be show o be:,2* / N(,2) (4-4) P [V / V ]. S(,2) 2 Usig he above resuls, he levels of he propery price idex,,*,*,2* (4-5) P ; P P ; P P P. P 2, P,, for =,,2 ca be defied as: Thus he price idex is se equal o i period ; i period, i equals he mached model price idex goig from period o, ad i period 2, i equals he produc of he prelimiary price idexes give i (4-3) ad (4-4). The Bailey, Muh ad Nourse (963) iovaio was o reparameerize he model described above ad o add a addiioal se of esimaig equaios for repea sales pairs i periods ad 2: i.e., for housig properies i S(,2). Their esimaig equaios wih hree periods of daa o repea sales are: ], (4-6) l[ V / V =π π + ε for S(,), 2 ] 2,2 (4-7) l[ V / V =π π + ε for S(,2), 2 ] 2,2 (4-8) l[ V / V =π π + ε for S(,2), 2 where ow we have π l P, π l P, ad π lp, wih he followig ormalizaio imposed (where addig a cosa o each π leaves he regressio uchaged): π = or P =. This leads o a model ha ca be esimaed usig leas squares regressio. Expoeiaig he leas squares esimaes for he 2 * parameers π ad π, deoed here by π ad π, leads o esimaes for he prelimiary idexes ad 2* P. The BMN esimaes for he housig price levels i periods -3 are: (4-9) P, P expπ, P expπ. * * 2* 2* The 3-period model geeralizes easily o he T-period case of Bailey, Muh ad Nourse (963). 2* 2 * P - 7 -
As origially proposed, he repea sales mehod ca oly be used o measure price level chage over ime, ad hus is o well suied for empirical sudies ha seek o exploi crosssecioal variaio by usig he absolue dollar values of dwelligs. 3 However, i he lieraure o spaial price level comparisos, Summers (973) proposed a hedoic regressio model where he oly explaaory variables are dummy variables for he coury ad he produc: he couryproduc-dummy or CP mehod. I box 3, we show he formal seps for how he repea sales ad he CP mehods are relaed for he case where complee mached model daa are available. Box 3. The CP Model wih Complee Mached Model aa Cosider a sample of N houses ( =,2, K,N ) each of which sold i each of he hree periods ( =,, 2 ): S (,,2). A sochasic model for he house prices, V, i each period ca be specified as follows: (4-2) V =α P expε, where P is he housig price idex level for period, ε =, K,N, α is a parameer ha reflecs he qualiy of housig ui relaive o average qualiy ad is a idepedely disribued, mea zero, cosa variace error erm. Takig logarihms of boh sides of (4-2) leads o he followig sysem of esimaig equaios for he N houses: V (4-3) l =β +π + ε, =, K,N, =,, 2. where β lα ad π l P. The model defied by (4-3) ad he ormalizaio (4-4), he leas squares (LS) esimaors for he model parameers saisfy he followig N+2 equaios: N N * = = N (4-4) l V = β + π, N 2 N * = = N (4-5) l V = β + π, ad 2 3 * * 2* * 2* (4-6) l V + l V + l V = β +π + π, =, K, N. Usig equaios (4-6) o elimiae he (4-7) = (/ N) l[v / V ], π = (/ N) l[v / V ]. β * * 2* π N N = = from (4-4) ad (4-5) yields soluios for he ukows: Afer expoeiaig hese esimaes, his complee iformaio CP model leads o he followig geomeric mea of he period relaive o he correspodig period values as he esimae for he period * housig price level, P, ad he geomeric mea of he period 2 values relaive o he correspodig 2* period values as he esimae for P : * N / N 2* = N = = / N (4-8) P [V / V ], P = [V / V ]. 2 2 Fially, he deails are show i box 4 for how he repea sales approach ca be modified o icorporae hedoic regressio correcios for chages i observed dwellig characerisics bewee price observaios. The resulig liear regressio model (equaios (4-9)-(4-22)) is he same as he wo coury CP model (wih icomplee iformaio). 32 Expoeiaig (4-27) reveals ha his hedoic regressio model leads o a period o price idex ha equals he 3 See, for example, Capozza, Hedersho, Mack, ad Mayer (22). 32 I is also ideical o he wo period Aizcorbe-Corrado-oms (2) dummy produc hedoic regressio model. - 8 -
equally weighed geomeric mea of he sellig prices i period divided by he geomeric mea of he sellig prices of he mached models i period. 33 I is basic form, he geeral ime dummy hedoic mehod ivolves regressig he logarihm of he propery sale price o he characerisics of he propery ad a ime dummy variable for each period spaed by he esimaio daa se (excep he omied base period). Oce he esimaio has bee compleed, he ime dummy coefficies ca be expoeiaed o creae a idex. Aleraively, usig iformaio o he characerisics of he properies sold, he daa ca be sraified ad a separae regressio ca be ru for each ime period for specified classes of resideial properies. Thus he hedoic regressio mehod could be used o produce a family of idexes. iewer, Heravi ad Silver (29) oulie aleraive formulaios ad esablish he relaioships amog hem. 34 7. The Srucures ad Lad ecomposiio Problem 35 Usually he logarihm of he purchase price is ake as he depede variable i real esae price models. While his specificaio accords wih he direcly observable propery price iformaio, i is icosise wih cerai aspecs of he srucure ad lad compoes of he price of a propery. Resideial real esae usually ivolves boh a srucure ad he lad ha he srucure is buil o (he sie). To model his composie, cosider a sample of dwellig uis purchased a he begiig of period. Suppose he purchase price of propery is p. The value for propery ca be regarded as he sum of he (ofe uobserved) cos per square meer for he srucure, imes he floor space of he srucure i square meers (deoed by A i box 5), plus he price per square meer of lad (ofe o direcly observed) imes he area of he sie i square meers (deoed by B i box 5). For period, he propery value ca be represeed as i equaio (6-) i box 5, ad for period, he value for his propery ca be represeed as i equaio (6-2). The srucure ad lad compoes are probably subjec o differe raes of iflaio ad depreciaio. Ideed, lad is ofe viewed as depreciaig lile if a all (hough here ca be depreciaio of sie ifrasrucure). The asse iflaio ad depreciaio effecs are embedded i he coefficies of A ad B. Esimaig equaios are give by (6-3) ad (6-4). If daa are also available for he characerisics of he srucure ad he lad, he he pair of equaios, (6-5) ad (6-6), ca be esimaed isead. This model is flexible ad provides a meas of decomposig a propery price idex io srucural ad lad compoes, hough he model is oliear. Moreover, he rae of depreciaio may vary across real ad ower-occupied propery. 33 I a series of papers, iewer (22, 23a, 23b, 23c, 24, 25a, 25b, 26a, 26b), iewer, Heravi ad Silver (27/2), Silver (23), ad Silver ad Heravi (25) show how aleraive specificaios ad weighs ca be used wihi he CP framework o derive a umber of kow idex umber formulas. See also de Haa (23), Silver (23) ad Silver ad Hervari (25). iewer (25b) shows ha he uweighed idexes ca be far from heir weighed couerpars. Thus i is impora o ru appropriaely weighed regressios. 34 Rao (28) oes ha he CP mehod -- he weighed versio i paricular -- is beig icreasigly used i derivig spaial comparisos due o is abiliy o hadle price quoaios. See, for example, Ae ad Meezes (22), Heso ad Ae (22), Rao (23, 25), ad eao, Friedma ad Alaas (24). 35 See iewer (23/2, 26/29). iscussios bewee Erwi iewer ad Ae Laferrère helped improve he preseaio of he model here. - 9 -
Box 4. The CP Model wih Icomplee Mached Model aa Nex a model is cosidered where o every house mus rade i each period for iformaio abou he house o be icluded i he aalysis daa se. I order o miimize oaioal complexiies, iewer (23a) provides he followig deails for he case of wo periods. Le S(,) be he se of housig uis ha sold i boh periods ad. Takig io accou he ormalizaio (4-), he esimaig equaios correspodig o hese houses are: (4-9) l V = β + u, for S(, ), (4-2) l V = β +π + u for S(, ). Le S(, ) deoe he se of housig uis i he arge populaio ha sold i period bu o i period. The esimaig equaios for hese observaios are: m (4-2) l V m = γ + u, for m S(,~ ), m where γ m is he logarihm of he qualiy adjusme facor for he mh housig ui ha sold i period bu o i period. Similarly, le S(, ) deoe he se of housig uis i he arge populaio ha sold i period bu o i period. The esimaig equaios for hese observaios are: k (4-22) l V k =δ + u for k S(,~), k where δ k is he logarihm of he qualiy adjusme facor for he kh dwellig for which price iformaio is available i period bu o. * * * * Le π, β, γ m ad δ k deoe he leas squares (LS) esimaes of he parameers π, β, γ m ad δ k ha appear i (4-9)-(4-22). The sacked vecor of depede variables for equaios (4-9)-(4-22) ca be wrie as he sum of he vecors of exogeous variables imes heir correspodig leas squares esimaes plus he vecor of leas squares residuals. As oed above, he ier produc of each exogeous vecor wih he vecor of LS residuals is zero. Thus he LS esimaors for he ukow parameers i he regressio model mus saisfy he followig equaios: S(,) k S(,~ ) k * * * = S(,) k S(~) k ) (4-23) l V + l V β + N(,) π + δ + N(, ~ π ; 2 * * (4-24) l V + l V = β + π, for S(, ) (4-25) l V m = γ * m, for m S(,~ ), (4-26) l V k = δ * k, for k S(,~ ), where N(,) is he umber of dwelligs ha raded i boh periods ad N(, ) is he umber ha sold i bu o. Equaios (4-26) ca be used o elimiae he i equaio (4-23), ad equaios (4-24) ca * β be used o elimiae he from equaio (4-23). The resulig equaio for π is: * (4-27) π = [/ N(,)] l[v / V ], S(,) which is he arihmeic average of he logarihms of he sales price raios for he wo periods. For he housig uis ha sold (or were reed) i, a more geeral hedoic regressio model is: K k k (4-28) l V = π + = z β + ε, S(). k ε is a idepedely disribued error erm wih mea ad cosa variace, sellig price or re of dwellig i period, z k * δ k * V * is he observed is he amou of characerisic k ha dwellig has, ad π equals he logarihm of he cosa qualiy price idex, P ; i.e., π =l P for =,,,T. The parameer β rasforms amous of characerisic k, z, io cosa qualiy uiliy uis for k =,,K. k k - 2 -
Box 5. Srucure ad Lad ecomposiio Suppose he oal cos, p, of a propery afer he srucure is compleed will equal he floor space area of he srucure, say A square meers, imes he buildig cos per square meer, α say, plus he cos of he lad, which will equal he cos per square meer, β say, imes he area of he lad sie, B. Now hik of a sample of properies of he same geeral ype, wih prices,, i period ad srucure areas ad lad areas B for =, K,N(), ad where hese prices are equal o coss of he above ype imes error erms η which have mea. This leads o a hedoic regressio model for period where α ad β are he parameers o be esimaed i he regressio: ] (6-) p = [ αa +βb η. Takig logarihms of boh sides of (6-) leads o he followig radiioal addiive errors regressio model: ] (6-2) l p = l[ αa +βb + ε, where he ew error erms, ε lη for =, K,N(), are assumed o have meas ad cosa variaces. For a subseque period, he price per square meer for he give ype of srucure will have chaged from α o αγ ad he lad cos per square meer will have chaged from β o βδ where γ is he period o price idex for he ype of srucure ad δ as he period o price idex for he lad ha is associaed wih his ype of srucure. For =, K,N(), he period couerpars o (6-) ad (6-2) are: ] (6-3) p = [ αγ A +βδ B η ad (6-4) l p = l[ αγ A +βδ B + ε, ] p where ε lη, he period propery prices are, ad he srucure ad lad areas are ad. iewer (26a) suggess ha equaios (6-2) ad (6-4) ca be ru as a sysem of oliear hedoic regressios. The mai parameers of ieres are γ ad δ, which ca be ierpreed as period price idexes (relaive o he correspodig period price levels of ) for he price of a square meer of his ype of srucure ad he price per meer squared of he uderlyig lad. This framework ca be geeralized o ecompass he radiioal array of characerisics used i real esae hedoic regressios. Suppose ha we ca associae wih each propery ha is rasaced i a lis of K price deermiig characerisics deermiig characerisics (6-4) are: X,X 2, K, XK Y,Y 2, K, YM (6-5) l p l{[ α + X α ]A + [ β + Y β ]B + ε, p A A B for he srucure ad a similar lis of M price for he ype of lad. The equaios ha geeralize (6-2) ad K M = k = k k m = m m }, K,N() (6-6) l p l{ γ [ α + X α ]A +δ [ β + Y β ]B + ε, = ad K M = k = k k m = m m } =, K,N(), where he parameers o be esimaed are ow he K+ qualiy of srucure parameers, α K, he, α,, αk M+ qualiy of lad parameers, β, β, K, βm, he period price idex for srucures parameer γ ad he period price idex for he lad uderlyig he srucures parameer δ. Noe ha [ α + X ] i (6- K k = k αk 5) ad (6-6) replaces he sigle srucures qualiy parameer α i (6-2) ad (6-4) ad [ M β + m = Y m βm ] i (6-5) ad (6-6) replaces he sigle lad qualiy parameer β i (6-2) ad (6-4). - 2 -
8. Cocludig Remarks The rece housig bubble has raised several quesios for he measureme of housig service iflaio. The mos fudameal quesio is, are res always he bes measure of he opporuiy cos of ower-occupied housig? Sice, as Verbrugge has show, res ofe diverge from coveioal user cos measures, he possibiliy has arise ha he aswer is o. Also, he markes for real ad ower occupied housig uis ca be quie isolaed from oe aoher. This is evide i Croe e al. s hedoic measures of housig services for oweroccupied housig. Heso ad Nakamura (29) show oo ha for more expesive homes i he Uied Saes, he esimaed re/price raio is subsaially lower, suggesig ha he housig services of more expesive homes are beig uderesimaed i he U.S. aioal expediures as hese are beig evaluaed ow. The equilibrium codiios uder which res equal user coss may o be fulfilled for susaied periods of ime, eiher because he real marke is hi i some areas ad price rages where here are subsaial proporios of ower occupied housig, or because of disequilibria due o oher facors icludig he ime required for ew housig cosrucio ad goverme regulaios goverig lad use ad buildig ha are believed o make he housig marke proe o bubbles. We have explored oe comprehesive measure of housig services ha may remai valid or may beer approximae he red rae of housig services cos iflaio durig codiios i which res ad user coss diverge. Our proposed measure of he opporuiy cos of housig services is he greaer of res ad user coss. This measure o oly arguably permis more accurae measureme of housig services i disequilibrium, bu also avoids he difficul problem of he possibiliy ha user cos i he shor ru may be egaive. I is worh remarkig ha he difficuly of measurig housig services arises from hree sources: () homes are uique, because of locaio, (2) homes are durable asses, ad (3) homeower rasacio ad housig supply adjusme coss are large. Wih large rasacio ad adjusme coss, prices may remai ou of equilibrium because arbirage is oo cosly. As durable asses, he rae of depreciaio ad he rae of ieres, eer orivially. Ad because of uiqueess, prices of differe uis are o easily comparable. We have explored he ierrelaioships of he repea sales ad hedoic mehods for dealig wih he comparabiliy problem. Ower-occupied housig services are i may couries he larges iem i he cosumer baske. I he Uied Saes cosumer price idex, i accous for 24 perce of he oal weigh. How we measure iflaio i his iem is highly cosequeial for our udersadig of macroecoomic dyamics, moeary policy, ad growh. Refereces Ashcraf, A.B. ad T. Schuerma (28), Udersadig he Securiizaio of Subprime Morgage Credi, New York Federal Reserve Bak, Saff Repor Number 38. hp://www.ewyorkfed.org/research/saff_repors/sr38.pdf Bailey, M.J., R.F. Muh ad H.O. Nourse (963), A Regressio Mehod for Real Esae Price Cosrucio, Joural of he America Saisical Associaio 58, 933-942. Bajari, P., C.L. Bekard ad J. Kraier (23), House Prices ad Cosumer Welfare, NBER Workig Paper No. W9783. hp://ssr.com/absrac=48285-22 -
Bauer, A., N. Halom, ad W. Peerma (24), Examiig Coribuios o Core Cosumer Iflaio Measures, workig paper 24-7, Federal Reserve Bak of Alaa. hp://www.frbalaa.org/filelegacydocs/wp47.pdf Caga, C.L. (27): Morgage Payme Rese he Issue ad he Impac, Firs America Core Logic Whie Paper, 9 March. Chrisese, L.R. ad.w. Jorgeso (969), The Measureme of U.S. Real Capial Ipu, 929-967, Review of Icome ad Wealh 5, 293-32. Chrisese, L.R. ad.w. Jorgeso (973), Measurig he Performace of he Privae Secor of he U.S. Ecoomy, 929-969, i M. Moss (ed.), Measurig Ecoomic ad Social Performace, Columbia Uiversiy Press, 233-35. Croe, T.M., L.I. Nakamura, ad R.P. Voih (29), Res Have Bee Risig, No Fallig, i he Poswar Period,, Review of Ecoomics ad Saisics, forhcomig. iewer, W.E. (974), Ieremporal Cosumer Theory ad he emad for urables, Ecoomerica 42, 497-56. iewer, W.E. (98), Aggregaio Problems i he Measureme of Capial, i. Usher (ed.), The Measureme of Capial, Uiversiy of Chicago Press, 433-528. iewer, W.E. (23/2), The Treame of Ower Occupied Housig ad Oher urables i a Cosumer Price Idex, iscussio Paper 3-8, eparme of Ecoomics, Uiversiy of Briish Columbia. hp://www.eco.ubc.ca/discpapers/dp38.pdf. Forhcomig i W.E. iewer, J. Greeless ad C. Hule (eds.) (2), Price Idex Coceps ad Measureme, NBER Sudies i Icome ad Wealh, Uiversiy of Chicago Press. iewer, W.E. (25a), The Measureme of Busiess Capial, Icome ad Performace, a uorial preseed a he Uiversiy Auooma of Barceloa, Spai, Sepember 2-22, 25. hp://www.eco.ubc.ca/diewer/barceloa.hm iewer, W.E. (25b), Issues i he Measureme of Capial Services, epreciaio, Asse Price Chages ad Ieres Raes, pp. 479-542 i Measurig Capial i he New Ecoomy, C. Corrado, J. Haliwager ad. Sichel (eds.), Chicago: Uiversiy of Chicago Press. iewer, W.E., (26/29), The Paris OEC-IMF Workshop o Real Esae Price Idexes: Coclusios ad fuure irecios. Paper preseed a he OEC-IMF Workshop o Real Esae Price Idexes held i Paris, 6-7 November 26. hp://www.eco.ubc.ca/diewer/dp7.pdf Published as chaper 6, pp. 87-6 i iewer, W.E., B.M. Balk,. Fixler, K.J. Fox ad A.O. Nakamura (29), Price ad Produciviy Measureme: Volume -- Housig. Trafford Press. Also available a www.vacouvervolumes.com/ ad www.idexmeasures.com. iewer, W. Erwi ad Alice O. Nakamura (29), Accouig for Housig i a CPI, chaper 2, pp. 7-32 i W.E. iewer, B.M. Balk,. Fixler, K.J. Fox ad A.O. Nakamura (29), Price ad Produciviy Measureme: Volume -- Housig. Trafford Press. Also available as a free e-publicaio a www.vacouvervolumes.com ad www.idexmeasures.com. oughery, A. ad R. Va Order (982), Iflaio, housig cos ad he cosumer price idex, America Ecoomic Review, 72, 54-65. reima, M.H. ad A. Peigo-Cross (24), Aleraive Mehods of Icreasig he Precisio of Weighed Repea Sales House Prices Idices, Joural of Real Esae, Fiace ad Ecoomics 28 (4), 229-37. Ellis, L. (28) The Housig Meldow: Why id I Happe i he Uied Saes? BIS Workig Papers No 259, Bak of Ieraioal Selemes. Fisher, I. (897), The Role of Capial i Ecoomic Theory, Ecoomic Joural 7, 34-367. Galli, J. (29), The Log-Ru Relaioship bewee House Prices ad Res, Real Esae Ecoomics, forhcomig. Garer, T.I. ad Verbrugge (29), Recocilig User Coss ad Real Equivalece: Evidece from he U.S. Cosumer Expediure Survey, Joural of Housig Ecoomics, forhcomig. Geesove,. (23), The Nomial Rigidiy of Aparme Res, Review of Ecoomics ad Saisics 85 (November), 844-853 Gilligham, R. ad W. Lae (982), Chagig he Treame of Sheler Coss for Homeowers i he CPI, Mohly Labor Review 5 (Jue 982), 9-4. Gordo, R.J. ad T. va Goehem (24), A Ceury of owward Bias i he Mos Impora Compoe of he CPI: The Case of Real Sheler, 94-23, forhcomig i E.R. Berd ad C.R. Hule (eds.), Hard-o- Measure Goods ad Services: Essays i Memory of Zvi Griliches, Sudies i Icome ad Wealh No. 67, Uiversiy of Chicago Press. hp://faculyweb.a.orhweser.edu/ecoomics/gordo/p36_forpub_4729.pdf - 23 -
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