Presented at the Fourth World Conference of the Spatial Econometrics Association, Purdue University, USA, June 9 12, 2010.

Transcription

1 Pesented at the Fouth old Confeence of the Spatal Econometcs Assocaton Pudue Unvest USA June 9. SPATIAL HIERARCHICAL BAYESIA MODEL FOR MISSIG DATA IMPUTATIO [eal daft] HAJIME SEYA atonal Insttute fo Envonmental Studes Cente fo Global Envonmental Reseach 6- Onogawa Tsukuba-sh Ibaak-ken Japan MORITO TSUTSUMI Depatment of Polc and Plannng Scences Unvest of Tsukuba -- Tennoda Tsukuba-sh Ibaak-ken Japan The applcaton of a hedonc appoach to eal estate data has plaed an mpotant ole n popet maket analss. Although the ecent dffuson of global postonng sstems GPS geogaphc nfomaton sstems GIS and emote sensng RS have povded access to detaled attbutes of popetes mssng data contnues to be a seous ssue whle conductng a hedonc analss of popetes. Knght et al. 998 had poposed a method fo mputng mssng data effectvel b emplong the Makov Chan Monte Calo MCMC method. Howeve the wok of Knght et al. 998 dd not deal wth spatal effects among data whch ae now wdel consdeed to be athe mpotant aspects of eal estate data. Theefoe the objectve of ths stud s to etend the wok of Knght et al. 998 b consdeng spatal dependence. In ode to consde spatal dependence ths stud adopts spatal statstcal nstead of spatal econometc models. Ths s because spatal statstcal models ae moe convenent fo conductng out-of-sample pedcton. e fomulate the hedonc egesson model as the heachal Baesan model b consdeng spatal dependence spatal heachal Baesan model and append the data mputaton steps to the MCMC smulaton n keepng wth the method poposed b Knght et al In addton ths stud conducts an empcal analss of the Japanese apatment ent maket and demonstates out-of-sample pedctons. As compaed to Knght et al. s 998 model the empcal esults ndcate a hghe pedctve accuac usng ou model. Ke wods: Mssng data Apatment ent Data augmentaton Heachcal Baesan model Spatal statstcs JEL: C R9

2 ITRODUCTIO The applcaton of the hedonc appoach developed b Rosen 974 to eal estate data has plaed an mpotant ole n popet maket analss. Although the ecent dffuson of global postonng sstems GPS geogaphc nfomaton sstems GIS and emote sensng RS have povded access to detaled attbutes of popetes mssng data contnues to be a seous ssue whle conductng a hedonc analss of popetes. Knght et al. 998 poposed a method fo mputng mssng data effectvel b usng the Makov Chan Monte Calo MCMC method. The advantage of Knght et al. s 998 method s ts ablt to dentf not onl the mssng dependent popet values but also eplanato vaables attbutes. Howeve Knght et al. s 998 method does not deal wth spatal effects among data whch ae now wdel consdeed to be athe mpotant aspects of eal estate data. Thee ae two majo eseach felds fo consdeng spatal dependence among data: Spatal Econometcs e.g. Anseln 988 and Spatal Statstcs e.g. Cesse 993. Recentl statstcal e.g. Anseln and Lozano-Gaca 9 and theoetcal economc e.g. Small and Stemetz 9 applcaton of these studes to hedonc analss often called the spatal hedonc appoach has been developed. Consequentl substantal empcal wok has been conducted whch has acheved emakable success Spatal Econometcs: e.g. Can 99; Anseln and Le Gallo 6; Tsutsum and Sea 8. Spatal Statstcs: e.g. Dubn 988; Valente et al. 5; Inoue et al. 7. Compaed both Spatal Econometcs and Statstcs: e.g. Páez et al. 7 Caldeón 9 Tsutsum and Sea 9. th espect to the elatonshp between both these appoaches and latest eseach dectons see Aba 6 Gffth 7 LeSage and Pace 9 Paéz 9 Anseln Pnkse and Slade and Gelfand et al.. Ths stud emplos the spatal statstcal nstead of spatal econometc appoach n ode to consde spatal dependence. Ths s because ou empcal analss ncludes spatal pedcton ntepolaton of the ent value at new stes and the spatal statstcal appoach s moe sutable fom ths pespectve e.g. Banejee et al. 4; Valente et al. 5. Fo studes concenng spatal pedcton b spatal econometc models see Anseln 998 Tsutsum et al. Bvand Kelejan and Pucha 7 and Gffth. A few attempts fo mputng mssng data consdeng spatal dependence have been made. Cetan emakable eceptons nclude Matn 984 LeSage and Pace 4 Gffth and Muakam and Tsutsum. LeSage and Pace 4 uses both the MCMC method E-M algothm and the othes use the latte algothm o smla. All these studes emploed spatal econometc models n ode to consde spatal dependence; moeove to the best of ou knowledge thee s no sgnfcant eseach that emplos the spatal statstcal appoach fo consdeng spatal dependence wth espect to mssng data mputaton. Hence the objectve of ths stud s to etend the wok of Knght et al. 998 b consdeng spatal dependence usng the spatal statstcal model. e fomulated the hedonc egesson model as the

3 heachal Baesan model b consdeng spatal dependence spatal heachal Baesan model and ncopoated the data mputaton steps to MCMC smulaton n keepng wth the method poposed b Knght et al Moeove ths stud conducted an empcal analss on the Japanese apatment ent maket and demonstated out-of-sample pedctons. As compaed to the Knght et al. s 998 model the empcal esults ndcated a hghe pedcton pefomance of ou model. Ths pape conssts of s sectons. The fst secton pesents the ntoducton and the second secton descbes the spatal statstcal model fom the heachcal Baesan pespectve. The thd secton ntoduces the wok of Knght et al. 998 and the fouth secton etends the wok of Knght et al. 998 b consdeng spatal dependence. The ffth secton dscusses the pefomance of the poposed model though an empcal analss of apatment ents n Toko Japan. Fnall the sth secton concludes ths pape. SPATIAL STATISTICAL MODEL Hedonc pces ae defned as the mplct pces of attbutes Rosen 974. Let the hedonc pce model be a lnea egesson model whch s pesented n the followng manne: I H whee s an n vecto of the popet values pe lot aea s an n k mat of the eplanato vaables assocated wth attbutes sa k vecto of the tend paametes and I s an n n dentt mat. ndcates nomal dstbuton. and ae paametes temed nugget patal-sll and ange espectvel. H s an n n coelaton mat whose j th elements H j ae gven b vald sotopc coelaton functons n whch depend onl on the dstance between the locatons of and j. e assume that the eponental fom fo the coelaton functon s gven b ep d j / whee d j s the dstance between the locatons of and j. ow let = be the set of paametes and assume that the pos on ae mutuall ndependent. In othe wods =. Subsequentl usng Baes theoem the posteo pobablt denst functon p.d.f denoted b s popotonal to f whee f s the lkelhood assocated wth eq.. The estmates have been obtaned fom the posteo dstbuton. Howeve the computaton of the lkelhood eques I + H whch ceates the poblem of mat nveson f n s a lage numbe. Theefoe t s convenent to wok wth a heachcal model Banejee et al. 4. In othe wods I s an n vecto whose elements ae gven b s an n H. posteo p.d.f s popotonal to the followng equaton: spatal andom effects vecto. The

4 f f. 3 Subsequentl we specf the pos of paametes as c T IGa b IGa b and Gaa b whee IGa b denotes the nvese gamma dstbuton wth mean b/a f a> and Ga a b denotes the gamma dstbuton wth mean a/b. Consdeng eq. 3 the full condtonal posteo dstbutons fo each paamete ae pesented n the followng manne: D d D whee T D c T d ; D d D whee H I D and d ; b n a IG ; H b n a IG ; H ep. 4 e estmated the model paamete usng the MCMC method e.g. Gelfand and Smth 99. It must be noted that all the full condtonals ecept that of ae of standad fom; theefoe t s smple to geneate samples fom the condtonal posteo dstbuton Gbbs sample. Howeve the full condtonal of the ange paamete s of non-standad fom whch ceates dffcultl n geneatng a sample. Ths stud adopts the Metopols-Hastng algothm andom walk chan n ode to geneate the sample as descbed below. Fst s geneated fom the followng equaton: * 5 whee s the step d of the MCMC teaton. Subsequentl the acceptance pobablt s computed n the followng manne: mn * * 6 and unfom dstbuton u s geneated on. s updated on the bass of the followng equaton: * * * u u. 7 e fed followng epeated tal and eo n ode to acheve the acceptance ate of appomatel 5% e.g. Robets 997. Spatal pedcton of the value at new out-of-sample stes s athe staghtfowad often

5 temed Baesan Kgng. Let = then pedctve dstbuton s pesented n the followng manne: d R R 8 whee R s the numbe of geneated samples s an m vecto of the popet values pe lot aea at the new stes and s an k m mat of the eplanato vaables at the new stes. ITRODUCTIO OF KIGHT ET AL. S 998 DATA AUGMETATIO METHOD Ths secton povdes a bef ovevew of the method poposed b Knght et al Fst consde the smple lnea egesson model that s pesented n the followng manne: n 9 whee s the popet value and ae eplanato vaables ae the paametes and obes ndependentl. Fst assume that thee s no mssng data. Subsequentl the lkelhood s ] / ep[ ; / n L whee. On the bass of Baes Theoem multplng the po dstbuton of wth lkelhood elds the posteo dstbuton of as ; L. Knght et al. 998 fed the po p.d.f of as wth and nvese gamma denst of.e. b a IG. Consequentl the full condtonal posteo dstbuton of each paamete s pesented n the followng manne: n n.5 b n a IG. th these full condtonals the estmates of the paametes could be obtaned usng the Gbbs sample. ow assume that and ae both mssng. Subsequentl the paamete set s augmented fom to. The full condtonals of and ae pesented n the followng manne:

6 whee ndcates all obsevatons ecept. At ths pont we can mpute the mssng data b smpl appendng the samplng fom eq. to the end of the Gbbs samplng steps. ETESIO OF KIGHT ET AL. S 998 METHOD BY COSIDERIG SPATIAL DEPEDECE In ths secton we ntoduce a spatal consdeaton to the wok of Knght et al Ths was smpl acheved b appendng the steps descbed n the pevous secton to eqs. 4 and 5. ow assume that both the popet value of aea p.e. p and l th attbute of aea q.e. lq ae mssng. Subsequentl we augment the paamete space fom to = p lq esultng n the full condtonal dstbuton of these paametes whch s eflected b the followng equatons: p p lq h hp wp h whee w p ndcates spatal andom effects. lq q wq hhq hl l l EMPIRICAL AALYSIS: DATA Fo the empcal analss the data set of apatment ent and dffeent tpes of attbutes wee povded b At Home Co. whch s a povde of eal estate maket nfomaton and a elevant suppot sevce agenc n Japan. The have establshed a netwok of eal estate nfomaton and ae afflated to appomatel 56 eal estate agences. The data set compsed nfomaton wth espect to the followng attbutes: poston coodnate lattude and longtude ent age floo aea stuctues such as enfoced concete oom tpe and numbe of ooms sewage gas pakng lot a condtone bathoom etc. The pesent stud uses a sample of 5 obsevatons fom the data. All of them ae fom Janua to Ma 6. Fgue ndcates the stud aea whch compses the 3 wads of Toko ncludng the thee cental wads of Choda Chuo and Mnato. Choda wad s stuated n the cente of the ct and n seveal was the cente of Japan. The Impeal Palace Det Toko Cental Staton buldngs of mnstes and a numbe of lage copoate headquates ae located n Choda. Chuo was hstocall the chef commecal cente of Toko patculal po to old a II. Mnato wad compses a numbe of embasses and hgh-class apatments whch ae manl ented b the foegn eecutves of oveseas fms.

7 Toko s thee cental wads FIGURE. Toko s 3 ads Includng the Thee Cental ads FIGURE. Stud Aea FIGURE. Obsevaton Ponts

8 Fgue ndcates the locatons of the obsevaton popetes and the ents n tems of en pe squae mete pe month ca. US dolla = 95 Japanese en n Ma. The obsevaton popet that s the most epensve n tems of ent pe squae mete pe month s located n Mnato wad and othe epensve obsevaton popetes ae towads the southwest. EMPIRICAL AALYSIS: MODEL The dependent vaable s the logathm of apatment ent [en/m pe month]. The followng ae the eplanato vaables that wee selected afte tal and eo: constant whch epesents the ntecept; tan whch epesents the tme taken to tavel fom the neaest staton to cental Toko b tan o subwa [mnutes] consdeng the aveage of the tme equed to tavel to Shnjuku staton and Toko o Otemach staton; BoT Toko whch epesents the Toe subwa that s opeated b the Bueau of Toko Metopoltan Govenment [dumm]; Odaku whch epesents the Odaku Electc Ralwa [dumm]; walk o bus whch epesents tme taken to tavel fom the apatment to neaest staton ethe walkng o b bus [mnutes]; floo aea log whch epesents the logathm of floo aea [m ]; age sq epesents the oot squae of age [ea]; enfoced concete whch epesents the enfoced concete stuctue [dumm]; nos. of ooms whch epesents the numbes of ooms; one-oom tpe whch epesents the ostensble one-oom apatment equpped wth a tn ktchen[dumm]; K-tpe whch epesents a one oom apatment wth an ndependent ktchen [dumm] pakng lot whch epesents the faclt of a pakng lot [dumm]; self-lockng whch epesents the self-lockng faclt [dumm]. In the cuent empcal analss we defned 5 obsevatons whch ae ndcated n fgue as the complete data. Followng ths we fst andoml selected v = [% 4% 7%] of the complete data. Subsequentl we assumed that one of the ent values o attbutes ecept fo categocal/dumm vaables ae mssng. Moe specfcall we deleted one of the vaables fom among ln apatment ent value tan walk bus floo aea log and age sq and eplaced t wth zeo. In the followng step we mputed the mssng data usng the data augmentaton method. Besdes these 5 obsevatons we used 5 obsevatons fo valdaton whch we temed valdaton data. On the bass of the mputed pseudo complete data the value of the valdaton data was pedcted usng Baesan Kgng and the oot mean squaed eo RMSE was calculated. The geogaphcal dstbuton of the complete and valdaton data s ndcated n fgue 3. The complete and valdaton data s the same as that used n Tsutsum et al. 7 whch compaed seveal paamete estmaton methods and pedctve pefomance of spatal statstcal and spatal econometc models. Consequentl the esult of ths eseach s compaable to that of Tsutsum et al. 7.

9 FIGURE 3. Obseved and Valdaton Data The pogam emploed n ths eseach s coded usng MATLAB 7.5. and we confm that ou code has successfull cleaed Geweke s 4 jont dstbuton test. In ode to compute the magnal lkelhood we emploed the Gelfand and De s 994 method. th espect to the ntal values fo the pos we adopted c = T =. I a = b = a = b = 5. 3 = a = 3 b =. The total numbe of teatons wee ; and the fst samples wee dscaded.e. bun-n. Table ndcates the esults of paamete estmaton no-mssng data case. The log-magnal lkelhood ndcates that the obsevaton ft mpoved consdeabl b emplong the spatal statstcal model. In Baesan statstcs t s common to conduct paamete nfeence b vefng whethe o not the 95 pecent cedble ntevals fo the assocated tend paametes eclude zeo. It must be noted that a majot of the tend paametes ae statstcall sgnfcant and sgns of the estmates ae ntutonall acceptable the t-value of the tend paametes have also been calculated fo efeence. e could not eclude cetan paametes that wee not statstcall sgnfcant fo eample floo aea fom the model n ode to pemt the compason of the esults of ths pape wth ou pevous wok ncludng Tsutsum et al. 7. The ange paamete whch ndcates appomatel one thd of the ntutve coelaton ange called effectve ange when the eponental functon s used was estmated at.9 km. Theefoe the effectve ange s appomatel 3 km whch s easonable fo the apatment maket n Toko s 3 wads. Fgue 4 and 5 ndcate the posteo dstbuton and samplng path of the ange paamete espectvel. The convegence of the MCMC samples had been tested usng Geweke s 99 method the esults of whch ae ndcated n Table. Table ndcates the p-value of Geweke s ch-squaed test and suggests that the samples wee successfull conveged to the posteo dstbutons.

10 TABLE. Result of Paamete Estmaton FIGURE 4. Posteo Dstbuton of Range Paamete FIGURE 5. Samplng Path of Range Paamete

11 TABLE. Result of Paamete Estmaton Table ndcates the RMSE of spatal pedcton. It s common to dscad the dataset wth mssng data; hence ths method was temed dscad method and was also adopted fo compasons. Fo all methods the RMSE obtaned on consdeng % mssng cases s compaable wth that obtaned wth complete data cases. Ths s because a data case contanng 35 obsevatons ncludng % mssng cases s suffcent fo paamete estmaton. Howeve when the ato of mssng cases s hgh.e. 4% o 7% the pefomance of ou method and that of Knght et al. s method 998 s supeo to that of the dscad method. In the event of 7% mssng cases the pefomance level of the dscad-spatal method RMSE =.78 s not ve dffeent fom the dscad-non-spatal one RMSE =.85. Ths s because modelng the ente spatal pocess wth onl 45 obsevatons s athe dffcult. On the othe hand the pefomance of ou method and that of Knght et al s method was satsfacto even n the event of 7% mssng cases. Moeove a consdeable mpovement n pedctve pefomance was notced when spatal dependence was consdeed. Evdentl ths s onl a smple eample and we wll have to vef the unvesal valdt of ths esult. COCLUSIOS Ths stud etended the wok of Knght et al. 998 b consdeng spatal dependence. The spatal pedcton to data valdaton was demonstated and the RMSE consdeabl mpoved when ou model was used. Ou method and Knght et al. s 998 method s advantageous n tems of hgh pedcton accuac smple pogammng onl two steps need to be appended to the MCMC sample and eas undestandng fo anbod who s famla wth Baesan statstcs. Theefoe we ae convnced that ths method ma be athe useful fo mputaton of mssng data dung eal estate polc makng o n the couse of busness. The potental aeas fo futue stud ae ndcated below:. Vefng the effectveness of ou method n a moe unvesal settng.e Monte Calo smulaton wth smulaton data set.. Clafng the effect of mssng covaates on statstcal chaactestcs of the kgng pedcto. 3. Imputng mssng categocal/dumm vaables. Knght et al. 998 has suggested a logt-tpe

12 functon fo ths pupose. 4. Etendng ou model to ansotopc and/o non-statona covaance models. 5. Etendng ou model to spatal econometc models. 6. Impovng the algothm b usng anothe samplng schema lke the slce sample Agawal and Gelfand 5. ACKOLEDGEMET The authos would lke to epess the gattude to At Home Co. Ltd. fo the collaboaton n povdng the apatment ate data. Moeove the authos have mmensel benefted fom the valuable comments povded b Pof. oel A. C. Cesse on ou eal daft whch was pesented at the ntenatonal smposum on statstcal analss of spato-tempoal data at the Unvest of Toko fom ovembe REFERECES Agawal D.K. and Gelfand A. 5. Slce samplng fo smulaton based fttng of spatal data models. Statstcs and Computng 5 : Anseln L Spatal econometcs: Methods and models. Kluwe Academc Publshes Dodecht The ethelands. Anseln L GIS eseach nfastuctue fo spatal analss of eal estate makets. Jounal of Housng Reseach 9 : Anseln L.. Tht eas of spatal econometcs. Papes n Regonal Scence 89 : 3 5. Anseln L. and Le Gallo J. 6. Intepolaton of a qualt measues n hedonc house pce models: Spatal aspects. Spatal Economc Analss : 3 5. Anseln L. and Lozano-Gaca. 9. Spatal hedonc models. In Patteson K. and Mlls T.C. Edtos Palgave handbook of econometcs. 6: 3 5 Palgave Macmllan. Aba G. 6. Spatal econometcs: Statstcal foundatons and applcatons to egonal convegence. Spnge-Velag Beln. Banejee S. Caln B.P. and Gelfand A.E. 4. Heachcal modelng and analss fo spatal data. Chapman & Hall/CRC. Bvand R.. Spatal econometcs functons n R: Classes and methods. Jounal of Geogaphcal Sstems 4 4: Caldeón G.F.-A. 9. Spatal egesson analss vs. Kgng methods fo spatal estmaton.

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