Spatial Econometric Models for Panel Data: Incorporating Spatial and Temporal Data

Transcription

1 Spaal Economerc Models for Panel Daa: Incorporang Spaal and Temporal Daa By Chrsopher Frazer, Graduae Suden Researcher, The Unversy of Texas a Ausn. 6.9 E. Cockrell Jr. Hall, Ausn, TX , smplexr@yahoo.com Kara M. Kockelman, Clare Boohe Luce Asssan Professor of Cvl Engneerng The Unversy of Texas a Ausn, 6.9 E. Cockrell Jr. Hall, Ausn, TX kkockelm@mal.uexas.edu, Phone: , FAX: (Correspondng Auhor) The followng paper s a pre-prn and he fnal publcaon can be found n Transporaon Research Record No. 902: 80-90, Presened a he 84h Annual Meeng of he Transporaon Research Board, January 2005 ABSTRACT Ces are consanly evolvng, complex sysems; and modelng hem, boh heorecally and emprcally, s a complcaed ask. However, undersandng he manner n whch developed regons change over me and space can be of grea mporance for ransporaon researchers and planners. In hs paper, mehodologes for modelng developed areas are developed whle ncorporang spaal and emporal effecs of he daa. The work emphaszes spaal relaonshps beween varous geographc, land-use, and demographc varables characerzng fne zones across regons. I derves and combnes land cover daa for he Ausn, Texas regon from a panel of saelle mages and U.S. Census of Populaon daa. Models for populaon, vehcle ownershp, and developed, resdenal, and agrculural land cover are esmaed; and he effecs of space and me on he models are shown o be sascally sgnfcan. Smulaons of populaon and land cover for he year 2020 help o llusrae he srenghs and lmaons of he models. INTRODUCTION Urban sysems are nrcae, mulfaceed and consanly evolvng. Ther evoluon s dcaed by a large number of nfluences, ncludng publc polcy, ndvdual preferences and acons, he physcal landscape, echnology and hsory. All of hese facors (and more) nerac n myrad ways. Dscernng how and why urban sysems evolve s, from he sar, an exremely dffcul ask. There s grea benef o uncoverng he dynamcs underlyng urban sysems. Undersandng he ways n whch geographc, economc, demographc, polcal and oher facors nerac s of neres o ransporaon engneers and land use planners, economss as well as hsorans, polcymakers and he publc. Models ha relably rack hese neracons are of grea neres o ransporaon planners, as hey llumnae how, among oher hngs, polcy mpacs land use and ravel paerns, welfare and developmen, congeson and ar qualy.

2 Parker, e. al. (2003) dscussed he wde range of many land-use/cover change (LUCC) models recenly developed. They poned ou ha, due o he complexy of he sysems encompassng land-use/cover, no sngle exsng model s of more use han ohers; hus, a wde range of models, from he heorecal o he emprcal, are beng nvesgaed by a varey of researchers (see, e.g., Candau (2002); Clarke and Gaydos (998); Parker, Berger and Manson (200)). In hs paper, a closer connecon beween he real world and he model, as opposed o largely heorecal work, s sough. Ths parallels some recen models, developed for use by plannng organzaons for regonal forecasng and polcymakng. The regonal models mos smlar o he work underaken here are UrbanSm, Wha If?, and CUF2. UrbanSm (Waddell 2002) mcro-smulaes he effecs of locaon, land use, and polcy decsons by households, workers, developers and polcymakers on he land use paerns and rens across a regon. Land use and developmen s modeled a he level of sngle parcels. Ohers are modeled a he level of user-defned grd cells whch have no lower bound. Kloserman s (999) Wha f? model of land use assgns land uses o a se of homogeneous zones n a boom-up fashon, derved from socoeconomc, geographc, ransporaon and zonng nformaon. Lands and Zhang s (998) Calforna Urban Fuures 2 (CUF2) model employs mulnomal models of land-use change per hecare (or oher un of observaon) o predc fuure land use paerns. One of he major drawbacks o many of hese models s ha hey fal o ncorporae and negrae he spaal and emporal correlaons ha are presen n urban sysems. Tha s, on an nuve level, would be expeced ha plos of land whch are close, n eher spaal or emporal dmensons, would have more smlares whch would nfluence or be represenave of her characerscs han hose whch are far away. Whereas panel daa echnques ha accoun for emporal correlaons are n wdespread use, he mehods descrbed n Anseln (988) and Elhors (2003) used o accoun for correlaons, or more correcly he auocorrelaons, n he spaal dmenson are less well known. There have been a varey of sudes accounng for spaal auocorrelaon. For example Case (992) examned he nfluence of neghbors on echnologcal changes on Indonesan farms, Coughln e al. (2003) looked a he effec of spaal dependence on sae loeres n he U.S., and Dubn (99) suded he spaal auocorrelaons of resdenal neghborhood quales. However, mos sudes ncorporang spaal auocorrelaon do no ncorporae emporal correlaons, and her focus s no amed a ransporaon-based applcaons. Researcher and planners would lke o oban as much nformaon as possble from he spaal and emporal characerscs of he urban landscape. A prmary goal of hs work s o develop mehodologes o analyze urban growh ha accoun for such characerscs and are of neres o ransporaon researchers and planners. These models are esed emprcally usng land-cover daa derved from saelle mages coupled wh U.S. Census daa. The followng secons deal he daa ses and her developmen, he appled mehodologes, and resuls for an Ausn, Texas applcaon. DATA DESCRIPTION The daa used n hs work s drawn prmarly from saelle and U.S. Census daa, whch, n her orgnal form, are spaally and emporally ncongruous. Ths secon dscusses hese daa sources, as well as he mehods used o negrae hem no a sngle daa se. I should be For addonal deph on sascal specfcaons, addonal model formulaons and resuls (ncludng sample selecon mehods and dfferenal equaon model approxmaons), readers may consul Frazer (2004).

3 noed ha he erm land cover s hroughou he ex as opposed o he more common erm land use. Ths s essenally because he daa derved from he vsual/specral quales of he land, raher han nformaon on he manner n whch humans acually use. Land Cover Daa Derved from Saelle Imagery Saelle daa offer excellen opporunes and consderable challenges. A serous and recurrng problem for modelng land use has been he lack of spaally dealed daa. Remoe sensng, magng echnology, and geographcal nformaon sysems (GIS) are makng accurae land cover maps far more accessble o he researcher, and o he publc. In parcular, global saelle magng, naed n he early 970s, provdes hghly dealed mages regularly. And mage analyss sofware can classfy hese by varous general caegores. GIS sofware combnes daa maps of varous ypes, dramacally faclang spaal analyss. The Uned Saes launched LandSa n 972. Passng over Ausn every 8 days, hs early saelle provdes mages wh 79 m 79 m pxel resoluon. LandSa 4 was launched n 982, and resuled n 85 km 85 km mages wh 30 m 30 m resoluon wh a repea orb cycle of 6 days. 984 s LandSa 5 and 999 s LandSa 7 have essenally dencal orb and mage characerscs o LandSa 4. These magng sysems work by scannng mulple passes (each represenng one pxel) over an area and recordng he reflecance of seven dsnc specral bands (Rchards and Ja 2000); sx of hese bands record wh 30 m 30 m resoluon, whle he sevenh, a hermal band, records wh 20 m 20 m resoluon (60 m 60 m for LandSa 7). The land-cover daa used n hs work was derved from mages aken by he LandSa 4, 5, and 7 saelle sysems. Four mages of Ausn, Texas, aken n he years 2000, 997, 99, and 983, were used. The mage secons used are all 48.5 km 55.8 km and have 30 m 30 m resoluon; each secon hus conans jus over hree mllon pxels of daa. The dervaon of land cover from he saelle mages was acheved by a mehod called supervsed mage classfcaon and was performed by Unversy of Texas - Ausn professor Dr. Barbara Parmener and sudens n a graduae geography course. Supervsed mage classfcaon bascally uses he saelle mage daa from areas of known land cover o creae a se of decson rules by whch he res of he mage can be classfed (Rchards and Ja 999). In he daa used here, each saelle pxel was classfed no one of nne land-cover ypes: waer, barren, fores/woodland, shrubland, herbaceous naural/sem-naural, herbaceous planed/culvaed, fallow, resdenal, or commercal/ndusral/ransporaon. In he conex of hs work, he second hrough ffh classfcaons are consdered unnhabed land, he sxh and sevenh are consdered agrculural land, and he fnal wo are developed land. Qualave comparsons of he land cover classfcaons wh aeral phoography showed he resuls o be accurae, hough no quanave analyss of he qualy of he classfcaon was carred ou. For more deals concernng boh he classfcaon process and possble ssues wh he daa, he reader s referred o Frazer (2004). Derved Land Cover Daa and Oher Daa Sources Two spaal sascs were compued based on he land-cover daa descrbed above. These are land-cover mx and land-cover enropy. Land-cover mx (from here on called mx) characerzes he dssmlary of he land-cover n a parcular area: For a gven pxel, mx s an ndex of adjacen pxels dssmlary; measures he level of homogeney beween a cenral pxel s use ype x ) and hose of s neghbors x ) (Kockelman 997, Cervero and Kockelman ( 0 (

4 997). For hs work, he neghborhood around a pxel was consdered o be he egh pxels mmedaely surroundng (see Fgure ). Mahemacally, mx s defned by mx( x 0 ) = 8 = δ 8 x0, x () where f x = x0 δ = x 0, x (2) 0 oherwse As an average measure of dssmlary, he mx ndex ranges from 0 o, wh a hgher numercal value correspondng o less smlary beween a gven pxel and s neghbors. x x x 2 3 x x 4 0 x 5 x 6 x 7 x 8 Fgure Reference dagram for pxel neghborhood used n calculaon of land-cover mx sasc. As a complemen o mx, land-cover enropy (from here on called enropy) measures he level of land-cover varey of a parcular neghborhood. Enropy s also called land-cover balance, and essenally provdes a measure for he level of heerogeney of land-cover n he neghborhood (Kockelman 997). Raher han comparng all he pxels n a neghborhood wh he cenral one, as s done n mx, nsead compares all of he pxels wh each oher. If here are J possble land-cover ypes whch a neghborhood may be made up of, hen enropy s defned by: J enropy ( x ) = Pj ln( Pj ) (3) ln( J ) j= where P j s he fracon of he neghborhood ha s land-cover ype j. Enropy also ranges from 0 o, wh a hgher value correspondng o a greaer level of neghborhood land-cover heerogeney. I equals when all land cover ypes exs n a zone and all her proporons are equal (.e., perfec balance n cover ypes). Because of hs non-cenralzed naure of he sasc, was calculaed for 300 m 300 m neghborhoods (whch correspond o he combnaon grd cells as descrbed n he nex secon) as opposed o he nne cell ones used for mx.

5 In addon o he land cover daa and s derved sascs, Census of Populaon daa was used. Sascs from boh he 00%-sample Census (SF) and he 7% sample (SF3) were used. These nclude populaon and household-level varables (such as household sze and he number of vehcles per household). Daa for Travs, Wllamson, Basrop and Hays Counes was colleced so as o compleely encompass he land-cover daa regon. Of course, he smalles areal un for Census daa s he block or block group, whch ypcally encompass dozens of 30 m x 30 m pxel-based cells. So daa had o be cleverly combned and hen allocaed o grd cells, as descrbed n he followng secon. Fnally, wo Eucldean dsance measures are used for analyss: dsance o he cenral busness dsrc (CBD) and dsance o he neares hghway. Daa Combnaon Mehods The fac ha he years of he Census daa do no algn wh he years of he saelle pcures, as well as he fac ha he Census block groups do no lne up wh any grd sysem, necessaes he use of varous mehods o reasonably combne he daa ses. Tha s, o use he varous daa sources colleced for hs work all ogeher, he daa mus all be regsered o he same emporal and spaal coordnae sysem. To spaally combne he daa, a grd ha combnes 00 of he pxel cells s used. Ths coarser grd s supermposed over he Census block groups, and he Census daa allocaed o each grd cell based on how much area each block group represens whn he cell. For acual coun varables, such as populaon, he fracon of he varable ha corresponded o he fracon of he block-group n he cell was ransferred; for he varables represenng averages over he blockgroup, such as average household ncome, he ransfer was done by (spaally) weghed summaon of he Census values. The new grd sysem has anoher benef n ha reduces he large land cover daa se. As noed earler, each land cover daa se has over 3 mllon pxels, whch s an excessve amoun of daa, especally when compared o he housand or so Census block groups. By usng a combnaon grd whose cells are exacly en pxels square (300 m 300 m), he land cover daa se was reduced by a couple orders of magnude, whle sll reanng sgnfcan resoluon of he regon s land cover paerns. Ths coarsenng of he grd sysem ransformed he land cover daa from a se of dsnc, sngle-valued land cover ypes o a proporons daa se (wheren each combnaon grd cell has a percenage of each land cover ype assocaed wh ). In order o algn he daa ses emporally, an approxmaon mehod was appled o he Census daa. Under he assumpon ha all Census varables roughly follow an exponenal growh paern wh me, an approxmaon of he form z e λ ( ) = α (4) s used for each varable a an aggregae level, wh parameers α and λ esmaed usng he 2000 and 990 Census fgures n a leas squares framework and z () represenng he average varable value n a grd cell a me (he smple exponenal form s movaed n Smh and Sncch 992). Averages for off-census years are hen calculaed and he values for he combnaon grd cells deermned from (4) by usng he devaons of each grd cell from he 2000 and 990 means. Tha s, for each grd cell, he value of he varable a me s gven by:

6 x (2000) + x (990) z ( ) = z( ) (5) x(2000) + x(990) where x () s he rue Census level of he varable n grd cell, and x () s he average across all grd cells. METHODOLOGY Spaal Lnear Regresson Model for Panel Daa The specfcaon used for modelng connuous varables n hs work s daa ses s he panel-daa spaal lnear regresson model. Examples of research usng hs model (hough n dfferen forms han ha used n hs work) nclude Dubn s (99) sudy of resdenal home values and he sudy of naonal homcde raes n Messner and Anseln (2002). In he conex of hs work, he general form of he model for an ndvdual cell (wh N oal cells and T oal me perods) s: y = β x + v + θ (6) where y s he dependen varable a me, v s an ndvdual-specfc effec assumed o be 2 normally dsrbued wh zero mean and varance σ v, and x s a vecor of exogenous explanaory varables, some of whch may be me lagged (see Frazer (2004) for a dscusson of exogeney ssues as hey relae o hs work). θ s an error erm whch, o capure spaal auocorrelaon, s specfed, n block marx form, as follows (Anseln 988): ( λw) ξ θ = λwθ + ξ θ = (7) 2 where ξ s a (TN ) vecor of whch every elemen s dsrbued as Normal(0, σ ) and W s a (TN TN) block dagonal marx wh T copes of he (N N) spaal wegh marx W ~ defned by: W ~ 0 f ( d = M f ( d 2 N ) ) f ( d 0 M f ( d 2 ) N 2 ) L L O L f ( d f ( d M 0 N 2N ) ) (8) where g( ) s a funcon and d j s he dsance beween cells and j. For hs work, an nverse squared-dsance measure was used n order o recognze greaer auocorrelaon presen among cells close o each oher, and a rapd reducon n such correlaon wh dsance. Thus, he equaon used s as follows (see Anseln 988 for a dscusson of oher funconal forms): 2 g ( d j ) = ( dj ) (9)

7 To esmae he model parameers, a combnaon of feasble generalzed leas squares regresson (FGLS) and maxmum-lkelhood esmaon (MLE) can be used (Elhors 2003). In he followng dervaon of he model, whch closely follows Elhors (200 and 2003), s frs noed ha he random effec can be realzed as a varable-parameers model, wh he consan varable, X =, havng a varable coeffcen β + v. Furhermore, β s paroned such ha β = [ β, β ], he egenvalues of W ~ are ω, he marx of he egenvecors of W ~ s Λ, and a parameer κ s defned such ha 2 κ 2 σ v =, (0) 2 σ Moreover, R s defned as an (N N) dagonal marx whose h dagonal elemen s gven by 2 Tκ + ( λω ) 2. Wh hese assumpons, he model s concenraed log-lkelhood funcon s gven by where NT ln L = 2 T [ ln( NT ) ln( c )] + = c 2 N 2 2 [ T ln( λω ) + () ln( + Tκ ( λω ) )] π () = 2 c = ( λw ~ ) Y Y N N = y ι X RΛ Y N X N N = N = y ι X N x ι β + N = x ι β (2) Here, Y s he (N ) vecor of observed y values a me, Y s he (N ) vecor of me averages across Y, X s he (N (K )) marx of exogenous varables mnus he consan erm, X s he (N (K )) marx of me averages across X, and ι s an (N ) vecor of 2 ones. Equaon s called he concenraed log-lkelhood funcon because β and σ have been facored ou of he equaon; hey can be recovered by and where β = (3) N N y β = = x N N T σ = = e e (4) 2 T

8 e ( λw ~ )[ Y Y ( X X ) β ] + RΛ[ Y β ι X ] = β (5) The e erm n (4) and (5) s he vecor of resduals or error esmaes ha correspond o (7) sξ erm. To esmae he parameers λ, κ 2, and β -, a wo-sep erave procedure s used (Elhors 2003). Frs, values for λ and κ 2 are chosen, hen β - s esmaed usng an ordnary leas squares * * roune of X on Y, where boh are sacked wh elemens gven by Y N N ( λw ~ ) Y Y y + RΛ Y y ι = (6) = = N N and X N N ( λw ~ ) X X x + RΛ X x = ι (7) = = N N * Nex, gven β -, λ and κ 2 are esmaed usng an MLE roune. The enre roune s eraed unl suable convergence s acheved. Panel Daa Spaal Logsc Regresson Model Because mus le whn he [0,] nerval, fraconal land-cover daa should no be modeled usng he spaal lnear regresson model descrbed above. However, a modfcaon of ha model can be appled ha allows for fraconal response n a farly sraghforward manner. The echnque used o model he proporon land-cover daa s a new echnque, represenng an exenson of he logsc regresson mehod (see, e.g., Greene (2000)). Ths mehod models bnary daa, so s appled here when modelng one land use ype versus anoher (for example developed vs. undeveloped ). Because of space consderaons, many of he mehodologcal deals are no ncluded here; hose neresed are referred o Frazer (2004). The echnque begns by usng he nverse of he logsc cumulave dsrbuon funcon (CDF): F ( P ) = ln P P (8) o ransform he proporons daa, P, o he (, ) nerval. Ths varable, wh ceran assumpons concernng he random effecs erm and approprae correcons for heeroscedascy (see Frazer (2004) for deals), can be modeled usng he panel daa spaal regresson echnque descrbed prevously. Usng earler defnons for X, v, and ξ, he fnal model form s: QF ( P) QXβ + v + ( λw) Qξ = (9) where Q s a varance-normalzng dagonal marx defned as: ( x β )( F( x β )) Q =, F (20)

9 wh F( ) beng he logsc CDF: e x β ( x β ) = x β F + e (2) Ths echnque works only for bnary proporons daa. Tha s, bnary dsncons such as developed versus undeveloped can be modeled, bu resdenal versus commercal versus undeveloped canno. In order o dsngush more han wo caegores of land-cover, hs echnque may be performed eravely. For example, f he resdenal proporon of cell n me s P Res, and he developed proporon s P Dev, hen he quany Res Dev P P = (22) P Res Dev can be modeled usng he mehods descrbed above. However, because Res Dev Dev ( ) Dev esmaes of ( ) P P, P from he developed versus undeveloped model resuls should be used o nsrumen he sub-model (.e., hs nverse probably should ac as an explanaory varable), snce leavng hese ou poenally would deprve he model of mporan nformaon. Dong so requres furher assumpons concernng he random effecs erm and a more complcaed verson of he varance-normalzng marx (equaon 20), bu he mehod s essenally he same as descrbed above (see Frazer (2004) for furher deals). Tme Adjusmen Because me dfferences beween successve saelle mages s no consan (one gap s hree years, one s sx years, and one s egh years), smply usng me-lagged varables whou accounng for hs dfference may lead o naccuraces and/or msleadng resuls. In order o accoun for hs, a me adjusmen facor s nroduced for he coeffcens of all me-lagged varables. If τ s he me dfference beween panel and he prevous panel, hen an esmaed parameer from he models represenng explanaory varable k and me perod s ransformed accordng o:, β k, ( k (35) β ) k a τ where a k s he me adjusmen facor. For varables ha are no me lagged, a k s equal o one; for me-lagged varables, a k s esmaed. To smplfy esmaon, a k s assumed o be he consan across all me-lagged varables n each model. MODEL RESULTS In hs secon he resuls are presened for applcaons of he spaal panel daa regresson model as appled o populaon and vehcles per household varables; as well as for land cover (developed, resdenal developed, and agrculural undeveloped) as modeled by

10 spaal logsc models for panel daa. Because of daa se sze, samplng had o be employed before model calbraon; hs echnque s dscussed as well. Lnear Regresson Model for Spaal Panel Daa Two dependen varables are modeled usng he spaal panel daa lnear regresson model; hey are populaon and he number of vehcles per household. Though no repored here due o space resrcons, models whou lagged varables or me adjusmen also have been esmaed, and he resuls sugges ha he models perform smlarly, generally wh only small changes for he effec of me lags and adjusmen (Frazer 2004). Due o compuaonal demands n fndng egenvecors and egenvalues of a spaal wegh marx nvolvng all observaons avalable (30,000 grd cells ranslaes o sze 30,000 30,000 marces), cell samplng s used o reduce he burden. All resuls repored are he means from 25 models run on 25 random samples of,000 observaons each. Wh he excepon of he parameers relang o random effecs and spaal auocorrelaon, he means are conssen esmaors of he populaon parameers (Greene 2000). The means of he sandard errors and - sascs are no conssen esmaes of hese secondary parameers, bu hey do provde an dea of sascal sgnfcance. A a 95% confdence level, some of he parameer esmaes for some of he samples do no dffer (n a sascal sense) from zero; however, hese are sll ncluded n he fnal models (see Table ) because n some of he samples hey were sascally sgnfcan (.e., -sasc >.64) and because he only rsk of leavng hese varables n he model s possble model over-specfcaon or over-nerpreaon. Also repored are elasces for he varables for he hree years modeled (he fnal year, 983, was dropped o perm use of melagged varables). The random-effecs and spaal auocorrelaon parameers are specfc o each random sample of 000 observaons, and hs mus be aken no accoun before usng he resuls repored below for predcons or smulaons., The reason for hs s ha he effecs only accoun for he error erms from a random sample of observaons, and no from he enre daa se. The populaon model uses he naural log of populaon as he response varable, n order o ensure non-negavy of predcons and o recognze he fac ha populaon may have an exponenal relaonshp wh some or all of he ndependen varables (as wh me, for example). The resuls of he spaal regresson model wh lagged ndependen varables and me adjusmen are presened n Table. The dsance measures are no me lagged because, a leas n he scale of hs work, hey are me nvaran. A square-roo of he dsance measure s used as an explanaory varable, snce s expeced ha here should be some added dampenng of s effec. (For example, he effec on cell populaon of movng one klomeer away from he CBD s expeced o be much more pronounced he closer ha cell s o he CBD; nuvely, hs s because he effec of a change n dsance (o he CBD or neares hghway) maers a a relave, as opposed o an absolue, level (Frazer and Kockelman 2003).) As expeced, populaon s predced o fall wh dsance o he CBD and rse wh enropy and mx sascs, and wh resdenal and commercal land coverage. Ineresngly, also s predced o rse slghly wh agrculural land coverage and wh dsance o he neares hghway (perhaps due o hghway exernales, parcular afer havng conrolled for a dsanceo-cbd varable, whch may accoun for many nework nensy effecs). From he repored elasces, s evden ha he dsance o he CBD s he varable wh he greaes mpac on

11 he model, followed by he dsance o he neares hghway. Ths ndcaes ha he locaon of a cells, as opposed o s land cover levels, s he mos mporan facor deermnng s populaon. More mporanly, s seen ha he parameers measurng he spaal auocorrelaon (λ), random effecs (κ), and he me adjusmen of lagged varables are all hghly sascally sgnfcan, as s he me adjusmen facor (esmaed o be 0.943). As expeced, he effec of spaal auocorrelaon s posve, whch ndcaes ha neghborng cells end o have smlar populaons. Table 2 presens he resuls from he vehcle ownershp model (for average vehcles per household per zone). Ownershp s esmaed o ncrease wh dsance o he CBD, dsance o he neares hghway, and land cover mx. I s esmaed o fall raher quckly as he fracon of land n commercal use ncreases, as one mgh expec (snce households may be smaller n more commercally developed locaons and rely less on vehcles for access o commercal servces and employmen). I also falls slghly wh resdenal and agrculural land coverages. Agan, he parameers represenng he effecs of spaal auocorrelaon and random effecs are hghly sascally sgnfcan. And, as wh he populaon model, he me adjusmen facor s esmaed o be less han one, mplyng ha he magnude of he effecs of pas land cover on he presen level of vehcle ownershp decrease wh me (see Frazer (2004) for a more dealed dscusson of ssues concernng and nerpreaons of he me adjusmen facor). Logsc Regresson Model for Spaal Panel Daa Three models of land cover proporons, based on wo bnary-spl levels (one condoned on he oher, for a oal of four land cover classfcaons), were run usng a logsc model for spaal panel daa. The frs spl s for he fracon of land ha s developed; he second condons on ha nformaon for he proporons ha are resdenal/non-resdenal (gven he proporon ha s developed) and agrculural/non-agrculural (gven he proporon ha s undeveloped). As n he esmaon of he prevously dscussed models, separae models were run for each of 25 randomly seleced ses of 000 cell observaons, and her esmaes averaged. The same caveas dscussed prevously, concernng he prmary parameer esmaes, sandard devaons, and -sascs, hold here as well. Tables 3 hrough 5 presen he average resuls from he hree models. I should be noed ha for he proporon of land ha s resdenal gven he proporon ha s developed (hereafer called he resdenal model), hree of he samples were hrown ou because he maxmum lkelhood procedure s Hessan calculaon faled (and exra-long esmaon, of 5 o 2 hours per sample) prevened re-esmaon of he models for hese samples). All of he models random effecs, spaal auocorrelaon, and me adjusmen parameers are sascally sgnfcan. Ineresngly, he average level of spaal correlaon s nearly dencal for all hree models, ndcang ha here may be smlar unobserved spaal nformaon across all models ha s no beng accouned for by he explanaory varables. As expeced, he fracon of land ha s developed s predced o fall wh dsance o he CBD, along wh ha ha s resdenal n naure. Developed land s predced o rse wh dsance o he neares hghway, however, probably o couner he effecs of he ncreasng dsance-o-cbd erm. Resdenal land falls wh hs dsance. For he agrculural model componen of hs wo-ered model sysem, agrculural land wll end o le farher from he CBD, bu closer o hghways, han non-agrculural, undeveloped land. I should be noed ha he repored levels for he land cover mx and enropy varables for hs model are heavly skewed by one of he 25 sample model resuls. Removng ha sample s

12 resuls from he averages causes he mx and enropy parameers o no only be smaller n magnude, bu also o change sgn, ndcang ha he sample may have nroduced sgnfcan esmaor bas. However, here s no mahemacal or sascal reason o drop he sample from he averages, so was lef n (see Frazer (2004) for more nformaon). SIMULATIONS FOR PREDICTION To es he praccal performance of he esmaed models, smulaons were run o develop predcons for populaon and developed land cover for Ausn s downown n he year A 5 km 5 km (or 50 cells 50 cells) secon of he CBD was seleced for applcaon, raher han he enre regon, n order o economze on calculaon mes. (The resuls presened here ook abou day each, for populaon and land cover predcons, due o he necessy of nverng a large number of,000,000 cell marces.) Though random effecs were used o esmae he me-consan pars of he models, a mehod more akn o fxed effecs s used o generae predcons for he populaon and developed land cover. The random effec mehod esmaes he parameers for a normal dsrbuon whch bes fs he me-consan pars of he dependen varables n he model. One naural mehod for usng hs nformaon n a predcon would be o ake a random draw from he esmaed dsrbuon for each cell o esmae he me-consan effec. However, hs dsregards mporan nformaon conaned n he daa used o esmae he models. Tha s, s possble o exrac he exac me-consan effec for every cell from whch he random-effec dsrbuon s esmaed. The esmae for he me-consan effec, denoed as v *, s deermned by usng he esmaes for he β coeffcens vecor, denoed as β * () (whch, for smplcy, s assumed o nclude he me-adjusmen facor from equaon 35), n he followng equaon: v * * * = y β x β ( ) (36) Here he overbar ndcaes an average over me. To generae predcons, he above approxmaon of he random effec, along wh he 2000 daa, he coeffcen esmaes usng he correc me-adjusmen facor (o predc ahead he desred number of years), and an approxmaon for he random error, θ * (dscussed below) are used ogeher: y = (37) Pred * * * * ( ) β + x,2000β ( ) + v + θ The process for smulang he spaal auocorrelaon requred mananng he samplng sraegy used n he model esmaons. To begn wh, a sngle random erm for each cell n he enre daa regon was drawn from he dsrbuon esmaed n he models o generae he error erms, ξ, from equaon 7. Then, 999 cells were randomly sampled from he enre daa regon for every one of he 2,500 downown cells, and her sampled error erms, ξ, used o accoun for spaal auocorrelaon. Because he calculaon of he spaal auocorrelaon for every cell requred he nverson of a marx, he process of generang he predcons ook a large amoun of compung me, whch s why he predcon regon was lmed o a cell regon. The resuls of he populaon predcon, along wh 2000 Census daa for reference, are presened n Fgure 2. Because of he logarhmc form of he model n whch small errors can grow exponenally, some of he predcons are unrealscally large (over 3000 persons for a

13 300 m x 300 m grd cell). Such predcons (here were 5 of hese) were removed and replaced wh averages of neghborng cell predcons for purposes of plong he populaon smulaon resuls. Que clearly, he presen dsrbuon of downown populaon n hs.5 km x.5 km neghborhood s preserved n he 20-year predcons. However, he populaon s, n general, expeced o decrease over he regon (oal populaon for he regon dropped from 309,36 n 2000 o 239,892 n 2020). Ths ndcaes ha hough he model self s able o accoun well for he populaon dsrbuon n he regon, he me dynamcs of populaon change are no beng correcly accouned for, a leas wh respec o predcons. Furher analyss s requred o deermne why hs non-nuve resul occurred, bu mos lkely s due o a msundersandng of how he me adjusmen facor acually affecs he model. In fuure research, dfferen ways o accoun for he me dfferences n he lagged varables wll be examned. Fgure 2. Populaon plo for downown Ausn: acual daa from 2000 (lef) versus average predcon for 2020 (rgh). Noe: Area of plo s 5 km x 5 km. Each ploed pon covers 300 m x 300 m. Darker areas represen hgher populaon levels. The resuls of he developed land cover predcons, along wh 2000 reference daa, are presened n Fgure 3. As wh he populaon predcons, he dsrbuon of he proporons of developed land cover are well mananed, bu he effecs of me seem o be ncorrecly accouned for. Tha s, he proporon of developed land cover s expeced o decrease from 2000 o 2020 (average developed proporon across he regon droppng from n 2000 o n 2020). Agan, hs ndcaes ha he way ha me s accouned for n he model s flawed n some respec, and furher analyss s requred o fnd ou exacly why hs s happenng.

14 Fgure 3 Proporon of developed land cover n he downown Ausn area: acual daa from 2000 (lef) versus average predcon for 2020 (rgh). Noe: Area of plo s 5 km x 5 km. Each ploed pon covers 300 m x 300 m; darker areas represen hgher proporons of developed land cover. From he smulaon resuls s obvous ha usng he models presened n hs work do no perform well n a predcve capably. Though hey capure he spaal dsrbuon of he varables well, hey do no accoun for he expeced growh over me of populaon and urban developmen. As menoned before, s unclear why hese resuls emerge as hey do, bu s mos lkely due o nadequaces assocaed wh he me adjusmen facor. One possble cause s he fac ha a sngle me adjusmen facor was used, as opposed o havng a separae one for each lagged varable. Oher ssues also may have affeced he predcons. An exenson of hs work mgh furher nvesgae he mehodology used o ncorporae spaal auocorrelaon n he predcons. Anoher ssue s he fac ceran, poenally mporan nformaon was lef ou of he model; for example, populaon levels n he proporon of developed land cover model. Anoher would be accounng for he possbly ha dfferen model forms mgh exs for areas wh dfferen characerscs; e.g., dsnc populaon models mgh exs for areas of hgh and low levels of developmen (see Frazer (2004) for examples of such models usng sample selecon mehods). Despe all of hese ssues, wha he smulaons do show s ha he models, despe her flaws n he emporal dmenson, perform very well n capurng he spaal dversy and dsrbuon of varables across he regon. Obvously fuure work on he models s requred before hey can acually be appled n a praccal seng, bu hese resuls provde a promsng sar owards ha end. CONCLUSIONS Ths paper presens a varey of nnovave models for land cover and oher daa mporan for ransporaon engneers, geographers and planners. The work rgorously recognzes boh space and me effecs by ncorporang spaal auocorrelaon, emporal random effecs, and adjusmens for dfferences n me lags no lnear regresson and logsc regresson model forms. Usng boh Census daa and land cover daa derved from saelle magery, models for populaon, average vehcles per household, and developed, resdenal, and agrculural land cover are developed. Because of compuaonal dffcules, a seres of samples

15 were used for esmaon. No only were he resuls of he models nformave, bu he spaal and emporal effecs were shown o be hghly sascally sgnfcan, suggesng ha her recognon and formal ncluson n he models s lkely o be of grea value. Posve spaal auocorrelaon shows ha, for example, areas of smlar populaon or land cover proporons have a endency o cluser. Also, he adjusmen facor for he dfferences n me lags, hough sascally sgnfcan, ndcaes ha he effecs of hese dfferences are no ha large (a leas no n he me scale of he daa). In he esmaed models, Census daa s no used as explanaory nformaon. The movaon behnd hs was ha he poenal error nroduced by he approxmaon for non- Census years could cloud evaluaon of model performance. Furhermore, a srucural equaons framework negrang he models s also no explored. Boh of hese ssues would serve as neresng nvesgaons for fuure research. Applyng he model resuls n a praccal applcaon (smulang populaon and developed land cover levels n 2020) exposes boh srenghs of he models and some poenal problems. Specfcally, he local spaal dversy of he regon s accouned for farly well n he predcons, however he effecs of me on he regon s developmen are no nuvely capured. Nowhsandng he ssues rased by predcon resuls, he models ably o explore neresng aspecs of he daa and rgorously accommodae panel daa and spaal neracons s of subsanal value. They provde mporan nformaon abou relaonshps among demographc and geographc varables a boh general and regonal levels. Ths nformaon can be of grea use for ransporaon researchers and planners; leads o an mproved undersandng of he nerrelaons whch affec he developmen of urban regons whch, n urn, can lead o more nformed and mproved polcy decsons. Moreover, he sascal mehodologes used n hs work for spaal panel daa analyss are largely new; hey can be vewed as seppng sones owards models ha more fully accoun for spaal and emporal heerogenees and effecs n ransporaon daa. Though hey sugges a need for fuure research (o more fully explore he power and praccaly of hese mehods), he resuls are very promsng. ACKNOWLEDGEMENTS The auhors are graeful for he suggesons of Dr. Darla Monroe and several anonymous revewers, as well as for he fnancal suppor of he secondary auhor s NSF CAREER Award. Ths maeral s based upon work suppored by he Naonal Scence Foundaon under Gran No

16 REFERENCES Anseln, Luc Spaal Economercs: Mehods and Models. Dordrech: Kluwer Academc Press. Candau, Jeanee Therese Temporal Calbraon Sensvy of he SLEUTH Urban Growh Model. Masers Thess. Unversy of Calforna, Sana Barbara. Cervero, Rober, and Kara Kockelman Travel Demand and he Three Ds: Densy, Dversy, and Desgn. Transporaon Research D 2(3): Clarke, Keh C., and Leonard Gaydos Loose-Couplng a Cellular Auomaon Model and GIS: Long-Term Urban Growh Predcon for San Francsco and Washngon/Balmore. Inernaonal Journal of Geographcal and Informaon Scence 2(7): Dubn, Robn A Spaal Auo Correlaon and Neghborhood Qualy. Regonal Scence and Urban Economcs 22: Elhors, J. Paul Panel Daa Models Exended o Spaal Error Auocorrelaon or a Spaally Lagged Dependen Varable. Unversy of Gronngen, Research Insue SOM Research Paper 0C05. Accessed March, 2004: hp:// Elhors, J. Paul Specfcaon and Esmaon of Spaal Panel Daa Models. Inernaonal Regonal Scence Revew 26: Frazer, Chrsopher Spaal Economerc Models for Land Use/Land Cover Daa: Theory and Applcaon usng Saelle Images for he Ausn, Texas Regon. Masers Thess. Deparmen of Cvl Engneerng. The Unversy of Texas a Ausn. Frazer, Chrs, and Kara Kockelman Ces and Saelle Imagery: Models for Regonal Change. Presened a he 2004 INFORMS conference n Alana Georga. Accessed March 3, 2004: hp:// Greene, Wllam Economerc Analyss. Upper Saddle Rver: Prence-Hall. Kloserman, R. E Wha f?: Collaborave Plannng Suppor Sysem. Envronmen and Plannng B 26: Kockelman, Kara M Travel Behavor as a Funcon of Accessbly, Land Use Mxng, and Land Balance: Evdence from he San Francsco Bay Area. Transporaon Research Record 607: Lands, J. and M. Zhang The Second Generaon of he Calforna Urban Fuures Model: Par : Model Logc and Theory. Envronmen and Plannng B 30:

17 Messner, Seve, and Luc Anseln Spaal Analyses of Homcde wh Areal Daa. Workng paper. Accessed July 2, 2004: hp://agec22.agecon.uuc.edu/users/anseln/papers/smla.pdf Parker, Dawn C., Seven M. Manson, Marco A. Janssen, Mahew J. Hoffmann, and Peer Deadman Mul-Agen Sysems for he Smulaon of Land-Use and Land-Cover Change: A Revew. Annals of he Assocaon of Amercan Geographers, 93(2): Parker, Dawn C., Thomas Berger, and Seven M. Manson, eds Agen Based Models of Land-Use and Land-Cover Change: Proceedngs of an Inernaonal Workshop, Ocober 4-7, 200. CIPEC Collaborave Repor CCR-3. Accessed July 0, 2004: hp:// Rchards, John A., and Xupng Ja Remoe Sensng Dgal Image Analyss. Berln: Sprnger-Verlag. Smh, Sanley K., and Terry Sncch Forecasng Sae and Household Populaons: Evaluang he Forecas Accuracy and Bas of Alernave Populaon Projecons for Saes. Inernaonal Journal of Forecasng 8: Waddell, Paul UrbanSm: Modelng Urban Developmen for Land use, Transporaon, and Envronmenal Plannng. The Journal of he Amercan Plannng Assocaon 68(3):

18 Esmaon Sample Properes Varable Bea S.E. T-sasc Sandard Max Mn Error 2000 Elasces Consan Square roo of Dsance o CBD Square roo of Dsance o Neares Hghway Proporon of Commercal Land Cover* Proporon of Resdenal Land Cover* ln(proporon of Rural Land Cover)* ln(land Cover Mx)* Land Cover Enropy* κ λ Tme Adjusmen Error Varance Random Effec Sandard Devaon.439 R-Squared Number of Vald Samples 25 Table. Resuls for spaal lnear regresson model of Y = ln(populaon).

19 Esmaon Sample Properes Elasces Varable Bea S.E. T-sasc Sandard Max Mn Error Consan Square roo of Dsance o CBD Square roo of Dsance o Neares Hghway ln(proporon of Commercal Land Cover)* Proporon of Resdenal Land Cover* Proporon of Rural Land Cover* E E E-04 Land Cover Mx* Land Cover Enropy* E κ λ Tme Adjusmen Error Varance Random Effec Sandard Devaon R-Squared Number of Vald Samples 25 Table 2. Resuls for spaal lnear regresson model of Y = average number of vehcles per household.

20 Esmaon Sample Properes Elasces Varable Bea S.E. T-sasc Sandard Max Mn Error Consan Square roo of Dsance o CBD Square roo of Dsance o Neares Hghway Land Cover Mx* Land Cover Enropy* κ λ Tme Adjusmen Error Varance.555 Random Effec Sandard Devaon R-Squared Number of Vald Samples 25 Table 3. Resuls from panel daa spaal logsc regresson model run on land cover proporon varables: Proporon of developed land cover.

21 Esmaon Sample Properes Elasces Varable Bea S.E. T-sasc Sandard Max Mn Error Insrumen Varable Consan Square roo of Dsance o CBD Square roo of Dsance o Neares Hghway Land Cover Mx* Land Cover Enropy* κ λ Tme Adjusmen Error Varance.488 Random Effec Sandard Devaon 0.53 R-Squared Number of Vald Samples 22 Table 4 Resuls from panel daa spaal logsc regresson model run on land cover proporon varables: Proporon of developed land cover ha s resdenal.

22 Esmaon Sample Properes Elasces Varable Bea S.E. T-sasc Sandard Max Mn Error Insrumen Varable Consan Square roo of Dsance o CBD Square roo of Dsance o Neares Hghway E E-04 Land Cover Mx* Land Cover Enropy* κ λ Tme Adjusmen Error Varance.253 Random Effec Sandard Devaon 0.28 R-Squared Number of Vald Samples 25 Table 5. Resuls from panel daa spaal logsc regresson model run on land cover proporon varables: Proporon of undeveloped land cover ha s rural.