and postulate that dependencies among the regression residuals (errors), u

Transcription

1 Part III. Areal Data Aalyss 6. Spatal Regresso Models for Areal Data Aalyss The prmary models of terest for areal data aalyss are regresso models. I the same way that geo-regresso models were used to study relatos amog cotuous-data attrbutes of selected pot locatos (such as the Calfora rafall example), the preset spatal regresso models are desged to study relatos amog attrbutes of areal uts (such as the Eglsh Mortalty example Secto 1.3 above). The ey dfferece s of course the uderlyg spatal structure of ths data. I the case of geo-regresso, the fudametal spatal assumpto was terms covarace statoarty, whch together wth mult-ormalty, eabled the full dstrbuto of spatal resduals to be modeled by mea of varograms ad ther assocated covarograms. I the preset case, ths statoarty assumpto s replaced by spatal autogressve hypotheses that are based o specfc choces of spatal weghts matrces, as developed Secto 5. Here we start wth the most fudametal spatal autogressve hypothess terms of regresso resduals themselves. 6.1 The Spatal Errors Model (SEM) The most drect aalogue to geo-regresso s the spatal regresso already developed Secto 3 above. I partcular, f we start wth the regresso model (3.1) above,.e., (6.1.1) Y 0, 1,.., 1 x u ad postulate that depedeces amog the regresso resduals (errors), u, at each areal ut are govered by the spatal autoregressve model (3.5) ad (3.6),.e., by (6.1.) u w u N, ~ (0, ), 1,.., for some choce of spatal weghts matrx, W ( w :, 1,.., ) [wth dag( W ) 0] the the resultg model summarzed matrx form by (3.) ad (3.9) as: (6.1.3) Y X u, u Wu N I, ~ (0, ) s ow desgated as the Spatal Errors Model (also deoted as the SE-model or smply SEM). 1 As metoed above, ths costtutes the most drect applcato of the spatal autogressve model Secto 3. I essece t s hypotheszed here that all spatal depedeces are amog the uobserved errors the model (ad hece the ame, SEM). I the case of the Eglsh Mortalty data for example, t s clear that whle the Jarma dex cludes may soco-ecoomc ad demographc factors fluecg rates of myocardal farctos, there are surely other factors volved. Moreover, sce may of 1 See footote 3 below for further dscusso of ths termology. ESE 50 III.6-1 Toy E. Smth

2 Part III. Areal Data Aalyss these excluded factors wll exhbt spatal depedeces, such depedeces wll ecessarly be reflected by the correspodg resdual errors, u, (6.1.3). Before cosderg other types of autoregressve depedeces, t s of terest to reformulate ths model as a stace of the Geeral Lear Regresso Model. Frst, f for otatoal coveece, we ow let (6.1.4) B I W the by expresso (3..5) above, we may solve for u terms of as follows: (6.1.5) u I W B N I 1 1 ( ), ~ (0, ) Thus by the Ivarace Theorem for mult-ormal dstrbutos, t follows at oce from the mult-ormalty of that u s also mult-ormal wth covarace gve by (6.1.6) cov( u) cov( B ) B cov( )( B ) B ( I )( B ) B ( B) ( BB ) V where the spatal covarace structure, V, s gve by 3 (6.1.7) V ( B B ) 1 Ths tur mples that (6.1.3) ca be rewrtte as (6.1.8) Y X u u N V, ~ (0, ) whch s see to be a stace of the Geeral Lear Regresso Model expresso (7.1.8) of Part II, where ths case the matrx C s replaced by V (6.1.7). Ths wll allow us to apply some of the GLS methods Secto of Part II to SE-models. Fally, there s a thrd equvalet way of wrtg ths SE-model whch s also useful for aalyss. If we smply substtute (6.1.5) drectly to (6.1.3) ad elmate u altogether, the ths same model ca be wrtte as (6.1.9) 1 Y X B N I, ~ (0, ) Sce all smultaeous relatos, u Wu, have bee elmated, expresso (6.1.9) s usually called the reduced form of (6.3) Here we have used the matrx dettes, ( A) ( A ), ad, A B ( BA), whch are establshed, respectvely, expressos (A3.1.0) ad (A3.1.18) of the Appedx. 3 Ths termology s motvated by the fact that all spatal aspects of covarace (6.1.6) are defed by V. ESE 50 III.6- Toy E. Smth

3 Part III. Areal Data Aalyss 6. The Spatal Lag Model (SLM) A alteratve lear model based o the spatal autoregressve model s obtaed by assumg that these autoregressve relatos are amog the depedet varables themselves. If we aga assume that the uderlyg spatal relatos amog areal uts are represetable by a spatal weghts matrx, W ( w :, 1,.., ) [wth dag( W ) 0 ], the the smplest way to wrte such a model terms of W s by modfyg expresso (6.1.1) as follows, (6..1) Y 0 w Y h h x, 1,.., 1 h where aga ~ N(0, ), 1,..,. Here the autoregressve term, h1wy h h, reflects possble depedeces of Y o values, Y h, other areal uts. A stadard example of (6..1) s terms of housg prces. If the relevat areal uts are say cty blocs wth a metropolta area, ad f Y s terpreted as the average prce (per square foot) of housg o bloc, the addto to other housg attrbutes ( x : 1,.., ), of bloc, such prces may well be flueced by prces surroudg blocs. So the relevat autoregressve relatos here are amog the housg prces, Y, ad ot the spatal resduals,. Such relatos are typcally called spatal lag relatos, whch motvates the ame spatal lag model (SLM) Smultaety Structure Before aalyzg ths model detal, t s mportat to emphasze oe fudametal dfferece betwee (6.1.1) ad (6..1). Sce the resduals are here assumed to be depedet, 5 oe mght at frst glace coclude that (6..1) s othg more tha a OLS model wth a added term, ( h1wy h h), where the uow spatal depedecy parameter,, s smply the relevat beta coeffcet. But the ey pots to otce are that () the Y h values are radom varables, ad moreover that () they appear o both sdes of the equato system (6..1),.e., that Y wll also appear equatos for Y h, wheever wh 0. Thus, the same way that opos ( u1,.., u ) amog households Fgure 3.1 volved smultaetes, the housg prces ( Y1,.., Y ) the preset llustrato also volve smultaetes. So ths s ot smply aother term a OLS model. 4 At ths pot, t should be emphaszed that (much le varograms versus semvarograms the Krgg models of Part II), there s o geeral agreemet regardg the ames of varous spatal regresso models. For example, whle we have reserved the term Spatal Autogressve Model (SAR) for the basc resdual process expresso (3.9) above, ths term s used by LeSage ad Pace (009) for the spatal lag model (SLM). Our preset termology follows that of the ope-source software, GEODA, (to be dscussed later) ad has the advatage of clarfyg where the basc spatal autoregressve model s beg appled,.e., to the error terms SEM ad to the depedet varable SLM. 5 Relaxatos of ths assumpto wll be cosdered the combed model of Secto below. ESE 50 III.6-3 Toy E. Smth

4 Part III. Areal Data Aalyss Ths ca be see more clearly by formalzg ths model matrx terms ad solvg for ts reduced form. By employg the same otato as (6.1.3), the Spatal Lag Model (SL-model or smply SLM) ca be wrtte as (6..) Y WY X N I, ~ (0, ) As a parallel to (6..1), we ca rewrte ths model by groupg Y terms (6..) as follows: (6..3) Y WY X ( I W) Y X BY X Y B X B 1 1 whch the yelds the correspodg reduced form of the SL-model: (6..4) 1 1 Y B X B N I, ~ (0, ) I ths reduced form, t should ow be clear that the spatal lag term, WY, (6..) s ot smply aother regresso term. Fally, oe ca also vew ths model as a stace of the Geeral Lear Regresso Model, though the correspodece s ot as smple as that of SEM. I partcular, f we ow treat the spatal depedecy parameter,, as a ow quatty, or more properly, f we codto (6..4) o a gve value of, the [ a maer smlar to the Cholesy trasformato expresso (7.1.16) of Part II] we ca treat (6..5) X 1 B X as a trasformed data set, ad aga use (6.1.5) through (6.1.7) to wrte (6..4) as (6..6) Y X u u N V, ~ (0, ) wth spatal covarace structure, V, aga gve by (6.1.7). The ey dfferece here s that s o loger smply a uow parameter the covarace matrx, V, but ow also appears X. So whle (6..6) does permt the GLS methods Secto ESE 50 III.6-4 Toy E. Smth

5 Part III. Areal Data Aalyss Part II to also be appled to SL-models, these applcatos are somewhat more restrctve tha for SE-models. 6.. Iterpretato of Beta Coeffcets Oe fal dfferece betwee SE-models ad SL-models that eeds to be emphaszed s the terpretato of the stadard beta coeffcets,, (6.1.8) versus (6..4) [or equvaletly, (6.1.9) versus (6..6)]. Recall that oe of the appealg features of OLS s the smple terpretato of beta coeffcets. For example, cosder a OLS verso of the housg prce example above, amely (6..7) Y 0, 1,.., 1 x wth ~ N(0, ), 1,..,. If say x 1 deotes the average age of housg o bloc (as a surrogate for structural qualty), the oe would expect that 1 s egatve. I partcular sce, (6..8) EY ( x1,.., x) 0, 1,.., 1 x the value of 1 should dcate the expected decrease mea housg prces o bloc resultg from a oe-year crease the average age of houses o bloc. More geerally, these margal chages ca be expressed as partal dervatves of the form: (6..9) x EY ( x1,.., x,.., x ), 1,..,, 1,.., ad are see to be precsely the correspodg coeffcet for varable x. Of course ths OLS model gores spatal depedeces betwee blocs. So f (6..7) s reformulated as a SE-model to accout for such depedeces, say of the form (6.1.8): (6..10) Y 0 x u, ( u1,.., u) ~ N(0, V ) 1 the sce Eu ( ) 0, 1,..,, t follows that (6..8) ad (6..9) cotue to hold. Thus, whle certa types of spatal depedeces have bee accouted for, the terpretato of betas (such as 1 above) cotues to hold. However, f the maor spatal depedeces are amog these prce levels themselves, so that a SL-model s more approprate, the the stuato s far more complex. Ths ca be see by observg from the reduced form (6..4), together wth the rpple 1 decomposto of ( I W) expresso (3.3.6) above that 6 6 Here t s mplctly assumed that the covergece codto, 1/, holds for ad W. W ESE 50 III.6-5 Toy E. Smth

6 Part III. Areal Data Aalyss (6..11) 1 1 E( Y X) B X ( I W) X ( I W W ) X X WX W X So the partal dervatve (6..9) caot eve be defed wthout specfyg all attrbutes o all blocs. Moreover, whle (6..8) mples that there are o teracto effects betwee blocs,.e., that the partal dervatves of E( Y x1,.., x,.., x ) wth respect to housg attrbutes o ay other bloc are detcally zero, ths s o loger true (6..11). For example, f the age of housg o bloc s creased, the ths ot oly has a drect effect o expected mea prces bloc, but also has drect effects o prces all other blocs. Moreover, such drect effects tur affect prces. So ths spatal rpple effect leads to complex terdepedeces that must be tae to accout whe terpretg each beta coeffcet. These effects ca be summarzed by aalyzg (6..11) more detal. To do so, we ow employ the followg otato. For ay m matrx, A ( a : 1,..,, 1,.., m), let A(, ) a deote the ( ) th elemet of A, ad let A(, ) deote the th colum of A. I these terms, (6..11) ca be decomposed as follows: (6..1) ( ) (, ) [ (, )] 1 1 EY X B X B X B X 1 1 (, ) (, ) (, ) 1 X h B h 1 h 1 x h 1 hb h 1 x h 1 1 h B h (, ) so that each th row of E( Y X ) ca be wrtte as (6..13) 1 ( ) (, ) 1 1 h h EY X x B h I terms of ths decomposto, t ow follows that the desred partal dervatves ca be obtaed drectly. Frst, as a parallel to (6..9) we see that (6..14) 1 x EY ( X) B (, ) So ths margal effect depeds ot ust o 1 B, whch has the more explct form but also o the th dagoal elemet of (6..15) 1 B W W (,) 1 (,) (,) 1 W (, ) ESE 50 III.6-6 Toy E. Smth

7 Part III. Areal Data Aalyss where the last le follows from the zero-dagoal assumpto o W. But sce W (,) together wth all hgher order effects are postve, t s clear that the effect of each s beg flated by these spatal effects, as descrbed formally above. Moreover t s also clear from (6..13) that expected mea prces are affected by housg attrbute chages other blocs. I partcular, for attrbute bloc h, t ow follows that (6..16) 1 x EY ( X) B ( h, ) h Total effects o EY ( X ) of attrbutes the same areal ut are desgated as drect effects by LeSage ad Pace (009, Secto.7.1), ad smlarly, the total effects of attrbutes dfferet areal uts are desgated as drect effects. For further aalyss of these effects see LeSage ad Pace (009). 6.3 Other Spatal Regresso Models Whle there are may varatos o the SE-model ad SL-model above, we focus oly o those that are of partcular terest for our purposes The Combed Model Whe developg the SL-model above, a questo that aturally arses s why all uobserved factors should be treated as spatally depedet. Clearly t s possble to have spatal autoregressve depedeces both amog the Y varables ad the resduals,. If we ow dstgush betwee these by lettg M ad deote the spatal weghts matrx ad spatal depedecy parameter for the spatal-error compoet, the oe may combe these two models as follows, 7 (6.3.1) Y WY X u umu N I,, ~ (0, ) wth correspodg reduced form gve by (6.3.) 1 ( I W) Y X ( I M) Y ( I W) X ( I W) ( I M) However, our prmary terest ths model wll be to costruct comparatve tests of SEM versus SLM as staces of the same model structure. Hece we shall focus o the specal case wth M W, 7 Ths model has bee desgated by Kelea ad Prucha (010) as the SARAR(1,1) model, stadg for Spatal Autoregressve Model wth Autoregressve dsturbaces of order (1,1). ESE 50 III.6-7 Toy E. Smth

8 Part III. Areal Data Aalyss (6.3.3) Y WY X u u Wu N I,, ~ (0, ), whch we ow desgate as the combed model, wth correspodg reduced form: (6.3.4) Y ( I W) X ( I W) ( I W) So for ay gve spatal weghts matrx, W, the correspodg SE-model (SL-model) s see to be the specal case of (6.3.3) wth 0 ( 0 ). Oe addtoal pot worth otg here s that whle ths combed model s mathematcally well defed, ad ca prcple be used to obta ot estmates of both ad, these ot estmates are practce ofte very ustable. I partcular, sce both ad serve as depedecy parameters for the same matrx, W, they fact play very smlar rolls (6.3.4). But, as wll be see Secto 10.4 below, ths stablty wll tur out to have lttle effect o the usefuless of ths model for comparg SEM ad SLM The Spatal Durb Model A secod model that wll prove useful for our comparsos of SEM ad SLM ca aga be motvated by the housg prce example above. I partcular, f housg prces, Y, bloc group are flueced by housg prces eghborg bloc groups, the t s ot ureasoable to suppose that they may be flueced by other housg attrbutes these bloc groups. If so, the a atural exteso of the SL-model (6..1) would be to clude these spatal effects as addtoal terms,.e., (6.3.5) 0 h 1 h1 1 Y w Y x w x, 1,.., h h h h Followg Asel (1988) ths exteded model s desgated as the Spatal Durb Model (also SD-model or smply SDM). Ths SD-model ca be wrtte matrx form by lettg ( 1,.., ). However, oe mportat addtoal dfferece s that (as all prevous models) the matrx, X, s defed to clude the tercept term (6.3.5). So here t s coveet to troduce the more specfc otato, 0 (6.3.6) X [1, X v] ad v where both X v[ ( x1,.., x)] ad v ow refer explctly to the explaatory varables. Wth ths addtoal otato, (6.3.5) ca be wrtte matrx form as follows: 8 8 It s of terest to ote here that may ways t seems more atural to use X for the x varables, ad to employ separate otato for the tercept. But whle some authors have chose to do so, cludg LeSage ad Pace (009) [compare (6.3.7) above wth ther expresso (.34)], the lear-model otato (Y X ) s so stadard that t seems awward at ths pot to attempt to troduce ew covetos. ESE 50 III.6-8 Toy E. Smth

9 Part III. Areal Data Aalyss (6.3.7) Y WY X WX N I 1, ~ (0, ) 0 v v v As poted out by LeSage ad Pace (009, Sectos., 6.1) ths model s also useful for capturg omtted explaatory varables that may be correlated wth the x varables. I ths sese, t may serve to mae the SL-model somewhat more robust. However, as developed more fully Secto 10.3 below, our ma terest ths model s that t provdes a alteratve method for comparg SLM ad SEM The Codtoal Autoregressve (CAR) Model There s oe addtoal spatal regresso model that should be metoed vew of ts wde applcato the lterature. Whle ths model s coceptually smlar to the SEmodel, t volves a fudametally dfferet approach from a statstcal vewpot. I terms of our housg prce example, rather tha modelg the ot dstrbuto of all housg prces ( Y1,.., Y ) amog bloc groups, ths approach focuses o the codtoal dstrbutos of each housg prce, Y, gve all the others. The advatage of ths approach s that t avods all of the smultaety ssues that we have thus far ecoutered. I partcular, sce all uvarate codtoal dstrbutos dervable from a mult-ormal dstrbuto are themselves ormal, ths approach starts off by assumg oly that the codtoal dstrbuto of each prce, Y, gve ay values ( yh : h ) of all other prces ( Yh : h ), s ormally dstrbuted. So these dstrbutos are completely determed by ther codtoal meas ad varaces. To costruct these momets, we start by rewrtg the reduced SE-model (6.1.9) as follows: (6.3.8) ( ) 1 1 Y X B Y X B B Y X ( I W)( Y X) Y X W( Y X) Y X W( Y X) But f we ow deote the th row of W by w ( w1,.., w) relato ca be wrtte as,, the the th le of ths (6.3.9) Y x w ( Y X ) x w ( Y x ) h h h h where the last equalty follows from the assumpto that w 0. Ths suggests that f we f we ow codto Y o gve values ( yh : h ) of ( Yh : h ), the the atural codtoal model of Y to cosder t the followg: ESE 50 III.6-9 Toy E. Smth

10 Part III. Areal Data Aalyss (6.3.10) Y ( y : h ) x w ( y x ), 1,.., h h h h h where aga ~ N(0, ), 1,..,. I ths form, t s ow mmedate that (6.3.11) EY [ ( y : h )] x w ( y x ), 1,.., h h h h h Moreover, sce Y ( yh: h ) (6.3.11) s smply a costat plus t also follows that Y ( yh: h ) must be ormally dstrbbuted wth the same varace as,.e., (6.3.1) Y y h var[ ( h: )], 1,.., Such codtoal models are usually desgated as Codtoal Autoregressve (CAR) models. The advatages of such codtoal formulatos are most evdet Bayesa spatal models, where stadard Gbbs samplg procedures for parameter estmato requre oly the specfcato of all codtoal dstrbutos. However, such Bayesa models are beyod the scope of ths NOTEBOOK. [For a excellet dscusso of CAR models a Bayesa cotext, see Baeree, Carl ad Gelfad (004, Secto 3.3).] Thus our preset aalyss wll focus o the Spatal Errors Model (SEM) ad the Spatal Lag Model (SLM), whch are by far the most commoly used spatal regresso models. I the ext secto, we shall develop the basc methods for estmatg the parameters of these models. Ths wll be followed Secto 8 wth a developmet of the stadard regresso dagostcs for these models. ESE 50 III.6-10 Toy E. Smth