2. SPATIALLY LAGGED DEPENDENT VARIABLES

Transcription

1 2. SPATIALLY LAGGED DEPENDENT VARIABLES In hs chaper, we descrbe a sascal model ha ncorporaes spaal dependence explcly by addng a spaally lagged dependen varable y on he rgh-hand sde of he regresson equaon. Ths model goes by many dfferen names. Anseln (1988) calls hs he spaal auoregressve model, bu hs ermnology s poenally confusng snce he erm auoregressve s used o denoe que dfferen spaal models n he geosascal leraure. For smplcy, we wll here call he spaally lagged y model, snce s man feaure s he presence of a spaally lagged dependen varable among he covaraes. The spaally lagged y model s approprae when we beleve ha he values of y n one un are drecly nfluenced by he values of y found n s neghbors. Ths nfluence s above and beyond oher covaraes specfc o. If we beleve ha y s no nfluenced drecly by he value of y as such among neghbors bu raher ha here s some spaally clusered feaure ha nfluences he value of y for and s neghbors bu s omed from he specfcaon, we may consder an alernave model wh spaally correlaed errors, whch we dscuss subsequenly. For he spaally lagged y model o be approprae, he dependen varable y mus be consdered as a connuous varable. In hs book, we do no examne he generally more complcaed case of bnary dependen varables. These are more complcaed snce hey ofen do no have a closed-form soluon and mus be esmaed wh erave echnques ousde he range of hs book (see Ward & Gledsch, 2002). 2.1 Regresson Wh Spaally Lagged Dependen Varables To movae and llusrae he spaally lagged y model, we reurn o our example of he dsrbuon of democracy around he world. We have seen ha he dsrbuon of democracy dsplays spaal cluserng n he sense ha counres are more lkely o have hgher values on he POLITY democracy score f hey are surrounded by counres ha also have hgh levels of democracy. Alhough some of he cluserng n democracy obvously could be due o spaal cluserng n GDP per capa, whch n urn s posvely relaed o democracy, we have shown ha he spaal cluserng n he democracy daa does no compleely dsappear when we condon on a counry s level of GDP per capa. The assumpon ha he errors ε of a model reang democracy as a funcon of GDP per capa are ndependen 35

2 36 can easly be examned by esng for possble spaal dependence n he resduals from he regresson; ha s, ε ˆ ( ˆ = y -y), usng he Moran I correlaon coeffcen and our specfed paern of connecves n he marx C, where saes n hs case are defned as conneced f hey are whn a 200-km dsance hreshold of one anoher. In hs nsance, we found srong evdence of resdual spaal correlaon. The Moran I sasc for hese resduals s 0.40, whch has an assocaed z score of approxmaely 8. 1 Ths s far from wha we would expec f he null hypohess of spaal ndependence were rue. Saed dfferenly, hs mples ha here s consderable posve assocaon beween he democracy level of a counry and ha of s geographcal neghbors, above and beyond wha we would expec from her levels of GDP per capa. Ths resul s farly ypcal, and wll ofen be he case ha ncludng spaally clusered covaraes alone wll no compleely remove spaal cluserng n he oucome of neres. Gven ha he dsrbuon of democracy sll dsplays spaal cluserng afer condonng on a counry s GDP per capa, we should look for possble ways o ncorporae hs spaal dependence n our prevous regresson model. As n he case of seral cluserng over me, we can hnk of spaal auocorrelaon eher as nusance or subsance. Spaal dependence leads o problems wh he regresson esmae ˆb for he effec of GDP per capa and s sandard errors, snce he errors canno be consdered o be ndependen among conneced uns. These problems n esmang he effec of GDP per capa on democracy can, n prncple, be addressed hrough alernave esmaors ha ake no accoun he spaal correlaon of he errors, ha s, he resdual varaon no capured by GDP per capa alone. Ths approach s ofen known as he spaal error model, an approach we dscuss subsequenly. However, our broader neres here s n wha nfluences democracy, no jus esmang he assocaon beween a counry s GDP per capa on s prospecs for democracy. If a counry s level of democracy appears o be assocaed wh s neghbors level of democracy, hs ells us somehng mporan abou he dsrbuon of democracy self and provdes an opporuny for learnng somehng abou possble nfluences from spaal dependence on prospecs and consrans on democracy. As such, a more plausble and neresng approach s o consder he spaal assocaon as a subsanve feaure of democracy raher han as a sascal nusance. The spaal assocaon observed here suggess ha we have dependence among observaons such ha he expeced value of democracy for a counry dffers noably dependng on he level of democracy n neghborng saes j. Insead of leng expeced democracy for a counry depend jus on GDP per capa, we devse a model where democracy s a funcon of

3 37 boh s own GDP per capa and he level of democracy among neghbors, defned by wy where he enres of he connecvy vecor w (.e., row from marx W) acqure nonzero values for all saes j ha are defned as conneced o. Recall agan ha he W connecvy marx s row sandardzed so ha each row n W sums o 1. Ths reasonng suggess a spaally lagged dependen varable model of he form y = + x + w y +ε [2.1] b0 b1 r, where a posve value for he parameer assocaed wh he spaal lag (r) ndcaes ha counres are expeced o have hgher democracy values f, on average, her neghbors have hgh democracy values. One can hnk of he spaally lagged y model as analogous o an auoregressve me seres model where emporal seral correlaon s addressed by ncludng a lagged dependen varable y on he rgh-hand sde when we esmae he effecs of oher rgh-hand sde covaraes (say x ) on y. The ˆb 1 coeffcen n he spaally lagged y model dffers from he coeffcen calculaed va he OLS regresson model n ha we are now assessng he effec of GDP per capa on he democracy level of a counry, whle conrollng for spaal dependence n y, or he exen o whch varaon n a counry s level of democracy can be accouned for by he value of y n oher conneced counres j. Hence, we wll need o ake no accoun he spaal ramfcaons when assessng he effec of changes n x. Tables 2.1 and 2.2 provde he esmaes from an OLS regresson on he level of democracy on he naural log of per capa GDP n 158 counres n 2002 wh and whou a spaal lag of y. We observe a large posve coeffcen for he log of GDP per capa of 1.68 n he OLS whou he spaally lagged y. In conras, n he spaally lagged y model, he esmaed coeffcen for he log of GDP per capa s 0.76, less han half of s orgnal sze, TABLE 2.1 OLS Whou Spaal Lag OLS ˆb SE( ˆb) Value Inercep ln GDP per capa N = 158 Log lkelhood (df = 3) = F = (df 1 = 1, df 2 = 156)

4 38 TABLE 2.2 OLS Wh Spaal Lag OLS ˆb SE( ˆb) Value Inercep ln GDP per capa r N = 158 Log lkelhood (df = 4) = F = (df 1 = 2, df 2 = 155) alhough connues o be far away from 0 by he convenonal sandards for sgnfcance ess. The esmae for he spaally lagged y erm s large and posve (0.76) and hghly sascally sgnfcan by sandard crera. Ths provdes suppor for he conjecure ha a counry s level of democracy covares wh he level of democracy among s geographcal neghbors. In subsanve erms, he model mples ha a counry s expeced level of democracy would be 7.6 pons lower f s neghbors had an average democracy score a he mnmum possble score (.e., 10) compared wh a neghbor average of 0, whch s close o he hsorcal average POLITY score snce Conversely, a counry wh he maxmum neghbor average democracy score of 10 would be expeced o be 7.6 pons more democrac relave o a counry wh a neghbor average of 0. These esmaes reflec he cluserng of democracy llusraed prevously. Alhough mos democraces end o have hgher GDP per capa, we do observe clusers of democracy n 2002 n areas where GDP per capa s no parcularly hgh, as n Lan Amerca, and cluserng of auocraces n areas wh hgh average GDP, as n he Gulf saes. Comparng he measures of he overall f for he model assumng ndependen observaons n Table 2.1 and he model wh he spaally lagged y n Table 2.2 ndcaes ha he model wh he spaally lagged y erm fs he daa noably beer. I has a hgher F sasc and a hgher log lkelhood han he model assumng ndependen observaons. Ths, n urn, renforces our belef ha he spaal lag of y adds somehng mporan o specfyng he dsrbuon of democracy, beyond wha we would expec from a counry s GDP per capa. Model heurscs alone do no provde he compellng reason for usng he spaal approach, however. The spaal approach s beer no because of he heurscs produces alone, bu because specfes a plausble form of he feedback or dependency among observaons.

5 39 A sandard ordnary leas squares regresson has he followng form: y = x b + e. If e s decomposed no a spaally lagged erm for he dependen varable whch s correlaed wh he dependen varable and an ndependen error erm, e = rw y + ε, hs leads o he formulaon for spaally lagged dependen varables: y = x b + rw y + ε. If, however, we specfy hs dfferenly, e = lw x + ε, we ge y = x b + lw x + ε, whch s a spaal error formulaon. We urn nex o an examnaon of he spaally lagged dependen varable model; he spaal error model s addressed n Chaper 3. 2 I s empng o nerpre he coeffcen esmae for GDP per capa n he model wh he spaally lagged y n Table 2.2 and compare hs drecly wh Table 2.1, suggesng a seemngly larger effec of GDP per capa. However, hs nerpreaon s no correc. The coeffcen esmaes have dfferen nerpreaons, as he model wh he spaally lagged y n Equaon 2.1 s an auoregressve specfcaon, so ha he coeffcen for he mpac of x now reflecs he shor-run mpac of x on y raher han he ne effec, as s he case of he coeffcen for x n he OLS model whou he spaally lagged y. Snce he value of y wll nfluence he level of democracy n oher saes y j and hese y j, n urn, feed back on o y, we need o ake no accoun he addonal effecs ha he shor mpac of x exers on y hrough s mpac on he level of democracy n oher saes. Ths s analogous o he nerpreaon of he coeffcen b for a covarae x n a me seres model where we have a emporal lag of he dependen varable y on he rgh-hand sde, for example, y x y = b + f +ε. In hs equaon, b ndcaes he mmedae effec of x on y. Bu hs wll, n urn, affec y n he followng me perod and he long-run effec of x mus hus ake no accoun he par of he ne effec ha works hrough he auoregressve par or he esmaed coeffcen for he mpac of he lag y. The long-run effec of x wll be b /(1- f). In a suaon where f s large, he long-run effec b /(1- f) can be subsanally larger han b.

6 40 Connung hs analogy, magne f we could ncrease he log of GDP per capa by one un n only a sngle counry, whch would have an mmedae mpac on ha counry s level of democracy of b 1. However, he model n Equaon 2.1 mples spaal dynamcs wh a feedback effec beween counres, where counry s level of democracy s also held o have an effec on s neghbors level of democracy. Hence, an ncrease n democracy ha affecs s level of democracy wll hen nfluence democracy n he neghbors of j. Conemporaneously, n urn, he neghbors neghbors wll also be affeced, hroughou all conneced counres. In general, all counres wll have some neghbors so ha evenually he nfluence of all counres wll be affeced. Bu noe ha Equaon 2.1 ncludes democracy for all counres n he sysem y, so f he democracy level of oher counres conneced o ncreases so wll he level of democracy n. In hs way, an exogenous shock o one observaon, such as our hough expermen, wll have a reverberang effec hroughou he sysem wh feedback among observaons and flow hrough he sysem as a seres of adjusmens unl seles on some new sable equlbrum (Cresse, 1993; Ln, Wu, & Lee, 2006). Raher han focusng on he coeffcen esmaes for x alone n a spaally lagged y model, s mporan o consder he equlbrum effecs. Unforunaely, he long-run effec for he spaally lagged y canno be saed n a form as smple as n he case of he long-run effec n he presence of a emporally lagged y. We wll reurn laer for how o characerze and esmae he equlbrum effec of covaraes n a spaally lagged y model. Frs, however, we urn o problems posed by he presence of he spaal lag of y on he rgh-hand sde of he equaon and he mpled endogeney problems relaed o conssen esmaon of he model n he ordnary leas squares seup. The followng secon reles on marx algebra and focuses on ssues of esmaon and why a maxmum lkelhood esmaor (MLE) may be preferable o esmang he spaally lagged y model. Snce usng he MLE by self does no requre an undersandng of all he deals n hs secon, readers who are no neresed n ssues of esmaon may skp he deals n hs secon and proceed mmedaely o he nex secon. 2.2 Esmang he Spaally Lagged y Model In a me seres model wh a emporally lagged y on he rgh-hand sde, he presence of he emporal lag y does no creae problems for esmaon wh OLS, provded here s no seral correlaon n he resduals of he regresson model. More precsely, OLS wh a lagged dependen