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Chapter 2
Spatial autoregressive models
Thischapterdiscussesindetailthespatialautoregressivemodelsintroduced
inChapter1.Aclassofspatialautoregressivemodelshavebeenintroduced
tomodelcross-sectionalspatialdatasamplestakingtheformshownin(2.1)
(Anselin,1988).
y = W
1
y+X+u
(2.1)
u = W
2
u+"
"  N(0;
2
I
n
)
Whereycontainsannx1vectorofcross-sectionaldependent variablesand
Xrepresentsannxkmatrixofexplanatoryvariables.W
1
andW
2
areknown
nxn spatialweightmatrices,usuallycontainingrst-ordercontiguityrela-
tionsorfunctions ofdistance. As s explainedinSection1.4.1, arst-order
contiguitymatrixhaszeros onthe maindiagonal,rowsthatcontainzeros
inpositionsassociatedwithnon-contiguousobservationalunitsandonesin
positionsreflectingneighboringunitsthatare(rst-order)contiguousbased
ononeofthecontiguitydenitions.
Fromthegeneralmodelin(2.1)wecanderivespecialmodelsbyimposing
restrictions.Forexample,settingX=0andW
2
=0producesarst-order
spatialautoregressivemodelshownin(2.2).
y = W
1
y+"
(2.2)
"  N(0;
2
I
n
)
Thismodelattemptstoexplainvariationinyasalinearcombinationof
contiguousor neighboringunitswithnootherexplanatoryvariables. The
30
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CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
31
modelistermedarstorderspatialautoregressionbecauseitsrepresentsa
spatialanalogytotherstorderautoregressivemodelfromtimeseriesanal-
ysis,y
t
=y
t−1
+"
t
,wheretotalrelianceisonthepastperiodobservations
toexplainvariationiny
t
.
SettingW
2
=0producesamixedregressive-spatialautoregressivemodel
shownin(2.3). Thismodelisanalogoustothelaggeddependent t variable
modelintimeseries. Herewehaveadditionalexplanatoryvariablesinthe
matrix X that t serve to explain variationiny over r the spatialsample of
observations.
y = W
1
y+X+"
(2.3)
"  N(0;
2
I
n
)
LettingW
1
=0resultsinaregressionmodelwithspatialautocorrelation
inthedisturbancesasshownin(2.4).
y = X+u
(2.4)
u = W
2
u+"
"  N(0;
2
I
n
)
Thischapterisorganizedintosections thatdiscuss andillustrateeach
ofthesespecialcasesofthespatialautoregressivemodelaswellasthemost
generalmodelformin(2.1).Section2.1dealswiththerst-orderspatialau-
toregressivemodelpresentedin(2.2). Themixedregressive-spatialautore-
gressivemodelistakenupinSection2.2.Section2.3takesuptheregression
modelcontainingspatialautocorrelationinthedisturbancesandillustrates
various testsfor spatialdependence usingregressionresiduals. Themost
generalmodel is s thefocus of Section2.4. Applied illustrationsofallthe
modelsareprovidedusingavarietyofspatialdatasets. Spatialeconomet-
ricslibraryfunctionsthatutilizeMATLABsparsematrixalgorithmsallow
ustoestimatemodels withover3,000observationsinaround100seconds
onaninexpensivedesktopcomputer.
2.1 The rst-order spatial AR model
Thismodelisseldomusedinappliedwork,butitservestomotivatesome
ofthe ideas that wedrawoninlater sectionsofthechapter. The model
whichwelabelFAR,takestheform:
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CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
32
y = Wy+"
(2.5)
"  N(0;
2
I
n
)
WherethespatialcontiguitymatrixW hasbeenstandardizedtohaverow
sumsofunityandthevariablevectoryisexpressedindeviationsfromthe
meansformtoeliminatetheconstantterminthemodel.
Toillustratetheproblemwithleast-squaresestimationofspatialautore-
gressivemodels,considerapplyingleast-squarestothemodelin(2.5)which
wouldproduceanestimateforthesingleparameterinthemodel:
^=(y
0
W
0
Wy)
−1
y
0
W
0
y
(2.6)
Can we showthat t this estimateis unbiased? If not, is it consistent?
Takingthesameapproachasinleast-squares,wesubstitutetheexpression
foryfromthemodelstatementandattempttoshowthatE(^)=toprove
unbiasedness.
E(^) = (y
0
W
0
Wy)
−1
y
0
W
0
(Wy+")
= +(y
0
W
0
Wy)
−1
y
0
W
0
"
(2.7)
Notethatthe least-squaresestimate isbiased,sincewecannotshowthat
E(^)=. TheusualargumentthattheexplanatoryvariablesmatrixXin
least-squaresisxedinrepeatedsamplingallowsonetopasstheexpecta-
tionoperator overterms like (y
0
W
0
Wy)
−1
y
0
W
0
andargue thatE(")= 0,
eliminatingthebiasterm.Herehowever,becauseofspatialdependence,we
cannotmakethecasethatWyisxedinrepeatedsampling.Thisalsorules
outmakingacaseforconsistencyoftheleast-squaresestimateof,because
theprobabilitylimit(plim)forthetermy
0
W
0
"isnotzero. Infact,Anselin
(1988)establishesthat:
plimN
−1
(y
0
W
0
")=plimN
−1
"
0
W(I−W)
−1
"
(2.8)
Thisisequaltozeroonlyinthetrivialcasewhereequalszeroandwehave
nospatialdependenceinthedatasample.
Giventhatleast-squareswillproducebiasedandinconsistentestimates
ofthespatialautoregressiveparameterinthismodel,howdoweproceed
to estimate? Themaximumlikelihoodestimatorfor requires thatwe
ndavalueofthatmaximizesthelikelihoodfunctionshownin(2.9).
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CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
33
L(yj;
2
)=
1
22
(n=2)
jI
n
−Wjexpf−
1
22
(y−Wy)
0
(y−Wy)g (2.9)
In order to simplify the maximizationproblem, we obtain a concen-
trated log likelihood function based on eliminatingthe parameter 
2
for
the variance of the disturbances. This is accomplished by substituting
^
2
= (1=n)(y−Wy)
0
(y−Wy) in the likelihood(2.9) and taking logs
whichyields:
Ln(L)/−
n
2
ln(y−Wy)
0
(y−Wy)+lnjI
n
−Wj
(2.10)
Thisexpressioncanbemaximizedwithrespecttousingasimplexunivari-
ateoptimizationroutine.Theestimatefortheparameter
2
canbeobtained
usingthevalueofthatmaximizesthelog-likelihoodfunction(say,~)in:
^
2
=(1=n)(y−~Wy)
0
(y−~Wy). Inthenextsection,wediscussasparse
matrixalgorithmapproachtoevaluatingthislikelihoodfunctionthatallows
ustosolveproblemsinvolvingthousandsofobservationsquicklywithsmall
amountsofcomputermemory.
Twoimplementationdetailsarisewiththisapproachtosolvingformaxi-
mumlikelihoodestimates.First,thereisaconstraintthatweneedtoimpose
ontheparameter. Thisparametercantakeonfeasiblevaluesintherange
(AnselinandFlorax,1994):
1=
min
<<1=
max
where
min
representstheminimumeigenvalueofthestandardizedspatial
contiguitymatrixW and
max
denotesthelargesteigenvalueofthismatrix.
Thissuggeststhatweneedtoconstrainouroptimizationproceduresearch
overvaluesofwithinthisrange.
Thesecondimplementationissueis thatthenumericalhessianmatrix
that wouldresult from agradient-basedoptimizationprocedure andpro-
videestimatesofdispersionfortheparametersisnotavailablewithsimplex
optimization. We canovercomethis problemintwoways. Forproblems
involving a small number of observations, we can use our r knowledge of
the theoreticalinformationmatrixtoproduce estimatesofdispersion. An
asymptoticvariancematrixbasedontheFisherinformationmatrixshown
belowfor theparameters  = (;
2
) can be usedtoprovide measures s of
dispersionfortheestimatesofand
2
.(seeAnselin,1980page50):
[I()]
−1
=−E[
@
2
L
@@
0
]
−1
(2.11)
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CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
34
This approach is s computationally impossible whendealing with large
scaleproblems involvingthousandsof observations. Inthesecases s wecan
evaluate the numericalhessianmatrixusingthemaximumlikelihoodesti-
mates of  and 
2
as well as s our sparse matrix function to compute the
likelihood.Wewilldemonstrateresultsfromusingbothoftheseapproaches
inthenextsection.
2.1.1 Thefar()function
Building on the software design set forth in section 1.5 for our spatial
econometricsfunctionlibrary,wehaveimplementedafunctionfartopro-
ducemaximumlikelihoodestimatesfortherst-orderspatialautoregressive
model.WerelyononthesparsematrixfunctionalityofMATLABsolarge-
scaleproblemscanbesolvedusingaminimumoftimeandcomputerRAM
memory. We demonstrate this functioninactionon adata set involving
3,107U.S.counties.
EstimatingtheFARmodelrequiresthatwendeigenvaluesforthelarge
nbynmatrixW,aswellasthedeterminantoftherelatednbyn matrix
(I
n
−W).Inaddition,matrixmultiplicationsinvolvingW and(I
n
−W)
arerequiredtocomputetheinformationmatrixusedtoproduceestimates
ofdispersion.
Weconstructedafunctionfarthatcanproduceestimatesfortherst-
orderspatialautoregressivemodelinacaseinvolving3,107observationsin
95secondsonamoderatelyfastinexpensivedesktopcomputer. TheMAT-
LABalgorithmsfordealingwithsparsematricesmakeitideallysuitedfor
spatialmodelingbecausespatialweightmatricesarealmostalwayssparse.
Anotherissuewe needtoaddress iscomputingmeasures ofdispersion
fortheestimatesand
2
inlargeestimationproblems. Asalreadynoted,
we cannot rely on the informationmatrixapproachbecause this involves
matrix operations on very large matrices. Anapproach that t we take to
producemeasuresofdispersionistonumericallyevaluatethehessianmatrix
usingthemaximumlikelihoodestimatesofand
2
.Theapproachbasically
producesanumericalapproximationtotheexpressionin(2.11). Akeyto
usingthisapproachistheabilitytoevaluatetheloglikelihoodfunctionusing
thesparsealgorithmstohandlelargematrices.
ItshouldbenotedthatPaceandBarry(1997)whenconfrontedwiththe
taskofprovidingmeasuresofdispersionforspatialautoregressiveestimates
basedonsparsealgorithmssuggestusinglikelihoodratioteststodetermine
thesignicanceoftheparameters.Theapproachtakenheremaysuerfrom
somenumericalinaccuracyrelativetomeasuresofdispersionbasedonthe
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CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
35
theoretical informationmatrix, but has s the advantage that users will be
presentedwithtraditionalt−statisticsonwhichtheycanbaseinferences.
Wewillhavemoretosayabouthowourapproachtosolvinglargespa-
tialautoregressive estimationproblems usingsparse matrixalgorithms in
MATLABcomparestoone proposedby PaceandBarry(1997),whenwe
applythefunctionfartoalargedatasetinthenextsection.
Documentationforthefunctionfar ispresentedbelow. This s function
waswrittentoperformonbothlargeandsmallproblems. Iftheproblem
issmall(involvinglessthan500observations),thefunctionfarcomputes
measuresofdispersionusingthetheoreticalinformationmatrix.Ifmoreob-
servationsareinvolved,thefunctiondeterminesthesemeasuresbycomput-
inganumericalhessianmatrix.(Usersmayneedtodecreasethenumberof
observationstolessthan500iftheyhavecomputerswithoutalargeamount
ofRAMmemory.)
PURPOSE: computes1st-order spatial autoregressive estimates
y = p*W*y y + e, using sparsematrix x algorithms
---------------------------------------------------
USAGE: results = far(y,W,rmin,rmax,convg,maxit)
where: y = dependentvariable e vector
W = standardizedcontiguitymatrix
rmin = (optional) ) minimum value of rho to use in search
rmax = (optional) ) maximum value of rho to use in search
convg = (optional) ) convergence criterion (default = 1e-8)
maxit = (optional) ) maximum # of iterations(default = 500)
---------------------------------------------------
RETURNS: a structure
results.meth = 'far'
results.rho
= rho
results.tstat = asymptotic c t-stat
results.yhat = yhat
results.resid = residuals
results.sige = sige e = (y-p*W*y)'*(y-p*W*y)/n
results.rsqr = rsquared
results.lik
= -log g likelihood
results.nobs = nobs
results.nvar = nvar r = 1
results.y
= y y data vector
results.iter
= # # of iterationstaken
results.romax = 1/max eigenvalue e of W (or r rmax if input)
results.romin = 1/min eigenvalue e of W (or r rmin if input)
--------------------------------------------------
Oneoptionweprovideallowstheusertosupplyminimumandmaximum
valuesofratherthanrelyontheeigenvaluesofW. Thismightbeusedif
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CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
36
wewishedtoconstraintheestimationresultstoarangeofsay0<<1.
Notealsothatthiswouldsavethetimeneededtocomputethemaximum
andminimumeigenvaluesofthelargeW matrix.
2.1.2 Appliedexamples
Givenourfunctionfarthatimplementsmaximumlikelihoodestimationof
smallandlargerst-orderspatialautoregressivemodels,weturnattentionto
illustratingtheuseofthefunctionwithsomespatialdatasets.Inaddition
to the estimationfunctions, we have functions prtand plt that provide
printedandgraphicalpresentationoftheestimationresults.
Example2.1providesanillustrationofusingthesefunctionstoestimate
arst-orderspatialautoregressivemodelforneighborhoodcrimefromthe
Anselin(1988)spatialdatasample.Notethatweconvertthevariablevector
containingcrimeincidentstodeviationsfromthemeansform.
% ----- Example 2.1 Using the far()function
load wmat.dat;
% standardized 1st-order contiguitymatrix
load anselin.dat; % load Anselin n (1988) Columbusneighborhood crime data
y = anselin(:,1);
ydev = y y - mean(y);
W = wmat;
vnames= strvcat('crime','rho');
res = = far(ydev,W); % do 1st-orderspatial l autoregression
prt(res,vnames);
% print t the output
plt(res,vnames);
% plot t actual vs predicted d and residuals
Thisexampleproducedthefollowingprintedoutputwiththegraphical
output presented inFigure 2.1. From the output t we would infer r that a
distinctspatialdependence amongthecrimeincidentsforthesampleof49
neighborhoodsexistssincetheparameterestimateforhasat−statisticof
4.259. Wewouldinterpretthisstatisticinthetypicalregressionfashionto
indicatethat the estimatedlies4.2standarddeviationsawayfromzero.
We alsoseethatthis modelexplains nearly44%of the variationincrime
incidentsindeviationsfromthemeansform.
First-order spatialautoregressivemodel Estimates
DependentVariable =
crime
R-squared
=
0.4390
sigma^2
= 153.8452
Nobs, Nvars
=
49,
1
log-likelihood =
-373.44669
# of iterations=
17
min and d max rho =
-1.5362,
1.0000
CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
37
***************************************************************
Variable
Coefficient
t-statistic
t-probability
rho
0.669775
4.259172
0.000095
0
5
10
15
20
25
30
35
40
45
50
-40
-20
0
20
40
FAR   Actual vs. Predicted
0
5
10
15
20
25
30
35
40
45
50
-60
-40
-20
0
20
40
Residuals
Figure2.1: Spatialautoregressivetandresiduals
Anothermorechallengingexampleinvolvesalargesampleof3,107ob-
servationsrepresentingcountiesinthecontinentalU.S.fromPaceandBarry
(1997). Theyexaminepresidentialelectionresults s for thislargesampleof
observationscoveringtheU.S.presidentialelectionof1980betweenCarter
and Reagan. The variable we wish to explainusing the rst-order spa-
tialautoregressivemodelis theproportionoftotalpossiblevotescastfor
bothcandidates. Only y persons 18 years andolder are eligibletovote,so
theproportionisbasedonthosevotingforbothcandidatesdividedbythe
populationover18yearsofage.
PaceandBarry(1997)suggestanalternativeapproachtothatimple-
mentedhereinthefunctionfar.Theyproposeovercomingthedicultywe
CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
38
faceinevaluatingthedeterminant(I−W)bycomputingthisdeterminant
onceoveragridofvaluesfortheparameterrangingfrom1=
min
to1=
max
priortoestimation.Theysuggestagridbasedon0.01incrementsforover
the feasiblerange. Giventhesepre-determinedvalues s forthedeterminant
(I−W),they point out thatonecanquicklyevaluatethelog-likelihood
functionforallvaluesofinthegridanddeterminetheoptimalvalueofas
thatwhichmaximizesthelikelihoodfunctionvalueoverthisgrid.Notethat
theirproposedapproachwouldinvolveevaluatingthedeterminantaround
200timesifthefeasible rangeofwas-1to1. In many y cases therange
is evengreater than thisandwouldrequireeven moreevaluations s of the
determinant. Incontrast,our r functionfarreports thatonly 17iterations
requiringloglikelihoodfunctionevaluationsinvolvingthedeterminantwere
neededtosolvefortheestimatesinthecaseoftheColumbusneighborhood
crimedataset.
Inaddition,considerthatonemightneedtoconstructanergridaround
theapproximatemaximumlikelihoodvalueofdeterminedfromtheinitial
gridsearch,whereasouruseoftheMATLABsimplex algorithmproduces
anestimatethatisaccuratetoanumberofdecimaldigits.
Aftersomediscussionofthecomputationalsavingsassociatedwiththe
useofsparsematrices,weillustratetheuseofourfunctionfarandcompare
ittotheapproachsuggestedbyPaceandBarry.
Arstpointtonote regardingsparsity isthat largeproblems suchas
thiswillinevitablyinvolveasparsespatialcontiguityweightingmatrix.This
becomesobviouswhenyouconsiderthecontiguitystructureofoursample
of3,107U.S.counties. Atmost,individualcountiesexhibitedonly8rst-
order(rookdenition)contiguityrelations,sotheremaining2,999entriesin
thisrowarezero. Theaveragenumberofcontiguityrelationshipsis4,soa
greatmanyoftheelementsinthematrixW arezero,whichisthedenition
ofasparsematrix.
Tounderstandhowsparsematrixalgorithmsconserveonstoragespace
andcomputermemory,considerthatweneedonlyrecordthenon-zeroele-
mentsofasparsematrixforstorage. Sincetheserepresentasmallfraction
ofthetotal3107x3107=9,653,449elementsintheweightmatrix,wesave
atremendousamountofcomputermemory. Infactforourexampleofthe
3,107U.S.counties,only12,429non-zeroelementswerefoundintherst-
orderspatialcontiguitymatrix,representingaverysmallfraction(farless
than1percent)ofthetotalelements.
MATLABprovidesa functionsparsethat canbe usedtoconstruct a
largesparsematrix by simply indicatingthe rowandcolumnpositionsof
non-zero elements andthevalue of the matrixelement forthese non-zero
CHAPTER2. SPATIALAUTOREGRESSIVEMODELS
39
rowandcolumnelements. Continuingwithourexample,wecanstorethe
rst-ordercontiguitymatrixinasingledatalecontaining12,429rowswith
3columnsthattaketheform:
row column value
Thisrepresentsaconsiderablesavingsincomputationalspacewhencom-
paredtostoringamatrixcontaining9,653,449elements. Ahandyutility
functioninMATLAB is spy y which allowsone toproducea specially y for-
mattedgraphshowingthesparsitystructureassociatedwithsparsematrices.
We demonstratebyexecutingspy(W) onour weight matrixW fromthe
PaceandBarrydataset, whichproduced the graphshowninFigure2.2.
Aswecanseefromthegure,mostofthenon-zeroelementsresidenearthe
diagonal.
Asanexampleofstoringasparserst-ordercontiguitymatrix,consider
example 2.2 below that reads data from the le `ford.dat' in sparse for-
matandusesthefunctionsparsetoconstructaworkingspatialcontiguity
matrixW. The examplealsoproduces agraphicaldisplayof the sparsity
structureusingtheMATLABfunctionspy.
% ----- Example 2.2 Using sparse matrix functions
load ford.dat; % 1st t order contiguity matrix
% storedin n sparse matrix form
ii = = ford(:,1);
jj = = ford(:,2);
ss = = ford(:,3);
clear ford;
% clear r out the matrix to save RAM memory
W = sparse(ii,jj,ss,3107,3107);
clear ii; clear jj; clear ss; % clear r out these vectors to save memory
spy(W);
Tocompareourfunctionfarwiththe approachproposedbyPaceand
Barry,weimplementedtheirapproachandprovidetimingresults.Wetake
amoreecientapproachtothegridsearchovervaluesoftheparameter
thansuggestedbyPaceandBarry.Ratherthansearchoveralargenumber
ofvaluesfor,webasedoursearchonalargeincrementof0.1foraninitial
gridof valuescoveringfrom1=
min
to 1=
max
. Giventhe determinant
of (I−W) calculatedusing sparse matrix algorithms s in MATLAB, we
evaluatedthenegativeloglikelihoodfunctionvaluesforthisgridofvalues
to nd the value that t minimizes the likelihood function. We then make
asecondpass basedona tightergridwithincrements of0.01aroundthe
optimalvaluefoundintherstpass.Athirdandnalpassisbasedonan
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