11th Economcs Summer Semnars Pamukkale Unversty Denzl Turkey
Appled Spatal Econometrcs Bernard Fngleton Unversty of Cambrdge UK
Am of Course To set out the basc approach of spatal econometrcs To llustrate the practces and usefulness of spatal econometrcs for appled economsts To hghlght some of the ptfalls
Structure of course Regresson and spatal dependence Resdual Spatal autocorrelaton Modellng spatal dependence Spatal lag model, Spatal error model,spatal Durbn model Estmaton methods Maxmum lkelhood estmaton (ML) Two stage least squares (2SLS) Generalzed Method of Moments (GMM) and Feasble Generalzed Spatal 2SLS (FGS2SLS) Spatal panel models GMM, FGS2SLS, random effects and spatal dependence Predcton Focus on computatonal aspects and how to do spatal econometrcs n an appled sense demonstraton programs n MATLAB
Sesson 1 The reasons for spatal econometrcs : Why spatal econometrcs? What s spatal econometrcs? Spatal versus tme seres
Why spatal econometrcs? Spatal economcs now wdely recognsed n the economcs/econometrcs manstream Krugman s Nobel prze for work on economc geography Importance of network economcs (eg Royal Economc Socety Easter 2009 School, on Auctons and Networks ) LSE ESRC Centre for Spatal Economcs Increasng polcy relevance : World Bank (2008), World Development Report 2009, World Bank, Washngton. Importantly, much nsght can be ganed by usng spatal econometrc tools n addton to more standard tme seres methods Tme seres methods and spatal econometrcs come together n the analyss of spatal panels, whch we wll look at towards the end of the course
What s spatal econometrcs? the theory and methodology approprate to the analyss of spatal seres relatng to the economy spatal seres means each varable s dstrbuted not n tme as n conventonal, manstream econometrcs, but n space.
Spatal versus tme seres DGP for tme seres y( t) y( t 1) ( t) (1) y(1) 0 2 ~ d(0, ) t 2... T
Spatal versus tme seres DGP for tme seres 1.5 1 0.5 0-0.5-1 -1.5 0 50 100 150 200 250
Spatal versus tme seres DGP for tme seres y Wy y s a T x 1 vector s a scalar parameter that s estmated s an T x 1 vector of dsturbances
DGP for tme seres y Wy W s a TxT matrx wth 1s on the mnor dagonal, thus for T = 10 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 W 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 The 1s ndcate locaton pars that are close to each other n tme
DGP for tme seres y Wy Provded Wy and are contemporaneously ndependent we can estmate by OLS and get consstent estmates, although there s small sample bas.
DGP for spatal seres In spatal econometrcs, we have an N x N W matrx N s the number of places. W = 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 1 1 0 N= 353 a porton of the W matrx for Luton(1), Md Bedfordshre(2), Bedford(3), South Bedfordshre(4), Bracknell Forest(5), Readng(6), Slough(7), West Berkshre(8), Wndsor and Madenhead(9), Wokngham(10) The 1s ndcate locaton pars that are close to each other n space
DGP for spatal seres Resdental property prces n England, 2001 Dstrct.shp 40703-89013 89013-129966 129966-176349 176349-274395 274395-639049 Fngleton B (2006) A cross-sectonal analyss of resdental property prces: the effects of ncome, commutng, schoolng, the housng stock and spatal nteracton n the Englsh regons' Papers n Regonal Scence 85 339-361 N= 353 We refer to these small areas As UALADs
DGP for spatal seres y Wy y s an N x 1 vector s a scalar parameter that s estmated s an N x 1 vector of dsturbances
DGP for spatal seres y Wy Ths s an almost dentcal set-up to the tme seres case And one mght thnk that t can also be consstently estmated by OLS But now there s one bg dfference we cannot estmate the spatal autoregresson by OLS and obtan consstent estmates of. Reason - Wy and are not ndependent. Wy determnes y but s also determned by y. But more about ths later
Regresson and spatal dependence Typcally n economcs we workng wth regresson models, thus y x t tk k t k But n spatal economcs typcally the analyss s cross-sectonal, so that y x k k k
Regresson and spatal dependence y x k k k y = Observed value of dependent varable y at locaton ( = 1,,N) x k = Observaton on explanatory varable x k at locaton, wth k = 1,,K k = regresson coeffcent for varable x k = random error term or dsturbance term at locaton Let us assume as n the classc regresson model that the errors smply represent unmodelled effects that appears to be random. We therefore commence by assumng 2 that E( ) 0, Var( ), E(, ) 0 for all,j. The assumpton s that the errors j are dentcally and ndependently dstrbuted. For the purposes of nference we mght specfy the error as a normal dstrbuton.
Regresson and spatal dependence Wrtng our model n matrx terms gves y y X X s an N x 1 vector s an N x k matrx s a k x 1 vector s an N x 1 vector 2 E( ) 0, E( ) I And spatal dependence manfests tself as spatally autocorrelated resduals ˆ y yˆ y X ˆ
Resdual Spatal autocorrelaton Ths term s analogous to autocorrelaton n tme seres, whch s when the resduals at ponts that are close to each other n tme/space are not ndependent. For nstance they may be more smlar than expected (postve autocorrelaton) for some reason. suggestng that somethng s wrong wth the model specfcaton that s assumng they are ndependent. For example the errors/dsturbances/resduals may contan the effects of omtted effects that vary systematcally across space.
Moran s I Based on W matrx A spatal weghts matrx s an N x N wth non-zero elements n each row for those columns j that are n some way neghbours of locaton The noton of neghbour s a very general one, t may mean that they are close together n terms of mles or drvng tme, or t may be dstance n some more abstract economc space or socal space that s not really connected to geographcal dstance. The smplest form of dstance mght be contguty, wth Wj= 1 f locatons and j are contguous, and Wj = 0 otherwse. Usually (but not necessarly) W s standardsed so that all the values n row are dvded by the sum of the row values.
We Calculatng Moran s I thnk of Moran s I as approxmately the correlaton between the two vectors Wˆ and. ˆ We can show ths for a 5 locaton analyss n graphcal form, known as a Moran scatterplot. W 0 0.5000 0.5000 0 0 0.3330 0 0.3330 0.3330 0 0.3330 0.3330 0 0.3330 0 0.2500 0.2500 0.2500 0 0.2500 0 0 0.5000 0.5000 0 ˆ -0.5000-0.3500 0.1000 0.2500 0.5000 0.25 0.2 0.15 data 1 lnear Hence -0.1250= 0.5 x -0.35 + 0.5 x 0.1. 0.1 0.05 ˆ W -0.1250-0.0500-0.1998-0.0625 0.1750 0-0.05-0.1-0.15-0.2-0.25-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 e
Average House prces n local authorty areas n England (UALADs) Resdental property prces n England, 2001 Dstrct.shp 40703-89013 89013-129966 129966-176349 176349-274395 274395-639049 N= 353
Calculatng Moran s I n practce Let us look at our map of house prces. Can we buld a model explanng ths varaton? Do we have spatally autocorrelated resduals? The presence of spatal autocorrelaton would suggest there s some specfcaton error, ether omtted spatally autocorrelated varable resdual heterogenety or a spatal autoregressve error process
Calculatng Moran s I n practce y X X X X X 1 1 2 2 3 3 4 4 5 5 y = mean resdental property prce n each of N local authorty areas X = 1, the constant, an N x 1 vector of 1s 1 X = total ncome n each local authorty area 2 X = ncome earned wthn commutng dstance of each local authorty area 3 X = local schoolng qualty n each local authorty area 4 X = stock of propertes n each local authorty area 5 y X X s a N x k matrx s a k x1 vector ˆ yx ˆ
Created by demo_0.m the value for Moran s I s 11.29 standard errors above expectaton. Expectaton s the expected value of I under the null hypothess of no resdual autocorrelaton. It s clear that there s very sgnfcant resdual autocorrelaton. Dependent varable y estmate t rato Constant (X 1 ) -571.874-6.47 Local ncome (X 2 ) 864.0059 10.02 Wthncommutngdstance ncome (X 3 ) 57.7055 14.08 Schoolng qualty (X 4 ) 175802.9235 7.74 Number of households (X 5 ) -0.7112-6.46 R 2 adjusted 0.567 Standard Error 42.113 Moran's I 0.39369 11.29 Degrees of freedom 348
Wev Calculatng Moran s I n practce What s W? W * W * 1 j W 2 j * j Wj j d Moran scatterplot 8 Ftted and observed relatonshp Wˆ versus ˆ 6 4 2 0-2 -4-2 0 2 4 6 8 10 e
Calculatng Moran s I n practce Unfortunately calculatng Moran s I s not easy wthout the help of software. I gve the formulae below just to show how dffcult ths s! I S 0 ˆ W ˆ/ S ˆˆ/ N W j We have seen all of these terms except S 0 If we row-standardse, so that each row sums to 1 then S0 N So then ˆ W ˆ I ˆˆ 0 j
Calculatng Moran s I n practce Gven I, we need to compare t wth what we would expect under the null hypothess of no resdual autocorrelaton E( I) tr( MW ) / ( N K) 1 M I X ( X X ) X 2 tr( MWMW ) tr( MWMW ) [ tr( MW )] Var( I) ( E( I)) ( N K)( N K 2) 2 These are the moments we would expect f the resduals were ndependent draws from a normal dstrbuton
Calculatng Moran s I n practce The test statstc s Z, whch has the followng dstrbuton under the null hypothess Z I E( I) ~ N (0,1) Var( I) f Z > 1.96 or Z < -1.96 then we reject the null of no spatal autocorrelaton and nfer that there s spatal autocorrelaton n the regresson resduals. In makng ths concluson, we should add the caveat that there s a 5% chance of a Type I error, false rejecton of the null In the case of our house prce data, I s 19.96 standard devatons above expectaton, a really clear ndcaton that there s postve resdual spatal autocorrelaton
Calculatng Moran s I n practce Postve spatal autocorrelaton s when resduals than are close to each other take smlar values. Negatve spatal autocorrelaton would be when the resduals (coded black for negatve and whte for postve) formed a chequer board pattern f the regons were squares. There are several alternatves to Moran s I, and Moran s I may also detect thngs other than spatally autocorrelated resdual Moran s I wll also tend to detect heteroscedastcty, that s when the resduals have dfferent varances rather than a common varance. However despte ths t s the most famous and commonly used method of detectng spatal autocorrelaton n regresson resduals.
Modellng spatal dependence Say we have a sgnfcant Moran s I statc, what next? We need to elmnate the spatal dependence one way to do ths s to ntroduce an autoregressve lag (spatal lag model) y X X s a N x k matrx s a k x1 vector s an N x 1 vector of errors y Wy X s a scalar parameter W s an Nx N matrx
Spatal lag model Here I lst the values of these varables for the frst 10 of the UALADs. dstrct uaname y Wy 1.0 Luton 87464 168313 2.0 Md_Bedfordshre 138856 151526 3.0 North_Bedfordshre 117530 137574 4.0 South_Bedfordshre 126650 157673 5.0 Bracknell_Forest 167633 200166 6.0 Readng 150094 186756 7.0 Slough 126361 222769 8.0 West_Berkshre 209543 170172 9.0 Wndsor_and_Madenhead 273033 183066 10.0 Wokngham 203059 205737 We can check whether Wy s a sgnfcant varable by addng t to our model y Wy X
Dependent varable y Constant (X 1 ) Local ncome (X 2 ) Spatal lag ML estmate t rato -541.135534-8.02 393.33 5.58 Wthncommutngdstance ncome (X 3 ) 27.45 6.89 Schoolng qualty (X 4 ) 149842.21 8.61 Number of households (X 5 ) -0.35-4.10 Spatal lag (Wy) 0.6089 14.90 R 2 adjusted 0.6330 Standard Error 32.13 Dependent varable y ols estmate t rato Constant (X 1 ) -571.874-6.47 Local ncome (X 2 ) 864.0059 10.02 Wthncommutngdstance ncome (X 3 ) 57.7055 14.08 Schoolng qualty (X 4 ) Number of households (X 5 ) R 2 adjusted 0.567 Standard Error 42.113 175802.9235 7.74-0.7112-6.46 Created by demo_0.m Degrees of freedom 347 Moran's I Degrees of freedom 348 0.39369 11.29 Fngleton B (2006) A cross-sectonal analyss of resdental property prces: the effects of ncome, commutng, schoolng, the housng stock and
Corrado L. & Fngleton B. (2012) Where s the economcs n spatal econometrcs? Journal of Regonal Scence 52 210-239 Drect, ndrect and total effects n spatal lag model Wth Wy, the true effect of a varable, whch typcally s not the same as β, as emphaszed by LeSage and Pace (2009) the effects on dependent varable of a unt change n an exogenous varable, the dervatve y/ X s not smply equal to the regresson coeffcent β the true dervatve also takes account of the spatal nterdependences and smultaneous feedback emboded n the model, leadng to a total effect that dffers somewhat (typcally) from β Ths dervatve s somewhat complcated because t depends on the ndvdual observatons but can be represented by a mean See also Corrado and Fngleton(2012)
Drect, ndrect and total effects n spatal lag model Total effect = drect effect + ndrect effect So we can partton the average total effect nto a drect and an ndrect component The average drect effect gves the effect of X on y when the locatons of X and y are the same drect effect s somewhat dfferent from β because at locaton r, a change n X affects y, whch then affects y at locaton s (s n.e. r) and so on, cascadng through all areas and comng back to produce an addtonal effect on y at r The dfference between the total effect and the drect effect s the average ndrect effect of a varable The average ndrect effect gves the effect of X on y when X and y are not n the same locaton
Drect, ndrect and total effects gven lagged dependent varable ML estmates : spatal lag model Varable Coeffcent Asymptot t-stat z-probablty const -541.135534-8.023904 0.000000 local_ncome 393.326396 5.577120 0.000000 commutng_ncome 27.450614 6.894253 0.000000 supply -0.353572-4.104351 0.000041 schoolng 149842.210059 8.613959 0.000000 rho 0.608979 14.898820 0.000000 Created by demo_0.m Drect Coeffcent t-stat t-prob local_ncome 439.507710 5.895381 0.000000 commutng_ncome 30.341573 7.244365 0.000000 supply -0.394318-4.179838 0.000037 schoolng 167313.290102 8.807929 0.000000 Indrect Coeffcent t-stat t-prob local_ncome 578.415728 5.010304 0.000001 commutng_ncome 39.749552 6.981685 0.000000 supply -0.520811-3.590056 0.000377 schoolng 222208.985696 4.900776 0.000001 Total Coeffcent t-stat t-prob local_ncome 1017.923438 5.870218 0.000000 commutng_ncome 70.091125 8.466974 0.000000 supply -0.915129-3.999705 0.000077 schoolng 389522.275797 6.582510 0.000000
Spatal error model a second way we may model the resdual dependence detected by Moran s I spatal autocorrelaton does not enter as an addtonal varable. But t s captured by the covarance structure of the errors The lnear regresson wth spatally autoregressve errors s the most relevant n many cases, snce spatal dependence n the error term s lkely to be present n most data sets for contguous spatal areas Usually n regresson models we assume that, whch mples 2 E( ) 0, E( ) I that for each area k the expected value of the error s 0, and the varance of the error dstrbuton s E. 2. We also assume no covarance across areas, so that ( j, k) 0 The error at j s unrelated to the error at k, for all j,k. Now when we model spatal error dependence, we assume that E(, ) 0. j k
an autoregressve error process N W j j j1 2 E( u) 0, E( uu ) I E( u, u ) 0 j k u In matrx notaton, the spatal error model s y X W u E( ) ( I W ) ( I W ) 2 1 1
Dependent varable y Constant (X 1 ) Local ncome (X 2 ) Spatal error ML estmate t rato -412.944662-5.87 291.17 3.15 Wthncommutngdstance ncome (X 3 ) 49.06 7.83 Schoolng qualty (X 4 ) 134152.03 7.29 Number of households (X 5 ) -0.29-3.03 Spatal error lambda 0.740976 19.53 R 2 adjusted 0.7453 Standard Error 32.07 Dependent varable y ols estmate t rato Constant (X 1 ) -571.874-6.47 Local ncome (X 2 ) 864.0059 10.02 Wthncommutngdstance ncome (X 3 ) 57.7055 14.08 Schoolng qualty (X 4 ) Number of households (X 5 ) R 2 adjusted 0.567 Standard Error 42.113 175802.9235 7.74-0.7112-6.46 Created by demo_0.m Degrees of freedom 347 Moran's I Degrees of freedom 348 0.39369 11.29 Fngleton B (2006) A cross-sectonal analyss of resdental property prces: the effects of ncome, commutng, schoolng, the housng stock and
The spatal Durbn model: a catch all spatal model Ths ncludes a spatal lag Wy and a set of spatally lagged exogenous regressors WX y Wy X WX y = the dependent varable, an N x 1 vector Wy = the spatal lag, an N x 1 vector X = an N x K matrx of regressors, wth the frst column equal to the constant = a K x 1 vector of regresson coeffcents = the spatal lag coeffcent = an N x1 vector of errors WX s the N by K matrx of exogenous lags resultng from the matrx product of W and X s the correspondng coeffcent vector. Restrctng the parameters of the spatal Durbn leads back to the spatal lag model or to the spatal error model
spatal Durbn model : ML estmates Varable Coeffcent Asymptot t-stat z-probablty const -513.835677-4.146915 0.000034 local_ncome -7.730616-0.083091 0.933780 commutng_ncome 40.795703 6.257112 0.000000 supply -0.103221-1.106877 0.268347 schoolng 134249.627896 7.733356 0.000000 Wlocal_ncome 974.661531 6.096601 0.000000 Wcommutng_ncome -25.325850-3.358633 0.000783 Wsupply -0.496569-3.109303 0.001875 Wschoolng 8596.323682 0.265708 0.790464 rho 0.621996 13.257551 0.000000 Rbar-squared = 0.6549 Standard Error = 873.1750^0.5 = 29.55 Created by demo_0.m
spatal lag y Wy X WX f 0 then y Wy X spatal error y Wy X WX f then y X and W u
Endogenety of the spatal lag y Wy X y depends on y k whch s part of Wy y k depends on y so s a cause of y va Wy and a response to y hence also
Endogenety of the spatal lag y Wy X y depends on all other y s, ncludng y k because they are wthn Wy. But y k also depends on all other y s, ncludng y because they are wthn Wy. So Wy determnes y and s determned by t. So we have to use the approprate lkelhood functon or 2sls to obtan consstent estmates. N y f ( W y ) j j j1 N y f ( W y ) k kj j j1
y x 0 1 1 to estmate by OLS t s convenent to multply through by x and sum over all x y x x 2 1 1 1 1 Rearrangng gves x y x 1 1 1 2 2 x 1 x 1
1 1 1 ˆ 1 OLS estmator s 2 2 1 2 x 1 x 1 x 1 hence gves E( ˆ ) only f x y x x y 1 ˆ 1 1 2 x 1 E 1 1 1 E 2 x 1 x x x 1 2 x 1 and rearrangng and takng expectatons 0 s E( ˆ ) 1
So far we have been talkng about bas, but t mght be argued that ths wll dsappear as the sample sze gets large. In other words an estmator can be based but consstent. In fact for our OLS estmator ths s not the case. Summng over a larger number of cases ncreases both the numerator and the denomnator n x So the bas cannot be removed by smply ncreasng the sze of the sample. In order for an estmator to be consstent, t should tend towards unbasedness as the sample sze goes to nfnty. Usng the OLS estmator ths wll not happen so the OLS estmator s both based and nconsstent. 1 x 2 1
yˆ ˆ ˆ X ˆ X... ˆ X 0 1 1 2 2 k1 k1 ˆ s an unbased estmator of f E( ˆ ) ˆ s a consstent estmator of f ˆ ths means that as the sample sze n ncreases then the probablty approaches 1 that ˆ les wthn the range c to c where c s a small constant > 0 the small p stands for 'converges n probablty' to as n goes to nfnty p
Consstent OLS estmaton requres 1 plm n ( Wy Wy) Q a fnte nonsngular matrx 1 plm n ( Wy ) 0 The frst constrant can be satsfed wth proper constrants on and W (more about ths later) BUT the second condton does not hold for spatal data Instead 1 1 1 plm n ( Wy ) plm n W ( I W ) The exstence of W leads to a quadratc form, so that the plm does not equal zero (unless ρ = 0)
Inconsstency : smulaton assume that we have 81 regons formng a 9 by 9 lattce (lke a chessboard). Assume that wth W = 1 f locatons and j are contguous, and W = 0 otherwse j regons are contguous f they share an edge W s standardsed so that all the values n row are dvded by the sum of the row values y Wy X y = the dependent varable, an N=81 x 1 vector Wy = the spatal lag, an N=81 x 1 vector X = an N x K=3 matrx of regressors, wth the frst column equal to the constant, other two columns sampled from a unform dstrbuton = a K x 1 vector of regresson coeffcents, values 1,4,5 = the spatal lag coeffcent, value 0.75 = an N x1 vector of errors, drawn from an N(0,1) dstrbuton j
Outcome of 500 Monte Carlo smulatons, n each case drawng The errors from an N(0,1) dstrbuton to obtan y, and then regressng y On Wy and X
Spatal error model y X W u E( ) ( I W ) ( I W ) u d I 2 ~ (0, ) ( ) 1 I W u 2 1 1 1 y X ( I W ) u
Spatal error model ˆ 1 2 1 ~ N(0, Sxx ) 1 ˆ Sxx X ( X ) / n OLS s.e. 2 2 1 Sxx 2 1 true s.e. Sxx X [( I W ) ( I W )] XSxx 2 1 1 1 Typcally the true standard error wll be greater than the OLS s.e. and therefore usng the OLS s.e. we wll often reject the null hypothess that β1 = 0 when we should not reject.