5.4 The spatial error model Regression model with spatially autocorrelated errors

Transcription

1 54 The spatial error model 54 Regressio model with spatiall autocorrelated errors I a multiple regressio model, the depedet variable Y depeds o k regressors X (=), X,, X k ad a disturbace ε: (4) is a x vector of the geo-refereced edogeous variable Y, X a xk observatio matrix of the regressors ad ε a x disturbace vector However, differet from the stadard regressio model (see sectio 4), the disturbaces ε i are ot assumed to be idepedetl ideticall distributed (iid) Istead ε i is assumed to follow a spatiall autocorrelated process I priciple, the error process ca be modelled b a spatiall autoregressive process, (55) ε λwε ν or a spatiall movig average process, where ε i (4) is replaced b (56) Xβ ε ε ν Wν, W is a x spatial weight matrix, ν a + vector iid radom variables ad λ a spatiall autoregressive parameter

2 The model (4) with the error processes (55) or (56) is called a regressio model with spatiall autocorrelated disturbaces Both spatial error processes are first-order processes Usuall the error depedece is modelled b (55) This modellig approach is also adopted here The model (4) with the error process (55) is called regressio model with spatiall autoregressive disturbaces Whe the spatial error model (4) ad (55) is estimated b the method of maximum likelihood, the ormal assumptio for the error term must be imposed: (57) ν ~ N(, I ) As the error process (55) ca be writte i the form (58) ε λwε ν ( I W) ε ν ε ( I W) ν the spatial error model (4) with (55) is give i depedece of the iid radom term (=data geerated process) b (59) Xβ ( I W) ν

3 From ecoometric aalsis of time-series it is well-kow that autocorrelated errors will ot affect the propert of ubiasedess of the OLS estimator for β i the stadard regressio model The same holds i case of spatial error autocorrelatio, where OLS still provides ubiased estimates of the regressio coefficiets However, the regressio coefficiets are iefficietl estimated as their stadard errors are biased OLS ol provides efficiet parameter estimates if the covariace matrix of the errors ε i is a scalar matrix (σ I i the stadard regressio model, see sectio 4) For the error term (59), however, the covariace matrix reads or Cov( ε) E( εε') E{[( I W) E{( I W) ( I W) νν'[( I W) I[( I W) ν][( I W) ]' ]'} ν]'} (54) Cov( ε) E( εε') ( I W) [( I W') ] I ecoometric models with o-scalar covariace matrices, geeralised leastsquares (GLS) is usuall recommeded for efficiet parameter estimatio

4 The iefficiec of OLS estimates of the regressio coefficiets would ivalidate statistical iferece i the spatial error model (4)/(55) or (59) Particularl, sigificace tests for OLS estimated regressio coefficiets are o more valid The ivalidit of sigificace tests arises from biased estimatio of the variaces ad stadard errors of the OLS estimates for β ad λ As alread oted, applicatio of the geeralised least-squares (GLS) method will accomplish efficiet parameter estimates Alterativel, the method of maximum likelihood (ML) ca be applied It leads to estimates of the regressio coefficiets ad the error variace that maximise the joit probabilit desit of the give sample I our expositio, we will adopt the maximum likelihood approach of parameter estimatio 4

5 54 Maximum likelihood estimatio i the spatial error model The spatial error model discussed i sectio 54 is give b the equatios (4) ad (55), (4) Xβ ε (55) ε λwε ν or equatio (54) (59) Xβ ( I W) ν The errors ν i are assumed to be idepedetl ormall distributed: (57) ν ~ N(, I) B solvig (59) for, (54) ν ( I W)( Xβ) we obtai the Jacobia (545) ν J I W 5

6 Thus, the log-likelihood fuctio for of the spatial error model is obtaied b addig the term l I-λW to the log likelihood fuctio of the stadard regressio model (B5) ad replacig u b accordig to (54): (54a) l L β,,σ, X lπ l σ l I W ν' ν Replacig the radom vector b (54) the log-likelihood fuctio reads σ (54b) l L β,,σ, X lπ l σ l I W σ ( Xβ)'( I W)'( Ι W)( Xβ) Maximisig the log likelihood fuctio, l L,is equal to miimisig the sum of trasformed squared errors,, corrected b the log of the Jacobia, l I-W O accout of this correctio the ML estimates will differ from the OLS estimates The coicide for where the spatial error model approaches the stadard regressio model [ Asmptotic properties like cosistec, asmptotic efficiec ad asmptotic ormalit of the ML estimates will ol hold if regularit coditios are satisfied Particularl the Jacobia J is required to be positive: J I W ] 6

7 A ML estimator for is obtaied b settig the first partial derivative of the loglikelihood fuctio (54b) with respect to equal to zero This estimator is equal with the geeralised least-squares estimator (GLS estimator), βˆ ML βˆ GLS, as it ca be thought as a least-squares estimator resultig from a regressio of * o X* (544) * = (I - W) ad (545) X* = (I - W) X whe the autoregressive parameter is kow: (546a) ˆ ˆ βml βgls ( X*' X*) X*' * or, more detailed, (546b) βˆ βˆ ML GLS [ X'( I λw)'( I λw) X] X'( I λw)'( I λw) [ As (I - W) is the spatial differece operator, the trasformatio matrix (I - W) ca be iterpreted as a geeralised spatial differece operator For the trasformatios (544) ad (545) cosist o formig spatial differeces of the ivolved variables] A first-order coditio resultig from the partial derivative of l L with respect to gives the ML estimator for the error variace (547a) σˆ ML ( e λwe)'( e λwe) or (547b) σˆ ML e '( I λw)'( I λw) e with the residual vector (548) e X ( β ˆ ˆ ML β GLS ) 7

8 Note, however, that of both ML estimators, βˆ ad ML βˆ GLS σˆ ML, are fuctios of the autoregressive parameter A ML estimator of ca be foud b maximisig the cocetrated log-likelihood fuctio (549) l Lc( ) C l[ e'( I λ W)'( I λ W) e] l I λ W that is obtaied b isertig the ML estimates (546b) ad (547a/b) for the parameters ad C is a costat l L c is a oliear fuctio of for which o aaltical solutio exists Thus, a ML estimator for has to be foud b umerical optimisatio However, the residual vector e i l L c depeds idirectl also o, as the calculatio of βˆ ML βˆ GLS requires a value of Therefore a iteratio procedure must be coceived where both parameters ad are updated coditioal o oe aother 8

9 Flowchart of ML estimatio i the spatial error model Step : OLS estimatio of o X OLS estimator: β ˆ ( X' X) X' Step : Compute OLS residual vector: e Xβˆ Step : Search for the autoregressive parameter λ that maximises l L c give ˆβ GLS ad e Step 4: GLS estimatio of * = (I - W) o X* = (I - W) X give λ GLS estimator (546b): βˆ GLS [ X'( I λw)'( I λw) X] X'( I λw)'( I λw) Step 5: Compute GLS residual vector: e Xˆβ GLS Step 6: Covergece criterio met? o Step es Step 7 Step 7: Compute ML estimator for the error variace give e (with ML estimator (547b): σˆ ML e '( I λ W)'( I λ W) e ˆβ GLS ) ad λ 9

10 Example: The data o output growth (X) ad productivit growth (Y) of the 5-regio example are used to illustrate ML estimatio i the spatial error model Regio 4 5 Output growth (X) Productivit growth (Y) The spatial error model relates the regioal productivit growth to ow regio s output growth ad spatiall autocorrelated disturbaces: (55) i xi i wij j j The spatiall lagged error term ma capture measuremet errors due to iadequate delieatio of regios as well as spatial effects geerated b uobserved attributes

11 ML estimatio i the spatial error model To illustrate ML estimatio i the spatial error model, we compute the oe-step ML estimates, ie we give the ML estimates resultig after oe iteratio Step : OLS estimatio of o X (see sectio 4) ˆ β ( X' X) X' Step : Compute OLS residual vector (see sectio 4) e XβˆOLS

12 l L( ) Step : Maximisatio of the cocetrated likelihood fuctio L C give ˆβ GLS to obtai a ML estimator ˆ ML for the autoregressive parameter ad e Figure: Log L c fuctio of the exteded spatial lag model 5 ML estimator for : ˆ ML 98 Log Lc fuctio at ˆ ML 98 : l L c * = 59 (eglectig the costat C)

13 Step 4: GLS estimatio of * = (I - W) o X* = (I - W) X usig the estimate for λ ( ) obtaied i step ˆ ML / / / / / / / / / / / 98 ) ( * W I / / / / / / / / / / / 98 ) ( * X W I X

14 Step 5: Compute GLS residual vector: * *' *) *' ( ˆ ˆ GLS ML X X X β β ˆGLS β X e Step 6: Covergece criterio met? Ol oe iteratio is carried out for illustrative purposes 4

15 Step 7: Compute ML estimator for the error variace give e (with ˆβ GLS ) ad λ σˆ ML e '( I λ W)'( I λ W) e ML estimated spatial error model of productivit growth (usig the oe-step estimators): i xi i 98 wij j j 5