GMM estimation of spatial autoregressive models with unknown heteroskedasticity

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1 Accepted Mauscript GMM estimatio of spatial autoregressive models with ukow heteroskedasticity Xu Li, Lug-fei Lee PII: S (09) DOI: 0.06/j.jecoom Referece: ECONOM 3288 To appear i: Joural of Ecoometrics Please cite this article as: Li, X., Lee, L.-f., GMM estimatio of spatial autoregressive models with ukow heteroskedasticity. Joural of Ecoometrics (2009), doi:0.06/j.jecoom This is a PDF file of a uedited mauscript that has bee accepted for publicatio. As a service to our customers we are providig this early versio of the mauscript. The mauscript will udergo copyeditig, typesettig, ad review of the resultig proof before it is published i its fial form. Please ote that durig the productio process errors may be discovered which could affect the cotet, ad all legal disclaimers that apply to the joural pertai.

2 GMM Estimatio of Spatial Autoregressive Models with Ukow Heteroskedasticity Xu Li Lug-fei Lee Departmet of Ecoomics Departmet of Ecoomics Tsighua Uiversity The Ohio State Uiversity Beijig, Columbus, Ohio 4320 P.R.Chia USA First draft: November 2005 Revised draft: December 2006 This draft: September 2007 Abstract I the presece of heteroskedastic disturbaces, the MLE for the SAR models without takig ito accout the heteroskedasticity is geerally icosistet. The 2SLS estimates ca have large variaces ad biases for cases where regressors do ot have strog effects. I cotrast, GMM estimators obtaied from certai momet coditios ca be robust. Asymptotically valid ifereces ca be draw with cosistetly estimated covariace matrices. Efficiecy ca be improved by costructig the optimal weighted estimatio. The approaches are applied to the study of couty teeage pregacy rates. The empirical results show a strog spatial covergece amog couty teeage pregacy rates. JEL Classificatio: C3, C5, C2 Keywords: spatial autoregressio, ukow heteroskedasticity, robustess, cosistet covariace matrix, GMM Correspodig address: Lug-fei Lee, Departmet of Ecoomics, The Ohio State Uiversity, 40 Arps Hall, 945 N. High St., Columbus, OH We appreciate havig fiacial support for our research from the NSF uder grat o , ad thak Patricia Reaga for helpful commets ad the data source for our empirical study. We are grateful to the guest editors ad three aoymous referees for valuable commets ad suggestios o a earlier versio of this paper. i

3 . Itroductio May ecoomic processes, for example, housig decisios, techology adoptio, uemploymet, welfare participatio, price decisios, etc., exhibit spatial patters. Recetly, spatial models that have a log history i regioal sciece ad geography have received substatial attetio i various areas of ecoomics, icludig urba, evirometal, labor, developmetal ad others. But the allowace of depedece betwee observatios complicates the estimatio procedure ad calls for some specialized techiques. The most popular spatial ecoometric model is the spatial autoregressive (SAR) model (e.g.,() i Sectio 2). For a stadard SAR model where the error terms are assumed to follow a ormal distributio N(0, σ 2 ), the most covetioal estimatio method is the maximum likelihood (ML). Sice there is a Jacobia term, the determiat of the S (λ) i the likelihood fuctio, the ML method etails sigificat computatioal complexities. Eve though some simplificatio or approximatio techiques have bee suggested, 2 the computatio ivolved may still be demadig, especially for large sample sizes ad geeral spatial weights matrices. Aother estimatio procedure is the two stage least square (2SLS) for the mixed regressive, spatial autoregressive model (Kelejia ad Prucha (998); Lee (2003)). The 2SLS estimator (2SLSE) has the virtue of computatioal simplicity but it is iefficiet relative to the maximum likelihood estimator (MLE) sice it focuses oly o the determiistic part of the model, leavig the iformatio cotaied i the (reduced form) error terms uexplored. Furthermore, it will be icosistet whe all the exogeous regressors are irrelevat. Kelejia ad Prucha (999) propose a Method of Momet (MOM) method for the regressio model with spatial autoregressive disturbaces based o correlatios of sample observatios. But their estimator is iefficiet as compared to the MLE. Lee (200) geeralizes the MOM method ito a systematic geeralized method of momets (GMM) procedure based o quadratic momet fuctios ad shows the existece of the best GMM estimator (GMME), which ca be asymptotically as efficiet as the MLE. I Lee (2007a), a GMM procedure that combies both advatages of computatioal simplicity ad efficiecy is itroduced for the estimatio of the mixed regressive, spatial autoregressive model. It is show that the GMME ca be asymptotically more efficiet tha the 2SLSE ad that the best GMME exists ad it has the same limitig distributio as the MLE. The basic idea is to combie quadratic momets with the liear momets, where the latter are based o the orthogoality of the exogeous regressors with the model disturbaces that geerates the 2SLSE. S (λ) = I λw, where W is the spatial weights matrix. Note that its dimesio is, which is large for large sample sizes. 2 See, for example, Ord (975), Smirov ad Aseli (200).

4 All these ML, MOM ad GMM estimators are, however, desiged for models with homoskedastic disturbaces. The homoskedastic assumptio may be restrictive i practice. I certai applicatios, we would expect the variaces of the error terms to be differet. For istace, cosider the aalysis of the spatial depedece i the uemploymet or crime rates of cotiguous states i the Uited States. As a rate variable is a result of aggregatio, heteroskedasticity may be preset. I the presece of social iteractios, the variace of the aggregated level data will be iflated, with a extet depedig o the stregth ad structure of the iteractios. I a study of cross-city crime rates, Glaeser et al., (996) show that the high variace of cross-city crime rates is largely caused by social iteractios amog idividuals. Therefore, the presece of social iteractios could complicate the variace structure of aggregated data, especially whe social iteractio patters deped ot oly o the populatio size i the city, but also o the distributio ad compositio of the populatio. LeSage (999) illustrates how the mea ad variace of home sellig prices chage as we move across observatios with differet distaces from the cetral busiess district. More discussios o spatial heteroskedasticity ca be foud i Aseli (988). I this paper, we cosider the case whe the error terms i the model are idepedet but with ukow heteroskedasticity. If variaces of the disturbaces or the exact structure of heteroskedasticity are kow, we may get rid of the heteroskedasticity by some appropriate trasformatios ad the apply the covetioal MLE or GMM techiques to the trasformed model. But oe may ot have accurate iformatio about the ature of the heteroskedasticity i a model ad may be usure of the specific structural form of the variaces. With ukow heteroskedasticity, we would like to kow the cosequeces for various estimators if the SAR model were estimated as if the disturbaces were i.i.d. As will be show without takig ito accout the heteroskedasticity, the MLE is geerally icosistet. I cotrast, the GMME obtaied from certai carefully desiged momet coditios ca be robust agaist ukow heteroskedasticity. Furthermore, oe may improve the efficiecy by costructig optimal weightig for the GMM estimatio eve whe the form of heteroskedasticity is ukow. Sectio 2 discusses the possible icosistecy property of the MLE ad derives its asymptotic bias for some special case. Robust GMM estimatio uder ukow heteroskedasticity is cosidered i Sectio 3. Its cosistecy ad asymptotic distributio are derived. Sectio 4 cosiders the optimal weightig of the robust GMM estimatio. Some extesive Mote Carlo studies illustrate possible degrees of bias for the various estimators i fiite samples i Sectio 5. Sectio 6 presets 2

5 specificatio tests o the testig of ukow heteroskedasticity, ad some Mote Carlo results o levels of sigificace ad powers of the Hausma-type ad Lagrage Multiplier (LM) test statistics. A empirical applicatio o couty teeage pregacy rates is provided i Sectio 7. Coclusios are draw i Sectio 8. The techical details are give i the Appedix. 2. Icosistecy of the MLE i the Presece of Heteroskedastic Disturbaces The model cosidered is the mixed regressive, spatial autoregressive model Y = λ 0 W Y + X β 0 + ɛ, () where X is a k matrix of ostochastic exogeous variables, W is a spatial weights matrix of kow costats with zero diagoal elemets, ad the elemets ɛ i s of the -dimesioal vector ɛ are idepedet with mea 0 ad variaces σi 2, i =,,. The spatial effect coefficiet λ 0 measures the average ifluece of eighborig observatios o Y, which usually lies betwee (, ) whe W is row-ormalized such that the sum of elemets of each row is uity. For a geeral W which is ot row-ormalized, the λ 0 will usually be assumed to be i a parameter space which guaratees that the determiat of (I λ 0 W ) is positive. There will be more discussio o the parameter space of λ 0 later o. The reduced form of the model is Y = S X β 0 + S ɛ, where S = I λ 0 W. For the SAR model i (), uder the assumptio of i.i.d. N(0, σ 2 0) disturbaces, the log likelihood for this stadard model is l L (δ) = 2 l(2π) 2 l σ2 + l S (λ) 2σ 2 ɛ (θ)ɛ (θ), (2) where δ = (λ, β, σ 2 ), θ = (λ, β ), S (λ) = I λw, ad ɛ (θ) = S (λ)y X β. Give λ, () becomes a regressio equatio of S (λ) o X, ad, the MLE of β is β (λ) = (X X ) X S (λ)y (3) ad the MLE of σ 2 as σ 2 (λ) = [S (λ)y X β (λ)] [S (λ)y X β (λ)] = Y S (λ)m S (λ)y,, where M = I X (X X ) X. The, we ca get the cocetrated log likelihood fuctio of λ, which is l L (λ) = 2 (l(2π) + ) 2 l σ2 (λ) + l S (λ). (4) The first order coditio for the cocetrated log likelihood fuctio is l L (λ) = λ σ (λ) 2 Y W M S (λ)y tr(w S (λ)). (5) 3

6 For cosistecy of the MLE λ, the ecessary coditio is plim l L (λ 0) λ = 0. But with heteroskedastic disturbaces, this coditio may ot be satisfied. Cosequetly, the cosistecy of the MLE is ot guarateed. I the presece of heteroskedasticity, at the true parameter λ 0, σ 2 (λ 0 ) = [S Y X β (λ 0 )] [S Y X β (λ 0 )] = ɛ M ɛ = So, σ 2 (λ 0 ) ad the average of σ 2 i, σ2 are asymptotically equivalet. 3 from equatios (5) ad (6), we have, at λ 0, σi 2 + o p (). (6) i= Let G = W S. The, l L (λ 0 ) = λ [ σ (λ 2 0 ) Y W M S Y tr(w S )] = ɛ G M ɛ + (X β 0 ) G M ɛ ɛ M ɛ ɛ M ɛ tr(g i= ) = G,iiσi 2 G + o p () i= σ2 i i= = [G,ii G ](σi 2 σ2 ) σ 2 + o p () = COV (G,ii, σi 2 ) σ 2 + o p (), (7) where G = tr(g ) = i= G,ii. Therefore, the limit of l L (λ 0) λ will be zero if ad oly if the covariace betwee the diagoal elemets of the matrix G, G,ii, i =,,, ad the idividual variaces σi 2, i =,,, is zero i the limit. I the heteroskedastic case, this coditio will be satisfied if almost all the diagoal elemets of the matrix G are equal. 4 It is of iterest to see whe we would have costat diagoal elemets i the G matrix for some special cases. Cosider a circular world where the uits are arraged o a circle such that the last uit y has eighbors y ad y, y has eighbors y 2 ad y, ad so forth. 5 If we assig equal weight to each eighbor of the same uit, the diagoal elemets of the resultig G matrix will be costat. The uits i a circular world ca have more eighbors, as log as each uit has the same umbers of eighbors ad with half of the eighbors lead ad the rest lag, the diagoal elemets of the G matrix will be the same. Aother special case is that W is a block-diagoal matrix with a idetical submatrix i the diagoal blocks ad zeros elsewhere. This correspods to the group iteractios sceario where all the group sizes are equal, ad each eighbor of the same uit is assiged equal weight. Whe these special spatial weights matrices are used, the MLE will still be cosistet i the presece of ukow heteroskedasticity. But for geeral spatial weights matrices, the cosistecy is ot esured. 3 The asymptotic argumets ca follow from the law of large umbers i the Appedix. I this sectio, we do ot provide the rigorous aalysis i order to save space. 4 It will be zero if ɛ i s are i.i.d., sice i that case σ 2 i = σ2, equatio (7) will coverge to zero regardless of the diagoal elemets of the matrix G. 5 Kelejia ad Prucha (999) use this type of weights matrix i their Mote Carlo study. 4

7 Followig the icosistecy of the MLE of λ 0, a cosequece is the icosistecy of the MLE of β 0. Because from (3), we have β ( λ) = β 0 + (λ 0 λ)(x X ) X G X β 0 + o p (), (8) which will ot coverge to β 0 i the limit if λ is ot cosistet. Thus, besides the computatioal burde it etails, the MLE for the SAR model with ukow heteroskedasticity is icosistet as log as the diagoal elemets of the matrix G are ot all equal. Because of the oliearity of λ i the cocetrated log likelihood fuctio, it is hard to make ay geeral coclusio about the asymptotic bias of λ. For the asymptotic bias of β ( λ) from (8), it is (λ 0 λ)(x X ) X (G X β 0 ). Thus, give the bias of λ, the asymptotic bias of β ( λ) is determied by the term (X X ) X (G X β 0 ), which is the OLSE of the coefficiet i the artificial regressio of G X β 0 o X. Thus, give the bias of λ, the relative asymptotic bias of β ( λ) depeds o the properties of X ad W. Cosider a special case, which is ofte used i empirical social iteractio studies. This is the case of group iteractios, where W is assumed to be a block-diagoal matrix, ad i each block, W r = m r (l m r l m r I mr ), r =,, R, where R is the umber of groups, m r is the group size for group r, l mr is the m r -dimesioal vector of oes, ad I mr is the m r -dimesioal idetity matrix. Note that the group sizes are ot all equal, ad for the asymptotic properties, we let the umber of groups R go to ifiity while maitaiig {m r } is bouded. This iteractio patter meas that there are o cross group iteractios ad a uit is equally affected by all the other members i the same group. Group could be village, class, ad the like. This group iteractio settig has bee studied by Case (99), Lee (2004, 2007c), amog others. Let s assume for all the groups, the x s are i.i.d. with mea µ ad variace Σ x for all observatios. ( ) I particular, i group r, let X (r) = (l mr, z (r) ), X (r) = (, z (r) ), µ = (, µ z ), ad 0 0 Σ x =, where z 0 Σ (r) = (z z r,, z m r,r) is the matrix of regressors excludig the itercept term ad z (r) = i= z ir. The after some calculatios we ca get X G X = ad (X X ) = m r mr R ( m r λ 0 m r r= λ 0 z (r) [ R r= ( m r mr i= z ir m r λ 0 z (r) z (r) mr i= z ir mr i= z ir z ir m r λ 0 z (r) mr m r +λ 0 )]. Note that i= (z ir z (r) ) (z ir z (r) ) lim {E( µ µ R X G X ) [ + R λ 0 λ 0 Σ x R m r ( )Σ x ]} = 0 (0) m r + λ 0 5 r= ) (9)

8 ( ) + ad (E( X X )) µz Σ = z µ z µ z Σ z Σ z µ z Σ. Thus, we ca get z lim R (E(X X )) E(X G X ) = lim ( λ 0 ( λ 0 R R m r λ 0 + R r= m r +λ 0 )µ z 0 ( R λ 0 R m r r= m r +λ 0 )I z ), () where I z is the (k )-dimesioal idetity matrix. Therefore, i this group iteractio settig with radomly distributed x s, if all the elemets i x except the costat term have zero mea, i.e., µ z = 0, the relative asymptotic bias of the itercept β 0 will be λ 0 times the bias of the MLE of λ 0. Also, except the itercept β 0, the MLE for all the other β 0 s have the same magitude of relative asymptotic bias, which is the term ( R λ 0 R m r r= m r +λ 0 ) times the bias of the MLE m r of λ 0. As ( R λ 0 R r= m r +λ 0 ) is less tha R ( λ ad 0) R is the average group size, the relative asymptotic bias of the itercept will be larger tha those of the other regressio coefficiets i β 0. I particular, if the average group size is moderately large, the biases of the coefficiets of regressors (rather tha the itercept term) ca be small. The precedig paragraph has cosidered the asymptotic bias of the MLE uder heteroskedasticity. Likewise, the MOM estimator suggested by Kelejia ad Prucha (999) is ot cosistet i the presece of ukow heteroskedasticity sice the momet coditios they proposed do ot have zero mea at the true parameters. The followig sectio discusses the feature of GMM estimatio ad possible robust estimatio. 3. GMM Estimatio Agaist Ukow Heteroskedasticity 3. A Brief Overview The cosistecy of the GMME i Lee (200, 2007a) with P from P which is a class of costat matrices P with tr(p ) = 0; or P 2, a subclass of P with Diag(P ) = 0, is based o the fudametal momet property that E(ɛ P ɛ ) = 0. If the ɛ i s have heteroskedastic variaces, E(ɛ P ɛ ) = tr[p E(ɛ ɛ )] will ot ecessarily be zero if P is from P \ P 2. Cosider the i th compoet of P ɛ, j= P,ijɛ j, which is clearly correlated with the correspodig compoet ɛ i of ɛ if P,ii 0. With homoskedastic disturbaces, the correlatios of P ɛ ad ɛ ca be caceled out as log as tr(p ) = 0. I the presece of heteroskedastic error terms, lettig tr(p ) = 0 may ot guaratee the correlatios betwee each compoet of P ɛ ad the correspodig compoet of ɛ exactly caceled out. Therefore, whe P is from P but ot P 2, P ɛ may be correlated with ɛ ad thus loses its validity as a istrumetal variable (IV) vector. I cotrast, if P is from P 2, E(ɛ P ɛ ) = 0 is true sice tr[p E(ɛ ɛ )] = tr[diag(p )E(ɛ ɛ )] = 0. We successfully maitai the ucorrelatio betwee P ɛ ad ɛ by excludig each compoet of ɛ from the correspodig term of P ɛ. Thus, i the presece of ukow heteroskedasticity, the GMM estimatio for the 6

9 SAR model will be based o P 2 but ot P. Lee (200) has oticed this possible robust property of usig quadratic momets with the matrix P s from P 2 but has ot provided ay rigorous theory. This paper follows up o this observatio ad will provide a rigorous theory ad ivestigate fiite sample properties i Mote Carlo studies for the SAR model. The MOM method suggested i Kelejia ad Prucha (999) uses essetially the two momets ɛ W ɛ ad ɛ (W W tr(w W) I )ɛ. While W has zero diagoal ad the momet ɛ W ɛ is robust agaist ukow heteroskedasticity, the other momet is ot as the diagoal of [W W tr(w W) I ] may ot be zero. A robust versio of this MOM method may replace the secod momet fuctio by ɛ (W W Diag(W W ))ɛ, where Diag(A) for a square matrix A deotes the diagoal matrix formed by the diagoal elemets of A Robust GMM Estimatio To aalyze rigorously the robust property of GMM estimatio with P 2, we adopt most regularity assumptios for GMM estimatio i Lee (200, 2007a) with proper modificatios to fit ito the heteroskedasticity settig. Iterested readers may refer to Lee (200, 2007a) for detailed discussios o related assumptios for the i.i.d. disturbaces case. 7 Assumptio. The ɛ i s are idepedet (0, σi 2 ) with fiite momets larger tha the fourth order such that E ɛ i 4+η for some η > 0 are uiformly bouded for all ad i. This assumptio implies the uiform boudedess of the variaces σ 2 i, the third momets, µ i,3 ad the fourth momets µ i,4 of ɛ i are also uiformly bouded for all ad i. Assumptio 2. The elemets of the k regressor matrix X are uiformly bouded costats, X has the full rak k, ad lim X X exists ad is osigular. Assumptio 3. The spatial weights matrices {W } ad the matrix {S } are uiformly bouded i absolute value i both row ad colum sums. This uiform boudedess assumptio limits the spatial depedeces amog the uits to a tractable degree ad is origiated by Kelejia ad Prucha (999). It rules out the uit root case (i time series as a special case). Let Q be a k matrix, where k k +, of IV s costructed from X ad W, such as X, W X, W 2 X, etc. The momet fuctios correspodig to the orthogoality coditios of X ad ɛ are Q ɛ (θ). But these liear momets reflect oly the iformatio i the determiistic part of 6 After the completio of this paper, we realize that Kelejia ad Prucha (2005) has exteded their approach i Kelejia ad Prucha (999) to cover the estimatio of the SAR model with spatial SAR process with ukow heteroskedaticity. Their approach for the SAR disturbace process has used the two momets ɛ W ɛ ad ɛ (W W Diag(W W)) ɛ, where ɛ is a estimated residual. For the SAR regressio equatio, they suggest the use of geeralized two stage least squares. 7 I this paper, we do ot cosider the large group iteractios case so as to simplify the presetatio. 7

10 W Y, leavig those i the stochastic part uexplored. This ca be see from the reduced form of the model. If λw < where is a matrix orm, we have (I λw ) = I +λw +λ 2 W 2 +, ad the reduced-form equatio becomes Y = S X β 0 + S ɛ = X β 0 + λ 0 W X β 0 + λ 2 0WX 2 β S ɛ. (2) It is obvious from (2) that formig IV vectors from fuctios of W ad X focuses oly o the iformatio i the ostochastic part E(W Y X ) of W Y. Lee (2007a) suggests the use of the momet coditios (P j ɛ (θ)) ɛ (θ) i additio to Q ɛ (θ). These additioal momets capture the correlatios across the spatial uits. They serve as the IV for G ɛ, the other compoet of W Y. 8 The matrices i P 2 (more geerally, P ) are assumed to have similar uiform boudedess property as i W ad S. Assumptio 4. The matrices P j s with Diag(P j ) = 0 are uiformly bouded i both row ad colum sums, ad elemets of Q are uiformly bouded. The set of momet fuctios for the GMM estimatio is as follows g (θ) = (P ɛ (θ),..., P m ɛ (θ), Q ) ɛ (θ) = (ɛ (θ)p ɛ (θ),..., ɛ (θ)p m ɛ (θ), ɛ (θ)q ). (3) Deote V ar(g (θ)) = Ω ad, for ay square matrix A, A s = A + A is the sum of A ad its traspose. Let Σ = Diag{σ, 2, σ}, 2 where σi 2 = E(ɛ2 i ), i =,,. Assumptio 5. Either (a) lim Q (G X β 0, X ) has the full rak (k + ), or (b) lim Q X has the full rak k, lim tr(σ G s P j ) 0 for some j, ad lim (tr(σ G s P ),..., tr(σ G s P m )) ad lim (tr(σ G P G ),..., tr(σ G P m G )) are liearly idepedet. This assumptio assures the idetificatio of θ 0 from the momet equatios E(g (θ 0 )) = 0 for sufficietly large. If G X β 0 ad X are liearly depedet, which icludes the case whe all exogeous variables X are irrelevat, the additioal momets i (b) will help to idetify θ 0. Ad the parameter space Θ of θ is assumed to have the followig property: Assumptio 6. The θ 0 is i the iterior of the parameter space Θ, which is a bouded subset of R k Note that W Y = G X β 0 + G ɛ. 9 For oliear extremum estimatio methods, such as the ML method, compactess o the parameter space Θ is usually eeded i order to apply some uiform laws of large umbers to demostrate cosistecy of extremum estimates (Amemiya 985). However, for our GMM approach with liear ad quadratic fuctios, θ appears oliearly i momet coditios i terms of polyomials. For S (λ), oly its value evaluated at cosistet estimates of λ 0 will be used. So for asymptotic aalysis, boudedess of Θ will be sufficiet. 8

11 The parameter space of λ is usually take to be (, ) whe W is a row-ormalized matrix. For the cases i which W is ot ormalized but its eigevalues are real with its largest eigevalue µ,max > 0 ad its smallest eigevalue µ,mi < 0, the parameter space ca be the iterval ( µ,mi, µ,max ) (Aseli 988). Kelejia ad Prucha (2005) allow complex eigevalues for W ad suggest the parameter space ( τ, τ ) where τ is the spectral radius of W. These parameter spaces are desiged to guaratee that the determiat of (I λw ) is positive. Kelejia ad Prucha (2005) also allow the parameters, icludig λ, to deped o as they are the resulted parameters after W beig rescaled by a ormalized factor which depeds o. If W is rescaled by the divisio with τ, the coefficiet λ (= τ λ) ca the be take as (, ). For our GMM estimatio, oe does ot eed to impose a specific parameter space for the miimizatio of the GMM objective fuctio because it is simply a polyomial fuctio of θ. So the regularity coditio i the precedig assumptio o the parameter space is solely for the theoretical purpose of provig cosistecy of the GMM estimator. As we do ot emphasize o ay scale ormalizatio of W, we simply cosider θ 0 beig a costat parameter vector. The followig propositio cocers about the asymptotic property of a GMM estimator i the geeral Hase GMM settig with a liear trasformatio a g (θ) of the momet fuctios g (θ), where a is a matrix with a full row rak greater tha or equal to the umber of parameters i θ. The a a i the GMM objective fuctio g (θ)a a g (θ) is a oegative defiite matrix, which represets a weightig matrix i this distace fuctio. This geeral framework motivates the issue of optimum weightig matrix. (200) to the heteroskedastic case. Propositio. Propositio below is a geeralizatio of Propositio 2. i Lee Suppose that diag(p j ) = 0 for j =,, m, ad Q is a k IV matrix so that lim a E(g (θ)) = 0 has a uique root at θ 0 i Θ. The, uder the stated assumptios -6 ad that lim a D exists ad has the full rak (k + ), the RGMME θ derived from mi θɛθ g (θ)a a g (θ) is a cosistet estimator of θ 0, ad ( θ θ 0 ) D N(0, Γ), where Γ = lim Ω = V ar(g (θ 0 )) = (D a a D ) D a a Ω a a D (D a a D ), (4) tr[σ P (Σ P ) s ] tr[σ P (Σ P 2 ) s ]... 0 tr[σ P 2 (Σ P ) s ] tr[σ P 2 (Σ P 2 ) s ] Q Σ Q 9

12 = i= i= j= P,ij(P,ij + P,ji )σi 2 σ2 j... 0 j= P 2,ij(P,ij + P,ji )σi 2 σ2 j Q Σ Q, (5) tr(σ P s G ) 0 D = E(g (θ 0 )) =.. θ tr(σ PmG s ) 0. (6) Q G X β 0 Q X The proof is similar to the i.i.d. case oce we realize that the uiform covergece of sample averages of relevat momet fuctios ca hold i the presece of heteroskedasticity ad the cetral limit theorem for liear-quadratic forms by Kelejia ad Prucha (999) allows for heteroskedastic disturbaces. The details of the proofs of all propositios are give i the Appedix. From Propositio, the RGMME obtaied from a arbitrary weightig matrix (with momet fuctios costructed from P 2 ) ca be cosistet (robust) agaist ukow heteroskedasticity. I particular, if we costruct the optimal GMM as i the i.i.d. case without takig ito accout the presece of heteroskedasticity, i.e., if we replace the weightig matrix a a by ( Ω ), where Ω is a estimator of Ω based o a iitial estimate of θ as if ɛ i s were i.i.d., the resultig GMME will still be cosistet ad asymptotically ormal. But the correct asymptotic covariace matrix will ot be the oe, (lim D Ω lim D ), i the i.i.d. case. Istead, it will take the messier form of (D Ω D ) D Ω Ω Ω D (D Ω D ), (7) where Ω is the probability limit of Ω, whose value depeds o the specific formula of Ω. Furthermore, as a special case of the GMM estimatio, the 2SLS estimatio with a = (0, (Q Q ) /2 ) ad a g (θ) = (Q Q ) /2 Q ɛ (θ) ca be cosistet from Propositio. 0 It ca also serve as the iitial cosistet estimator i our GMM estimatio. I order to make asymptotically valid ifereces from the RGMME, we eed to fid a cosistet estimator of the asymptotic variace as give i (4). As i White (980), we ca cosistetly estimate the part Q Q i Ω i (5) without beig able to estimate, which ivolves ukows, cosistetly. The tricky part is the estimatio of the other elemets associated with the quadratic momet fuctios. Those elemets cosist of times a sum of 2 terms. However, the uiform boudedess property of P esures the covergece of these sums. The followig propositio ca be used to provide a cosistet estimator for the covariace matrix Ω. 0 Assumptio 5(a) is crucial for the cosistecy of the 2SLSE. 0

13 Propositio 2. Uder the assumed regularity coditios, ( D D ) = o P () ad ( Ω Ω ) = o P (), where D ad Ω are, respectively, estimators of D ad Ω with θ 0 replaced by a cosistet iitial estimator θ ad Σ by Σ, where Σ = Diag{ ɛ 2,, ɛ 2 } ad ɛ i s are the residuals of the model with θ 0 estimated by θ. 4. Optimal RGMM Estimator From the precedig sectio, we see that the cosistecy of the RGMME is, i geeral, ot affected by the choice of the weightig matrix, but its asymptotic variace is. By usig a wrog weightig matrix, we ll still get the cosistet estimator but the estimator may ot be efficiet. By the geeralized Schwartz iequality, the optimal weightig matrix for the GMM estimatio with the momet fuctios g (θ) is Ω, the iverse of the covariace matrix for the momet fuctios g (θ 0 ). Propositio 3 shows that, with a cosistet estimator Ω, the feasible optimal RGMME obtaied from mi θɛθ g (θ) Ω g (θ) will be cosistet ad asymptotically ormal with variace (lim D Ω D ). The variace matrix Ω is assumed to satisfy some covetioal regularity coditios. Assumptio 7. The lim Ω exists ad is osigular. Propositio 3. Suppose that ( Ω ) ( Ω ) = o p (), the the feasible optimal ORGMME θ o, derived from mi θɛθ g (θ) Ω g (θ) has the asymptotic distributio ( θo, θ 0 ) D N(0, (lim D Ω D ) ). (8) Similarly, a cosistet estimator for the asymptotic covariace matrix is ( D Ω D ). The optimal ORGMME here refers to the RGMME based o the optimal weightig with specified momet fuctios. I the i.i.d. disturbaces case, the best choices P from P 2 ad Q are available, which are, respectively, kow as (G Diag(G )) ad (G X β 0, X ). However, for the case with ukow heteroskedasticity, the best selectio of P ad Q may ot be available. This is so because tr(p s G Σ ) 0 D =.. tr(pmg s Σ ) 0 Q G X β 0 Q X If the P ad Q used ivolve the ukow parameters λ 0 ad β 0, the feasible RGMM estimatio will be carried out with λ 0 ad β 0 replaced by some iitial cosistet estimators λ, β. The resultig feasible RGMME will have the same limitig distributio. The proof is similar to the i.i.d. case thus is omitted here. Details ca be foud i Propositio 2.3 i Lee (200).

14 ad Ω = tr(σ P (Σ P ) s ) tr(σ P (Σ P 2 ) s )... 0 tr(σ P 2 (Σ P ) s ) tr(σ P 2 (Σ P 2 ) s ) tr(σ P m (Σ P ) s ) tr(σ P m (Σ P 2 ) s ) Q Σ Q ivolve the ukow Σ. If a best selectio were available, they would ivolve the matrix Σ but the latter has a ukow form. I practice, the selectio of cosistetly estimated (G Diag(G )) ad (G X β 0, X ) might be a desirable strategy. Remark: The results i Propositios ad 3 are derived for the spatial sceario where each of the spatial uits iteracts with oly a few eighborig oes. This is the typical case i spatial models. However, some models with social iteractios, i particular, ivolvig all members i a group settig, ivolve large group iteractios. The large group iteractios case has bee studied i Lee (2004) for the ML estimatio, ad Lee (2007c) for a coditioal ML approach. For the GMM estimatio, it is i Lee (2007a) for the SAR model with homoskedastic disturbaces. To simplify presetatios, we have ot cosidered the large group iteractios case i this paper. However, it will be of iterest to have some remarks o this sceario. I the large group iteractios sceario (Lee 2004, 2007b, 2007c), a spatial uit may be iflueced by may eighborig uits, but each of its eighbors ifluece will be uiformly small i the sese that elemets of W = (w,ij ) are of order O( h ) uiformly i all, i ad j, where h as. Similar results of Propositios ad 3 ca hold with some proper modificatios ad additios of the assumed regularity coditios. For the large group iteractios case, while h, it shall be assumed that lim h (9) = 0 i order to obtai cosistet estimates. Assumptio 4 eeds to be stregtheed i that elemets of P j s are of order O( h ) uiformly i i, j ad so that their magitudes are compatible with those of elemets of W. With Assumptio 5(a) i additio to the (modified) Assumptios -4, the results i Propositio will be valid. The results i Propositio 3 will also be valid if Assumptio 6 is replaced by that lim h Ω exists ad osigular. Note that uder Assumptio 5(a), the quadratic momets will be domiated by the liear momets i the GMM estimatio ad the GMM estimates will be asymptotically equivalet to the 2SLS estimates uder the large group iteractios. (Lee 2007b). However, whe Assumptio 5(a) fails i that G X β 0 ad X are liearly depedet, the quadratic momets will be useful. Whe G X β 0 ad X are multicolliear, there would be o (extra) IV variable available for W Y or liear momets. The λ 0 ca oly be estimated via the 2

15 quadratic momets uder the modified Assumptio 5(b): lim h tr(σ G s P j ) 0 for some j, ad lim [ h tr(σ G s P ),, h tr(σ G s P m )] ad lim [ h tr(σ G P G ),, h tr(σ G P m G )] are liearly idepedet. The diverget rate of h to ifiity shall satisfy the coditio lim h + 2 δ = 0 for some δ > 0 such that E ɛ,i 4+2δ are uiformly bouded i all ad i. This stregtheed coditio is eeded i order to apply the geeralized CLT for liear ad quadratic form i Lee (2004). For this case, while the GMM estimates ca be cosistet, their rates of covergece will be of order O( h ), which is lower tha the order of the case without multicolliearity. Iterested readers ca cosult Lee (2007b) for more details. 5. Mote Carlo Study Some Mote Carlo experimets are desiged to study the fiite sample properties of the various robust ad o-robust estimators. We focus o the case of group iteractios. The data geeratig process is as follows. There are two regressors i additio to the itercept term, which are geerated as x ir, N(3, ) ad x ir,2 U(, 2). The size of each group is determied by a uiform U(3, 20) variable (roud to the closest iteger), so the mea group size is about. The error terms are ormally distributed with mea zero ad their variaces vary across groups. We cosider several variace structures with special attetio o this particular desig: for each group, if group size is greater tha 0, the the variace is costructed to be the same as group size, otherwise, the variace is the square of the iverse of the group size (V-D). This desig V-D emphasizes a oliear variace structure. The variace fuctio is decreasig ad the icreasig. Aother simpler variace desig assumes that the variace is the iverse of group size (V-D2). For compariso purpose, the correspodig baselie homoskedastic case has disturbaces beig i.i.d. N(0, σ 2 ), where σ 2 is the mea of the variaces of the heteroskedastic errors. For each of the variace desigs, several sets of true parameters are cosidered. Parameter desig (P-D) has θ 0 = (λ 0, β 0, β 20, β 30 ) = (0.2, 0.8, 0.2,.5), ad desig 2 (P-D2) has θ 0 = (λ 0, β 0, β 20, β 30 ) = (0.2, 0.2, 0.2, 0.). The stochastic part of the model with P-D2 becomes relatively more domiat tha that of P-D, sice the determiistic regressio part of the model has the smaller coefficiets o the X s. We expect that it would be difficult to deal with P-D2 by the 2SLS approach as its regressors have much smaller effects o Y. I additio for λ 0 = 0.2, we also cosider a stroger iteractio effect model with λ 0 = 0.6. The parameter desig P-D3 has θ 0 = (λ 0, β 0, β 20, β 30 ) = (0.6, 0.8, 0.2,.5), ad P-D4 has θ 0 = (λ 0, β 0, β 20, β 30 ) = (0.6, 0.2, 0.2, 0.). 2 2 I additio to λ 0, we also pay attetio to x ad its coefficiets. We are iterested i comparig the 2SLS ad the robust GMM estimates. The 2SLS estimates might be sesitive to x ad its coefficiets, sice the 2SLS forms estimatio based oly o the determiistic part of the model, which is determied by the importace of x. 3

16 The models are estimated by the method of maximum likelihood (ML); the o-robust GMM (GMM) with P = (G tr(g) I ) ad IV matrix (G X β, X ); the robust GMM (RGMM) with P = (G Diag(G )) ad IV matrix (G X β, X ). 3 Both the GMM ad RGMM approaches will require a iitial estimate i the evaluatio of G (ad β i G X β). The iitial estimate used ca be from a simple 2SLS or a simple first step GMM. The simple first step GMM (SGMM) uses P = W ad the liearly idepedet colums of (W X, X ) as IV s without a weightig matrix. For the simple 2SLS (2SLS), the IV s used are simply the liearly idepedet colums of (W X, X ). Ad for the weightig matrices i the GMM ad RGMM approaches, we use the variace formulas for the i.i.d. case. For the RGMM approach, the optimal weightig based o the robust variace formula uder ukow heteroskedasticity will also be cosidered, which is the ORGMM. Whe IV matrix W 2 X i additio to (W X, X ) are used i a 2SLS estimatio, it is oted as 2SLS-2 estimatio. The feasible best 2SLS with the IV matrix (G X β, X ), evaluated at the simple 2SLSE, will be deoted by B2SLS. For the feasible GMM ad RGMM, the SGMME is usually used as the iitial estimate of G. Whe the simple 2SLSE is used istead, the correspodig approaches will be deoted as GMM(2sl) ad RGMM(2sl). For each case, the results reported are based o 000 Mote Carlo replicatios. The umbers of groups R are 00 ad For the estimates of each coefficiet, we report the empirical mea (Mea), the correspodig bias (Bias), the empirical stadard error (SD), ad the root mea square error (RMSE). Table summarizes the results from V-D with P-D. The case with small coefficiets of β 0 s i P-D2 is reported i Table 2. The estimates reported i these two tables focus o the MLE, o-robust GMME, RGMME, ORGMME, ad 2SLSE. We compare the fiite sample biases of these robust ad o-robust estimates, ad their relative efficiecy i terms of SD ad RMSE. Table 3 supplemets the results i Tables ad 2 with additioal estimators, such as the 2SLS-2, B2SLS, SGMM, GMM(2sl) ad RGMM(2sl) estimators, for compariso purposes. To ecoomize the presetatio, oly results for R = 00 are reported i Table 3, 4 ad 5. Table 4 presets the results with P-D3 ad P-D4, where λ 0 = 0.6. Results for the variace desig V-D2 with the four parameter sets are reported i Table 5. The saliet features of results for various estimators are summarized i the followig list: For the i.i.d. disturbaces case, the MLE has some biases i λ 0 ad the itercept term β 0 whe R = 00. These biases become small whe R icreases to 200. With heteroskedastic disturbaces, 3 The matrices correspod to the best P ad Q i the i.i.d. case. 4 We have also experimeted with R = 50. Because of space limitatio, those results are ot reported here but they ca be foud i the workig paper versio of this paper. 4

17 the MLE ca be biased i λ 0 ad β 0 eve i large sample R = 200. The bias of the estimate of λ 0 is dowward. However, those biases are ot statistically sigificat eve with R = 200. The estimate of the itercept term is biased upward. The estimates of the regressio coefficiets β 20 ad β 30 are ubiased eve for the heteroskedastic cases. These patters hold i Tables ad 2 for both P-D ad P-D2 with large or small coefficiets β 0 s for V-D. The features of the biases of the MLE of λ 0 hold with P-D3 ad P-D4 i Table 4 uder the same desig V-D. With V-D2 (ad all P-D, P-D2, P-D3, ad P-D4) i Table 5, the MLE s are essetially ubiased for all the parameters, eve whe there are heteroskedastic disturbaces. I terms of bias, the GMME has similar patters as the MLE. I terms of magitudes of the biases, some may be slightly better tha those of the MLE but are mostly similar. For the RGMM, the RGMME s are essetially ubiased for all the cases (i Tables, 2, 4, ad 5 ). The 2SLSE s are cosistet i theory. However, its fiite sample performace i terms of bias ca vary, depedig o the patter of variaces of the disturbaces ad the parameter values. With P-D2 ad P-D4 uder V-D, where β 0 s are small, the 2SLSE s for λ 0 ad β 0 ca have large biases eve for R = 200 (i Tables 2 ad 4). These are also accompaied by relatively large SD s. This is so regardless whether the disturbaces are i.i.d. or heteroskedastic. For the other parameter desigs with larger β 0 s (P-D i Table, P-D3 i Table 4 or V-D2 i Table 5), the performace of the 2SLSE s i terms of bias is satisfactory. This 2SLS uses (W X, X ) as IV s. For the desig P-D2 with V-D, the 2SLS-2 uses additioal IV s WX 2 may reduce the bias oly a little i Table 3. The 2SLSE s for λ 0 ad β 0 have the largest SD ad RMSE compared with those of the MLE s ad the various GMME s (uder V-D i Tables, 2 ad 4, ad uder V-D2 i Table 5, for all parameter desigs). With the additioal IV s WX 2 i 2SLS-2 (i Table 3), the SD ad RMSE ca be slightly reduced. I these fiite samples, the SD ad RMSE of the B2SLSE ca eve be larger tha those of the 2SLSE. Uder V-D, whe the coefficiets β 0 s are small, the biases ad SD s of the various 2SLSE s for λ 0 ad β 0 are too large to be acceptable. Whe the 2SLSE is poor, it has cosequeces for the GMM ad RGMM approaches if it is used as a iitial estimate for G ad G X β. I Table 3 with P-D2 i V-D, the GMME(2sl) ad RGMME(2sl) are poor as they have large biases ad SD s i λ 0 ad β 0. Whe the 2SLSE s are satisfactory for P-D, the GMME(2sl) ad RGMME(2sl) i Table 3 are comparable with the correspodig GMME ad RGMME i Table (i both Mea ad SD). I terms of SD ad RMSE, the GMME ad MLE are similar uder all the desigs (as reported i 5

18 Tables, 2, 4, ad 5). The SD s of the GMME ad MLE of λ 0 uder heteroskedasticity are slightly larger tha those uder i.i.d. disturbaces for V-D. With V-D, the RMSE s of the MLE ad GMME of λ 0 uder heteroskedastic misspecificatio are larger tha those of the correctly specified i.i.d. cases. The correspodig RMSE s for the itercept term are larger but to a smaller degree. For V-D2 (i Table 5), those SD s ad RMSE s are mostly similar for all parameter desigs. As for a compariso of the SGMME i Table 3 with the GMME i Tables ad 2, the SGMME s are less efficiet i λ 0 ad β 0. 5 The RGMME does ot seem to lose efficiecy compared with the GMME as their SD s ad RMSE s are similar uder i.i.d. disturbaces i these fiite samples, eve though the RGMME might be theoretically less asymptotically efficiet tha the GMME. This is so for all the results i Tables, 2, 4 ad 5 with all the variace ad parameter desigs. Uder heteroskedaticity, there is o obvious domiated patter i terms of SD compariso of the RGMME with the GMME. I terms of RMSE, with R = 200, the RMSE s of the RGMME s of λ 0 ad β 0 are slightly smaller tha those of the GMME s (i Tables, 2 ad 4). 6 Table 5, there is o differece betwee these two estimators. For V-D2 i The ORGMM is the RGMM which uses the robust heteroskedastic variace of the momets as the optimal weightig matrix. Comparig the results of ORGMME with those of RGMME, the results are similar overall. It does ot seem that optimal weightig with a robust variace uder ukow heteroskedaticity would improve efficiecy i these fiite samples. 6. Tests for Heteroskedasticity 6. The LM Test for Heteroskedasticity The possible presece of heteroskedasticity ca be tested with the Breusch-Paga LM test (Breusch ad Paga 979), usig estimated residuals ɛ i s of the model from MLE or GMME. The Breusch-Paga LM test assumes the alterative hypothesis σ 2 i = f(α + z i α 2 ), where z i is a vector of p-dimesioal exogeous variables ad f is a cotiuously differetiable fuctio. However, due to the local ature of the LM test, oe does ot eed to specify the fuctioal form of f. So the fuctioal restrictio o this test is simply a liear idex structure α + z i α 2 o the form of ukow heteroskedasticity. Uder the ull hypothesis H 0, α 2 = 0. Let Z be the (p+) matrix of observatios o (, z i ) ad let d be the -dimesioal vector of d i = test statistic is 2 d Z (Z Z ) d, which is asymptotically χ 2 (p) uder H 0. ɛ2 i ɛ ɛ /. The the LM 5 Additioal results of the SGMME i the settigs of Tables 4 ad 5 ca be foud i the workig paper versio. 6 For R=50, there are a few cases where the MLE or GMME have smaller RMSEs tha those of RGMME. These occur whe RGMME happes to have a relatively larger SD. 6

19 6.2 The Hausma-type Tests Alterative statistics may be based o the compariso of robust estimates agaist estimates which are asymptotically efficiet uder H 0. These are the Hausma-type test statistics (Hausma 978), which seem atural as the 2SLSE ad RGMME are robust ad the MLE ad GMME are asymptotically efficiet uder H 0 for our model. assumptio of a liear idex form for the variace fuctio. The Hausma-type test does ot eed the The mai idea of the Hausma-type test is to compare two estimators θ ad θ, with θ beig asymptotically efficiet uder the ull hypothesis H 0, but icosistet uder the alterative H, while θ is cosistet uder both H 0 ad H. The Hausma-type test statistic is ( θ θ ) V ar( θ θ ) ( θ θ ) = ( θ θ ) [V ar( θ ) V ar( θ )] ( θ θ ) D χ 2 (m), where [V ar( θ ) V ar( θ )] is a geeralized iverse of the matrix [V ar( θ ) V ar( θ )] with m beig its rak (see, e.g., Ruud (2000)). Asymptotically, this statistic is ivariat with respect to the choice of a geeralized iverse. Whe ɛ i s are i.i.d. ormal, the MLE is asymptotically efficiet. So is the best GMME θ obtaied by settig P = (G tr(g) I ) ad Q = (G X β 0, X ), as it is asymptotically equivalet to the MLE whe ɛ i s are i.i.d. ormal. Uder H 0, the asymptotic variace matrix of the MLE (or GMME) is V ar( θ ) = Σ, where ( ) tr[(g tr(g) I ) s G ] + (G σ Σ = 0 2 X β 0 ) (G X β 0 ) (G σ0 2 X β 0 ) X X (G X β 0 ) X. (20) σ X 0 2 σ 2 0 The correspodig RGMME θ has Q = (G X β 0, X ) but P = (G Diag(G )), which is cosistet uder both H 0 ad H, but is ot asymptotically efficiet uder H 0. So is the B2SLSE with Q = (G X β 0, X ). The RGMME θ has the asymptotic variace matrix V ar( θ ) = Σ 2 where ( ) tr[(g Diag(G )) s G ] + (G σ0 Σ 2 = 2 X β 0 ) (G X β 0 ) (G σ0 2 X β 0 ) X X (G X β 0 ) X, (2) σ X 0 2 σ 2 0 ad the B2SLSE θ,b has its asymptotic variace V ar( θ,b ) = Σ b, where Σ b, = ( ) (G X β 0 ) (G X β 0 ) (G X β 0 ) X σ0 2 X (G X β 0 ) X X. (22) Uder the alterative H of heteroskedasticity, as the MLE ad GMME θ are icosistet but the B2SLSE θ,b ad RGMME θ are cosistet, these estimators ca be used to form the Hausma-type test statistics. 7

20 The differece i variace matrices, [V ar( θ ) V ar( θ )], may or may ot have full rak. To ivestigate the rak of [V ar( θ ) V ar( θ )] ad/or [V ar( θ,b ) V ar( θ )], the expressio V ar( θ ) V ar( θ ) = V ar( θ )[V ar( θ ) V ar( θ ) ]V ar( θ ) is useful as V ar( θ ) ad V ar( θ ) are ivertible. The rak of this differece i variace matrices is that of [V ar( θ ) V ar( θ ) ], i.e., the rak of the matrix( of the differece i the precisio) matrices. From (20) ad (2), V ar( θ ) V ar( θ ) tr[(diag(g ) tr(g) = I ) s G ] 0, ad, with (22), V ar( θ ) ( ) 0 0 V ar( θ,b ) tr[(g tr(g) = I ) s G ] 0, both of which have rak oe. Therefore, a geeralized iverse of the differece i variace matrices of MLE (or GMME) vs RGMME ca be 0 0 ( ) [V ar( θ ) V ar( θ )] = V ar( θ ) tr [(Diag(G ) tr(g) I ) s G ] 0 V ar( θ ), (23) 0 0 ad that of the MLE (or GMME) vs B2SLSE is ( ) [V ar( θ,b ) V ar( θ )] = V ar( θ,b ) tr [(G tr(g) I ) s G ] 0 V ar( θ ). (24) 0 0 Aother geeralized iverse ca be derived with the eigevalue ad eigevector decompositio of the matrix [V ar( θ ) V ar( θ )]. As this matrix has rak oe from our precedig aalysis, let µ > 0 be the sigle ozero eigevalue ad let the correspodig orthoormal eigevector matrix be Γ. The correspodig geeralized iverse of [V ar( θ ) V ar( θ )] is Γ Λ Γ where Λ is a diagoal matrix cosistig of µ ad zeros o the diagoal elemets. This geeralized iverse is umerically o-egative defiite ad is the Moore-Perose geeralized iverse. 7 The Hausma-type tests by comparig MLE (or GMME) vs RGMME, ad MLE (or GMME) vs B2SLSE are both asymptotically χ 2 (). 6.3 Mote Carlo Results for the Tests Table 6 presets the results of the Hausma-type ad LM tests for heteroskedasticity i the SAR model. The Mote Carlo experimetal desigs are V-D with P-D ad P-D2. The correspodig ML, GMM ad RGMM estimates are those i Tables ad 2, ad the B2SLSE is i Table 3. The left pael of the table shows the results for the homoskedasticity cases, ad the right pael shows those for the heteroskedasticity cases. I each pael, the first two colums preset, respectively, the results for the Hausma-type tests, usig MLE vs B2SLSE ad MLE vs RGMME. The results for the two LM tests, oe based o MLE, the other o GMME, are show i the last two colums of each pael. The alterative hypothesis for the LM tests is σ 2 i = f(α 0 +z i α), with z i beig the group 7 O the other had, the geeralized iverses i (23) ad (24) are ot symmetric. With a fiite sample, the geeralized iverse based o the eigevalue ad eigevector has the umerical advatage i that the derived asymptotic χ 2 test statistics will always be o-egative. 8

21 size. 8 As discussed i the previous subsectio, it is ot ecessary to specify the fuctioal form of f. The Hausma-type tests use both the Moore-Perose geeralized iverse ad the geeralized iverses i (23) ad (24). The correspodig results are similar. 9 The Hausma-type test usig MLE vs B2SLSE has o power for the sample sizes R = 50 to 200. Eve though its empirical levels are higher tha the theoretical oes, its powers are ot eve larger tha the empirical levels. For the Hausma-type test of MLE vs RGMME, its empirical levels are very large, showig over-rejectio of the ull hypothesis. It does have power eve after adjustig the proper level of sigificace, but its large empirical levels will reder this test useless. These pheomea ca be uderstood by ivestigatig the geeralized iverse formulas i (23) ad (24) ad the small biases of the correspodig estimates. For the Hausma-type test usig MLE vs RGMME, the test statistic is iflated by the variace differece term tr[diag(g ) tr(g) I ) s G ]. I the samples for the Mote Carlo study, this term happes to be very small, with mea ragig from 0.26 to.06 for all cases. These are small eve though the trace operatio is a summatio over terms. Thus, it might produce a big umber whe its iverse is ivolved, which is explicit i (23). O the cotrary, for the Hausma-type test usig MLE vs B2SLSE, the correspodig variace differece term has mea value ragig from 50 to 670, which would give a small umber after iversio. Overall, the Hausma-type tests are ot reliable. I cotrast, the LM tests perform very well. The empirical levels are close to the theoretical oes ad they have excellet powers Applicatio to Couty Teeage Pregacy Rates Teeage pregacy is oe of the cotexts where social iteractio effects are believed to be most importat. Jecks ad Mayer (990), for example, coclude that, eighborhoods ad classmates probably have a stroger effect o sexual behavior tha o cogitive skills, school erollmet decisios, or eve crimial activity. May studies, icludig Hoga ad Kitagawa (985), Crae (99), Case ad Katz (99) ad Evas et al., (992), aalyze eighborhood effects i teeage pregacy by usig micro-data. It would be of iterest to study the spatial effects at more aggregated levels ad see how couty teeage pregacy rates are affected by each other. We suspect the possible presece of ukow heteoskedasticity i this aggregated data. Therefore, we apply the RGMM 8 I the variace desig V-D, the group size variable i the variace fuctio is oliear ad complicated. So the liear idex specificatio of the variace for the LM test provides oly a approximatio to the true variace fuctio. Our itetio is to see whether a liear idex approximatio ca capture the alterative i its power fuctio, sice i practice we may ot kow the exact variace fuctio. 9 The results of the Hausma-type tests reported i Table 6 are those with the Moore-Perose geeralized iverse. 20 This may idicate that the liear idex approximatio of the oliear variace fuctio is valuable. The liear approximatio does capture the group size variable i the variace fuctio. 9