Spatial Nonstationarity and Spurious Regression: The Case with Row-Normalized Spatial Weights Matrix

Transcription

1 Spatial Nostatioarity ad Spurious Regressio: The Case with Row-Normalized Spatial Weights Matrix March 26, 29 Abstract This paper ivestigates the spurious regressio i the spatial settig where the regressat ad regressors may be geerated from possible ostatioary spatial autoregressive processes. Uder the ear uit root speci catio with a row-ormalized spatial weights matrix, it is show that the possible spurious regressio pheomea i the spatial settig are relatively weaker tha those i the ostatioary time series sceario. The regressio estimates might or might ot coverge to. The divergece might occur oly whe the regressat has a ear uit root much closer to uity tha that of the regressor. For the t ad F statistics, there could be over-rejectio of the ull of ucorrelatedess uder certai situatios, but they do ot diverge. However, the coe ciet of determiatio R 2 coverges to, which provides strog evidece of the spurious regressio eve whe t ad F statistics are large. The Mora I test may reject the ull of o spatial depedece i the least squares residual. Simulatio results about di eret statistics are i lie with the theoretical results we derive i this paper. JEL classi catio: C3; C23; R5 Keywords: Near Uit Root, Spatial Nostatioarity, Spurious Regressio

2 Itroductio Spatial ecoometrics deals with spatial correlatios amog ecoomic uits. The spatial autoregressive (SAR) model by Cli ad Ord (973) has received the most attetio i ecoomics. Recetly, there is a growig iterest i extedig the uit root ad coitegratio pheomea i the time series to the spatial settig. This is because the SAR model has sometimes bee regarded as a geeralizatio of a autoregressive model i the time series to the spatial settig. Figleto (999) has ivestigated implicatios of the uit root i the SAR model ad has detected spurious regressios via Mote Carlo simulatios. I order to geerate uit root spatial data, he itroduces a ucoected cetral uit such that the spatial weights matrix has a zero row. I that settig, uit root ca be preset ad the system is a equilibrium oe. This settig is aalogous to havig a iitial observatio i the time series autoregressive model. Mur ad Trivez (23) follow up Figleto (999) by allowig a determiistic itercept, which ca take large values i the data geeratig process (DGP). By arguig that the itercept may geerate a spatial tred variable i the reduced form equatio, they ivestigate possible spurious regressio features by Mote Carlo simulatios. They d that the geerated determiistic tred ca also cause spurious regressio. Lauridse ad Kosfeld (26, 27) develop a two step LM test to distiguish the possible two sources of spurious regressio: oe is the uit root i the DGPs of the regressat ad/or regressors, ad the other is the spatial error i the regressio. However, the assumptio of a ucoected uit might be too strog ad have limited applicatios i empirical studies. I empirical work, it is commo to have a row-ormalized spatial weights matrix with a few exceptios. I this paper, we ivestigate the spurious regressio i the spatial settig with attetio to a weights matrix with row-ormalizatio, where there is o ucoected cetral uit. While the spatial weights matrix i Figleto (999) would ot be row-ormalized, the row-ormalizatio of the weights matrix is allowed i the studies of Lauridse ad Kosfeld (26, 27). As poited out by Mur ad Trivez (23), the possible spurious regressio pheomeo i Figleto (999) might likely be geerated by large

3 variaces of the process istead of o-circularity, as the itroduced ucoected cetral uit might ot geerate accumulated oise disturbaces for other spatial uits. Hece, the spurious regressio pheomeo could be geerated by large variaces i the process istead of o-circularity. Because of the popularity of the row-ormalized weights matrix, we ited to ivestigate possible spurious regressio i this spatial settig. For a spatial weights matrix beig row-ormalized, the spatial e ect caot be equal to oe (Ord 975). Therefore, we cosider the case where the spatial e ect is close to oe. I the time series, the ear uit root feature turs out to be similar to that of the exact uit root case i terms of asymptotic aalysis (see Phillips (987)). I the spatial settig, if the true spatial e ect is ear uity, the spatial depedece across uits will be strog ad the variaces of the depedet variables will become large eve though the spatial weights matrix is row ad colum sum bouded. Lee ad Yu (27) derive the asymptotic properties of the istrumetal variable (IV) ad quasi-maximum likelihood (QML) estimators whe the true spatial e ect ca go to uity at ay rate. I this paper, we ivestigate the spurious regressio whe we have possible ostatioary compoets i the DGP of the regressat ad/or the regressors, where the ostatioarity is caused by ear uit roots. I the time series, Grager ad Newbold (974) report the spurious regressio by Mote Carlo. Phillips (986) provides aalytical results of liear regressios ivolvig geeral itegrated radom processes, icludig the spurious regressio of Grager ad Newbold (974) ad the coitegratio regressio of Grager ad Egle (985). With a row-ormalized weights matrix ad a ear uit root spatial e ect, this paper ivestigates aalytically whether spurious regressio pheomea could occur or ot i the spatial settig. We d that the possible spurious regressio pheomea i the spatial settig are much weaker tha those of the itegrated time series. The least square estimates of the coe ciets i a spurious regressio might or might ot coverge to. They might be diverget oly whe the regressat has a ear uit root much closer to uity tha those of the regressor. Eve so, the correspodig t-statistic is asymptotically ormal ad does ot diverge. The Mora I or LM test for spatial correlatio i the least squares residual will show 2

4 a sigi cat correlatio. However, the determiat of coe ciet R 2 will always coverge to i probability. This implies that whe high values of t or F were observed, low values of R 2 would idicate the spurious regressio i the spatial settig. The aalytical results are give ad simulatio results for ite samples are provided to support the theoretical implicatios. The rest of this paper is orgaized as follows. Sectio 2 speci es the DGP of the regressat ad regressors. Sectio 3 ivestigates the possible cosequeces of spurious regressio i terms of the least squares estimates, the associated t or F statistics, R 2, ad Mora I ad LM test statistics (to test the spatial e ect i the regressio residuals). Simulatio results are reported i Sectio 4 ad coclusios are made i Sectio 5. Several lemmas are provided i the Appedices. 2 The DGP Cosider the cross sectioal ( rst order) SAR models Y = W Y + Z + ; (2.) Y 2 = 2 W 2 Y 2 + Z ;. Y m = m W m Y m + Z m m + m ; where, for j = ; :::; m, Y j is vector of depedet variables, Z j is k j ostochastic exogeous variables, W j is a ostochastic spatial weights matrix, ad the disturbace of the -dimesioal vector j is i.i.d. (; 2 j ). I spatial ecoometric literature, W jy j is usually referred to as a spatial lag of Y j. Suppose that W j is diagoalizable with eigevalues d ij such that either d ij = or jd ij j < 2 for i = ; :::;. Furthermore, suppose that there are m j uit eigevalues, ad the remaiig ( m j ) eigevalues are all uiformly bouded away from i the absolute value for all. For ay value i its parameter space, ca be reparameterized as =. The ear uit root case refers to the situatio that, for the true spatial e ect parameter j = j, j goes to i ity as goes to i ity. Thus, as goes 3

5 to i ity, j approaches. The j speci es, i its geeral form, how fast j approaches uity as the sample size icreases 3. Note that because the SAR model is speci ed to be a equilibrium model, it is ot meaigful to assume that j =. With our speci catio of the ear uit root case, as j <, this model is still a equilibrium model for ay ite. This is because (I j W j ) is ivertible. With W j beig diagoalizable, let R j be the eigevector matrix such that W j = R j D j R j, (2.2) where D j = diagf mj ; d mj+; ; d j g is the diagoal eigevalue matrix, mj is m j vector of oes ad diagfa g is a diagoal matrix formed by elemets of a row vector a. This implies that (I j W j ) = R j (I j D j )R j. As j jj < ad jd ij j for all i, (I j D j ) is ivertible ad, so is (I j W j ). Thus, with our ear uit root speci catio, for ay ite, this model is still a well-de ed equilibrium model. For the eigevalue matrix D j, it ca be decomposed ito two parts as D j = J j + ~ D j where J j = diagf mj ; ; ; g ad ~ D j = diagf; ; ; d mj+;j; ; d j g. The J j cosists of all the m j uit eigevalues ad ~ D j cosists of all the eigevalues with their absolute values less tha oe. Accordigly, W j ca also be decomposed ito two parts: W j = W u j + ~ W j, where W u j = R jj j R j ad ~ W = R j ~ Dj R j. Deote S j = I j W j, from (2.), the equilibrium vector Y j is Y j = S j (Z j j + j ). For S j = I j W j = R j (I j D j )R j, we have Sj = R j(i j D j ) R j. Because (I j D j ) is a diagoal matrix, (I j D j ) = j j J j + (I j ~ Dj ) as is derived i Lee ad Yu (27). Hece, deotig G j = W j S j, we have S j = j j W u j + (I j ~ Wj ), (2.3) G j = j j W u j + W j (I j ~ Wj ), because W j Wj u = R jd j J j R j = W j u. Therefore, from (2.3), we have the followig represetatio for 4

6 ay value of j i terms of j, Y j = j Y u j + ~ Y j ; (2.4) where Y u j = j W u j(z j j + j ), ad ~Y j = (I j ~ Wj ) (Z j j + j ). Equatio (2.3) is revealig i that, whe j is ear uity, S j is ill coditioed ad its iverse has the large factor j, which implies that the rst term of Y j o the right had side of (2.4) may preset a ustable compoet with icreasig variaces as j approaches i ity. Regardless whether j is close to oe or ot, the secod term of Y j i (2.4) is a stable oe. If j is close to oe, the implied variace of Y j ca be large because the rst compoet has the factor j, which ca be explosive whe j is ear the uit root. Also, Y j ad W j Y j have the same ustable compoet because W j Yj u = Y j u, ad, cosequetly, (I W j )Y j becomes stable 4. Thus, Y j ad W j Y j feature spatial ostatioarity i the cross sectioal settig whe j is ear uity. To aalyze the model, we make the followig assumptios. Assumptio W j is diagoalizable, row sum ad colum sum bouded 5 (for short, UB), ad has m j uit roots, ad the remaiig eigevalues d ij, i = m j + ; ;, are uiformly bouded away from i the absolute value for all ad i. Assumptio 2 j = j ad j j j <. j is positive, which might remai ite or approach i ity as goes to i ity. Assumptio 3 Elemets of disturbaces f j g of the jth equatio i (2.) are i:i:d with zero mea, variace 2 j ad its higher tha fourth momet exists. Also, i ad j are idepedet for i 6= j. 5

7 Assumptio 4 (I j ~ Wj ) ad W u j are UB. Assumptio says that W j is diagoalizable so that we ca decompose the process ito a stable part ad a possible ustable part. As a example, a weights matrix row-ormalized from a symmetric matrix is diagoalizable; ad all its eigevalues are real, less tha or equal to oe i the absolute value ad its largest eigevalue is always (see Ord (975)). Assumptio 2 speci es that the true spatial e ect ca be ear uity ad how close it is ear uity whe j approaches i ity. It also allows the stable case where j remais ite 6. Assumptio 3 is a stadard assumptio. I this paper, it is to cotrol the stochastic boudedess of j, so that the (possible) large variace ostatioary behavior of Y j comes oly through the ear uit j. Assumptio 4 is to guaratee that the stable ad (possible) ustable parts of S j i (2.3) are UB after beig rescaled. I the stadard case with the true beig strictly less tha oe, a importat assumptio due to Kelejia ad Prucha (998) is that S ( ) is UB. For the (possible) ear uit root case, S j ( j) will ot be uiformly bouded i either row or colum sum orms due to the explosive factor j for the ustable compoet i (2.3). By takig out the j factor, the magitude of the remaiig matrices will ot grow with. For our subsequet aalysis, we shall focus o the settig that the ustable compoet Yj u does ot vaish asymptotically. This would be the case if W j u were ot domiated asymptotically by ~W j i the sese that the share of uit eigevalues, m j =, would ot vaish as approaches i ity for the ear uit root process. 3 Spurious Regressio To study the possible spurious regressio, we will focus o the case where there are o exogeous variables icluded i (2.) as i Figleto (999). Hece, the DGP of the variables of iterest are Y = W Y + ; (3.).. Y m = m W m Y m + m ; 6

8 where j is i:i:d:, ad idepedet of i for i 6= j. Deote Y ; = (Y 2 ; ; Y m ), X = [l ; Y ; ] where l is the colum vector of oes, is a scalar ad = ( 2 ; ; m ) is a (m ) vector. We will ivestigate the OLS estimate of (; ) i the regressio of Y o l ad Y ; as if Y = l + Y ; + V = X + V, (3.2) where V is the residual vector. The followig Lemma 3. is useful for the asymptotic properties of the OLS estimates of ad ad their related statistics. Deote Y j = Y j, so that Y ; = (Y2; ; Ym). Also, deote S j = 2 S j j j S j for j = ; :::; m. Hece, as S j = j j W u j + (I j ~ Wj ) from (2.3) where W u j ad (I j ~ Wj ) are UB by Assumptio 4, S j is UB regardless of whether j is ite or large. Lemma 3. Let X = [l ; Y ;]. Uder Assumptios, 2, 3 ad 4, for i; j = ; :::; m; ( 2 Y i Yj j = tr(s j) + O p ( O p ( p ) p ) if i = j if i 6= j ; (3.3) ad p X Y = p l Y Y2Y. Y my 8 < 2 where m;ij = lim! l S l if i = j = 2 j : 2 lim! tr(s js ) if i = j > otherwise well de ed. Proof. See Appedix B. C A d! N(; m); (3.4), uder the assumptio that those limits were We ote that the variace matrix m is a diagoal matrix because the di eret SAR processes i (3.2) are mutually idepedet. 3. Estimates The OLS estimates for ad i (3.2) are ^ ^ = (X X ) (X Y ). For the DGPs i (3.), because the variables of the di eret processes are idepedet, oe would expect that the liear regressio 7

9 of Y o X = [l ; Y 2 ; ; Y m ] should yield isigi cat coe ciets ^ ; otherwise, we might have some degree of spurious regressios. Deote m = 2m where 2m = C A, m we have p ^ ^ = m X X p X Y. (3.5) From Lemma 3., we have X X = D;xx + O p ( p ) where D;xx diagf; 2 2 tr(s 2); :::; 2 m tr(s m)g, ad p X Y will be asymptotically ormally distributed with the limitig variace matrix m = 2 diagf lim! l S l ; 2 2 lim tr(s 2S ); :::; 2 m lim! Assumptio 5 lim! tr(s j) 6= for j = ; 2; :::; m.! tr(s ms )g. Deote D xx lim! D ;xx = diagf; 2 2 lim! tr(s 2); :::; 2 m lim! tr(s m)g. As X X D xx, Assumptio 5 speci es that the rescaled X X is of full rak i the limit. Hece, p! p m (^ ; ^ ) d! N(; [D xx ] m[d xx] ). (3.6) Therefore, whether ^ j, where j = 2; :::m, coverges i probability to or ot will deped o the factors i p m. The situatios ca be divided ito three cases: Case (): whe p j!, ^ j is jp cosistet ad jp ^ j will be asymptotically ormally distributed. Case (2): whe j p! c where c is a positive ite costat, ^ j is asymptotically ormally distributed. Because its limitig distributio is ot degeerate, the estimate ^ j does ot coverge to. Case (3): whe j p!, ^ j will diverge to i ity, i.e., it is ot stochastically bouded. 8

10 Thus, ^ j will ot coverge to whe lim! j p 6=. I the spurious regressio of itegrated time series, the least squares estimated coe ciets have odegeerate distributios (Phillips, 986). For our spatial situatio, this feature may also appear. We ote that this spurious feature o ^ j j = 2; :::; m, appears oly whe approaches oe faster tha j i the sese that p j coverges to a positive costat or diverges to i ity. Ituitively, this meas that is much closer to oe tha j for j 6=. If both ad j approach oe with a similar rate, case () will apply ad ^ j will coverge to ad p ^j is asymptotically ormal with zero mea ad a ite limitig variace; also, the spurious regressio feature of odegeerate limitig distributio for ^ j will ot occur. For ^, we ca see from (3.6) that it is diverget as log as approaches oe. For the estimate ^ 2 of the variace of V, deote e as the regressio residual e = (I P )Y where P = X (X X ) X. We see that e e = Y Y Y P Y. The followig lemma is useful to get the order of ^ 2. Lemma 3.2 Uder Assumptios, 2, 3 ad 4, for ay ostochastic UB square matrix B, Proof. See Appedix B. Y B P Y = O p ( ): (3.7) From Lemma 3.2, we have 2 Y P Y Y Y = 2 tr(s ) + O p ( p ). Hece, 2 ^ 2 = e e 2 m = Y P Y = O p ( ); also, from Lemma 3., we have = 2 tr(s ) + O p ( p ). (3.8) Thus, ^ 2 will be diverget at the 2 rate whe goes to i ity. The divergece of ^ 2 is expected simply due to the large variace of the regressat Y. 3.2 t, F Statistics ad R 2 The t-statistic for each j where j = 2; :::; m, is t j = ^ j ^ p [(X X ) ] jj = p q ^ [( j ^ j X X ) ] jj. As X X = D;xx + O p ( p ), we have ( X X) = [D;xx] + O p ( p ) because D;xx is osigular i 9

11 the limit from Assumptio 5. Hece, from (3.5) ad p X Y is O p () from Lemma 3., we have t j = f[d ;xx] g jj p Yj Y + O p ( p ) ^ (f[d =2 ;xx] g jj + O p ( p )). As ^ = q tr(s ) + O p ( p ) from (3.8), we have t j = f[d ;xx] =2 g jj q tr(s ) p YjY + O p ( p )! d tr(s j S ) N ; lim. (3.9)! tr(s j ) tr(s ) Hece, we see that the asymptotic distributio of the idividual t-statistics is ormal, but its variace is, i geeral, di eret from. For the special case that W j = W is a symmetric matrix ad both j ad coverge to, i.e., both are ear uit roots, the oe ca easily show, usig the property j S j is approximated by W u j ad R j = R j, that the limitig variace lim! tr(sjs) tr(s j)tr(s ) = lim! m. I this case, the variace is greater tha oe as the umber of uit eigevalues m is less tha. 7 Deote A 22 as obtaied from A with the rst row ad colum deleted for ay square matrix A. For the F test of the overall sigi cace of regressors, i.e., H : 2 = ::: = k =, the test statistic is F = m ^ f^ 2 [(X X ) ] 22 g ^. By usig the rescaled variables, p F = 2m^ [( p (m ) ^2 2 X X) ] 22 2m^. We have [( X X ) ] 22 = f( X X ) 22 ( X X ) 2 [( X X ) ] ( X X ) 2 g from the iverse of a partitioed matrix, where ( X X ) pq is the correspodig partitioed matrix for p = ; 2 ad q = ; 2. As ( X X ) =, ( X X ) 2 = [( X X) 2 ] = O p ( p ) ad ( X X) 22 = [D;xx] 22 + O p ( p ) from Lemma 3., [( X X) ] 22 = f[d;xx] 22 g + O p ( p ). Similarly, from (3.5), we have p 2m^ = f[d;xx] 22 g p Y ;Y + O p ( p ). Hece, F = (m ) 2 tr(s ) p Y ;Y f[d;xx] 22 g p Y ;Y + O p ( p ). Hece, from Lemma 3., F d = (m ) U mu m,

12 where U m is a (m ) vector ad its elemet (u 2m ; :::; u mm ) are idepedet ormal where for j = 2; :::; m, u jm N tr(s j S ) ; lim.! tr(s j ) tr(s ) This implies that, the F -statistic is asymptotically the average of m idepedet square of ormal radom variables, which might have di eret variace 8. Hece, the F -statistic multiplied by (m ) is ot asymptotically chi-square distributed with (m ) degrees of freedom i geeral. Additioally, we ca see that F d = P m j=2 (m ) t2 j. (3.) For the coe ciet of determiatio, we have R 2 = e e=( 2 ) Y M Y=( 2 ) = Y PY Y Y ( l Y )2 ( l Y )2 e e Y M Y where M = I l l. Hece, R 2 =. As Y P Y = O p ( ) from Lemma 3.2, l Y = O p ( p ) from Lemma 3., ad Y Y = 2 tr(s ) + O p ( p ) with lim! tr(s ) 6= from Assumptio 5, we have R 2 = Y P Y Y Y ( l Y) 2 ( l Y = O p ( )2 )! p. (3.) These imply that the t-statistic ad F -statistic may ot be reliable to test the ull of j = for j = 2; :::; m because their asymptotic distributios are ot the usual oes for the covetioal t ad F statistics. However, from (3.), we ca see that R 2 is a good idicator of the isigi cace of j. 3.3 Mora I ad LM Tests of Spatial Error i OLS I additio to the testig of, it is of iterest to test the spatial e ect i the disturbaces of the OLS regressio. The Mora I test statistic is I Mora = S e We e e with S = P i= P j= w ;ij ad e = (I P )Y, where W is a ostochastic UB spatial weights matrix. Whe W is row-ormalized, S = so that I Mora = e We e. Hece, e I Mora = e W e = e e Deote Sj = js j. Hece, S j = j S j Y W Y Y W P Y Y Y Y P W Y + Y P W P Y P. Y Y is UB by (2.3) ad Assumptio 4. For the umerator of I Mora, from Lemma 3., we have 2 Y W Y = 2 tr(s W S ) + O p( p ); also, from Lemma

13 3.2, the remaiig three terms are O p ( ) after beig rescaled by 2. Similarly for the deomiator, we have 2 Y Y = 2 tr(s S ) + O p( p ) ad Y P Y = O p ( ). Hece, I Mora = tr(s W S ) tr(s S ) + O p ( p ). (3.2) The case W = W is of special iterest. Because W u j W u j = W u j ad ~ W j W u j =, they imply that tr(wj uw jwj u u ) = tr(wj W j u ). Hece, whe!, the domiat term i S is W u j so that I Mora p!. Istead of the Mora I test statistic, the associated LM test is LM = e p W e ^ 2 tr =2 (W 2 + WW ) = [ tr(w 2 + WW )] =2 m I Mora, (3.3) which diverges to i ity at the p rate as!, regardless of whether j is large or ot as log as I Mora does ot coverge to. 3.4 Costat Terms i the DGP of Y j s Whe there are costat terms icluded i the DGP (3.) so that Y j s have ozero meas, we show that they do ot chage the estimates of j. This is so eve the costat terms may take o large values (or diverget) as i Mur ad Trivez (23). For otatioal purposes, we use Y c j as the couterpart of Y j so that Y c = W Y c + c l + ; (3.4). Y c m = m W m Y c m + c m l + m ; where c j is a sequece of ostochastic scalars, for j = ; :::; m. Deote Y c ; = (Y c 2; ; Y c m), c as a scalar ad c = ( c 2; ; c m) a (m ) vector. Cosider the OLS regressio of Y c o l ad Y c ; as if Y c = c l + Y c ; c + V c, (3.5) 2

14 where V c is the residual vector. Rather tha developig a lemma similar to Lemma 3. where c j could possibly diverge, we ca compare (3.5) with (3.2). Whe W j is row-ormalized, we have Y c j = S j (c jl + j ) = j c j l + S j j = j c j l + Y j. Hece, (3.5) ca be re-writte as Y = ( c c + P m j=2 jc j c j)l + Y ; c + V c. Compared with (3.2), we ca see that c c + P m j=2 jc j j is reparameterized as ad c =, where V c is the same as V after this reparameterizatio. This implies that we will have the same estimates of as i (3.2) eve though a costat term (possibly large) is icluded i the DGP of Y j. Cosequetly, the t-statistic ad F -statistic ivolvig estimated c would be the same as those for the estimate i (3.2). Note that c = + c P m j=2 jc j j. As jp ^ j = O p () ad p ^ = O p (), we have p ^ c = O p (max(; p c ; c 2 ; :::; c m )). However, iferece about the costat term i the OLS regressio might ot be of much iterest. 3.5 Ituitios It might be helpful to see the ituitio behid the results we have derived. For illustratio, cosider the simple OLS regressio of Y o l ad Y 2, via Y o l ad Y 2 (so that m = 2) as doe i the previous sectios. These two problems are related because mi ; (Y l Y 2 ) (Y l Y 2 ) = 2 mi ; ( Y l Y ) ( Y l = 2 mi ; (Y l Y2 ) (Y l Y2 ), Y ) where = ad = 2. Hece, ^ = ^ ad ^ = ^. Similarly, 2 ^ 2 = = 2 2 (Y ^ l Y 2^ ) (Y ^ l Y 2^ ) 2 (Y ^ l Y2^ ) (Y ^ l Y2^ ) = 2 ^ 2, 3

15 where ^ 2 = 2 (Y ^ l Y 2^ ) (Y ^ l Y 2^ ). Hece, as the regressio of Y o l ad Y 2 is a stadard OLS regressio with the regular O() order, we ca get the implied orders of ^, ^ ad ^ 2. Additioally, it follows that ad R 2 = ( 2)^ 2 =(Y l Y l ) (Y l Y l ) = 2 ( 2)^ 2 =[ 2 (Y = ( 2)^ 2 =[(Y l Y l Y l ) (Y ly l )] ly l )] = R 2, l ) (Y l t = [(Y Y 2 2 l ) l (Y Y 2 2 l )] ^ =2 ^ = [( Y 2 2 = [(Y 2 l l Y Y l ) ( Y 2 l ) (Y 2 2 l Y 2 2 ly 2 l )] l )] =2 =2 ^ ^ 2 ^ ^ = t. We would expect R 2 to coverge to as Y ad Y 2 are idepedet. t would ot diverge as Y ad Y 2 have the regular O p () order. This explais why the R 2 is close to ad t is O p (). The asymptotic distributio of t would ot ecessarily have the covetioal asymptotic N(; ) distributio because elemets of Y might ot be idepedet with a homoskedastic variace. The proper asymptotic distributios of, t, etc., are derived as i previous sectios. 4 Mote Carlo We ru a small Mote Carlo to check possible spurious regressio i the spatial settig. The DGPs are Y = W Y + ; Y 2 = 2 W Y ; where ad 2 are idepedet N(; I ), ad we have the same weights matrix W. Both ad 2 are ear uity, but might have di eret values. We choose :95 ad :999 as possible values so that j = 2 or 4

16 . For W, we rst geerate a (row-ormalized) 4 4 quee matrix ad the costruct a block diagoal matrix. Hece, we have mj = =4 so that we have a sigi cat portio of the ustable compoet. We use 25 blocks so that we have = 5. The regressio equatio is speci ed as Y = l + Y 2 + V, (4.) where the repetitio is. After we ru the regressio of (4.), we calculate the t-statistics for. As Y ad Y 2 are idepedet, covetioal cases for regressio should have low frequecy of rejectig the ull of = so that most of the t-statistics should lie withi the rage ( 2; 2) with approximately 5% level of sigi cace for a two-sided test. If we have a very high frequecy of rejectig the ull hypothesis as the t-statistics are large i the absolute value, we might have a spurious regressio i the spatial settig. Tables -3 are the empirical desities of relevat statistics whe = 5. Table presets the empirical desities of ad the correspodig t-statistic. Table 2 presets the empirical desities of ^ 2, R 2, ad Table 3 presets the empirical desities of I Mora ad LM test statistics. To have a more-detailed look at the empirical desity of the t-statistics, we also report the frequecies of the t-statistics i Table 7. From Tables -3, we ca see that () For the estimates of, they will diverge oly whe = ad 2 = 2, while the t-statistics do ot diverge, with a fat tail compared to the stadard ormal distributio 9. (2) For ^ 2, they diverge, ad will have huge values whe = ; for R 2, they are close to for all cases. (3) For Mora I test statistics, it is close to ; ad the LM test statistics are of the order p. All the results ()-(3) are cosistet with the theoretical predictio, i.e., value of ^ is i (3.5) such that ^ = O p ( 2 p ) with zero mea, the t-statistic is i (3.9) which are asymptotically ormal but ot ecessarily N(; ), the value of ^ 2 is i (3.8) with ^ 2 = O p ( 2 ), ad R 2, Mora I ad LM test statistics are i (3.), (3.2) ad (3.3). I our simulatio result, the t-statistic has a fat tail compared to the N(; ) distributio. This is so, because i our simulatio, tr(s 2S ) tr(s 2)tr(S ) is about2 4, which implies that the t has 5

17 approximately the stadard deviatio of 2. This is cosistet with the empirical desity i Table. We ru additioal simulatios with a smaller to see the ite sample behavior of the estimates ad the statistics of iterest. The frequecy of t-statistics for = is reported i Table 7 3. Fially, we ru the simulatio where the umber of uit roots i W is small. We choose the 4949 weights matrix used i Aseli (988) so that mj = =49. Tables 4-6 are the couterparts of Tables -3 where we use blocks of the Aseli (988) s matrix so that = 49. With Aseli (988) s weights matrix, we have tr(s 2S ) tr(s 2)tr(S ) = 48:9754 whe = 2 = :999 (whe = :95 ad 2 = :999, or vice versa, tr(s 2S ) tr(s 2)tr(S ) = 25:4925). This explais that we have a larger variace for the t compared to the cases usig the quee matrix. 5 Coclusio Tables -6 here Table 7 here This paper ivestigates possible spurious regressio pheomea i the spatial settig where the regressat ad/or regressors are geerated from possible ostatioary SAR processes. With a row-ormalized spatial weights matrix (rather tha oe with a ucoected uit), it is show that the possible spurious regressio pheomea i the spatial settig are relatively weaker tha those i the ostatioary time series sceario. The OLS estimates of the regressio coe ciets might or might ot coverge to. The divergece cases occur oly whe the ear uit root of the regressat is much closer to tha those of the regressors. For the t ad F statistics, there could be over-rejectio of the ull of ucorrelatedess uder certai situatios, but they do ot diverge. The Mora I test may reject the ull of o spatial depedece with the least squares residuals. However, the coe ciet of determiatio R 2 coverges to, which provides strog evidece o a spurious regressio. Simulatio results about di eret statistics are i lie with the theoretical results we derive i this paper. 6

18 I the time series, Grager ad Newbold (974) state that low value of the Durbi-Watso statistic might imply the spurious regressio. Phillips (986) derives the asymptotic distributio of relevat statistics. It is show that the Durbi-Watso statistic coverges i probability to, while the regressio R 2 has a o-degeerate limitig distributio as T!. Also, the t ad F statistics will diverge. I the spatial settig for the ear uit root case where the spatial weights matrix is row-ormalized, we have the followig observatios: (a) The t statistic will ot diverge. However, it is ot asymptotically a stadard ormal statistic because the asymptotic variace of the statistic is di eret from. Hece, eve though we might have some degree of spurious regressio i terms of the t ad F statistics, they are ot as strog as the time series couterpart where the t ad F statistics diverge at T =2 ad T rate respectively; (b) The R 2! p, which is di eret from the time series literature. This implies that low R 2 with high t value ca provide a good idicator for the spurious regressio i the spatial settig; (c) The Mora I test is O p () ad will coverge to whe!. The Mora I test statistic (which is the Durbi-Watso statistic i the time series) might be used as a idicator for the misspeci catio of a regressio relatio with homogeous disturbaces i the spatial settig. Our results are di eret from Figleto (999), which has a special speci catio of the spatial weights matrix. I Figleto (999) where the spatial e ect ca be, a ucoected cetral uit is itroduced. I our speci catio, we use a row-ormalized spatial weights matrix, which is used i empirical applicatios ad has a log history for the SAR model (Ord 975). With the ostatioarity of the spatial data comig from the ear uit root speci catio, we d that the spurious regressio is weaker tha the oe i Figleto (999). I a spatial settig of a spurious regressio, a sigi cat Mora I or LM test statistic (to test the spatial error i the OLS regressio) may be caused by the regressat beig a spatial correlatio process. Uder the speci catio with a ucoected cetral uit i the spatial weights matrix as i Figleto (999), the high value of the Mora I or LM test statistic ca be caused by the presece of a uit root i the DGP of variables 7

19 i the regressio. Similarly, i our speci catio of row-ormalized weights matrix ad ostatioarity, the high value of the Mora I or LM test statistic ca also be caused by the ear uit root i the DGP of the regressat. Based o the results we derived, we ca combie the Mora I or LM test with the R 2 measure to have better idicators to detect possible spurious regressio i the spatial settig where the spatial weights matrix is row-ormalized. Lauridse ad Kosfeld (26, 27) develop a two step LM test to distiguish two possible sources of spurious regressio: oe is the uit root i the DGPs of the regressat, ad the other is the spatial error process i the spurious regressio. Their procedure is essetially testig whether the spatial process of the regressat is stable or ostable. Hece, their procedure may also be applicable to our case of ear uit root. However, the detailed aalysis of that test procedure is beyod the scope of this paper ad shall be left for future ivestigatio. Notes The estimatio ad testig for spatial depedece i cross sectioal data ca be foud i Aseli (988, 992), Kelejia ad Robiso (993), Cressie (993), Aseli ad Florax (995), Aseli ad Rey (997), Aseli ad Bera (998), Kelejia ad Prucha (998, 2, 27) ad Lee (23, 24, 27), amog others. 2 Whe W j is row ormalized from a symmetric matrix, W j is diagoalizable. See Lemma A. i Yu et al. (27). A weights matrix row ormalized has real eigevalues, with all its eigevalues less tha or equal to oe i the absolute value ad its largest eigevalue always (see Ord (975)). Hece, W j beig diagoalizable with the speci ed eigevalues, is a slight geeralizatio of W j beig row-ormalized from a symmetric matrix. 3 I a time series ear uit root model, the deviatio from the uit root is measured through a ocetrality parameter, c where the AR() coe ciet is usually speci ed as exp(c=) or with c beig the ocetrality parameter (see Phillips (987)). For the ear uit root i the SAR model, ca take a geeral form as log as it is icreasig i, which does ot eed to be speci ed i empirical applicatios. 4 This property might be useful for the estimatio. However, we do ot explore the use of this spatial di erece operator, I W j, i this paper. 5 We say a (sequece of ) matrix P is uiformly bouded i row ad colum sum orms if sup kp k < ad sup kp k <, where kp k sup i P j= jp ij;j is the row sum orm ad kp k = sup j P i= jp ij; j is the colum sum orm. 6 As our asymptotic aalyses below ca allow both the ear uit root ad stable cases, this geerality provides a ui ed framework for our study. 7 Whe both ad j are ear uit root, oe ca easily evaluate the variace of t j i (3.9) via W u ad W j u, ad the t-statistic ca be adjusted to be asymptotically N(; ) distributed. For a geeral case, oe eeds to estimate values i order to adjust such a variace. 8 Equivaletly, it is asymptotically a weighted sum of (m ) idepedet 2 () radom variables. 9 For the estimates of, the values are large, ad they diverge whe =, while the t-statistics do ot diverge, but with a fat tail compared to the stadard ormal distributio. As the iferece of is ot of much iterest, we do ot report the relevat statistics. I a additioal simulatio where we set = 2 so that is ot close to, I Mora is smaller with the mea value :483 for 2 = 2 ad :4828 for 2 =. Hece, the Mora I statistic is close to oly whe is large, which is implied by (3.2). Usig the 4 by 4 quee matrix, we ca icrease the umber of blocks from to 25 to see the e ect of the sample size. From the results (due to the space limit, the tables are ot preseted), the e ect of the sample size o ^, t ad Mora I statistics are ot apparet. However, ^ 2 ad the LM statistic are icreasig i ad R2 is decreasig i. 8

20 tr(s 2 S ) tr(s 2 )tr(s ) 2 The values of 2 = :95, ad = 2 = : The other results are similar. Due to the space limit, we do ot report them i the tables. are , ad 4 respectively whe = :95 ad 2 = :999, = :999 ad 9

21 Appedices A Some Lemmas Lemma A. Deote B = [b ;gh ] g;h= a ostochastic UB matrix. Uder Assumptio 3, for i; j = ; :::; m, ( 2 j ib j = tr(b ) + O p ( O p ( p ) p ) if i = j if i 6= j : Lemma A.2 Deote B j = [b j;gh ] g;h= a ostochastic UB matrix for j = 2; :::; m, ad deote C j = [c j;g ] g= a costat vector. Uder Assumptio 3, =2 Q; p P m j=2 ( jb j + C j) d! N(; ); where Q; = 2 P m j=2 2 j tr(b j B j) + 2 P m j=2 C j C j. Proof for Lemma A.: Deote j4 as the fourth momet of elemets of j. For i = j, as E j B j = 2 j tr(b ) ad var( j B j ) = ( j4 3 4 j ) P g= b2 ;gg + 4 j (trb (B +B )) = O(), we have j B j = 2 j tr(b ) + O p ( p ). For i 6= j, as E i B j = ad var( i B j ) = 2 i 2 j tr(b B ) = O(), we have i B j = O p ( p ). Proof for Lemma A.2: Kelejia ad Prucha (27) exted the cetral limit theorem i Kelejia ad Prucha (2) to a system of quadratic forms. For P m j=2 j B j, deote = ( ; ; 2; ; :::; m; ; :::; ; ; 2; ; :::; m; ) = ( ;; :::; ;) ; where ;i = ( ;i ; 2;i ; :::; m;i ). Thus, P m j=2 j B j = A where A is a ( m) ( m) matrix such that A = [a ;gh ] g;h= where a ;gh b 2;gh b m;gh A is a m m square matrix. Also, P m j=2 C j = where = (c ; ; (m ) ; :::; c m; ; (m ) ) is a ( m) vector. Hece, P m j=2 ( jb j + C j) = A +, 2

22 where var( p A + p ) = Q;. As A is UB because m is ite, ad elemets of is bouded, we have the result from Theorem A. i Kelejia ad Prucha (27). B Proof for Lemma 3. ad 3.2 Proof for Lemma 3.: From (2.4), Y j = j Y u j + ~ Y j where Y u j = jw u j j ad ~ Y j = (I j ~ Wj ) j. Hece, Y j = j W u j j + j ~Y j. Whe j!, the domiat term of Y j is jw u j j. Whe j is a ite costat (or a coverget sequece), Y j is just j S j j. By Assumptio 4 ad Lemma A., Y j Y j = 2 j tr(s j) + O p ( p ) ad Y i Y j = O p( p ) for i 6= j. This proves the rst part. For the secod part, p l Y = p l S ad p Y j Y = p j j S j S. As j S j is UB from Assumptio 4, we have the result from Lemma A.2. Proof for Lemma 3.2: As P = X (X X ) X, we have Y B P Y = Y B X (X X ) X Y = Y B X (X X ) X Y = Y B X( X X) X Y. For Y B X, as X = [l ; Y ;] which does ot iclude Y, we have Y B X = O p ( p ) as E( Y B X ) = ad var( Y B X ) = O( ). From Lemma 3., X X = D;xx + O p ( p ) ad X Y = O p ( p ) where D;xx is O(). Hece, Y B P Y = Y B X( X X) X Y = O p ( ). 2

23 Refereces Aseli, L. (988) Spatial Ecoometrics: Methods ad Models, The Netherlads, Kluwer Academic. Aseli, L. (992) Space ad applied ecoometrics, Aseli (ed.), Special Issue, Regioal Sciece ad Urba Ecoomics, 22. Aseli, L. ad A.K. Bera (998) Spatial depedece i liear regressio models with a itroductio to spatial ecoometrics, i A. Ullah ad D.E.A. Giles (ed.) Hadbook of Applied Ecoomics Statistics. New York: Marcel Dekker, Aseli, L. ad R. Florax (995) New Directios i Spatial Ecoometrics, Berli: Spriger-Verlag. Aseli, L. ad S. Rey (997) Spatial ecoometrics, i Aseli, L. ad S. Rey (ed.), Special Issue Iteratioal Regioal Sciece Review, 2. Cli, A.D. ad J.K. Ord (973) Spatial Autocorrelatio, Lodo,Pio Ltd. Cressie, N. (993) Statistics for Spatial Data, New York, Wiley. Figleto, B. (999) Spurious spatial regressio: some Mote Carlo results with a spatial uit root ad spatial coitegratio, Joural of Regioal Sciece, 39, -9. Grager, C.W.J. ad P. Newbold (974) Spurious regressios i ecoometrics, Joural of Ecoometrics 2, -2. Grager, C.W.J. ad R.F. Egle (985) Dyamic model speci catio with equilibrium costraits: Coitegratio ad error correctio, Discussio paper o. 85-8, UCSD. Kelejia, H.H. ad I.R. Prucha (998) A geeralized spatial two-stage least squares procedure for estimatig a spatial autoregressive model with autoregressive disturbace, Joural of Real Estate Fiace ad Ecoomics, 7:, Kelejia H.H. ad I.R. Prucha (2) O the asymptotic distributio of the Mora I test statistic with applicatios, Joural of Ecoometrics, 4, Kelejia H.H. ad I.R. Prucha (27) Speci catio ad estimatio of spatial autoregressive models with 22

24 autoregressive ad heteroskedastic disturbaces. Workig Paper, Uiversity of Marylad. Kelejia, H.H. ad D. Robiso (993) A suggested method of estimatio for spatial iterdepedet models with autocorrelated errors, ad a applicatio to a couty expediture model, Papers i Regioal Sciece, 72, Lauridse, J. ad R. Kosfeld (26) A test for spurious spatial regressio, spatial ostatioarity, ad spatial coitegratio, Papers i Regioal Sciece, 85, Lauridse, J. ad R. Kosfeld (27) Spatial coitegratio ad heteroscedasticity, Joural of Geography Systems, 9, Lee, L.F. (23) Best spatial two-stage least squares estimator for a spatial autoregressive model with autoregressive disturbaces, Ecoometric Reviews, 22, Lee, L.F. (24) Asymptotic distributios of quasi-maximum likelihood estimators for spatial ecoometric models, Ecoometrica, 72, Lee, L.F. (27) GMM ad 2SLS estimatio of mixed regressive, spatial autoregressive models, Joural of Ecoometrics, 37, Lee, L.F., J. Yu (27) Near uit root i the spatial autoregressive model, Workig Paper, The Ohio State Uiversity. Mur, J. ad F.J. Trivez (23) Uit roots ad determiistic treds i spatial ecoometric models, Iteratioal Regioal Sciece Review, 26, Ord, J.K. (975) Estimatio methods for models of spatial iteractio, Joural of the America Statistical Associatio, 7, Phillips, P.C.B. (986) Uderstadig spurious regressios i ecoometrics, Joural of Ecoometrics, 33, Phillips, P.C.B. (987) Towards a ui ed asymptotic theory for autoregressios, Biometrika, 74, Phillips, P.C.B. (988) Regressio theory for ear-itegrated time series, Ecoometrica, 56,

25 Yu, J., R. de Jog ad L.F. Lee (27) Quasi-maximum likelihood estimators for spatial dyamic pael data with xed e ects whe both ad T are large: a ostatioary case, Workig Paper, The Ohio State Uiversity. 24

26 Table : Empirical desity about ^, 25 blocks of 4*4 quee matrix, =5 ^ t-statistics From rst to third row are = :95 ad 2 = :999, = :999 ad 2 = :95, ad = 2 = :

27 Table 2: Empirical desity about ^ 2 ad R 2, 25 blocks of 4*4 quee matrix, =5 ^ 2 R From rst to third row are = :95 ad 2 = :999, = :999 ad 2 = :95, ad = 2 = :

28 Table 3: Empirical desity about Mora I ad LM, 25 blocks of 4*4 quee matrix, =5 Mora I LM From rst to third row are = :95 ad 2 = :999, = :999 ad 2 = :95, ad = 2 = :

29 Table 4: Empirical desity about ^, blocks of Aseli (988) s matrix, =49 ^ t-statistics From rst to third row are = :95 ad 2 = :999, = :999 ad 2 = :95, ad = 2 = :

30 Table 5: Empirical desity about ^ 2 ad R 2, blocks of Aseli (988) s matrix, =49 ^ 2 R From rst to third row are = :95 ad 2 = :999, = :999 ad 2 = :95, ad = 2 = :

31 Table 6: Empirical desity about Mora I ad LM, blocks of Aseli (988) s matrix, =49 Mora I LM From rst to third row are = :95 ad 2 = :999, = :999 ad 2 = :95, ad = 2 = :999. 3

32 Table 7: t-statistics for ^, rep= = from the quee matrix = 98 from Aseli (988) = :95 = :999 = :999 = :95 = :999 = :999 2 = :999 2 = :95 2 = :999 2 = :999 2 = :95 2 = :999 below to to to to to to to to to to to to to to above = 5 from the quee matrix = 49 from Aseli (988) = :95 = :999 = :999 = :95 = :999 = :999 2 = :999 2 = :95 2 = :999 2 = :999 2 = :95 2 = :999 below to to to to to to to to to to to to to to above 4 3