On the Bootstrap for Spatial Econometric Models

Transcription

1 O the Bootstrap for Spatial Ecoometric Models Fei Ji a, Lug-fei Lee a a Departmet of Ecoomics, The Ohio State Uiversity, Columbus, OH 4310 USA Abstract This paper is cocered with the use of the bootstrap for spatial ecoometric models. We show that the bootstrap for spatial ecoometric models ca be studied based o liear-quadratic (LQ forms of disturbaces. By provig the uiform covergece of the cumulative distributio fuctio for LQ forms to that of ormal distributios, we show that the bootstrap is geerally cosistet for test statistics that ca be approximated by LQ forms, which iclude Mora s I, Cox-type ad spatial J-type test statistics. Possible asymptotic refiemets of the bootstrap for spatial ecoometric models may be studied based o some asymptotic expasios for LQ forms. We discuss two cases: whe the disturbaces are ormal, we directly show the existece of Edgeworth expasios for LQ forms ad apply the result to show that the bootstrap for Mora s I ca provide asymptotic refiemets; whe the disturbaces are ot ormal, we show the existece of a oe-term asymptotic expasio of LQ forms based o martigales, which sheds light o the secod-order correctess of the bootstrap for LQ forms. Keywords: Bootstrap, spatial, cosistecy, asymptotic refiemet, liear-quadratic form JEL classificatio: C1, C83, R15 1. Itroductio The bootstrap is a statistical procedure that estimates the distributios of estimators or test statistics by resamplig the data. Its approximatios ca be at least as good as those from the first-order asymptotic theory uder mild coditios. Thus it ca be used as a alterative whe evaluatig the asymptotic distributios is difficult. A more appealig feature of the bootstrap is that it is ofte more accurate i fiite samples tha the asymptotic theory, i.e., it ca provide asymptotic refiemets. The bootstrap is frequetly used to correct the bias of estimators, estimate the critical values for hypothesis tests, costruct cofidece itervals, etc. Useful survey papers o the bootstrap iclude, amog others, DiCiccio ad Efro (1996, MacKio (00, Daviso et al. (003, ad Horowitz (001, 003. The bootstrap has bee discussed ad implemeted by may researchers for models i spatial ecoometrics. Aseli (1988, 1990 discusses the bootstrap estimatio i spatial autoregressive (SAR models, which addresses: ji.10@osu.edu (Fei Ji, lee.1777@osu.edu (Lug-fei Lee Job market paper (Fei Ji October 4, 01

2 is implemeted by Ca (199. Figleto (008 ad Figleto ad Le Gallo (008 use the bootstrap to test the sigificace of the movig average parameter i models with spatial movig average disturbaces. Li et al. (011 ivestigate the properties of bootstrapped Mora s I uder heterogeeous ad o-ormal disturbaces with a Mote Carlo study. Figleto ad Burridge (010 ad Burridge (01 fid that the bootstrap ca essetially remove the size distortio of the spatial J test i Kelejia (008 i Mote Carlo studies. Yag (011 proposes the residual bootstrap for LM tests of spatial depedece. Ji ad Lee (01 employ the bootstrap to remove the size distortio of Cox-type tests for SAR models with SAR disturbaces (SARAR models for short. Su ad Yag (008 suggest a bootstrap procedure that leads to a robust estimate of a variace-covariace matrix. Yag (01 proposes a bootstrap procedure to correct the bias ad variace of quasi-maximum likelihood estimators (QMLE for SAR models. Mochuk et al. (011 compare several bootstrap methods i Mote Carlo studies for costructig cofidece itervals i a spatial error model. Although there have bee may applicatios of the bootstrap i spatial ecoometric models icludig Mote Carlo studies i the precedig papers, its validity for these models has ot bee formally justified. The objective of this paper is to establish the cosistecy of the bootstrap for several test statistics i spatial ecoometric models ad provide a prelimiary discussio of possible asymptotic refiemets. We shall show that may estimators i spatial ecoometric models ca be approximated by liear-quadratic (LQ forms of the disturbaces, ad test statistics are either approximated by or closely related to LQ forms, due to the presece of spatial depedece. The bootstrap i spatial ecoometric models thus ca be studied based o LQ forms i geeral. Kelejia ad Prucha (001 prove a cetral limit theorem for LQ forms usig a cetral limit theorem for martigale differece arrays. We shall show that the covergece of the cumulative distributio fuctio (CDF for a LQ form is uiform uder the same coditios. Usig this uiform covergece, the bootstrap ca geerally be show to be cosistet for statistics that ca be approximated by a LQ form. We apply the result to show the cosistecy of the bootstrap for Mora s I ad the spatial J-type tests (Kelejia ad Piras, 011. We shall also discuss possible asymptotic refiemets of the bootstrap for spatial ecoometric models based o the LQ forms. For o-spatial ecoometric models, the bootstrap is ofte cosidered for the statistics that are smooth fuctios of sample averages of idepedet radom vectors, see, e.g., Hall (1997, or statioary depedet radom vectors, see, e.g., Götze ad Hipp (1983, 1994, for which the Edgeworth expasios are well established. The existed Edgeworth expasios for the radom vectors ca be used to prove the cosistecy ad asymptotic refiemets of the bootstrap. The framework does ot apply to LQ forms, which caot be writte as simple averages of disturbaces or their cross-products. For statistics i spatial ecoometric models, we may ivestigate whether the bootstrap ca provide asymptotic refiemets by cosiderig Edgeworth expasios of LQ forms. Such expasios, however, have ot be proved to exist i the literature. For LQ forms with ormal disturbaces, we shall show the existece of Edgeworth expasios

3 ad apply the result to show that the bootstrap ca provide asymptotic refiemets for Mora s I; for LQ forms with o-ormal disturbaces, we verify a asymptotic expasio of LQ forms based o martigales (Myklad, The Edgeworth expasio for a LQ form is established by usig a smoothig iequality that bouds the gap betwee two fuctios with the related Fourier trasforms. The special feature of the square matrix ivolved i a LQ form for spatial ecoometric models, i.e., the boudedess i both row ad colum sum orms, ca be used to obtai the order of the boud. The asymptotic expasio based o martigales is ot i a poitwise topology but sheds light o the bootstrap. It implies the secod-order correctess of the bootstrap for LQ forms i the sese of the covergece i Myklad ( The rest of the paper is orgaized as follows: Sectio demostrates a close relatioship betwee LQ forms ad estimators ad test statistics i spatial ecoometric models; Sectio 3 first shows the uiform covergece of the CDF for LQ forms ad the applies the result to show the bootstrap is cosistet for Mora s I ad spatial J-type tests; Sectio 4 establishes the Edgeworth expasio of a LQ form with ormal disturbaces, which is applied to show the secod-order correctess of the bootstrap for Mora s I, ad establishes the asymptotic expasio i Myklad (1993 for LQ forms with o-ormal disturbaces; Sectio 5 cocludes. Lemmas ad proofs are collected i the appedices.. Statistics i Spatial Ecoometrics ad LQ Forms I this sectio, we show that several estimators for spatial ecoometric models ca be approximated by LQ forms of disturbaces, ad may test statistics ca be approximated by or relate closely to LQ forms. As a result, we may study the bootstrap for spatial ecoometric models based o LQ forms. As the SARAR model is a popular ad geeral spatial model, which cotais both the spatial lag (SAR model ad spatial error (SE model as special cases, our discussio will maily focus o this model. A SARAR model is specified as y = λw y + X β + u, u = ρm u + ɛ, ɛ = (ɛ 1,..., ɛ, (1 where is the sample size, y is a -dimesioal vector of observatios o the depedet variable, X is a k x matrix of exogeous variables, W ad M are spatial weights matrices with zero diagoals, ɛ i s are i.i.d. with mea zero ad variace σ, ad θ = (λ, ρ, β, σ = (γ, σ is a vector of parameters. Let θ 0 be the true parameter vector, S (λ = I λw ad R (ρ = I ρm with I beig the -dimesioal 1 For LQ forms with o-ormal disturbaces, directly ivestigatig Edgeworth expasios ca be hard. Götze et al. (007 establishes a oe term Edgeworth expasio for a quadratic form. Their proof is based o a symmetrizatio iequality ad the differetial iequality method. The quadratic matrix i Götze et al. (007 has some special feature ot shared by the square matrix i the LQ form here. With a square matrix bouded i both row ad colum sum orms i the quadratic form, the expasio established usig similar methods may ot geerate a remaider term of a desirable order. I additio, the geeralizatio to a LQ form is ot straightforward. 3

4 idetity matrix. Deote S = S (λ 0 ad R = R (ρ 0 for short. The SARAR model ests the SAR ad SE models. The SAR model is (1 with i.i.d. disturbaces, i.e., ρ = 0, ad the SE model is (1 without the spatially lagged term of the depedet variable, i.e., λ = 0. The spatial Durbi model has a additioal term (+W X ζ o the r.h.s. of the equatio for y of the SAR model. As W X ca be take as a exogeous variable matrix, with some additioal idetificatio cosideratio i some cases, the aalysis for a spatial Durbi model is similar to that for a SAR model. For estimators of the SARAR model, the derivatives of the correspodig criterio fuctios evaluated at the true parameter vector are ofte LQ forms of the disturbaces, rather tha just liear forms, due to the presece of spatial depedece. As a result, these estimators ca be approximated by a LQ form. Lee (004 has proved the cosistecy ad asymptotic ormality of the QMLE for a SAR model without SAR disturbaces. The aalysis ca be exteded to the SARAR model (1 as i Ji ad Lee (01, from which we have 1 (ˆθ θ 0 = ( E L (θ L (θ 0 θ θ + o P (1, ( θ where ˆθ is the QMLE ad L (θ deotes the log likelihood fuctio of the model. Every elemet of the vector 1 L (θ 0 θ is liear i the disturbaces or of the LQ form (ɛ A ɛ σ 0 tr(a + b ɛ /, (3 where A is a -dimesioal square matrix ad b is a -dimesioal vector. Thus every elemet of (ˆθ θ 0 ca be approximated asymptotically by a liear combiatio of LQ forms, which is still a LQ form with the same ɛ. For the geeralized method of momets (GMM estimator, from Lee (001, 007, (E g (ˆγ γ 0 = ( (γ 0 a a g (γ 0 γ ( 1( g E E (γ 0 a γ a g (γ 0 + o P (1, γ where ˆγ is the GMM estimator of γ, a is a matrix with full colum rak greater tha or equal to (k x +, ad g (γ = ( ɛ (γd 1 ɛ (γ,..., ɛ (γd m ɛ (γ, ɛ (γq / with ɛ (γ = R (ρ[s (λy X β], D i s beig -dimesioal square matrices with zero traces ad Q beig a matrix of istrumetal variables costructed as fuctios of X, W ad M i a two-stage least squares (SLS approach. Every elemet of g (γ 0 is a quadratic or liear form of the disturbaces, the every elemet of (ˆγ γ 0 ca be approximated by a LQ form of the disturbaces. The geeralized spatial SLS (GSSLS approach i Kelejia ad Prucha (1998 first estimates (λ, β usig oly liear momets, the derives estimates of ρ ad σ based o quadratic momets usig the residuals from the first step, ad fially updates the estimate of (λ, β by a GSLS takig ito accout the covariace structure. As (λ, β is estimated usig oly liear momets, its estimator ca be approximated by a liear form of the disturbaces, but the estimator of ρ is approximated by a LQ form because quadratic momets are used. The estimators discussed above ca be used to implemet hypothesis tests such as the classical Wald, likelihood ratio ad Lagragia multiplier (LM tests i the likelihood framework, or by the Wald test, 4

5 the distace test, ad the gradiet test i the GMM framework. These asymptotically equivalet tests are based o the asymptotic ormality of the estimators. As a result, they ca be studied based o LQ forms. I additio to the classical hypothesis tests, Mora s I test (Mora, 1950; Cliff ad Ord, 1973, 1981 is a popular test for spatial depedece, ad tests for o-ested hypotheses, such as the spatial J-type tests (Kelejia, 008; Kelejia ad Piras, 011 ad Cox-type tests (Ji ad Lee, 01, have bee proposed for testig the selectio of various spatial weights matrices i spatial models. The Mora I statistic is l M l ˆɛ M ˆɛ ˆɛ ˆɛ, where l is a -dimesioal vector of oes ad ˆɛ is the residual vector from the least squares estimatio. The test is based o the asymptotic ormality of the stadardized test statistic by deductig the mea ad dividig by the stadard deviatio. Burridge (1980 shows that for the SE model with ormal disturbaces or the spatial movig average model y = X β + u, u = ρm ɛ + ɛ, ɛ N(0, σ I, the LM test statistic is proportioal to the Mora I statistic, which is I = tr(m + M M ˆɛ M ˆɛ ˆɛ ˆɛ. (4 Let H = I X (X X 1 X. Uder the ull hypothesis of o spatial depedece, Eq. (4 becomes I = = ɛ H M H ɛ tr(m + M M ɛ H ɛ ɛ H M H ɛ σ0 tr(m H tr(m tr(m + M M ( k x σ0 + H tr(m + M M k x ( ɛ H M H ɛ ɛ H ɛ ( k x σ0 tr(m + M M ( k x σ0. ɛ H ɛ (5 Uder some regularity assumptios, the last two terms o the r.h.s. of Eq. (5 have the order O( 1/, thus the LM or Mora I statistic ca be approximated by a quadratic form of the disturbaces. Whe the ull hypothesis is the SARAR model with ormal disturbaces, (4 ad (5 ca still be used to test for spatial depedece. Kelejia ad Prucha (001 propose a geeralized Mora s I test for which the test statistic equals a quadratic form of some regressio residuals divided by a ormalizig factor. Their regularity coditios guaratee that the test statistic ca be approximated by a LQ form. The spatial J-type tests for testig oe spatial ecoometric model agaist aother oe are based o augmetig the ull model by usig a predictor from the alterative model. The augmeted model is estimated by the SLS i Kelejia (008 or Kelejia ad Piras (011 ad the they test whether the coefficiet of the predictor is statistically differet from zero or ot. Due to the SLS estimatio, the test statistic is oly a liear form of the disturbaces plus a term that coverges to zero i probability. But a 5

6 liear form is just a special case of the more geeral LQ form, so the test statistic ca also be studied usig LQ forms. The more efficiet GMM estimatio with both liear ad quadratic momets for the augmeted model may sigificatly improve the power of the spatial J-type tests (Ji ad Lee, 01. The test statistics with the GMM estimatio are approximated by LQ forms. Ji ad Lee (01 derive the Cox-type specificatio tests for SARAR models. The Cox-type tests are based o the log likelihood ratios for the ull ad alterative models with a proper adjustmet for the asymptotically ozero mea. While the first order asymptotic expasio of estimators ca be approximated by LQ forms, the adjusted log likelihood ratio itself is a LQ form at give parameters. As a result, the Cox test statistic, equal to the adjusted log likelihood ratio divided by its stadard error, is the sum of a LQ form ad a remaider term where the remaider coverges to zero i probability. Our study focuses o the bootstrap for test statistics which ca be approximated by LQ forms, icludig Mora s I ad spatial J-type test statistics. 3. Cosistecy of the Bootstrap I this sectio, we first preset a geeral result o the cosistecy of the bootstrap for statistics that may be approximated by LQ forms. The we apply the result to the Mora I statistic i Eq. (5 ad J-type test statistics for SARAR models. Cosider a statistic t for a spatial ecoometric model which is asymptotically ormal with mea zero. The t would ivolve spatial weights matrices, exogeous variables ad depedet variables. The depedet variables i t ca be replaced by their reduced forms as fuctios of disturbaces ɛ = (ɛ 1,..., ɛ, exogeous variables ad the true parameter vector θ 0. The t may also ivolve the estimator ˆθ of θ 0 ad the estimator ˆς of other momet parameter vector ς 0 for ɛ i. To compute a bootstrapped versio of t, a proper bootstrap procedure eeds to be cosidered. The spatially depedet variable usually caot be resampled directly, because doig so would destroy the iheret depedece structure. Istead, the residual bootstrap ca be used as we usually assume that the disturbaces ɛ i s are i.i.d.. We may first derive a cosistet estimator of parameters i a spatial ecoometric model ad compute the residual vector ˆɛ. The ˆɛ may ot have a zero mea, so we deduct its empirical mea from the vector to obtai ɛ = (I 1 l l ˆɛ. Next, sample with replacemet times from the elemets of ɛ to obtai a vector ɛ. 3 The a pseudo data vector y o the depedet variable ca be computed by usig the reduced form with the parameter ˆθ ad disturbaces ɛ. For example, for the SARAR model (1, we have y = S 1 (ˆλ ( X ˆβ + R 1 (ˆρ ɛ. Freedma (1981 shows the ecessity of receterig for regressio models. For the SARAR model (1, if X cotais l correspodig to a itercept term i the model, the the residuals from the quasi-maximum likelihood estimatio have mea zero ad there is o eed to receter. 3 That is, geerate the bootstrap error terms from the empirical distributio fuctio of the recetered residuals. 6

7 Estimatig θ usig y yields ˆθ ad a residual vector ˆɛ. The bootstrapped versio of t, t, is the statistic obtaied from replacig ɛ, θ 0, ˆθ ad ˆς i t by, respectively, ɛ, ˆθ, ˆθ ad ˆς, where ˆς is a vector of sample momets of ˆɛ that correspod to the momet parameters i ς 0. Let σ 0, µ 3 ad µ 4 be, respectively, the secod, third ad four momets of the zero-mea i.i.d. disturbaces ɛ i s, A = [a,ij ] be a ostochastic matrix, b = (b 1,..., b be a -dimesioal ostochastic vector, ad c = 1/( ɛ A ɛ σ 0 tr(a + b ɛ be a LQ form with mea zero ad variace σ c = 1[ σ 4 0 tr(a + σ 0b b + ( (µ4 3σ 4 0a,ii + µ 3a,ii b i ]. We assume that t ca be approximated by c /σ c such that d = t c /σ c coverges to zero i probability. Let c = 1/( ɛ A ɛ σ tr(a + b ɛ with variace σ c = 1[ σ 4 tr(a + σ b b + ( (µ 4 3σ 4 a,ii + ] µ 3a,ii b i coditioal o the bootstrap samplig process, where σ Defie d = t c /σ c. We assume the followig coditios about c. = 1 ɛ ɛ, µ 3 = 1 ɛ3 i ad µ 4 = 1 ɛ4 i. Assumptio 1. The ɛ i s i ɛ = (ɛ 1,..., ɛ are i.i.d. (0, σ 0 ad E ɛ i 4(1+δ < for some δ > 0. Assumptio. The sequece of symmetric matrices {A } are bouded i both row ad colum sum orms, 4 ad elemets of the vectors {b } satisfy sup 1 b i (1+δ <. Assumptio 3. The sequece {σ c } is bouded away from zero. The A ad b are fuctios of spatial weights matrices ad exogeous variables. As spatial weights matrices are ofte assumed to be bouded i both row ad colum sum orms ad the elemets of exogeous variables are assumed to be bouded costats (Kelejia ad Prucha, 1998; Lee, 004, it is reasoable to impose Assumptio. Kelejia ad Prucha (001 have proved the asymptotic ormality of c /σ c uder Assumptios 1 3. Uder the same coditios, we ca have the uiform covergece of the CDF for c /σ c to that for a stadard ormal variable as subsequetly show. As i Kelejia ad Prucha (001, we write c a sum of martigale differeces, the theorems i Heyde ad Brow (1970 ad Haeusler (1988 o the departure of c /σ c from the stadard ormal distributio are applicable. Let Φ(x be the CDF for a stadard ormal radom variable, P ad E be, respectively, the probability distributio ad expectatio iduced by the bootstrap samplig process, ad let K a ad K b be costats such that for ay, sup 1 j a,ij K a, sup 1 i Theorem 1. Uder Assumptios 1 3, a,ij K a, ad 1 b i (1+η K b for 1 < η δ. sup P(c /σ c x Φ(x r, (6 x R sup P (c /σc x Φ(x r, (7 x R 4 As ɛ Aɛ = ɛ (A + A ɛ/, it is w.l.o.g. to assume the symmetry of A. 7

8 sup P ( c /σc + d x P ( c /σ c + d x r + P( d > τ + r + P ( d > τ + 1/ π 1/ τ, x R (8 sup P ( (c /σc + d e x P ( (c /σ c + d e x r + P( d > τ + r + P ( d > τ x R + 1/ π 1/ τ + sup Φ(x/e Φ(x/e (9, x R where τ is ay positive term depedig oly o, e is a positive ostochastic term depedig o, θ 0 ad momet parameters of ɛ i, r = Kσ (1+δ/(3+δ c δ/(3+δ ((K a δ( K a E ɛ i σ 0 +δ + +δ K a (E ɛ i +δ + K b E ɛ i +δ δ( σ 4 0K 4 a(µ 4 σ σ 8 0K 4 a + σ 0K a(µ 3K a + σ 4 0K b (K a µ 3 σ 0K 3 a( µ 3 K a + σ 0K b (1+δ/ 1/(3+δ with K beig a costat depedig oly o δ, r is a term obtaied from replacig the populatio momet parameters of ɛ i i r with the correspodig sample momets of ɛ i, ad e is a term obtaied from replacig θ 0 ad populatio momet parameters of ɛ i i e by, respectively, ˆθ ad correspodig sample momets of ɛ i. The l.h.s. of (6 is the Kolmogorov-Smirov distace betwee the CDFs of two radom variables. The iequality gives a rate of covergece, O( δ/(3+δ, of the CDF of c /σ c to that of a stadard ormal radom variable. The larger is δ, i.e., the higher momets of ɛ i assumed to exist, the faster is the covergece. The covergece rate approaches O( 1/, the rate for a sample average of i.i.d. radom variables, as δ becomes larger. A similar result for the bootstrapped versio of c /σ c is give i (7. The result i (8 is show by usig (6 ad (7. To prove the cosistecy of the bootstrapped t, we may show that the r.h.s. of (8 coverges to zero i probability. This type of covergece with respect to the Kolmogorov-Smirov distace implies the asymptotic cosistecy of cofidece itervals. If we ca show that the sample momets of ɛ i coverge i probability to the relevat populatio momets of ɛ i, the the cotiuous mappig theorem implies that r is of order O P ( δ/(3+δ. The remaider term d is ofte of order O P ( 1/, thus we may let τ = O( α for some 0 < α < 1/. It oly remais to show that P ( d > τ coverges to zero i probability. For asymptotically ormal statistics with o-uit variaces, e.g., various estimators, we may rescale terms i (8 to obtai (9, which ca be more coveiet for the proof of cosistecy. Now we apply the results i Theorem 1 to show the cosistecy of bootstrapped Mora s I ad spatial J-type test statistics for SARAR models Mora s I To show the cosistecy of the bootstrap for Mora s I i Eq. (5, we write I i the form o the l.h.s. of (9. Note that the variace of ɛ H M H ɛ is σ0 4 tr ( H M H (M + M whe ɛ N(0, σ0i, we may let c = 1/( ɛ H M H ɛ σ0 tr(m H, (10 8

9 σc = 1 σ0 4 tr[h M H (M + M ], (11 e = tr[h M H (M + M ], k x tr(m + M M (1 d = I /e c /σ c. (13 Let I be the bootstrapped I. The I, ad the correspodig c, σ c, e ad d are derived as described earlier. Propositio 1. Uder H 0 ad Assumptios I1 I4 i Appedix A.1, the Mora I statistic i Eq. (5 satisfies sup x R P (I x P(I x = o P (1. The above propositio is the case where ɛ N(0, σ 0I, which guaratees that I i (5 is asymptotically stadard ormal. Whe the i.i.d. disturbaces are ot ormal, I is still asymptotically ormal but with a o-uit variace i geeral, sice the variace of ɛ H M H ɛ is (µ 4 3σ 4 0 (H M H ii + σ 4 0 tr[h M H (M + M ]. To make the test statistic robust to the distributio of the disturbaces, we cosider the followig statistic where ˆσ c I = ɛ H M H ɛ, (14 ˆσc = 1 (ˆµ 4 3ˆσ 4 (H M H ii + 1ˆσ 4 tr[h M H (M + M ] with ˆµ 3 = 1 ˆɛ3 i ad ˆµ 4 = 1 ˆɛ4 i. The I is asymptotically stadard ormal. We use (8 to show the cosistecy of the bootstrap for I. Now let c = 1/( ɛ H M H ɛ σ0 tr(m H, (15 σc = 1 (µ 4 3σ0 4 (H M H ii + 1 σ0 4 tr[h M H (M + M ], (16 d = I c /σ c. (17 Deote the bootstrapped I by I. Correspodigly, we have c, σ c ad d. Propositio. Uder H 0 ad Assumptios I1 I3 ad I4 i Appedix A.1, sup x R P (I x P(I x = o P ( Spatial J-type Tests I this subsectio, we show the cosistecy of the bootstrapped spatial J-type tests for SARAR models (Kelejia ad Piras, 011. Cosider the problem of testig oe SARAR model agaist aother oe: H 0 : y = λ 1 W 1 y + X 1 β 1 + u 1, u 1 = ρ 1 M 1 u 1 + ɛ 1, ɛ 1 = (ɛ 1,1,..., ɛ 1,, (18 H 1 : y = λ W y + X β + u, u = ρ M u + ɛ, ɛ = (ɛ,1,..., ɛ,, (19 9

10 where ɛ 1,i s are i.i.d. (0, σ1 ad ɛ,i s are i.i.d. (0, σ. Other terms i the above models, with subscripts idicatig differet models, have similar meaigs as those for the model (1. For i = 1,, let θ i = (λ i, ρ i, β i, σ i, S i (λ i = I λ i W i, R i (ρ i = I ρ i M i. The true parameter vector for the model (18 is θ 10. The idea of the J-type tests is to augmet the ull model usig a predictor ŷ for the depedet variable from the alterative model ad test whether the coefficiet of the predictor is sigificatly differet from zero. I specific, the augmeted model is R 1 (ρ 1 y = λ 1 R 1 (ρ 1 W 1 y + R 1 (ρ 1 X 1 β 1 + αr 1 (ρ 1 ŷ + ɛ, (0 Note that a spatial Cochrae-Orcutt trasformatio has bee used for the efficiecy of the predictor ŷ. Give a estimator ˆθ for the alterative model, a predictor of y ca be ˆλ W y +X ˆβ from the r.h.s. of the equatio for y i (19 or S 1 (ˆλ X ˆβ from the reduced form. 5 I Kelejia ad Piras (011, a GSSLS estimator ρ 1 is plugged i (0 ad ŷ is also computed usig the GSSLS estimator, the (0 is estimated by the SLS. Alteratively, we ca use the QMLE to compute ŷ ad the estimate ρ 1 joitly with λ 1, β 1 ad α i (0 by the GMM. Uder the ull hypothesis, each estimator of α is asymptotically ormal ad the test is based o such a distributio. We first ivestigate the case with the estimatio method i Kelejia ad Piras (011, ad the study the case with the alterative estimatio method. The spatial SLS estimatio of a SARAR model (Kelejia ad Prucha, 1998, (18 or (19, ivolves several steps: γ i = (λ i, β i is first estimated by the SLS, the the residuals are used to estimate ξ i = (ρ i, σ i by a GMM with quadratic momets of the form E(ɛ i D ij,ɛ i = σ i0 tr(d ij,, where D ij, is a -dimesioal square matrix ad σi0 is the true secod momet whe the ith SARAR model geerates the data, ad fially the estimates of λ i ad β i are updated by the SLS estimatio of the Cochrae-Orcutt trasformed spatial model, for i = 1,. Kelejia ad Prucha (1998 use the matrices I, M i ad M i M i for their quadratic momets i the secod step. Let Z i = (W i y, X i, P A = A (A A 1 A for ay full rak matrix A with row dimesio, Υ i be the istrumets for the first step estimatio, ˇγ i be the first step SLS estimator of γ i, ˆξ i be the estimator of ξ i i the secod step, Ξ i be the istrumets for the fial step ad ˆγ i be the estimator of γ i from the fial step. 6 With these otatios, we have ˇγ i = (Z i P Υ i Z i 1 Z i P Υ i y, the objective fuctio of the secod step i the spatial SLS is g (ξ i ; ˇγ i g (ξ i ; ˇγ i, where g (ξ i ; ˇγ i = 1 [ɛ i (ρ i; ˇγ i ɛ i (ρ i ; ˇγ i σ i, ɛ i (ρ i; ˇγ i M i M iɛ i (ρ i ; ˇγ i σ i tr(m i M i, ɛ i (ρ i; ˇγ i M i ɛ i (ρ i ; ˇγ i ] with ɛ i (ρ i ; ˇγ i = R i (ρ i [S i (ˇλ i y X i ˇβi ], ad ˆγ i = [Z i R i (ˆρ ip Ξi R i (ˆρ i Z i ] 1 Z i R i (ˆρ ip Ξi R i (ˆρ i y. For the estimatio of (0, the istrumets ca be from both models, so they ca be geerated from 5 The aalyses for the two predictors are similar. I the followig part, we oly focus o the predictor ˆλ W y + X ˆβ for simplicity. 6 The Υ i ca be geerated from W i ad X i, say the liear idepedet colums of X i, W i X i ad W i X i, ad Ξ i ca be geerated from W i, M i ad X i, say the liear idepedet colums of X i, W i X i, W i X i, M i X i ad M i X i. 10

11 X 1, X, W 1, W, M 1 ad M. By the Frisch-Waugh-Lovell theorem o partitioed regressios, ˆα = [( P R 1 (ˆρ 1 ŷ (I P V(ˆρ 1P R 1 (ˆρ 1 ŷ ] 1 ( P R 1 (ˆρ 1 ŷ (I P V(ˆρ 1R 1 (ˆρ 1 y = [ŷ R 1(ˆρ 1 P (I P V(ˆρ 1P R 1 (ˆρ 1 ŷ ] 1 ŷ R 1(ˆρ 1 P (I P V(ˆρ 1R 1 (ˆρ 1 R 1 1 ɛ 1, where V (ˆρ 1 = P R 1 (ˆρ 1 Z 1. As R 1 (ˆρ 1 R1 1 = I + (ρ 10 ˆρ 1 M 1 R 1, the spatial J test statistic J 1 = ˆα /ˆσˆα = ˆα [ŷ R 1(ˆρ 1 P (I P V(ˆρ 1P R 1 (ˆρ 1 ŷ ] 1/ /ˆσ 1, ( 1 (1 where ˆσ 1 = 1ˆɛ 1ˆɛ 1 with ˆɛ 1 = R 1 (ˆρ 1 [S 1 (ˆλ 1 y X 1 ˆβ1 ], is asymptotically stadard ormal uder the ull hypothesis ad the assumptio that 1 R 1 (ˆρ 1 ŷ coverges to a o-zero limit i probability alog with other regularity coditios. The assumptio o ŷ is o the whole term 1 R 1 (ˆρ 1 ŷ, but remais implicit o the specific behavior of ˆγ uder the ull hypothesis. As we would like to study the cosistecy of the bootstrapped spatial J tests, there is a eed to ivestigate the remaider term of the spatial J test statistic after beig approximated by a liear form of the disturbaces. This ca be doe by usig the pseudo-true values. The alterative model may have differet fuctioal forms or variables from those for the ull model, thus the estimator for the alterative model geerally would ot coverge to the true parameter value of the ull model. But we ca ofte fid a sequece of o-stochastic vectors, i.e., pseudo-true values, such that the differece betwee the estimator ad the pseudo-true value coverges to zero i probability. As the spatial SLS ivolves three steps, we have a pseudo-true value i each step. I the first step, as ˇγ i = (Z i P Υ i Z i 1 Z i P Υ i y, the pseudo-true value γ i,1 ca be γ i,1 = (E Z i P Υ i E Z i 1 E Z i P Υ i E y. 7 As show i Lemma 9, 1/ (ˆγ i γ i,1 = O P (1. The i step, the pseudo-true value ξ i,1 ca be ξ i,1 = arg mi ξi 1 E g (ξ i ; γ i,1 E g (ξ i ; γ i,1. I the last step, the pseudo-true value γ i,1 is γ i,1 = [(R i E Z i P Ξi R i E Z i ] 1 (R i E Z i P Ξi R i E y, where R i deotes R i ( ρ i,1 for short. Let σ ˆα = σ 10[ γ,1 E(Z R 1P (I P V P R 1 E(Z γ,1 ] 1 ad ᾱ = σ 10 σˆα γ,1 E(Z R 1P (I P V ɛ 1 with V = P R 1 E Z 1. The as show i the proof of Propositio 3, J 1 = ᾱ / σˆα + o P (1. Although J 1 is approximated by a liear form of the disturbaces, the bootstrap for J 1 ca be proved to be cosistet usig a LQ form. 8 Correspodig to the bootstrapped data vector y, 9 let ˇγ i = (Z i P Υ i Z i 1 Z i P Υ i y with Z i = (W iy, X i, ˆξ i = arg mi ξ i g (ξ i ; ˇγ i g (ξ i ; ˇγ i, where g (ξ i ; ˇγ i = 1 [ɛ i (ρ i; ˇγ i ɛ i (ρ i; ˇγ i σ i, ɛ i (ρ i; ˇγ i M i M iɛ i (ρ i; ˇγ i σ i tr(m i M i, ɛ i (ρ i; ˇγ i M iɛ i (ρ i; ˇγ i ] with ɛ i (ρ i; ˇγ i = R i(ρ i [S i (ˇλ i y X i ˇβ i ], ad ˆγ i = [(R i(ˆρ i Z i P Ξ R i (ˆρ i Z i ] 1 (R i (ˆρ i Z i P Ξ R i (ˆρ i y. The the bootstrapped spatial J test 7 For geerality, we use the term pseudo-true value for both i = 1 ad i =. Note that γ 1,1 = γ A alterative is to use the Mallows metric as i regressio models. See Freedma ( Here y = S 1 (ˆλ 1 [X 1 ˆβ1 + R 1 1 (ˆρ 1ɛ 1 ], where ɛ 1 is a -dimesioal vector of radom samples from the elemets of (I l l /ˆɛ 1, with ˆɛ 1 beig the residual vector from the GSSLS estimatio of the model (18. 11

12 statistic is J 1 = ˆα /ˆσ ˆα = ˆα [ŷ R 1(ˆρ 1P (I P V (ˆρ 1 P R 1 (ˆρ 1ŷ ] 1/ /ˆσ 1, (3 where ŷ = Z ˆγ, ˆσ 1 = 1 ɛ 1(ˆρ 1; ˆγ 1ɛ 1(ˆρ 1; ˆγ 1, V (ˆρ 1 = P R 1 (ˆρ 1Z 1, ad ˆα = [ŷ R 1(ˆρ 1P (I P V (ˆρ 1 P R 1 (ˆρ 1ŷ ] 1 ŷ R 1(ˆρ 1P (I P V (ˆρ 1 R 1 (ˆρ 1R 1 1 (ˆρ 1ɛ 1. Propositio 3. Uder H 0 ad the assumptios i Appedix A., sup x R P (J 1 x P(J 1 x = o P (1. Cosider ow the alterative estimatio method of the augmeted model (0. Let θ = ( γ, σ be the QMLE of the model (19 with θ,1 beig the pseudo-true value uder H 0. For the estimatio of (0, we ca use both liear momets ad quadratic momets for the GMM. Let D 1,..., D m be -dimesioal square matrices with zero traces for the quadratic momets ad be the istrumetal matrix used i the SLS estimatio approach. The D i s ca be costructed from W 1, M 1, W ad M. The momet vector is g (ψ; γ = 1 (ɛ (ψ; γ D 1 ɛ (ψ; γ,..., ɛ (ψ; γ D m ɛ (ψ; γ, ɛ (ψ; γ, where ψ = (λ 1, ρ 1, β 1, α ad ɛ (ψ; γ = R 1 (ρ 1 [S 1 (λ 1 y X 1 β 1 α( λ W y + X β ]. The true parameter vector of ψ is ψ 0 = (λ 10, ρ 10, β 10, 0. A geeral objective fuctio of the GMM is g (ψ; γ a a g (ψ; γ, where {a } is a sequece of full rak matrices that coverges to a costat matrix a 0. By the geeralized Cauchy-Schwarz iequality, the optimal weightig matrix is the variace-covariace (VC matrix Ω of 1/ g (ψ 0 ; γ. For the feasible optimal GMM, a first step cosistet estimator ˇψ ca be derived from miimizig g (ψ; γ g (ψ; γ, the a estimator ˆψ ca be the miimizer of g (ψ; γ ˆΩ 1 g (ψ; γ, where ˆΩ is the matrix obtaied by replacig the ψ 0 ad other momet parameters of ɛ 1,i i Ω by, respectively, ˇψ ad the correspodig sample momets of the first-step residuals. Uder some regularity coditios, ˆψ is cosistet for ψ 0 ad 1/ ( ˆψ ψ 0 is asymptotically ormal with limitig VC matrix lim [E G (ψ 0 ; γ Ω 1 statistic be E G (ψ 0 ; γ ] 1, where G (ψ; γ = g(ψ;γ ψ. The we may let the spatial J test J = 1/ e ψ ˆψ /[e ψ(g ( ˆψ ; γ ˆΩ 1 G ( ˆψ ; γ 1 e ψ ] 1/, (4 where e ψ is a vector with legth equal to that of ψ, whose last elemet is 1 ad other elemets are zero. As show i Sectio, J ca be approximated by a LQ form as every elemet of g (ψ 0 ; γ is a liear or quadratic form of ɛ 1. With the bootstrapped data vector y, 10 let θ be the QMLE of the model (19, the momet vector for the GMM estimatio of (0 be g (ψ; γ = 1 (ɛ (ψ; γ D 1 ɛ (ψ; γ,..., ɛ (ψ; γ D m ɛ (ψ; γ, ɛ (ψ; γ with ɛ (ψ; γ = R 1 (ρ 1 [S 1 (λ 1 y X 1 β 1 α( λ W y + 10 Here we may let y = S 1 ( λ 1 [X 1 β1 + R 1 1 ( ρ 1ɛ 1 ], where θ 1 = ( λ 1, ρ 1, β 1, σ 1 is the QMLE of the model (18, ad ɛ 1 is a -dimesioal vector of radom samples from the elemets of (I ll /ˆɛ 1, with ˆɛ 1 beig the residual vector from the QML estimatio of the model (18. 1

13 X β ], ˇψ ad ˆψ be the first-step ad secod-step estimators i the feasible optimal GMM approach respectively, G (ψ; γ = g (ψ;γ ψ, ad ˆΩ be the matrix obtaied by replacig the estimators i ˆΩ by the correspodig oes with y. The the bootstrapped J is J = 1/ e ψ ˆψ /[e ψ(g ( ˆψ ; γ ˆΩ 1 G ( ˆψ ; γ 1 e ψ ] 1/. (5 Propositio 4. Uder H 0 ad the assumptios i Appedix A.3, sup x R P (J x P(J x = o P (1. 4. Asymptotic Refiemets The Edgeworth expasio has bee well established for a smooth fuctio of sample averages of idepedet radom vectors ad/or statioary depedet radom vectors. It provides a useful tool to prove that the bootstrap may provide asymptotic refiemets. The LQ forms for spatial ecoometric models ivolve spatial weights matrices ad caot be writte as simple sample averages of disturbaces or their cross-products. If we would like to ivestigate possible asymptotic refiemets of the bootstrap i spatial ecoometric models usig some expasios, we eed to justify the validity for such expasios first. Whe the disturbaces i a LQ form are ormally distributed, Edgeworth expasios ca be established without much difficulty. But whe the disturbaces are ot ormal, directly ivestigatig possible expasios ca be hard. A alterative approach is to decompose a LQ form ito the sum of martigale differeces ad the study the expasios for martigales. Myklad (1993 establishes a asymptotic expasio for martigales, but the expasios are ot i a poitwise topology. Below we discuss the cases of ormal ad o-ormal disturbaces separately Normal Disturbaces Whe the disturbaces i a LQ form c /σ c = 1/( ɛ A ɛ σ0 tr(a + b ɛ /σc are i.i.d. ormal, we ca easily derive its characteristic fuctio. By a smoothig iequality i Feller (1970, the differece betwee two fuctios has a upper boud geerated from the Fourier trasforms relatig to these two fuctios. The iequality is used to establish the Berry-Essee boud for the error i the approximatio of the ormal distributio or the Edgeworth expasio to the true distributio for a sample mea of i.i.d. disturbaces. It ca also be used to establish the Edgeworth expasio of a LQ form. Let f (k (x be the kth order derivative of a fuctio f(x. We ca use the boudedess i both row ad colum su orms of the matrix A to boud the derivatives of the characteristic fuctio for a LQ form. Theorem. Uder Assumptios ad 3, whe ɛ N(0, σ 0I, sup P(c /σ c x [Φ(x + κ (1 x Φ (1 (x] = O( 1, (6 x R 13

14 where κ κ = 3/ σ 3 c sup P (c /σc x [Φ(x + κ (1 x Φ (1 (x] = O P ( 1, (7 x R = 3/ σ 3 c [4σ 6 0 tr(a 3 /3 + σ 4 0b A b ] = O( 1/ with σ c [4σ 6 tr(a 3 /3 + σ 4 b A b ] = O P ( 1/ with σ c = 1 [σ 4 0 tr(a + σ 0b b ] ad = 1 [σ 4 tr(a + σ b b ], ad for r 3, there exist real polyomials P 3 (x,..., P r (x with bouded coefficiets such that sup P(c /σ c x R x Φ(x Φ (1 (x r (i / P i (x = O( (r 1/. (8 i=3 Eqs. (6 ad (7 ca be used to show that the bootstrap ca provide asymptotic refiemets for some statistics that ca be approximated by a LQ form. Eq. (8 presets a geeral high order expasio for the CDF of a LQ form. Note that κ has a relatively simple form. Istead of bootstrappig test statistics, we may correct the bias distortio for test statistics that ca be approximated by a LQ form. 11 The above theorem ca be applied to show that the bootstrap for Mora s I is more accurate tha the first-order asymptotic theory. Propositio 5. Uder H 0 ad Assumptios I1 I4 i Appedix A.1, the Mora I statistic i Eq. (5 satisfies P (I x P(I x = O P ( No-ormal Disturbaces For LQ forms with o-ormal disturbaces, a theorem o asymptotic expasios for martigales i Myklad (1993 ca be applied to establish a expasio, which the author calls the Edgeworth expasio for martigales. The coditios eeded are maily imposed o the variatio measures associated with martigales, e.g., the optioal kth-order variatio, which is defied as the sum of the kth powers of the martigale differeces. Oe coditio is the cetral limit theorem which relates to the optioal secod-order variatios. The c /σ c ca be decomposed as the sum of martigale differeces that are quadratic i the disturbaces. We eed the existece of E ɛ i 4(1+δ for some δ > 0 to show the asymptotic ormality of c /σ c, which is based a cetral limit theorem for martigales. For the cetral limit theorem relatig to the optioal secod-order variatios, higher momets for ɛ i are required to exist. Correspodigly, a stroger coditio o b is also assumed. Assumptio 1. The ɛ i s i ɛ = (ɛ 1,..., ɛ are i.i.d. (0, σ0 ad E ɛ i 8(1+δ < for some δ > 0. Assumptio. The sequece of symmetric matrices {A } are bouded i both row ad colum sum orms ad the elemets of the vectors {b } satisfy sup 1 b i 4(1+δ <. 11 Robiso ad Rossi (010 have cosidered a fiite sample correctio of Mora s I test for a pure SAR model. They have ot show the validity of their expasio for the CDF of Mora s I test statistic, which is i terms of the CDF for a chi-square distributio. 14

15 Theorem 3. Uder Assumptios 1, ad 3, we have + h(x df (x = + h(x dφ(x / E [( ψ o (Y + ψ p (Y h ( (Y ] + o( 1/, (9 where F (x = P(c /σ c x, Y is the ormal radom variable that c /σ c coverges to, ad expressios for ψ o (Y ad ψ p (Y are give i (C.17 (C.0, uiformly o a set l of fuctios h which are twice differetiable, with h, h (1 ad h ( uiformly bouded, ad with {h (, h l} beig equicotiuous a.e. Lebesgue. Deote the covergece i (9 by o ( 1/ (Myklad, 1993, the F (x = Φ(x /( ψ (1 o (x + ψ (1 p (x [ψ o (x + ψ p (x]x Φ (1 (x + o ( 1/. (30 As poited out by Myklad (1993, the expasio geerally does ot hold whe h is a idicator fuctio of a iterval, so it is a smoothed expasio. Note that ψ o (x ad ψ p (x are liear i x, the ψ (1 o (x + ψ p (1 (x [ψ o (x+ψ p (x]x = (1 x [ψ o (1 (x+ψ (1 (x]. I the special case that ɛ i s are i.i.d. ormal, we p ca verify that 1 6 1/ [ψ o (1 (x + ψ p (1 (x [ψ o (x + ψ p (x]x] = (1 x lim κ, thus (30 has similar terms as the usual oe-term Edgeworth expasio (6. 5. Coclusio I this paper, we cosider the use of the bootstrap i spatial ecoometric models. We show that the bootstrap for estimators ad test statistics i spatial ecoometric models ca be studied based o LQ forms. We have established the uiform covergece of the CDF for a LQ form to that of the stadard ormal radom variable. Based o this result, we show that the bootstrap is cosistet for Mora s I ad spatial J-type test statistics. As possible asymptotic refiemets for the bootstrap are usually show by usig some asymptotic expasios, we discuss expasios for LQ forms: for ormal disturbaces, we have established the Edgeworth expasios for LQ forms ad applied the result to show the secod-order correctess of the bootstrap for Mora s I; for o-ormal disturbaces, we have established a asymptotic expasio based o martigales. There are some extesios which ca be of iterest for future research. Some asymptotic chi-square tests i spatial ecoometrics, e.g., hypothesis tests with multiple costraits, are costructed from vectors of LQ forms. The curret uiform covergece result, which is oly about a sigle LQ form, does ot cover vectors of LQ forms. It is of iterest to establish the uiform covergece result for vectors of LQ forms so that the bootstrap ca be show to be cosistet for asymptotic chi-square tests. It also remais to show high order expasios of a vector of LQ forms for asymptotic refiemets of the bootstrap. 15

16 Appedix A. Assumptios Appedix A.1. Assumptios for Mora s I Assumptio I1. The sequece of matrices {M } have zero diagoals ad are bouded i both row ad colum sum orms. Assumptio I. The sequece of full rak matrices {X } have uiformly bouded costat elemets, ad lim 1 X X exists ad is osigular. Assumptio I3. The sequece {( 1 [(µ 4 3σ 4 0 (H M H ii + σ4 0 tr((m + M ]} is bouded away from zero. Assumptio I4. The disturbace vector ɛ N(0, σ 0I. Assumptio I4. The ɛ i s i ɛ = (ɛ 1,..., ɛ are i.i.d. ad E ɛ 8 i <. The variace of 1/ ɛ H M H ɛ is guarateed to be bouded away from zero i Assumptio I3, as 1 tr[h M H (M + M ] = ( 1 tr[(m + M ] + o(1 by Lemma 1. Whe the disturbaces are ot assumed to be ormal, I geerally ivolves the estimated fourth momet of ɛ i. To prove the cosistecy of the bootstrapped I usig Theorem 1, we eed to kow the rate of covergece of the estimated fourth momet to the true oe, thus a strog coditio o ɛ i is imposed i Assumptio I4. Appedix A.. Assumptios for the Spatial J Tests: J 1 Assumptio J1. The ɛ 1,i s are i.i.d. (0, σ 10 ad the momet E(ɛ 4 1,i exists. Assumptio J. The matrices X 1 ad X have full raks ad uiformly bouded costats. The limits lim 1 X 1X 1 ad lim 1 X X exist ad are osigular. Assumptio J3. Matrices S 1 ad R 1 are osigular. Assumptio J4. The sequeces of matrices {W 1 }, {M 1 }, {R1 1 } ad {S 1 1 } are bouded i both row ad colum sum orms. The {W 1 } ad {M 1 } have zero diagoals. Assumptio J5. The 1 Υ 1Υ 1, 1 Ξ 1Ξ 1, 1 Υ 1(W 1 S 1 1 X 1β 10, X 1 ad 1 Ξ 1R 1 (W 1 S 1 1 X 1β 10, X 1 coverge to full rak matrices. Assumptio J6. The miimum eigevalue of the matrix 1 tr(m 1M 1 σ 10 tr(m 1 R 1 1 σ 10 tr(r 1 1 M 1M 1 R 1 1 σ 10 tr(m 1M 1R 1 1 σ 10 tr(r 1 1 M 1M 1R σ 10 tr[(m 1 + M 1M 1 R 1 1 ] σ 10 tr(r 1 1 M 1M 1R 1 1 is bouded away from zero, λ 1 < 1, ρ 1 < 1 ad 0 < σ 1 < c for some c > 0. 16

17 Assumptio J7. The 1 Υ Υ, 1 Ξ Ξ, 1 Υ (W S 1 1 X 1β 10, X ad 1 Ξ R (W S 1 1 X 1β 10, X coverge to full rak matrices. Assumptio J8. For ay η > 0, there exists κ > 0 such that, whe ξ ξ,1 > η, 1 [E g (ξ ; γ,1 E g (ξ ; γ,1 E g ( ξ,1 ; γ,1 E g ( ξ,1 ; γ,1 ] > κ for all large eough. The ξ,1 is i the iterior of the compact parameter space of ξ. Assumptio J9. The 1 ad 1 R 1 (W S 1 1 X 1β 10, X γ,1 coverge to full rak matrices. Assumptios J1 J6 are similar to those i Kelejia ad Prucha (1998. Assumptio J7 is for the estimators ˇγ ad ˆγ, similar to Assumptio J5 for ˇγ 1 ad ˆγ 1. Assumptio J8 states the idetificatio uiqueess coditio for ξ,1. The coditio for the estimatio of the augmeted model (0, Assumptio J9, is stated i terms of the pseudo-true value γ,1. Appedix A.3. Assumptios for the Spatial J Tests: J Let L 1 (θ 1 be the log likelihood fuctio of the model (18, L (θ be the log likelihood fuctio of the model (19, ad θ i,1 = arg max L i (θ i ; θ 10 with L i (θ i ; θ 10 = E L i (θ i uder H 0, for i = 1,. Maximizig L i (θ i ad L i (θ i ; θ 1 for give β i ad σi yields fuctios L i(φ i ad L i (φ i ; θ 1 respectively, where φ i = (λ i, ρ i. Assumptio J10. The ɛ 1,i s are i.i.d. (0, σ10 ad the momet E(ɛ 8 1,i exists. Assumptio J11. The matrices X 1 ad X have full raks ad uiformly bouded costats. The limits lim 1 X 1X 1 ad lim 1 X X exist ad are osigular. Assumptio J1. Matrices S 1 ad R 1 are osigular. Assumptio J13. The sequeces of matrices {W 1 }, {M 1 }, {R1 1 }, {S 1 1 }, {W } ad {M } are bouded i both row ad colum sum orms. The {W 1 }, {M 1 }, {W } ad {M } have zero diagoals. Assumptio J14. Each sequece of matrices {S 1 1 (λ 1}, {R 1 1 (ρ 1}, {S 1 (λ } ad {R 1 (ρ } is bouded i either row or colum sum orm uiformly i the compact parameter space. The λ 10, ρ 10, λ,1 ad ρ,1 are i the iteriors of their parameter spaces. Assumptio J15. The limits lim 1 X 1R 1(ρ 1 R 1 (ρ 1 X 1 ad lim 1 X R (ρ R (ρ X exist ad are osigular for ay ρ 1 ad ρ i their respective parameter spaces. The smallest eigevalues of R 1(ρ 1 R 1 (ρ 1 ad R (ρ R (ρ are bouded away from zero uiformly o their respective parameter spaces. 17

18 Assumptio J16. For the idetificatio of the model (18, either (i lim 1 [l σ 10S 1 1 R 1 1 R 1 1 S 1 1 l σ 1,a(φ 1 S 1 1 (λ 1R 1 1 (ρ 1R 1 1 (ρ 1S 1 1 (λ 1 ] exists ad is ozero for ay φ 1 φ 10, where σ 1,a(φ 1 = σ 10 tr[r 1 1 S 1 1 S 1(λ 1 R 1(ρ 1 R 1 (ρ 1 S 1 (λ 1 S 1 1 R 1 1 ], or (ii lim 1 (Q 1X 1 β 10, X 1 (Q 1 X 1 β 10, X 1 exists ad is osigular, ad lim 1 [l σ 10S 1 1 R 1 1 R 1 1 S 1 1 l σ 1,a(λ 10, ρ 1 S 1 1 R 1 1 (ρ 1R 1 1 (ρ 1S 1 1 ] exists ad is ozero for ay ρ 1 ρ 10, where Q 1 = W 1 S1 1. For the model (19, for η > 0, there exists κ > 0 such that, whe φ φ,1 > η, 1( L ( φ,1 ; θ 10 L,1 (φ ; θ 10 > κ for ay large eough. 1 Assumptio J17. The limits lim L1(φ 10;θ 10 φ 1 φ 1 exist ad are osigular. 1 ad lim L( φ,1;θ 10 φ φ Assumptio J18. The limit of 1 tr[r 1 1 S 1 1 S R R S S 1 1 R 1 1 ] or 1 (X 1 β 10 S 1 1 S R H R S S 1 1 X 1β 10 exists ad is o-zero. Assumptio J19. Either (i lim 1 R 1 (ρ 1 Γ, where Γ = (W 1 S 1 1 X 1β 10, X 1, λ,1 W S 1 1 X 1β 10 + X β,1, has full rak k x1 + for each possible ρ 1 i its parameter space, ad the momet equatios tr[r 1 1 R 1(ρ 1 P im R 1 (ρ 1 R 1 1 ] = 0, for i = 1,..., m, have the uique solutio at ρ 10, or (ii lim 1 R 1 (ρ 1 X 1 has full rak k x1 for each possible ρ 1 i its parameter space, ad the momet equatios tr[r 1 1 S 1 1 (S 1(λ 1 α λ,1 W R 1(ρ 1 P im R 1 (ρ 1 (S 1 (λ 1 α λ,1 W S 1 have the uique solutio at the true parameter values. 1 R 1 1 ] = 0, for i = 1,..., m, Assumptios J10 J18 are directly from Ji ad Lee (01 with the exceptio of Assumptio J10. A strog coditio is eeded i Assumptio J10 as explaied i Appedix A.1 for Assumptio I4. Assumptio J19 is the idetificatio uiqueess coditio of the GMM estimatio for the augmeted model (0, which resembles a coditio for the GMM estimatio of high order SARAR models i Lee ad Liu (010. Appedix B. Lemmas Appedix B.1. Elemetary Lemmas Lemmas 1 4 ca be foud i, e.g., Li ad Lee (010. Lemma 1. Suppose that matrices {A } are bouded i both row ad colum sum orms. Elemets of k matrices {X } are uiformly bouded ad lim 1 X X exists ad is osigular. Let H = I X (X X 1 X. The {H } are bouded i both row ad colum sum orms ad tr(h A = tr(a + O(1. Lemma. Suppose that A = [a,ij ] ad B = [b,ij ] are matrices ad ɛ i s i ɛ = (ɛ 1,..., ɛ are i.i.d. with mea zero ad variace σ 0. The, (1 E(ɛ ɛ A ɛ = E(ɛ 3 i (a,11,..., a,, ad ( E(ɛ A ɛ ɛ B ɛ = [E(ɛ 4 i 3σ4 0] a,iib,ii + σ 4 0 tr(a tr(b + σ 4 0 tr[a (B + B ]. 18

19 Lemma 3. Suppose that matrices {A } are bouded i both row ad colum sum orms, elemets of the k matrices {C } are uiformly bouded, ad ɛ i s i ɛ = (ɛ 1,..., ɛ are idepedet (0, σ i. The sequeces {σ i } ad {E(ɛ4 i } are bouded. The ɛ A ɛ = O P (, E(ɛ A ɛ = O(, 1 [ɛ A ɛ E(ɛ A ɛ ] = o P (1 ad 1/ C A ɛ = O P (1. Lemma 4. Suppose that {A } is a sequece of symmetric matrices with row ad colum sum orms bouded ad b = (b 1,..., b is a -dimesioal colum vector such that sup 1 b i +η1 < for some η 1 > 0. Furthermore, suppose that ɛ 1,, ɛ are mutually idepedet with zero meas ad the momets E( ɛ i 4+η for some η > 0 exist ad are uiformly bouded for all ad i. Let σ Q be the variace of Q where Q = ɛ A ɛ + b ɛ tr(a Σ with Σ beig a diagoal matrix with E ɛ i s o its diagoal. Assume that 1 σ Q is bouded away from zero. The Q σ Q d N(0, 1. Lemmas 5 8 are for the SARAR model (1, where ɛ i s i ɛ = (ɛ 1,..., ɛ are i.i.d. with mea zero, variace σ 0, third momet µ 3 ad fiite fourth momet µ 4, ad ˆɛ = R (ˆρ [S (ˆλ y X ˆβ ] with ˆθ beig 1/ -cosistet, i.e., 1/ (ˆθ θ 0 = O P (1. The ɛ, y ad ˆθ are derived as described i Sectio 3. Let be the Euclidea orm of a vector. Lemma 5. Let P l = [p l,ij ] be matrices which are bouded i row sum orms, for l = 1,..., s. If sup,j E ɛ j s <, the 1 s l=1 p l,ijɛ j = O P (1. Proof. For s = 1, the result is immediate. So cosider s > 1. For s > 1, there exists a fiite r such that 1 r + 1 s = 1. Hölder s iequality implies that p l,ij ɛ j p l,ij 1 r pl,ij 1 s ɛj [ where c = sup l=1,,s P l. It follows that Hece, l=1 ( p l,ij 1 r r ] 1 r [ ( p l,ij 1 s ɛj s ] 1 s s c 1 r [ p l,ij ɛ j s ] 1 1 s c r [ ( p l,ij ɛ j s ] 1 s, l=1 s s E( p l,ij ɛ j c s r ( l=1 s s p l,ij ɛ j c s r [ ( p l,ij ɛ j s ]. l=1 p l,ij sup,j l=1 E ɛ j s sc 1+ s r sup,j E ɛ j s = sc s sup E ɛ j s = O(1.,j The result of stochastic boudedess follows from Markov s iequality. Lemma 6. For ay iteger r, if E ɛ i r <, E ɛ r i = E ɛ r i + o P (1, 1 ˆɛr i = E ɛr i + o P (1, E ɛ i r = E ɛ i r +o P (1 ad 1 ˆɛ i r = E ɛ i r +o P (1. If E ɛ r i <, 1/ [E ɛ r i E ɛr i ] = O P (1 ad 1/ [ 1 ˆɛr i E ɛr i ] = O P (1. 19

20 Proof. Let J = I 1 l l. As y = S 1 (X β 0 + R 1 ɛ, ɛ = J ˆɛ = J ( R + (ρ 0 ˆρ M ( S y X β 0 + (λ 0 ˆλ W y + X (β 0 ˆβ = ɛ l ɛ l + (λ 0 ˆλ ( J R + (ρ 0 ˆρ J M W S 1 X β 0 + ( J R X + (ρ 0 ˆρ J M X (β0 ˆβ (B.1 Write ɛ + (λ 0 ˆλ ( J R + (ρ 0 ˆρ J M W S 1 R 1 ɛ + (ρ 0 ˆρ J M R 1 ɛ. = ɛ + r ζ 1,jp j + s ζ,jq j ɛ, where p j = [p j,i ] is a -dimesioal vector with bouded costat elemets, Q j = [q j,il ] is a matrix with bouded row ad colum sum orms, ad ζ 1,j ad ζ,j s are equal to l ɛ /, λ 0 ˆλ, ρ 0 ˆρ, elemets of β 0 ˆβ or their products. The ζ 1,j = O P ( 1/ ad ζ,j = O P ( 1/. The ɛ r i ca be expaded by the multiomial theorem, which states that (x 1 + +x m r = ( r k 1,...,k m k 1,...,k m x k xkm m, where ( r k 1,...,k m is a multiomial coefficiet ad the summatio is take over all sequeces of oegative iteger idices k 1 through k m such that their sum is r. The we have a expasio form for 1 ɛ r i 1 ɛr i, where each term i the expasio has the product form T 1 T with T 1 beig products of ζ 1,j ad ζ,j s ad T ot ivolvig ζ 1,j ad ζ,j s. The T is either bouded or stochastically bouded by Lemma 5. It follows that E ɛ r i = E ɛr i + o P (1 by the law of large umbers ad 1/ [E ɛ r i E ɛr i ] = O P (1 by Chebyshev s iequality. Other results are similarly derived. Lemma 7. Let P l = [p l,ij ] be matrices with bouded row sum orms for l = 1,..., s, the P ( 1 s k=1 p k,ijɛ j > η = O P (1 for η > 0, if E ɛ i s <. Proof. The proof is similar to that for Lemma 5 except for the applicatio of Lemma 6. Lemma 8. For η > 0 ad a iteger r, P ( 1 ˆɛ r i E ɛ r i > η = o P (1 if E ɛ i r < ad P ( ˆθ ˆθ > κ = o P (1 for κ > 0, ad P ( a 1 ˆɛ r i E ɛ r i > η = o P (1 for 0 a < 1/ if E ɛ i r < ad P ( a ˆθ ˆθ > κ = o P (1 for κ > 0. Proof. As y = S 1 Write ˆɛ (ˆλ (X ˆβ + R 1 (ˆρ ɛ, ˆɛ = ( R (ˆρ + (ˆρ ˆρ M ( S (ˆλ y X ˆβ + (ˆλ ˆλ W y + X ( ˆβ ˆβ = ɛ + (ˆλ ˆλ ( R (ˆρ + (ˆρ ˆρ M W S 1 (ˆλ X ˆβ + ( R (ˆρ X + (ˆρ ˆρ M X ( ˆβ ˆβ + (ˆλ ˆλ ( R (ˆρ + (ˆρ ˆρ M W S 1 (ˆλ R 1 (ˆρ ɛ + (ˆρ ˆρ M R 1 (ˆρ ɛ. = ɛ + r ζ 1,jp j + s ζ,jq j ɛ, where p j = [p j,i ] is a -dimesioal vector with bouded costat elemets, Q j = [q j,il ] is a matrix with bouded row ad colum sum orms, ad 0