GMM estimation of spatial panels

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1 MRA Munich ersonal ReEc Archive GMM estimation of spatial panels Francesco Moscone and Elisa Tosetti Brunel University 7. April 009 Online at MRA aper No. 637, posted 8. July 009 :4 UTC

2 GMM estimation of spatial panels F. Moscone y Brunel Business School E. Tosetti z University of Cambridge Abstract We consider Generalized Method of Moments (GMM) estimation of a regression model with spatially correlated errors. We propose some new moment conditions, and derive the asymptotic distribution of the GMM based on them. The analysis is supported by a small Monte Carlo exercise. Keywords: Generalized Method of Moments, spatial econometrics. JEL Code: C, C5. Introduction GMM estimation of spatial regression models has been originally advanced by Kelejian and rucha (999). They suggested three moment conditions that exploit the properties of disturbances implied by a standard set of assumptions. Substantial work has followed their original study. Druska and Horrace (004) have considered GMM estimation of a panel regression with time dummies and time-varying spatial weights. Lee and Liu (006a) suggested a set of linear and quadratic moment conditions in the errors with inner matrices satisfying certain regularity properties; Lee and Liu (006b) have extended this framework to estimate regression models with higher-order spatial lags. Fingleton (008a) and Fingleton (008b) proposed a GMM estimator for spatial regression models with an endogenous spatial lag and moving average errors. Kelejian and rucha (008) have generalized their original work to allow heteroskedasticity and spatial lags in the dependent variable. This has been extended by Kapoor et al. (007) to estimate a spatial panel regression with individual-speci c error components. We focus on GMM estimation of a regression model where the error follows a spatial autoregressive (SAR) process. We show that there are more moments than those currently exploited in the literature, and derive the asymptotic distribution of the GMM based on such moments. We perform a small Monte Carlo exercise to compare the properties of GMM estimators based on di erent sets of moments. The framework Consider the model expressed in stacked form y t = X t + u t ; t = ; :::; T; () u t = Su t + " t ; () where y t = (y t ; :::; y Nt ) 0,X t = (x t ; :::; x Nt ) 0,u t = (u t ; :::; u Nt ) 0, " t = (" t ; :::; " Nt ) 0 and S is N N spatial weights matrix. We assume: The authors acknowledge nancial support from ESRC (ref. no. RES ). We have bene ted from comments by the partecipants of the III World Spatial Econometrics Association. y francesco.moscone@brunel.ac.uk. z et68@cam.ac.uk.

3 Assumption : " it IIDN(0; ), with K <, for i = ; :::; N; t = ; :::; T. Assumption : X t and " t 0 are independently distributed for all t; t 0. As N and/or T T t= X0 tx t M, where M is nite and non-singular. Assumption 3: S has zero diagonal elements; S and (I N S) have bounded row and column norms. Assumption 4: ; S, where S S = max fj i (S)jg. in Normality and constant variance of " it stated in Assumption are only taken for ease of exposition, and our results can be readily extended to the case of non-normal, heteroskedastic variables (see Kelejian and rucha (008) on this). Assumption 4 implies that () can be rewritten as u t = R" t ;where R = (I N S). Let be the OLS estimator of. Under Assumptions -3, is consistent for, as N and/or T tends to in nity, but, for 6= 0, is not e cient. E cient estimation of can be achieved by estimating the parameters in equation (), namely and, and then apply feasible GLS. 3 GMM estimation of SAR processes Let 0 = 0 ; 0 0 be the true parameter vector for (). Kelejian and rucha (999) suggest the following moments for estimating 0 M () = E " 0 t" t = 0; (3) M () = E M 3 () = E t= " 0 ts 0 S" t t= N T r(s0 S) = 0; (4) " 0 ts" t = 0: (5) t= Moment (3) is implied by the constant variance of " t ; (4) exploits the variance of the spatial lag, S" t ; (5) is based on the covariance between " t and S" t. From (), the following additional moments can be suggested: M 4 () = E u 0 tu t N T r RR0 = 0; (6) M 5 () = E M 6 () = E M 7 () = E M 8 () = E M 9 () = E t= u 0 ts 0 Su t t= u 0 tsu t t= u 0 t" t t= u 0 ts 0 S" t t= u 0 ts" t t= N T r (R0 S 0 SR) = 0; (7) N T r (R0 SR) = 0; (8) T r(r) = 0; (9) N N T r (R0 S 0 S) = 0; (0) N T r(r0 S) = 0: () Moments (6)-(7) exploit the variance of u t and Su t, respectively; (8), (9) and () are based on the covariance of u t with Su t, " t and S" t, respectively; (0) exploits the covariance between the spatial lags Su t and S" t. Remark Under 0 = 0, moment (3) would be identical to (6) and (9); (4) would coincide with (7) and (), and (5) would be the same as (8) and (0). Hence, when 0 is zero or close to zero we expect the additional moments (6)-() to be redundant.

4 In this paper we intend to study the properties of the GMM estimator based on subsets of conditions (3)-(). We rst observe that conditions (3)-() contain the following expressions " 0 ta`" t ; ` = ; :::; 9; () t= A = I N ; A = S 0 S; A 3 = S; A 4 = R 0 R; A 5 = R 0 S 0 SR; A 6 = R 0 SR; A 7 = R 0 ; A 8 = R 0 S 0 S; A 9 = R 0 S; A` having bounded row and column norms. Under Assumption, the mean and variance of () are Cov " E V ar " 0 ta`" t t= " 0 ta`" t = N T r (A`) ; ` = ; :::; r; (3) t= " 0 ta`" t t= # " 0 ta h " t t= = N T 4 T r A ` + A 0`A` ; (4) = N T 4 T r (A`A h + A 0`A h ) ; ` 6= h; (5) Let u it = y it 0 x it and " it = u it N j= s ij u jt. The sample analogues of (3)-() can be obtained by replacing " t by " t and u t by u t. Let M () = [M () ; :::; M r ()] 0 be a vector containing r 9 moments among (3)-(), and M (; ") = [M ; (; ") ; :::; M ;r (; ")] 0 be the corresponding sample moments. Given that u it (and hence " it ) is based on a consistent estimate of, under Assumptions -4 the hypotheses of Theorem in Kelejian and rucha (00), and Theorem A and Lemma C in Kelejian and rucha (008) are satis ed and p [M (; ") E [M (; ")]] 0; as N and/or T (6) ( ) = p [M (; ") M (; ")] 0; as N and/or T (7) ( ) = V r () = M (; ") a N(0; I r ); as N and/or T (8) where V r () = E M (; ") M (; ") 0 is assumed to be non-singular, i.e. r (V r ()) K > 0. V r () has (4) on its main diagonal, and (5) as o -diagonal elements. The above results hold for N and/or T going to in nity. The asymptotic in T can be proved by applying standard multivariate law of large numbers and central limit theorem, since under Assumptions - " 0 ta`" t (and " 0 ta`" t ), for t = ; :::; T; are IID. The above results have been used by Kelejian and rucha (008) and Kelejian and rucha (999) to prove asymptotic normality of the GMM based on (3)-(5). We now show that this result continues to hold if the GMM is based on any subsets of (3)-() such that the covariance matrix of the corresponding sample moments is non-singular. 3. Estimation Suppose we select r moments from (3)-(), such that V r () is non-singular, and let M (; ") be the vector of their sample analogues. The GMM estimator = ; 0 is the solution to the following optimization problem = arg min fm (; ") 0 Q M (; ")g ; (9) where is the parameter space, and Q is a rr, positive-de nite weighting matrix satisfying Q p Q:The following theorem establishes the asymptotic distribution of. See Ullah (004). These results hold under normality of " it, but they can be easily extended to the non-normal case. Notice that under Assumption and 4 is a compact interval. 3

5 Theorem Under Assumptions -4, in (9) is consistent for 0 and, as N and/or T, ( ) = 0 a N 0; (D 0 QD) D 0 QQ ( 0 ) QD (D 0 QD) ; (0) where D = D ( 0 ; ") = M ( 0 ; "). The e cient GMM estimator can be obtained by imposing Q = Q ( 0 ), where Q ( 0 ) = fe [ M ( 0 ; ")M 0 ( 0 ; ")]g () is the optimal weighting matrix. Notice that, under Assumption, Q () = V (), and therefore Q () has as (`; h) element expression (5) multiplied by. In practise, Q and D are evaluated at point estimates, Q and D ; ". In the Appendix we sketch the proof of consistency of and derive D; and refer to Kelejian and rucha (008), Kelejian and rucha (999) for further details on consistency of GMM estimators of spatial models. We do not report the proof of asymptotic normality of since, once established (6)-(8), this is identical to that in Kelejian and rucha (008). When conditions (3)-(5) are employed in (9) Q ( 0 ) is 0 Q () = 4 N N T r 0 S) 0 T r (S 0 S) T r h(s 0 S) i T r S 0 S 0 T r S 0 S T r S + S 0 S Since enters in Q () only as a scale factor, we can compute in a single step by minimizing (9). However, in general, and do enter in the formula for Q (). In this case estimation can proceed adopting a two-stage iterative procedure where in the rst stage we minimize (9) using Q = I r, and OLS residuals u it, and in the second stage, we employ to compute Q and use it in (9). Once estimated, e cient estimation of can be obtained by applying feasible GLS. We next run a Monte Carlo exercise to evaluate and compare the small sample properties of GMM estimators based on subsets of (3)-(). 4 Monte Carlo experiments We consider: y it = + x ;it + x ;it + u it ; i = ; :::; N; t = ; :::; T; x`;it = 0:6x`;it + `it ; `;it IIDN(0; 0:6 ); NX u it = s ij u jt + " it ; " it IIDN(0; ): j= The values of x`;it and u it are drawn for each i and t, and at each replication. S is a regular lattice where each unit has two adjacent neighbours and set s ij = if i and j are adjacent and s ij = 0 otherwise; S is row-standardized. We experimented with = 0:0; 0:4; 0:8; and provide results for the following estimators of = ; 0 (adopting ()): GMM, based on (3)-(5); () GMM, based on (6)-(8); () GMM, based on (9)-(); and (3) GMM, based on (3)-(). Estimation of is performed on u it = y it x ;it x ;it. We assess the performance of estimators by computing their bias, RMSE, size and power (at 5% signi cance level). We ran ; 000 replications for all pairs N = 0; 0; 50; T = 5; 0. Table shows results for estimators of. For purpose of comparison, we also provide the quasi-ml estimator C A : GMM of, ML. The bias and RMSE of GMM decrease as N and/or T get large, for all values of. The size of is close to the nominal 5% level for = 0:0; 0:4, for all N; T larger than 0; while it deviates from the 5% level when T = 5. When = 0:8, the empirical rejection frequencies are slightly larger than the nominal 5% level. We observe that, for a given pair of N and T, larger values of are associated to smaller RMSEs and higher 4

6 power of GMM. A similar pattern can be observed for the GMM estimator based on other sets of conditions (i.e., ;GMM, ;GMM and 3;GMM ), and for ML. However, some important di erences in the performance of these estimators can be noted. First, ;GMM performs overall better than GMM : its bias (in absolute value) and RMSE are lower than those for GMM, for all values of, and the size is very close to 5%, for = 0:0; 0:4. In the case = 0:8, ;GMM is slightly oversized when N and T are small. Notice that results for ;GMM are very close to outputs for ML in all cases considered. The performance of ;GMM and 3;GMM is similar to that of ;GMM when the = 0:0; 0:4. For = 0:8, ;GMM presents some distortions and is characterized by rejections frequencies larger than 5%, ranging between 6:0% and 4:80%. 3;GMM has bias and RMSE similar to ;GMM, while its size deviates from the 5% level. An explanation for this result is that some moments used in computing 3;GMM might be highly correlated, leading to a nearly-singular Q matrix. 5 Conclusions We have introduced new moments in a GMM estimation of a spatial regression model. Given that when = 0 some of the suggested moments are redundant, we have proposed to use only a subset of the moments in the estimation procedure. Our Monte Carlo experiments point at conditions (9)-() as those that yield the best performance of the GMM estimator. References Druska, V. and W. C. Horrace (004). Generalized moments estimation for spatial panels: indonesian rice farming. American Journal of Agricultural Economics 86, Fingleton, B. (008a). A generalized method of moments estimator for a spatial model with moving average errors, with application to real estate prices. Empirical Economics 34, 35½U57. Fingleton, B. (008b). A generalized method of moments estimator for a spatial panel model with an endogenous spatial lag and spatial moving average errors. Spatial Economic Analysis 3, Kapoor, M., H. H. Kelejian, and I. rucha (007). anel data models with spatially correlated error components. Journal of Econometrics 40, Kelejian, H. H. and I. rucha (999). A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, Kelejian, H. H. and I. rucha (00). On the asymptotic distribution of the moran i test with applications. Journal of Econometrics 04, Kelejian, H. H. and I. rucha (008). Speci cation and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Forthcoming, Journal of Econometrics. Lee, L. F. and X. Liu (006a). E cient GMM estimation of a SAR model with autoregressive disturbances. Mimeo. Lee, L. F. and X. Liu (006b). E cient GMM estimation of high order spatial autoregressive models. Mimeo. Ullah, A. (004). Finite Sample Econometrics. Oxford: Oxford University ress. 6 Appendix Let M () = [M () ; :::; M r ()] 0 and M (; ") = [M ; (; ") ; :::; M ;r (; ")] 0, and R (; ") = M (; ") 0 Q () M (; ") ; Z () = M () 0 Q ( 0 ) M () : Consistency of the GMM can be showed by proving: (I) Identi cation uniqueness: for all N; T, and for K > 0 : inf :k 0 k K jz () Z ( 0 )j > 0: (II) Uniform convergence: lim N;T sup jr (; ") Z ()j = 0 5

7 To prove (I), note that M () = ' () D; M (; ") = G (") ' () Dg (") ; where (here we provide ; ' () ; D; when M () contains (3)-(), but we recall that our analysis is based on subsets of these moments) T 3 E t= u0 T tsu t E t= u ts 0 Su t T E t= u0 ts 0 S T u t E t= u0 ts 0 S u t N T r (S0 S) T E t= u0 ts 0 T Su t E t= u0 ts 0 Su t = ; T E t= u0 tsu t T 6 E 4 t= u0 ts 0 S u t T E t= u0 ts u t h ' () = N T r RR0 N T r(r0 S 0 SR) N T r (R0 SR) N T r(r) N T r (R0 S 0 S) N T r(r0 S) D = h 0 I 3 I 3 I 3 ; = T E t= u0 T tu t E t= u0 ts 0 i 0 T Su t E t= u0 tsu t ; and G ("), g (") are the sample analogues of consistency requires the following assumption:,. Following Kelejian and rucha (999), the proof of i 0 ; 0 Assumption 5: is non-singular, i.e. its smallest eigenvalue r ( 0 ) > 0. We have, for k 0 k K > 0, M () 0 Q ( 0 ) M () M ( 0 ) 0 Q ( 0 ) M ( 0 ) = [' () ' ( 0 )] 0 0 Q ( 0 ) [' () ' ( 0 )] r (Q 0 ( 0 )) r [' () ' ( 0 )] 0 [' () ' ( 0 )] > 0 If moments (6)-(8) alone are included in the analysis, we need to take the following identi cability conditions: Assumption 6: ' r () ' r ( 0 ) 6= 0; for all such that k 0 k K > 0; and r = 4; :::; 9. To prove (II), let = [G (") ; Dg (")] and = [ ; D], and notice that R (; ") Z () [ 0 Q () 0 Q p () ] k' ()k 0 since, under Assumptions -4, and given (6), p, and the elements of ' () are bounded. 6. The D matrix D = 6 4 where R is evaluated at : T t= u0 ts 0" t T t= u0 ts 0 S" t N T r (S0 S) T t= u0 ts 0 S" t N "0 ts u t 0 N T r RSRR0 + RR 0 S 0 R 0 N T r RR0 N T r R0 S 0 R 0 S 0 SR + R 0 S 0 SRSR N T r(r0 S 0 SR) N T r R0 S 0 R 0 SR + R 0 SRSR N T r (R0 SR) T t= u0 tsu t N T r (RSR) N T r(r) T t= u0 ts u t N T r (R0 S 0 R 0 S) N T r(r0 S) T t= u0 ts 0 S u t N T r (R0 S 0 R 0 S 0 S) N T r (R0 S 0 S)

8 Table : Small sample properties (X00) of GMM and ML estimates of = 0:0 = 0:4 = 0:8 N T Bias RMSE Size ower Bias RMSE Size ower Bias RMSE Size ower ML estimation GMM ;GMM ;GMM ;GMM