1 Overview. 2 Data Files. 3 Estimation Programs

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1 1 Overview The programs made available on this web page are sample programs for the computation of estimators introduced in Kelejian and Prucha (1999). In particular we provide sample programs for the estimation of the following regression model the disturbances are generated from a (Cliff-Ord type) first order spatial autoregressive model: y n = X n β + u n (1) u n = ρw n u n + ε n ρ < 1 (2) y n is the n 1 vector of observations on the dependent variable X n is the n k matrix of observation on k exogenous variables W n is an n n spatial weighting matrix of known constants β is the k 1 vector of regression parameters ρ is the spatial autoregressive parameter u n is the n 1 vector of regression disturbances ε n is an n 1 vector of innovations mean zero and variance σ 2 ε. The variables W n u n is typically referred to as the spatial lag of u n. A spatial Cochrane-Orcutt type transformation of the model yields y n (ρ) =(I n ρw n )y n and X n (ρ) =(I n ρw n )X n. y n (ρ) =X n (ρ)β + ε n (3) Note: We provide two sets of sample programs one for TSP and one for Stata. In the following we only describe the use of the TSP programs. The use of the Stata programs is analogous. Note: The consistency result for the generalized moments (GM) estimator for ρ given in the paper only requires that the disturbances can be estimated n 1/2 -consistently and thus the result also applies to situations y n is generated by a more general model than (1). The expression for the asymptotic variance covariance matrix of the GM estimator introduced below is specific to the above model. It is essentially obtained as a special case of a more general result for a wider class of models given in Kelejian and Prucha (2004). 2 Data Files The sample program involves two exogenous variables and an idealized spatial weighting matrix. This matrix corresponds to the case each unit has one neighbor ahead and one neighbor behind in a wrap around world and the row sums of the weighting matrix are normalized to one; for a more detailed description of this idealized matrix see the Monte Carlo section of the paper. The sample size is taken to be 100. The actual estimation programs assume that the data for the exogenous variables and spatial weighting matrix are stored in files named VAR1.DAT and MMAT.DAT respectively. 3 Estimation Programs The main estimation program is contained in the file PROGRAM1.TSP. This program calls three subroutines contained in the files GMPROC1.TSP GLSPROC1.TSP and VGMPROC1.TSP. Those subroutines compute the GM estimator for ρ the feasible GLS estimator for β and its asymptotic variance covariance matrix and the asymptotic variance of the GM estimator for ρ respectively. The program PROGRAM1.TSP first reads in the data for the exogenous variables and spatial weighting matrix from the files VAR1.DAT and MMAT.DAT. The actual estimation of the parameters of the model (1)-(2) is then performed in four steps. 1

2 Step 1: In the first step we estimate the regression model in (1) using ordinary least squares (OLS) to obtain bβ OLS = (XnX 0 n ) 1 Xny 0 n bu n = y n X nβols b. Step 2: In the second step the spatial autoregressive parameter ρ and σ 2 ε are estimated via the generalized moments estimator introduced in Kelejian and Prucha (1999) utilizing the residuals obtained via the first step. More specifically the estimators of ρ and σ 2 ε eρ n and eσ 2 εn aredefined as the nonlinear least squares estimators corresponding to the regression g n = G n ρ ρ 2 + n (4) σ 2 ε n can be viewed as a vector of regression residuals 2bu 0 nu b n bu 0 nu b n n G n = 1 n 2bu 0 nu b n u b 0 nu b n Tr(Wn 0 W n) bu 0 b nu n + bu 0 b nu n bu 0 b nu n 0 g n = 1 n bu 0 nbu n bu 0 n b u n bu 0 n b u n and bu n = W n bu n and b u n = W 2 n bu n. The code for computing the GM estimators is contained in the file GMPROC1.TSP. Step 3: In the third step we first apply a spatial Cochrane-Orcutt type transformation to the original regression model (1) based on eρ n. The transformed model is as given in (3) but ρ replaced by eρ n.we then reestimate β from the transformed model using OLS. This yields the following feasible GLS estimator bβ FGLS =(X 0 n (eρ n )X n (eρ n )) 1 X 0 n (eρ n )y n (eρ n ) (5) X n (eρ n )=(I n eρ n W n )X n and y n (eρ n )=(I n eρ n W n )y n. Thevariancecovariancematrixof b β FGLS is estimated as ˇσ 2 ε(x 0 n (eρ n )X n (eρ n )) 1 ˇσ 2 ε = n 1ˇε 0 nˇε n ˇε n = y n (eρ n ) X n (eρ n ) β b FGLS =(I n eρ n W n )ǔ n ǔ n = y n X n b βfgls. This estimator b β FGLS and the estimator for it s variance covariance matrix is computed in GLSPROC1.TSP. Step 4: In the fourth step we compute an estimate of the variance of eρ n. As discussed in more detail in the appendix based on results in Kelejian and Prucha (2004) it is seen that eρ n. N(0 e Ω ρn /n) eω ρn =( e J 0 n e J n ) 1 e J 0 n e Ψn e Jn ( e J 0 n e J n ) 1 = e J 0 n e Ψ n e Jn /( e J 0 n e J n ) 2 2

3 and = b 1 n u u b n a n bu n 0nbu n bu 0 nbu n + bu 0 b nu n ) c n = 1 1/2 1+a 2 n a n = n 1 Tr{WnW 0 n } ej n 2 1 c b 0 n u u b an bu 0 bu b u 0 n b u n 1 2eρ n and h i eψ n = eψrsn 2 2 rs=12 eψ rsn = ˇσ 4 ε(2n) 1 tr A rn + A 0 rn Asn + A 0 sn A 1n = c n [WnW 0 n a n I n ] A 2n = W n. We note that ˇσ 2 ε can be replaced by any other consistent estimator such as eσ 2 ε. Also observe that in computing eψ rsn we can utilize that tr A 1n + A 0 1n A1n + A 0 1n = 4c 2 n[tr(wnw 0 n WnW 0 n ) na 2 n] tr A 1n + A 0 1n A2n + A 0 2n = tr A 2n + A 0 2n A1n + A 0 1n tr A 2n + A 0 2n A2n + A 0 2n = 4c n tr(w 0 nw n W n ) = 2[tr(W n W n )+na n ]. The code for the computation of the asymptotic variance e Ω ρn /n iscontainedinthefile VGMPROC1.TSP. 3

4 A Appendix: Asymptotic Distribution of eρ n In establishing the asymptotic distribution of the GM estimator eρ n we first observe in the following that this estimator can be defined equivalently as the nonlinear least squares estimator of a regression corresponding to two moment conditions. As such the estimator can then be seen to be a special case of a more general class of estimators considered in Kelejian and Prucha (2004). Given this observation the asymptotic distribution of eρ n can now be easily obtained as a special case of asymptotic normality results developed in this later paper. A.1 Original Form of Moment Conditions The GM estimators for ρ and σ 2 ε in Kelejian and Prucha (1999) are based on the three moment conditions E[ 1 n ε0 nε n ] = σ 2 ε (A.1) E[ 1 n ε0 nε n ] = σ 2 εtr{w 0 nw n } E[ 1 n ε0 nε n ] = 0 ε n = W n ε n. Let u n = W n u n and u n = Wnu 2 n then upon substitution of ε n = u n ρu n and ε n = u n ρu n into (A.1) we obtain γ n = Γ n ρ ρ 2 (A.2) Γ n = 1 n γ n = 1 n σ 2 ε 2E(u 0 nu n ) E(u 0 nu n ) n 2E(u 0 nu n ) E(u 0 nu n ) Tr(WnW 0 n ) E(u 0 nu n + u 0 nu n ) E(u 0 nu n ) 0 E(u0 nu n ) E(u 0 nu n ) E(u 0 nu n ). The GM estimators eρ n and eσ 2 εn are defined as the nonlinear least squares estimators corresponding to the regression (4).. We note that (4) represents the sample analogue of (A.2) and this ample analogue was obtained by suppressing in (A.2) the expectations operator and by replacing u n u n u n by bu n bu n u b n respectively. A.2 Alternative Form 1 of Moment Conditions Let c n = 1 1/2 1+a 2 n a n = n 1 Tr{WnW 0 n } 4

5 then substitution of the first moment condition in (A.1) into the second and multiplication of the second moment conditions by c n yields the following two moment conditions: c n E[ 1 n ε0 nε n ] = c n a n E[ 1 n ε0 nε n ] (A.3) E[ 1 n ε0 nε n ] = 0. Upon substitution of ε n = u n ρu n and ε n = u n ρu n into (A.3) we obtain ρ γ n = Γ n ρ 2 (A.4) Γ n = 1 n γ n = 1 n " i 2c n he(u 0 nu n ) a n E(u 0 nu n ) i c n he(u 0 nu n ) a n E(u 0 nu n ) E(u 0 nu n + u 0 nu n ) E(u 0 nu n ) cn [E(u 0 nu n ) a n E(u 0 nu n )] E(u 0. nu n ) # Now consider the sample analogue of (A.4): g n = G n ρ ρ 2 + n (A.5) bu G n = 1 2c n 0n u b n a n bu n 0nbu n bu 0 nbu n + bu 0 b nu n ) " h g n = 1 cn u b 0 b i # nu n a n bu 0 nbu n n bu 0. nbu n c bu 0 b n u an bu 0 bu b u 0 n b u n and the 2 1 vector n can again be viewed as a vector of regression residuals. It is not difficult although a bit tedious to check that the nonlinear least squares estimator corresponding to the regression (A.5) is identical to the GM estimator eρ n. A.3 Alternative Form 2 of Moment Conditions To connect the GM estimator eρ n to the class of estimators considered in Kelejian and Prucha (2004) observe that we can rewrite the moment conditions (A.3) as n 1 ε0 E n A 1n ε n ε 0 =0. na 2n ε n A 1n = c n [WnW 0 n a n I n ] A 2n = W n. 5

6 Correspondingly we can also rewrite the matrix Γ n and vector γ n in (A.4) as Γ n = 1 2Eu 0 nwna 0 1n u n Eu 0 nwna 0 1n W n u n n Eu 0 nwn(a 0 2n + A 0 2n )u n Eu 0 nwna 0 2n W n u n γ n = 1 Eu 0 n A 1n u n n Eu 0 na 2n u n and G n and vector g n in (A.5) as G n = 1 2bu 0 nwna 0 1n bu n bu 0 nwna 0 1n W n bu n n bu 0 nwn(a 0 2n + A 0 2n)bu n bu 0 nwna 0 2n W n bu n g n = 1 bu 0 n A 1n bu n n bu 0. na 2n bu n A.4 Asymtotic Normality of eρ n We have shown above that eρ n can also be viewed as the nonlinear least squares estimator corresponding to (A.5). Using furthermore the alternative form of the moment conditions and expressions for Γ n γ n and G n g n given in the previous subsection we can now use Theorem 2 in Kelejian and Prucha (2004) to establish the asymptotic distribution of eρ n. In particular it follows from that Theorem 2 that under the regularity conditions maintained by that theorem we have: n 1/2 (eρ n ρ) =(J 0 nj n ) 1 J 0 nψ 1/2 n ξ n + o p (1) and ξ n J n = Γ n 1 2ρ d N(0I2 ).Furthermoren 1/2 (eρ n ρ) =O p (1) and Ω ρn =(J 0 nj n ) 1 J 0 nψ n J n (J 0 nj n ) 1 const > 0. The above result implies that the difference between the cumulative distribution functions of n 1/2 (eρ n ρ) and N 0 Ω ρn converges pointwise to zero which justifies the use of the latter distribution as an approximation of the former. 1 It follows furthermore from Theorem 3 in Kelejian and Prucha (2004) that Ω e ρn Ω ρn = o p (1) andthus eρ n. N h 0 e Ω ρn /n i. We can now test the hypothesis of zero spatial correlation i.e. H 0 : ρ =0bycomparing eρ n [ e Ω ρn /n] 1/2 the fractiles of the standardized normal distribution. 1 This follows from Corollary F4 in Pötscher and Prucha (1997). Compare also the discussion on pp in that reference. 6

7 References [1] Kelejian H. H. and I. R. Prucha A Generalized Moments of Estimator for the Autoregressive Parameter in a Spatial Model International Economic Review [2] Kelejian H. H. and I. R. Prucha Specification and Estimation of Spatial Autoregressive Models Autoregressive and Heteroskedastic Disturbances Department of Economics University of Maryland Revised October 2004 mimeo. [3] Pötscher B.M. and Prucha I.R. Dynamic Nonlinear Econometric Models Asymptotic Theory. New York: Springer Verlag