Spatial Econometric Analysis Using GAUSS

Spatial Econometric Analysis Using GAUSS 6 Kuan-Pin Lin Portland State Univerisity Model Estimation Spatial Error Model Spatial AR(1) y X yXI y X (I W ) 1 yXI WyXI E ( | X, W ) 0 2 Var ( | X,W ) I Var (yXI | X,W ) 2 2 (I W ) '(I W ) 1 Cov(WyXI, ) W (I W ) 0

1 Model Estimation Spatial Error Model Spatial MA(1) y X yXI y X (I W ) yXI W E ( | X, W ) 0 Var ( | X, W ) 2 I Var (yXI | X, W ) 2 (I W )(I W ) ' Model Estimation Spatial Error Model Spatial ARMA(1,1) y X yXI y X (I W ) 1 (I W ) yXI WyXI W E ( | X,W ) 0 Var ( | X, W ) 2 I Var (yXI | X, W ) 2 (I W ) 1 (I W )(I W ) '(I W ) 1 ' Spatial Error AR(1) Model Maximum Likelihood Estimation Normal

Density Function 2 (I W )y (I W ) X ~ normal iid (0, I) f (y ) f ( ) f ( ) I W y ' n 1 ' f () exp 2 2 2 2 (I W )(y X) Spatial Error AR(1) Model Maximum Likelihood Estimation Log-Likelihood Function n n L(, , ; y, X, W ) ln(2 ) ln( 2 ) 2

2 (y X) '(I W ) '(I W )( y X) ln I W 2 2 2 n ln I W i 1 ln(1 i ) 1 , 2 ,..., n are eigenvalues of W Stability : 1/ min 1/ max 1 Spatial Error AR(1) Model Maximum Likelihood Estimation Quasi Maximum Likelihood (QML) Estimator ( ', , 2 ) ' max arg L(, , 2 ; y, X, W ) ) L ( ) (

Var ' 1 ) L( ) L( ) L( ' ' ' 2 (I W )(y X), n 2 2 1 Spatial Error AR(1) Model Maximum Likelihood Estimation Generalization to consider spatial MA(1) and

spatial ARMA(1,1) is straightforward. n n yXI ' J ' JyXI 2 L ln(2 ) ln( ) ln J 2 2 2 2 yXI y X J SPAR(1) (I-W) SPMA(1) (I+W)-1 SPARMA(1,1) (I+W)-1(I-W) Crime Equation Anselin (1988) [anselin.8] Spatial Error Model: AR, MA, ARMA

(Crime Rate) = + (Family Income) + (Housing Value) + = W + or = W + SPAR(1) QML Parameter SPAR(1) QML s.e 0.56178 0.14142 SPMA(1) QML Parameter SPMA(1) QML s.e 0.79909 0.24514

-0.94131 0.43774 -0.92181 0.41823 -0.30225 0.16214 -0.28739 0.14551 59.893 5.0994 59.253 5.4177 L

-183.38 -183.07 Crime Equation Anselin (1988) [anselin.8] QML Estimator: SPLAG(1) vs. SPAR(1) SPAR(1) QML Parameter SPAR(1) QML s.e 0.56178 0.14142 SPLAG(1) QML Parameter SPLAG(1)

QML s.e 0.43101 0.12962 -0.94131 0.43774 -1.0316 0.42108 -0.30225 0.16214 -0.26593 0.17309 59.893

5.0994 45.080 6.4051 L -183.38 -182.39 Spatial Error AR(1) Model Generalized Method of Moments Moment Functions (Kelejian and Prucha, 1998) y X yXI yXI WyXI yXI y X ( X ' X) 1 X ' y yXI WyXI W E (' ) 2I E ( ' ) W [ E (' )]W ' 2WW ' E ( ' ) W [ E (' )] 2W Spatial Error AR(1) Model Generalized Method of Moments

Sample Moment Functions 1 n 2 2 trace(WW ') i 1 i n 1 n 2 trace(W ) 0 i 1 i i n 1 n 2 i 1i2 n n i i j 1 wij j

n i j 1 wij j Spatial Error AR(1) Model Generalized Method of Moments Nonlinear GMM: 1 Parameter, 2 Equations min arg Q ( ) where Q ( ) m( ) ' m( ) is positive definite 1 n m( ) i 1 mi ( ) n mi ( ) m( ; yi , xi' , , W ) i2 i2trace(WW ') ii i 1, 2,..., n Spatial Error AR(1) Model Generalized Method of Moments Nonlinear GMM: 1 Parameter, 2 Equations Let G ( ) m( ) Q( )

2G ( ) m( ) 0 2Q( ) 2G ( ) G ( ) positive definite ' 1 Var ( ) G ( )' G ( ) G ( ) ' Var (m( )) 'G ( ) G ( ) ' G ( ) 1 Spatial Error AR(1) Model Generalized Method of Moments Minimum Distance (MD) Estimator min arg Q( ) where Q ( ) m( ) ' m( ) 1 ' ' ' Var ( ) G ( ) G ( ) G ( ) Var ( m( )) G ( ) G ( ) G ( ) Efficient GMM Estimator

min arg Q( ) where Q( ) m( ) '[Var (m( ))] 1 m( ) ' 1 Var ( ) G ( ) Var (m( )) G ( ) 1 1 Spatial Error AR(1) Model Generalized Method of Moments Estimation of the variance-covariance matrix of moment functions Var (m( )) E (m( )m( ) ') Var (m( )) can be consistently estimated by 1 n n ' k m (

) m j ( ) 2 i 1 j 1 ij i n where mi ( ) m( ; yi , xi' , ,W ) kij K (dij / d ) or kij K (d ij / d max ) dij kij (kii 1, kij 0) Model Estimation Spatial Error Model Spatial AR(1) Model y X yXI yXI WyXI Estimate and simultaneously: QML Estimate and iteratively: GMM/GLS OLS GMM

GLS Crime Equation Anselin (1988) [anselin.9] Spatial Error AR(1) Model (Crime Rate) = + (Family Income) + (Housing Value) + = W + GMM vs. QML Estimator GMM Parameter GMM s.e QML Parameter QML s.e 0.54904 0.10596

0.56179 0.14142 -0.95537 0.33081 -0.94131 0.43774 -0.30193 0.09017 -0.30225 0.16214 60.096 5.3245

59.893 5.0994 Q 0.06979 References H. Kelejian and I. R. Prucha,1998. A Generalized Spatial Two-stage Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbance. Journal of Real Estate Finance and Economics, 17, 99-121. L.F.Lee 2003. Best Spatial Two-stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances. Econometrics Reviews, 22, 307-335. L.F. Lee, 2007. GMM and 2SLS Estimation of Mixed Regressive Spatial Autoregressive Models. Journal of Econometrics, 137, 489514.