1. Spatial Econometric Analysis4
Kuan-Pin Lin Portland State Univerisity

2. Model Estimation Spatial Error Model
Spatial AR(1)

3. Model Estimation Spatial Error Model
Spatial MA(1)

4. Model Estimation Spatial Error Model
Spatial ARMA(1,1)

5. Spatial Error AR(1) Model Maximum Likelihood Estimation
Normal Density Function

6. Spatial Error AR(1) Model Maximum Likelihood Estimation
Log-Likelihood Function

7. Spatial Error AR(1) Model Maximum Likelihood Estimation
Quasi Maximum Likelihood (QML) Estimator

8. Spatial Error AR(1) Model Maximum Likelihood Estimation
Generalization to consider spatial MA(1) and spatial ARMA(1,1) is straightforward. J
SPAR(1)
(I-rW)
SPMA(1)
(I+qW)-1
SPARMA(1,1)
(I+qW)-1(I-rW)

9. Crime Equation Anselin (1988)
Spatial Error Model: AR, MA, ARMA (Crime Rate) = a + b (Family Income) + g (Housing Value) + e e = r We + u, or e = q Wu + u
SPAR(1)
QML
Parameter
s.e
SPMA(1) r q b g a
59.893
5.0994
59.253
5.4177 L

10. Crime Equation Anselin (1988)
QML Estimator: SPLAG(1) vs. SPAR(1)
SPAR(1)
QML
Parameter
s.e
SPLAG(1) r l b g a
59.893
5.0994
45.080
6.4051 L

11. Spatial Error AR(1) Model Generalized Method of Moments
Moment Functions (Kelejian and Prucha, 1998)

12. Spatial Error AR(1) Model Generalized Method of Moments
Sample Moment Functions

13. Spatial Error AR(1) Model Generalized Method of Moments
Nonlinear GMM: 1 Parameter, 2 Equations

14. Spatial Error AR(1) Model Generalized Method of Moments
Nonlinear GMM: 1 Parameter, 2 Equations

15. Spatial Error AR(1) Model Generalized Method of Moments
Minimum Distance (MD) Estimator
Efficient GMM Estimator

16. Spatial Error AR(1) Model Generalized Method of Moments
Estimation of the variance-covariance matrix of moment functions

17. Model Estimation Spatial Error Model
Spatial AR(1) Model
Estimate b and r simultaneously: QML
Estimate b and r iteratively: GMM/GLS
OLS
GMM
GLS

18. Crime Equation Anselin (1988)
Spatial Error AR(1) Model (Crime Rate) = a + b (Family Income) + g (Housing Value) + e e = r We + u
GMM vs. QML Estimator
GMM Parameter
GMM
s.e
QML Parameter
QML r b g a
60.096
5.3245
59.893
5.0994 Q

19. 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,
L.F.Lee，2003. Best Spatial Two-stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances. Econometrics Reviews, 22,
L.F. Lee, GMM and 2SLS Estimation of Mixed Regressive Spatial Autoregressive Models. Journal of Econometrics, 137,