1. Spatial Econometric Analysis5
Kuan-Pin Lin Portland State Univerisity

2. Spatial Autoregressive Model with Autoregressive Disturbances
SARAR(1,1) = SPLAG(1)+SPAR(1)

3. Spatial Autoregressive Model with Moving Average Disturbances
SARMA(1,1) = SPLAG(1)+SPMA(1)

4. Spatial Autoregressive Model with ARMA Disturbances
SARARMA(1,1,1) = SPLAG(1)+SPAR(1)+SPMA(1)

5. Model Estimation Maximum Likelihood Estimation
Log-Likelihood Function
SPLAG(1) + … J
SPAR(1)
(I-rW)
SPMA(1)
(I+qW)-1
SPARMA(1,1)
(I+qW)-1(I-rW)

6. Model Estimation Maximum Likelihood Estimation
Quasi Maximum Likelihood (QML) Estimator

7. Model Estimation SARAR(1,1)

8. Model Estimation SARAR(1,1): Generalized Method of Moments
Moment Functions (Kelejian and Prucha, 1998, 2009)

9. Model Estimation SARAR(1,1): Generalized Method of Moments
Sample moment functions are the same two equations of one parameter r as in the spatial error AR(1) model.
The efficient GMM estimator follows exactly the same as the spatial error AR(1) model with the IV estimator of the spatial lag model.

10. Model Estimation SARAR(1,1)
The Model
Estimate l, b and r simultaneously: QML
Estimate l, b and r iteratively: IV/GMM/GLS
IV or 2SLS
GMM
GLS

11. Crime Equation Anselin (1988)
SARAR(1) Model (Crime Rate) = a + b (Family Income) + g (Housing Value) + + l W (Crime rate) + e , e = r We + u
GMM vs. QML Estimator
GMM Parameter
GMM
s.e
QML Parameter
QML l r b g a
43.916
10.738
47.784
6.9048
Q/L
2.6706

12. Applications Geographically Weighted Regression (GWR)
Spatial Heterogeneity
Spatial Autocorrelation
Limited Dependent Variables
Spatial Probit and Spatial Tobit Models
Spatial Inference
Spatial Prediction
Best Predictors
Spatial Model Comparison

13. References
K.P. Bell, N.E. Bockstael, 2000, Applying the Generalized-Moments Estimation to Spatial Problems Involving Microlevel Daqta, Review of Economic s and Statistics, 82,
H. Kelejian, and I. R. Prucha, 2010, Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances. Journal of Econometrics, 157,
Das, D., H. Kelejian, and I.R. Prucha, Small Sample Properties of Estimators of Spatial Autoregressive Models with Autoregressive Disturbances. Papers in Regional Science, 82, 1-26.
L.F. Lee, GMM and 2SLS Estimation of Mixed Regressive Spatial Autoregressive Models. Journal of Econometrics, 137,