1. Spatial Econometric Analysis2
Kuan-Pin Lin Portland State University

2. Spatial Econometric Models
Spatial Exogenous Model
Spatial Lag Model
Spatial Mixed Model
Spatial Error Model
Spatial AR(1)
Spatial MA(1)
Spatial ARMA(1,1)
Spatial Error Components Model

3. Spatial Exogenous Model Lagged Explanatory Variables
The Model

4. Spatial Lag Model Lagged Dependent Variable
The Model

5. Spatial Mixed Model
The Model

6. Spatial Error Models
Spatial AR(1)
Spatial MA(1)
Spatial ARMA(1,1)

7. Spatial Error Components Model
The Model

8. Spatial Econometric Models
The General Model: SARAR(1,1)
Allowing spatial weights matrix to be different in the regression and in the error. The special case is W = M.

9. Spatial Model Specification Tests
Moran Test
Moran’s I Test Statistic
Asymptotic Theory
Bootstrap Method
LM Test and Robust LM Test
Spatial Error Model
Spatial Lag Model

10. Hypothesis Testing
The Basic Model

11. Moran-Based Test Statistics
Moran’s I Index
Can not distinguish between spatial lag or spatial error

12. LM-Based Test Statistics
LM Test Statistic for Spatial Error
Can not distinguish between spatial AR or spatial MA

13. LM-Based Test Statistics
LM Test Statistic for Spatial Lag

14. LM-Based Test Statistics
Robust LM Test Statistic for Spatial Error
Robust LM Test Statistic for Spatial Lag

15. LM-Based Test Statistics
Joint LM Test for Spatial Correlation (Spatial Lag and Spatial Error)

16. Hypothesis Testing Example
Crime Equation
(Crime Rate) = a + b (Family Income) + g (Housing Value) + e (numbers in parentheses are p-values of the tests)
Moran-I
LM-err
LM-lag
Robust LM-err
Robust
Hetero.
Crime Rate
5.6753
(0.000)
26.902
Family Income
4.6624
17.841
Housing Value
2.1529
(0.031)
3.3727
(0.066) e
2.954
(0.003)
5.723
(0.017)
9.363
(0.002)
0.0795
(0.778)
3.72
(0.054)
1.058
(0.589)

17. References
L. Anselin, and A. K. Bera, R. J.G.M. Florax, and M. Yoon (1996), “Simple Diagnostic Tests for Spatial Dependence,” Regional Science and Urban Economics, 26,
L. Anselin, and H. Kelejian (1997), “Testing for Spatial Autocorrelation in the Presence of Endogenous Regressors,” International Regional Science Review, 20, 153–182.
L. Anselin, and S. Rey (1991), “Properties of Tests for Spatial Dependence in Linear Regression Models,” Geographical Analysis, 23,
H. Kelejian, and I.R. Prucha (2001)., “On the Asymptotic Distribution of Moran I Test Statistic with Applications,” Journal Econometrics, 104,