Spatial Econometric Analysis Using GAUSS

7

Kuan-Pin LinPortland State Univerisity

Spatial Autoregressive Model with Autoregressive Disturbances

SARAR(1,1) = SPLAG(1)+SPAR(1)

W y y Xβ ε

W ε ε υ

2

( | , ) 0

( | , ) ( ')

E W

Var W E

υ X

υ X υυ I

min max: 1/ , 1/ 1Stability

Spatial Autoregressive Model with Moving Average Disturbances

SARMA(1,1) = SPLAG(1)+SPMA(1)

W y y Xβ ε

W ε υ υ

2

( | , ) 0

( | , ) ( ')

E W

Var W E

υ X

υ X υυ I

min max: 1/ , 1/ 1Stability

Spatial Autoregressive Model with ARMA Disturbances

SARARMA(1,1,1)= SPLAG(1)+SPAR(1)+SPMA(1)

W y y Xβ εW W ε ε υ υ

2

( | , ) 0

( | , ) ( ')

E W

Var W E

υ X

υ X υυ I

min max: 1/ , , 1/ 1Stability

Model EstimationMaximum Likelihood Estimation

Log-Likelihood Function

22

' '( ; , , ) ln(2 ) ln( )

2 2 2ln ln , ( )

n n J JL W

I W J I W

ε εy X

ε y Xβ

SPLAG(1) + … J

SPAR(1) (I-W)

SPMA(1) (I+W)-1

SPARMA(1,1) (I+W)-1(I-W)

Model EstimationMaximum Likelihood Estimation

Quasi Maximum Likelihood (QML) Estimator

1 12 2

ˆ max arg ( ; , , )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )ˆˆ ( )' ' '

L W

L L L LVar

y X

Model EstimationSARAR(1,1)

1( )

, ' '

W W

W W

y y Xβ ε y Zδ I υ

ε ε υ Z y X δ β

2

12

2 1 1

2 1

( | , ) 0

( | , )

( | , ) ( ) '( )

( , ) ( ) ( ) 0

( , ) ( ) 0

E W

Var W

Var W W W

Cov W W W W

Cov W W W

υ X

υ X I

ε X I I

y ε I I

ε υ I

Model EstimationSARAR(1,1): Generalized Method of Moments

Moment Functions (Kelejian and Prucha, 1998, 2009)

ˆˆ ˆˆ

ˆ

ˆ ˆ

W IV estimator

W

W

ε y y Xββ

υ ε ε

υ υ' 2

' ' ' 2

' ' 2

( )

( ) [ ( )] '

( ) [ ( )]

E

E W E W WW

E W E W

υυ I

υυ υυ

υυ υυ

Model EstimationSARAR(1,1): Generalized Method of Moments

Sample moment functions are the same two equations of one parameter 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.

Model EstimationSARAR(1,1)

The Model

Estimate , and simultaneously: QML Estimate , and iteratively: IV/GMM/GLS

IV or 2SLS GMM GLS

( )[( ) ]W I W I W

W

y y Xβ ε y Xβ υ

ε ε υ

ˆ ˆ,

ˆ

ˆ ˆ,

β

β

Crime EquationAnselin (1988) [anselin.10, anselin.11]

SARAR(1) Model(Crime Rate) = + (Family Income) + (Housing Value) + + W (Crime rate) + , = W +

GMM vs. QML Estimator

GMM Parameter

GMMs.e

QML Parameter

QMLs.e

0.45602 0.17491 0.36806 0.14947

-0.1221 0.13571 0.16669 0.17286

-1.0438 0.37611 -1.0259 0.44610

-0.2537 0.08706 -0.28165 0.18534

43.916 10.738 47.784 6.9048

Q/L 2.6706 -182.23

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

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, 72-82.

H. Kelejian, and I. R. Prucha, 2009. Specification and Estimation of Spatial Autoregressive Models with Autoregressive and Heteroskedastic Disturbances. Journal of Econometrics, forthcoming.

Das, D., H. Kelejian, and I.R. Prucha, 2003. Small Sample Properties of Estimators of Spatial Autoregressive Models with Autoregressive Disturbances. Papers in Regional Science, 82, 1-26.

L.F. Lee, 2007. GMM and 2SLS Estimation of Mixed Regressive Spatial Autoregressive Models. Journal of Econometrics, 137, 489-514.