Transcript Spatial Econometric Analysis Kuan-Pin Lin Portland State Univerisity Spatial Autoregressive Model with Autoregressive Disturbances  SARAR(1,1) = SPLAG(1)+SPAR(1) y  Wy  Xβ  ε ε 

Spatial Econometric Analysis
5
Kuan-Pin Lin
Portland State Univerisity
Spatial Autoregressive Model with
Autoregressive Disturbances

SARAR(1,1) = SPLAG(1)+SPAR(1)
y  Wy  Xβ  ε
ε  Wε  υ
E (υ | X,W )  0
Var (υ | X,W )  E (υυ ')     I
2
Stability : 1/ min  ,   1/ max  1
Spatial Autoregressive Model with
Moving Average Disturbances

SARMA(1,1) = SPLAG(1)+SPMA(1)
y  Wy  Xβ  ε
ε  Wυ  υ
E (υ | X,W )  0
Var (υ | X,W )  E (υυ ')     2I
Stability : 1/ min  ,  1/ max  1
Spatial Autoregressive Model with
ARMA Disturbances

SARARMA(1,1,1)
= SPLAG(1)+SPAR(1)+SPMA(1)
y  Wy  Xβ  ε
ε  Wε  Wυ  υ
E (υ | X,W )  0
Var (υ | X,W )  E (υυ ')     2I
Stability : 1/ min  ,  ,  1/ max  1
Model Estimation
Maximum Likelihood Estimation

Log-Likelihood Function
n
n
ε ' J ' Jε
2
L(; y, X,W )   ln(2 )  ln( ) 
2
2
2
2
 ln I  W  ln J , ε  ( I  W )y  Xβ
SPLAG(1) + …
J
SPAR(1)
(I-W)
SPMA(1)
(I+W)-1
SPARMA(1,1)
(I+W)-1(I-W)
Model Estimation
Maximum Likelihood Estimation

Quasi Maximum Likelihood (QML) Estimator
ˆ  max arg L(; y, X, W )

ˆ )


L
(

ˆ )  
ˆ (
Var

  ' 
2
1
ˆ )   L(
ˆ )     2 L(
ˆ )
 L(


 

     '     ' 
1
Model Estimation
SARAR(1,1)
y  Wy  Xβ  ε  y  Zδ  (I  W ) υ
1
Z  Wy
ε  Wε  υ
X  , δ    β ' '
E (υ | X, W )  0
Var (υ | X, W )   2 I
Var (ε | X, W )  
2
(I  W ) '(I  W )
1
Cov(Wy , ε)   2W (I  W ) 1 (I  W ) 1  0
Cov(Wε, υ)   2W (I  W ) 1  0
Model Estimation
SARAR(1,1): Generalized Method of Moments

Moment Functions
(Kelejian and Prucha, 1998, 2009)
ˆ 
εˆ  y  ˆWy  Xβˆ     IV estimator
 βˆ 
υ  εˆ  Wεˆ
'
2
E
(
υυ
)


I
υ  Wυ
E ( υυ ' )  W [ E (υυ' )]W '   2WW '
E ( υυ' )  W [ E (υυ' )]   2W
Model Estimation
SARAR(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 Estimation
SARAR(1,1)
The Model
y  Wy  Xβ  ε  ( I  W )[( I  W )y  Xβ]  υ
ε  Wε  υ



Estimate , b and  simultaneously: QML
Estimate , b and  iteratively: IV/GMM/GLS
ˆ , ˆ
 IV or 2SLS  β
 GMM
 ˆ 

GLS

ˆ
ˆ
 β,  
Crime Equation
Anselin (1988)

SARAR(1) Model
(Crime Rate) = a + b (Family Income) + g (Housing Value) +
+  W (Crime rate) + e , e =  We + u

GMM vs. QML Estimator
GMM
Parameter
GMM
s.e
QML
Parameter
QML
s.e

0.45602
0.17491
0.36806
0.14947

-0.1221
0.13571
0.16669
0.17286
b
-1.0438
0.37611
-1.0259
0.44610
g
-0.2537
0.08706
-0.28165
0.18534
a
43.916
10.738
47.784
6.9048
Q/L
2.6706
-182.23
Applications

Geographically Weighted Regression (GWR)



Limited Dependent Variables



Spatial Heterogeneity
Spatial Autocorrelation
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, 2010, Specification and Estimation of
Spatial Autoregressive Models with Autoregressive and
Heteroskedastic Disturbances. Journal of Econometrics, 157, 53-67.
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, 489514.