Transcript Spatial Econometric Analysis Kuan-Pin Lin Portland State University 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.

Spatial Econometric Analysis
2
Kuan-Pin Lin
Portland State University
Spatial Econometric Models
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


Spatial Exogenous Model
Spatial Lag Model
Spatial Mixed Model
Spatial Error Model
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


Spatial AR(1)
Spatial MA(1)
Spatial ARMA(1,1)
Spatial Error Components Model
Spatial Exogenous Model
Lagged Explanatory Variables

The Model
y  Xβ  WXγ  ε
E (ε | X, W )  0
2

 I
n
'
  wij x j  Var (ε | X, W )  E (εε ')  

WX   j 1

i  1, 2,..., n 
Spatial Lag Model
Lagged Dependent Variable

The Model
y  Wy  Xβ  ε
E (ε | X, W )  0
  wij y j 

Wy   j 1
i  1, 2,..., n 
 2 I
Var (ε | X, W )  E (εε ')  

n
 (I  W )y  Xβ  ε
 y  (I  W ) 1 Xβ  (I  W ) 1 ε
Var (y )   2 [(I  W ) '(I  W )]1
Cov(Wy, ε)   2W (I  W ) 1  0
Spatial Mixed Model

The Model
y  Wy  Xβ  WXγ  ε
 2 I
E (ε | X, W )  0, Var (ε | X, W )  E (εε ')  

 (I  W )y  Xβ  WXγ  ε
 y  (I  W ) 1 ( Xβ  WXγ )  (I  W ) 1 ε
Var (y )   2 [(I  W ) '(I  W )]1
Cov(Wy, ε)   2W (I  W ) 1  0
Spatial Error Models



ε  Wε  υ
Spatial AR(1)
ε  Wυ  υ
Spatial MA(1)
Spatial ARMA(1,1) ε  Wε  Wυ  υ
E (υ | X,W )  0
Var (υ | X,W )  E (υυ ')    2I
Spatial Error Components Model

The Model
ε  Wψ  υ
E (ε)  0
Var (ε)   WW '   I
2
2
E (ψ)  E (υ)  0, Cov(ψ, υ)  0
E (ψψ ')     I, E (υυ ')     I
2
2
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.
y  Wy  Xβ  WXγ  ε
ε   Mε   Mυ  υ
E (υ | X,W , M )  0
Var (υ | X,W , M )  E(υυ ')     I
2
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
Hypothesis Testing

The Basic Model
y  Xβ  ε
ε  Wε  υ or
ε  Wυ  υ
E (υ | X, W)  0
Var (υ | X, W)  2I
υ ~ normal iid (0,2I)
H 0 :   0 or   0
H1 : (not H 0 )
Moran-Based Test Statistics

Moran’s I Index
εˆ 'Wεˆ εˆ 'Wεˆ
I

~ normal iid ( E ( I ),V ( I ))
2
εˆ ' εˆ
nˆ

εˆ  y  Xβˆ
βˆ  ( X ' X) X ' y
Can not distinguish between spatial lag or spatial error
trace( MW )
E(I ) 
, where M  I  X( X ' X) 1 X
nK
trace(MWMW ' )  trace[( MW )2 ]  [trace( MW )]2
V (I ) 
 E ( I )2
(n  K )(n  K  2)
LM-Based Test Statistics

LM Test Statistic for Spatial Error
2
1  εˆ Wεˆ 
LM  Error   2  ~  2 (1)
b  ˆ 
'
yˆ  Xβˆ , ˆ 2  εˆ 'εˆ / n
b  trace(WW  W 'W )

Can not distinguish between spatial AR or spatial MA
LM-Based Test Statistics

LM Test Statistic for Spatial Lag
2
1  εˆ Wy 
2
LM  Lag 
~

(1)
 2 
ˆ  b  ˆ  
'
(Wyˆ )' M (Wyˆ )
ˆ 
ˆ 2
b  trace(WW  W 'W )
LM-Based Test Statistics

Robust LM Test Statistic for Spatial Error
2
 εˆ Wεˆ
b εˆ Wy   1 1 
2
LM  Error   2 

~

(1)

2  
 ˆ  ˆ  b ˆ    ˆ b 
'
'
*

Robust LM Test Statistic for Spatial Lag
1
LM  Lag 
ˆ
*
2
2
 εˆ Wyˆ 
1  εˆ Wy εˆ Wεˆ 
2


~

(1)
 2 
 2
2 
ˆ  
 ˆ   ˆ  ˆ 
'
'
'
LM-Based Test Statistics

Joint LM Test for Spatial Correlation
(Spatial Lag and Spatial Error)
2
2
 1   εˆ Wεˆ   1   εˆ Wyˆ 
LM     2      2  ~  2 (2)
 b   ˆ   ˆ   ˆ 
 ( LM  Error )  ( LM  Lag * )
'
'
 ( LM  Lag )  ( LM  Error * )
Hypothesis Testing
Example

Crime Equation

(Crime Rate) = a + b (Family Income) + g (Housing Value) + 
(numbers in parentheses are p-values of the tests)
Moran-I
LM-err
LM-lag
Crime
Rate
5.6753
(0.000)
26.902
(0.000)
26.902
(0.000)
Family
Income
4.6624
(0.000)
17.841
(0.000)
17.841
(0.000)
Housing
Value
2.1529
(0.031)
3.3727
(0.066)
3.3727
(0.066)

2.954
(0.003)
5.723
(0.017)
9.363
(0.002)
Robust
LM-err
Robust
LM-lag
Hetero.
0.0795
(0.778)
3.72
(0.054)
1.058
(0.589)
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, 77-104.
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, 112-131.
H. Kelejian, and I.R. Prucha (2001)., “On the Asymptotic Distribution
of Moran I Test Statistic with Applications,” Journal Econometrics,
104, 219-257.