Spatial Econometric Analysis Using GAUSS 4 Kuan-Pin Lin Portland State University
Dec 26, 2015
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
Spatial Exogenous ModelLagged Explanatory Variables
The Model
'
1
1,2,...,
n
ij jjw
Wi n
xX
W y Xβ Xγ ε
2
( | , ) 0
( | , ) ( ')
E W
Var W E
ε X
Iε X εε
Spatial Lag ModelLagged Dependent Variable
The ModelW y y Xβ ε
1
1,2,...,
n
ij jjw y
Wi n
y
2
( | , ) 0
( | , ) ( ')
E W
Var W E
ε X
Iε X εε
1 1
2 1
2 1
( )
( ) ( )
( ) [( ) '( )]
( , ) ( ) 0
W
W W
Var W W
Cov W W W
I y Xβ ε
y I Xβ I ε
y I I
y ε I
Spatial Mixed Model
The Model
2
( | , ) 0, ( | , ) ( ')
W W
E W Var W E
y y Xβ Xγ ε
Iε X ε X εε
1 1
2 1
2 1
( )
( ) ( ) ( )
( ) [( ) '( )]
( , ) ( ) 0
W W
W W W
Var W W
Cov W W W
I y Xβ Xγ ε
y I Xβ Xγ I ε
y I I
y ε I
Spatial Error Models
Spatial AR(1) Spatial MA(1) Spatial ARMA(1,1)
W ε ε υ
W ε υ υW W ε ε υ υ
2
( | , ) 0
( | , ) ( ')
E W
Var W E
υ X
υ X υυ I
Spatial Error Components Model
The Model
W ε ψ υ
2 2
( ) ( ) 0, ( , )
( ') , ( ')
E E Cov
E E
ψ υ ψ υ 0
ψψ I υυ I
2 2
( ) 0
( ) '
E
Var WW
ε
ε I
Spatial Econometric Models
The General Model
W W y y Xβ Xγ ε
W W ε ε υ υ
2
( | , ) 0
( | , ) ( ')
E W
Var W E
υ X
υ X υυ I
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
W or
W
y Xβ ε
ε ε υ
ε υ υ
2
( | , ) 0
( | , )
E
Var
υ X W
υ X W I
0
1 0
: 0 0
: ( )
H or
H not H
2~ (0, )normal iid υ I
Moran-Based Test Statistics
Moran’s I Index
Can not distinguish between spatial lag or spatial error
2
ˆ ˆ ˆ ˆ' '~ ( ( ), ( ))
ˆ ˆ ˆ'
W WI normal iid E I V I
n ε ε ε ε
ε ε
1( )( ) , ( ' )
trace MWE I where M
n K
I X X X X
' 2 22( ) [( ) ] [ ( )]
( ) ( )( )( 2)
trace MWMW trace MW trace MWV I E I
n K n K
ˆˆ
ˆ ( ' ) '
ε y Xβ
β X X X y
LM-Based Test Statistics
LM Test Statistic for Spatial Error
Can not distinguish between spatial AR or spatial MA
2'
22
2 '
ˆ ˆˆ
~ (1)
ˆˆ ˆ ˆ ˆ, /
( ' )
W
LM ErrorT
n
T trace WW W W
ε ε
y Xβ ε ε
LM-Based Test Statistics
LM Test Statistic for Spatial Lag2'
22
' 22
ˆˆ
~ (1)
1ˆ ˆ ˆ( ) ( )
ˆ
W
LM LagnJ
J Wy M Wy Tn
ε y
LM-Based Test Statistics
Robust LM Test Statistic for Spatial Error
Robust LM Test Statistic for Spatial Lag
2' '
2 2* 2
ˆ ˆ ˆˆ ˆ
~ (1)1
W T WnJ
LM ErrorT
TnJ
ε ε ε y
2' '
2 2* 2
ˆ ˆ ˆˆ ˆ
~ (1)
W W
LM LagnJ T
ε y ε ε
Hypothesis TestingExample
Crime Equation (anselin.3) (Crime Rate) = + (Family Income) + (Housing Value) +
(numbers in parentheses are p-values of the tests)
Moran-I LM-err LM-lagRobust LM-err
RobustLM-lag Hetero.
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)
0.0795(0.778)
3.72(0.054)
1.058(0.589)
Hypothesis TestingExample
China Output 2006 (china.6) ln(GDP) = + ln(L) + ln(K) +
(numbers in parentheses are p-values of the tests)
Moran-I LM-err LM-lagRobust LM-err
RobustLM-lag Hetero.
ln(GDP) 1.949(0.052)
2.359(0.125)
2.359(0.125)
ln(L) 1.946(0.052)
2.351(0.125)
2.351(0.125)
ln(K) 2.387(0.017)
3.7658(0.052)
3.7658(0.052)
1.534(0.125)
0.972(0.324)
0.005(0.942)
1.094(0.296)
0.127(0.721)
1.719(0.423)
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.