spatial econometric analysis using gauss 10 kuan-pin lin portland state university
DESCRIPTION
Spatial Panel Data Models Assumptions Fixed Effects Random Effects Spatial Error Model: A=I or =0 Spatial Lag Model: B=I or =0 Panel Data Model: A=B=ITRANSCRIPT
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Spatial Econometric Analysis Using GAUSS
10
Kuan-Pin LinPortland State University
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Spatial Panel Data Models The General Model
1 1
( )( )
( )[ ( )( )]( ), ( )
T
T T
T T T
N N
WW
A Bwhere A W B W
y I y Xβ εε I ε i u v
y I Xβ I i u vI I
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Spatial Panel Data Models AssumptionsFixed EffectsRandom Effects
Spatial Error Model: A=I or =0Spatial Lag Model: B=I or =0Panel Data Model: A=B=I
2 1( | , ) ( ' )v TVar W B B ε X I
2 2 1
( | , )
( ) ( ' )u T v T
Var W
B B
ε XJ I
( | , ) 0E W ε X
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Spatial Panel Data Models Example: U. S. Productivity (48 States, 17 Years)
Panel Data Model ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + u + v
Spatial Lag Model ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor)+ 4(Unemp)
+ λW ln(GSP) + u + v
Spatial Error Model ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + W e eu + v
Spatial Mixed Model ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) +
λW ln(GSP) + W e eu + v
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Model Estimation Based on panel data models (pooled, fixed
effects, random effects), we consider: Spatial Error Model Spatial Lag Model Spatial Mixed Model
Model Estimation Generalized Least Squares (IV/GLS) Generalized Method of Moments (GMM/GLS) Maximum Likelihood Estimation
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Spatial Lag Model Estimation
The Model: SPLAG(1)
OLS is biased and inconsistent.
( )
(( ) , ) 0
T
T
T
W
Cov W
y I y Xβ εε i u v
I y ε
( )
' 'T W
Z I y X
δ βy Zδ ε
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Spatial Lag Model Estimation Fixed Effects
2 2( ) ( ) ( )T
v NT vVar Var Var
y Zδ i u v y Zδ vε v I v Q
, ,( )T T N
where
y = Qy Z = QZ v = QvQ I J I
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Spatial Lag Model Estimation Fixed Effects: IV or 2SLS
Instrumental Variables
Two-Stage Least Squares
1 2 1
2
ˆ ˆ ˆˆ ˆ ˆ( ' ) ' , ( ) ( ' )ˆ ˆ ˆ ˆˆ ' / ( 1),
v
v
Var
N T
δ Z Z Z y δ Z Z
v v v y Zδ
1ˆ ( ' ) 'Z H H H H Z
2
( | ) 0, ( , ) 0
, T
E Cov
where W
v H Z H
H X WX W X W I
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Spatial Lag Model EstimationRandom Effects
2 2( ) ( )T
u T v T NVar
y Zδ εε i u v
ε Ω J I I
2 2 2 2 21 1,
( ) ,v u v
T T N T N
T
where
1
1
Ω Q QQ I J I Q J I
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Spatial Lag Model Estimation Random Effects: IV/GLS
Instrumental Variables
Two-Stage Generalized Least Squares
1 1 1
1 1
ˆ ˆ ˆ( ' ) 'ˆ ˆ( ) ( ' )Var
δ Z Ω Z Z Ω y
δ Z Ω Z
1ˆ ( ' ) 'Z H H H H Z
2
( | ) 0, ( , ) 0
, T
E Cov
where W
ε H Z H
H X WX W X W I
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Spatial Lag Model Estimation Random Effects: IV/GLS
Feasible Generalized Least SquaresEstimate v
2 and u2 from the fixed effects model:
FGLS for random effects model:1 1 1 1 1ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ( ' ) ' , ( ) ( ' )Var δ Z Ω Z Z Ω y δ Z Ω Z
11
2 2
2 2 1 2 2
ˆ ˆˆ ˆ ˆ ˆ ˆ( ' ) ' , , /
ˆ ˆ ˆ ˆˆ ˆ' / ( 1), ' /ˆ ˆˆ ˆ ˆ ˆ( ) , ( )
TFE FE i itt
v u
u T v T N u T v T N
v v T
N T N
δ Z Z Z y v y Zδ
v v v v
Ω J I I Ω J I I
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Spatial Error Model Estimation The Model: SPAR(1)
Fixed Effects Random Effects
( )T
T
Wi
y Xβ εε I ε ee u v
2 2 2 2 21 1( ,
( ) ,v u v
T T N T N
Var T
1
1
e) Q QQ I J I Q J I
2( ) ( ) v NTVar Var e v I
1 1( ) ( ) ( )( ) 'T T
N
Var B Var Bwhere B W
ε I e II
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Spatial Error Model EstimationFixed Effects
Moment Functions 2
2
( ' ) / ( 1)
( ' ) / ( 1) ( ' ) /
( ' ) / ( 1) 0
v
v
E N T
E N T trace W W N
E N T
v v
v v
v v
* *
* *
, ( )
[ ( )] , [ ( )]T
T N T N
where W
W W
v y - X β v I v
y I I y X I I X
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Spatial Error Model Estimation Fixed Effects
The Model: SPAR(1)
Estimate and iteratively: GMM/GLS OLS GMM GLS
* *
( )T W
y Xβ εy X β v
ε I ε v
* *
ˆ
ˆ ˆ ˆ( ) ( ) ( )ˆˆ ˆ( ) ( )
T W
y Xβ ε β
ε β I ε β v
y X β v β
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Spatial Error Model Estimation Random Effects
Moment Functions (Kapoor, Kelejian and Prucha, 2006)
2
2
( ' ) / ( 1)
( ' ) / ( 1) ( ' ) /( ' ) / ( 1) 0
v
v
E N T
E N T trace W W NE N T
e Qe
e Qee Qe
21 1
21 1
1
( ' ) /
( ' ) / ( ' ) /( ' ) / 0, ( )T
E N
E N trace W W NE N where W
e Q e
e Q ee Q e e I e
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Spatial Error Model Estimation Random Effects
The Model: SPAR(1)
Estimate and iteratively: GMM/GLS OLS GMM GLS
* *
( )TT
T
W
y Xβ εy X β e
ε I ε ee i u v
e i u v
* *
ˆ
ˆ ˆ ˆ( ) ( ) ( )ˆˆ ˆ( ) ( )
T W
y Xβ ε β
ε β I ε β e
y X β e β
* *[ ( )] , [ ( )]T N T Nwhere W W y I I y X I I X
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Spatial Mixed Model Estimation
The Model: SARAR(1,1)
1
( )( )
( )( )
( ) , ' '
T
T T
T T
T
N
WW i
B
where W
and B W
y I y Xβ εε I ε u v
y Zδ I i u vZ I y X δ βI
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Spatial Mixed Model Estimation Two-Stage EstimationSample moment functions are the same as in the
spatial error AR(1) model. The efficient GMM estimator follows exactly the same as the spatial error AR(1) model.
The transformed model which removes spatial error AR(1) correlation is estimated the same way as the spatial lag model using IV and GLS.
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Spatial Mixed Model Estimation Fixed Effects
The Model: SPARAR(1,1)
* * *
( )( )
( )( )
( )
TT
TT
T
T
WW
WW
W
y I y Xβ εy I y Xβ ε
ε I ε eε I ε v
e i u v
y I y X β v
* *
,..., ( )
[ ( )] , [ ( )]T N
T N T N
where
W W
y Qy Q I J I
y I I y X I I X
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Spatial Mixed Model Estimation Fixed Effects
Estimate and iteratively: GMM/GLS IV/2SLS GMM GLS * * *
ˆ ˆ( ) ,ˆ ˆ ˆ ˆ ˆ( , ) ( ) ( , )
ˆ ˆˆ ˆ ˆ( ) ( ) ( ) ( ) ,
T
T
T
W
W
W
y I y Xβ ε β
ε β I ε β v
y I y X β v β
* *( ) [ ( )] , ( ) [ ( )]T N T Nwhere W W y I I y X I I X
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Spatial Mixed Model Estimation Random Effects
The Model: SPARAR(1,1)
* * *
( )( )
( )
T
T
T
T
T
WW
W
y I y Xβ εε I ε ee i u v
y I y X β ee i u v
* *[ ( )] , [ ( )]T N T Nwhere W W y I I y X I I X
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Spatial Mixed Model Estimation Random Effects
Estimate and iteratively: GMM/GLS IV/2SLS GMM GLS
2 2
* * *
2 2
ˆ ˆ( ) ,ˆ ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( ) ( , ) , ,
ˆ ˆˆ ˆ ˆ( ) ( ) ( ) ( ) ,ˆ ˆ ˆ( , )
T
T v u
T
v u
W
W
W
with
y I y Xβ ε β
ε β I ε β e
y I y X β e β
* *( ) [ ( )] , ( ) [ ( )]T N T Nwhere W W y I I y X I I X
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Example: U. S. ProductivityBaltagi (2008) [munnell.5]
Spatial Panel Data Model: GMM/GLS (Spatial Error) ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + =ρW + e, e = iu + v
FixedEffects s.e
RandomEffects s.e
0.005 0.026 0.031 0.023
0.202* 0.024 0.273* 0.021
3 0.782* 0.029 0.736* 0.025
4 -0.002* 0.001 -0.005* 0.001
0 - - 2.222* 0.136
ρ 0.578* 0.046 0.321* 0.060
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Example: U. S. ProductivityBaltagi (2008) [munnell.5]
Spatial Panel Data Model: GMM/GLS (Spatial Mixed) ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + λW ln(GSP) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
-0.010 0.026 0.040 0.024
0.185* 0.025 0.259* 0.022
3 0.756* 0.029 0.728* 0.026
4 -0.003* 0.001 -0.005* 0.001
0 - - 2.031* 0.174
λ 0.093* 0.024 0.030* 0.015
ρ 0.488* 0.051 0.312* 0.059
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Another ExampleChina Provincial Productivity [china.9]
Spatial Panel Data Model: GMM/GLS (Spatial Error) ln(Q) = + ln(L) + ln(K) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
0.2928 0.073 0.4898 0.062
0.0282 0.017 0.0090 0.017
- - 2.6298 0.587
ρ 0.5013 0.059 0.6424 0.071
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Another ExampleChina Provincial Productivity [china.9]
Spatial Panel Data Model: GMM/GLS (Spatial Mixed) ln(Q) = + ln(L) + ln(K) + W ln(Q) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
0.256 0.080 0.481 0.076
0.022 0.019 0.013 0.015
- - 6.513 2.394
λ 0.287 0.189 1.203 0.059
ρ 0.267 0.074 -0.475 0.239
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Maximum Likelihood Estimation Error Components
AssumptionsFixed Effects:Random Effects:
2~ ( , )v NTN v 0 I
2 2 '
~ ( , ),
,T N
u T v T T T T
N
e i u v 0 Ω Ω I
J I J i i
2 2~ ( , ), ~ ( , ),v NT u NN N t v 0 I u 0 I
T e i u + v
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Maximum Likelihood EstimationFixed Effects
Log-Likelihood Function
2 2
2
( , , , ) ln(2 ) ln( )2 2
' ln | | ln | |2
( ), ( )
( , , | , , ) ( )( ) ( )
v v
v
N N
T T T
NT NTL
T A T B
where A I W B I W
W I B I A I B
β
e e
e e β y X y Xβ
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Maximum Likelihood EstimationFixed Effects
Log-Likelihood Function (Lee and Yu, 2010)
Where z* is the transformation of z using the orthogonal eigenvector matrix of Q.
2 2
'* *
2
* * * *
( 1) ( 1)( , , , ) ln(2 ) ln( )2 2
( 1) ln | | ( 1) ln | |2
( ), ( )
( , , | , , ) ( )( ) ( )
v v
v
N N
T T T
N T N TL
T A T B
where A I W B I W
W I B I A I B
β
e e
e e β y X y X β
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Maximum Likelihood EstimationRandom Effects
Log-Likelihood Function2 2
'1
2 2 2 2
( , , , , )
1ln(2 ) ln | | ( ) ln | | ln | |2 2 2
( , )( ), ( )( , , | , , ) ( )( ) ( )
u v
N
T v u v u
N N
T T T
L
NT N I T A T B
where I JA I W B I W
W I B I A I B
β
e e
e e β y X y Xβ
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Example: U. S. ProductivityBaltagi (2008) [munnell.4]
Spatial Panel Data Model: QML (Spatial Lag) ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + λW ln(GSP) + , = iu + v
FixedEffects s.e
RandomEffects s.e
-0.047 0.026 0.013 0.028
0.187* 0.025 0.226* 0.025
3 0.625* 0.029 0.671* 0.029
4 -0.005* 0.0009 -0.006* 0.0009
0 - - 1.658* 0.166
λ 0.275* 0.022 0.162* 0.029
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Example: U. S. ProductivityBaltagi (2008) [munnell.4]
Spatial Panel Data Model: QML (Spatial Error) ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
0.005 0.026 0.045 0.027
0.205* 0.025 0.246* 0.023
3 0.782* 0.029 0.743* 0.027
4 -0.002* 0.001 -0.004* 0.001
0 - - 2.325 0.155
ρ 0.557* 0.034 0.527* 0.033
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Example: U. S. ProductivityBaltagi (2008) [munnell.4]
Spatial Panel Data Model: QML (Spatial Mixed) ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) + 4(Unemp) + λW ln(GSP) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
-0.010 0.027 0.044 0.023
0.191* 0.025 0.249* 0.023
3 0.755* 0.031 0.742* 0.027
4 -0.003* 0.001 -0.004* 0.001
0 - - 2.289* 0.212
λ 0.089 0.031 0.004 0.017
ρ 0.455* 0.052 0.522* 0.038
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Another ExampleChina Provincial Productivity [china.8]
Spatial Panel Data Model: QML (Spatial Lag) ln(Q) = + ln(L) + ln(K) + W ln(Q) + = iu + v
FixedEffects s.e
RandomEffects s.e
0.2203 0.0707 0.3794 0.074
0.0177 0.0163 -0.0046 0.016
- - 0.9081 0.626
λ 0.4361 0.0557 0.3941 0.055
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Another ExampleChina Provincial Productivity [china.8]
Spatial Panel Data Model: QML (Spatial Error) ln(Q) = + ln(L) + ln(K) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
0.2969 0.073 0.4928 0.077
0.0297 0.017 0.0091 0.017
- - 2.6548 0.657
ρ 0.4521 0.058 0.4364 0.055
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Another ExampleChina Provincial Productivity [china.8]
Spatial Panel Data Model: QML (Spatial Mixed) ln(Q) = + ln(L) + ln(K) + W ln(Q) + =ρW + e , e = iu + v
FixedEffects s.e
RandomEffects s.e
0.143 0.058 0.247 0.062
0.004 0.013 -0.014 0.013
- - -0.119 0.496
λ 0.731 0.058 0.712 0.064
ρ -0.571 0.136 -0.563 0.145
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References Elhorst, J. P. (2003). Specification and estimation of
spatial panel data models, International Regional Science Review 26, 244-268.
Kapoor M., Kelejian, H. and I. R. Prucha, “Panel Data Models with Spatially Correlated Error Components,” Journal of Econometrics, 140, 2006: 97-130.
Lee, L. F., and J. Yu, “Estimation of Spatial Autoregressive Panel Data Models with Fixed Effects,” Journal of Econometrics 154, 2010: 165-185.