Spatial Econometric Analysis Using GAUSS

10

Kuan-Pin LinPortland State University

Spatial Panel Data Models

The General Model

1 1

( )

( )

( )[ ( )( )]

( ), ( )

T

T T

T T T

N N

W

W

A B

where A W B W

y I y Xβ ε

ε I ε i u v

y I Xβ I i u v

I I

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 X

J I

( | , ) 0E W ε X

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

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

Spatial Lag Model Estimation

The Model: SPLAG(1)

OLS is biased and inconsistent.

( )

(( ) , ) 0

T

T

T

W

Cov W

y I y Xβ ε

ε i u v

I y ε

( )

' '

T W

Z I y X

δ β

y Zδ ε

Spatial Lag Model Estimation

Fixed Effects

2 2( ) ( ) ( )

T

v NT vVar Var Var

y Zδ i u v y Zδ v

ε v I v Q

, ,

( )T T N

where

y = Qy Z = QZ v = Qv

Q I J I

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

Spatial Lag Model Estimation

Random Effects

2 2( ) ( )

T

u T v T NVar

y Zδ ε

ε 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 Q

Q I J I Q J I

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

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

1

1

2 2

2 2 1 2 2

ˆ ˆˆ ˆ ˆ ˆ ˆ( ' ) ' , , /

ˆ ˆ ˆ ˆˆ ˆ' / ( 1), ' /

ˆ ˆˆ ˆ ˆ ˆ( ) , ( )

T

FE 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

Spatial Error Model Estimation

The Model: SPAR(1)

Fixed Effects Random Effects

( )T

T

W

i

y Xβ ε

ε I ε e

e u v

2 2 2 2 21 1( ,

( ) ,v u v

T T N T N

Var T

1

1

e) Q Q

Q I J I Q J I

2( ) ( ) v NTVar Var e v I

1 1( ) ( ) ( )( ) 'T T

N

Var B Var B

where B W

ε I e I

I

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

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

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 N

E N T

e Qe

e Qe

e Qe2

1 1

21 1

1

( ' ) /

( ' ) / ( ' ) /

( ' ) / 0, ( )T

E N

E N trace W W N

E N where W

e Q e

e Q e

e Q e e I e

Spatial Error Model Estimation Random Effects

The Model: SPAR(1)

Estimate and iteratively: GMM/GLS OLS GMM GLS

* *

( )T

TT

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

Spatial Mixed Model Estimation

The Model: SARAR(1,1)

1

( )

( )

( )( )

( ) , ' '

T

T T

T T

T

N

W

W i

B

where W

and B W

y I y Xβ ε

ε I ε u v

y Zδ I i u v

Z I y X δ β

I

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.

Spatial Mixed Model Estimation Fixed Effects

The Model: SPARAR(1,1)

* * *

( )( )

( )( )

( )

T

TT

TT

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

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

Spatial Mixed Model Estimation Random Effects

The Model: SPARAR(1,1)

* * *

( )

( )

( )

T

T

T

T

T

W

W

W

y I y Xβ ε

ε I ε e

e i u v

y I y X β e

e i u v

* *[ ( )] , [ ( )]T N T Nwhere W W y I I y X I I X

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

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

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

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

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

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

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

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

Maximum Likelihood EstimationRandom Effects

Log-Likelihood Function

2 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 NI T A T B

where I J

A I W B I W

W I B I A I B

β

e e

e e β y X y Xβ

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

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

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

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

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

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

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.