Spatial Econometric Analysis Using GAUSS

9

Kuan-Pin LinPortland State University

Spatial Panel Data Analysis

The Model Representation Based on panel data models (pooled, fixed effects,

random effects), we consider: Spatial Lag Model Spatial Error Model

Spatial Error AR(1) Spatial Error MA(1) Spatial Error Components

Spatial Mixed Model T-first representation is more convenient to incorporate

parametric spatial panel data analysis

Spatial Panel Data Models

Pooled Fixed Effects Random Effects

Spatial Lag

Model

Spatial error

Model

Spatial Mixed Model

Spatial Panel Data Models

Spatial Lag Model

1

( )

( )( )

( )

t t t t T

t t T

T T

N

W W

A

where A W

y y X β ε y I y Xβ ε

ε u v ε i u v

y I Xβ i u v

I

Spatial Panel Data Models

Spatial Error Model: AR(1), KKP (2006)

1

( )

( )( ), ( )

t t t

t t t T

t t T

T T N

W W

B where B W

y X β ε y Xβ ε

ε ε e ε I ε e

e u v e i u v

y Xβ I i u v I

Spatial Panel Data Models

Spatial Error Model: AR(1)Alternative Specification, Anselin (1988)

1

( )

( ) , ( )

t t t

Tt t

Tt t t

T T N

WW

B where B W

y X β ε y Xβ ε

ε i u eε u e

e I e ve e v

y Xβ i u I v I

Spatial Panel Data Models

Spatial Error Model: MA(1)

( )( )

( )( ), ( )

t t t

t t t t t N t

t t

T T N

W W

C where C W

y X β ε

ε e e y X β I u v

e u v

y Xβ I i u v I

Spatial Panel Data Models

Spatial Error Components

( )

( )

t t t T N

t t t T

T N T

W W

W

y X β u e y Xβ i u e

e ζ ξ e I ζ ξ

y Xβ i u I ζ ξ

Spatial Panel Data Models

Spatial Seemingly Unrelated Regressions

1,2,...,

t t t t t t

t t t t t t

t t

W

W W

t T

y y X β ε

ε ε e e

e u v

Spatial Panel Data Models

Spatial Mixed Model: Anselin AR(1)

1 1

( )

( )

( )[ ( ) ]

t t t t

t t t

T T

T

T T T

W

W

W

W

A B

y y X β u e

e e v

y I y Xβ i u e

e I e v

y I Xβ i u I v

Spatial Panel Data Models

Spatial Mixed Model: KKP AR(1)

1 1

( )

( )

( )[ ( )( )]

t t t t

t t t

T

T T

T T T

W

W

W

W

A B

y y X β ε

ε ε u v

y I y Xβ ε

ε I ε i u v

y I Xβ I i u v

Spatial Model Specification Tests

Based on panel data models (pooled, fixed effects, random effects), we consider: Moran Test LM Test and Robust LM Test

Spatial Error ModelSpatial Lag Model

Spatial Model Specification Tests

Pooled Fixed Effects Random Effects

Moran’s I

LM-Error

LM-Lag

Robust

LM-Error

Robust

LM-Lag

Moran’s I Test StatisticPooled Model

Moran’s I Index

Can not distinguish between spatial lag or spatial error

2

ˆ ˆ ˆ ˆ' '~ ( ( ), ( ))

ˆ ˆ ˆ'

,T

I normal iid E I V In

W n NT

ε Wε ε Wε

ε ε

W I

1( )( ) , ( ' )

traceE I where

n K

MW

M I X X X X

' 2 22( ) [( ) ] [ ( )]

( ) ( )( )( 2)

trace trace traceV I E I

n K n K

MWMW MW MW

ˆˆ

ˆ ( ' ) '

ε y Xβ

β X X X y

Moran’s I Test StatisticFixed Effects Model

Moran’s I Index

Can not distinguish between spatial lag or spatial error

2

ˆ ˆ ˆ ˆ' '~ ( ( ), ( ))

ˆ ˆ ˆ'

, ( 1)T

I normal iid E I V In

W n N T

ε Wε ε Wε

ε ε

W I

1( )( ) , ( ' )

traceE I where

n K

MW

M I X X X X

' 2 22( ) [( ) ] [ ( )]

( ) ( )( )( 2)

trace trace traceV I E I

n K n K

MWMW MW MW

1

ˆ ˆ

ˆ ( ' ) '

ε y Xβ

β X X X y

LM-Error Test Statistic Pooled Model

LM Test Statistic for Spatial Error

2'

22

2 '

ˆ ˆ( )ˆ

~ (1). ( ' )

ˆˆ ˆ ˆ ˆ, /

T W

LM ErrorT trace WW W W

NT

ε I ε

ε y Xβ ε ε

LM-Error Test StatisticFixed Effects Model

LM Test Statistic for Spatial Error

2'

22

2 '

ˆ ˆ( )ˆ

~ (1)( 1). ( ' )

ˆ ˆ ˆ ˆˆ, / ( 1)

T W

LM ErrorT trace WW W W

N T

ε I ε

ε y Xβ ε ε

LM-Error Test StatisticRandom Effects Model

LM Test Statistic for Spatial Error

See Baltagi, B. H., S. H. Song, W. Koh (2003).

2

22 2

121

4

41

ˆ ˆ' ( ) ( )

( 1) ( ' )

vv

v

RLM Error

B

R Q W J W

B T trace WW W W

ε ε

LM-Lag Test StatisticPooled Model

LM Test Statistic for Spatial Lag

2'

22

2 '

' 2

ˆˆ

~ (1)

ˆˆ ˆ ˆ ˆ, / ,

ˆ ˆ ˆ( ) ( ) / . ( ' )

T

LM LagB

W NT

B y y T trace WW W W

εWy

W I ε ε ε y Xβ

W M W

LM-Lag Test StatisticFixed Effects Model

LM Test Statistic for Spatial Lag

2'

22

2 '

' 2

ˆ

ˆ~ (1)

ˆ ˆ ˆ ˆˆ, / ( 1),

ˆ ˆ ˆ( ) ( ) / ( 1). ( ' )

T

LM LagB

W N T

B y y T trace WW W W

εWy

W I ε ε ε y Xβ

W M W

LM-Lag Test StatisticRandom Effects Model

LM Test Statistic for Spatial Lag

See Baltagi, B., Liu, L. (2008).

2 21

2 21

1 1

2 21

ˆ '[ ( ) ( )] /

. ( ' )

ˆ ˆ'[ ( ' ) ( ' )]

ˆ[ ' ]

ˆ'[ ( ' ) ( )]

v

v

v

LM Lag Q W J W b

b T trace WW W W

Q W W J W W

Q W W J W

ε y

y y

A X X A

A X y

Hypothesis TestingExample

U. S. Productivity (munnell.3) ln(GSP) = + ln(Public) + 2ln(Private) + 3ln(Labor) +

4(Unemp) + (numbers in parentheses are p-values of the tests)

Moran-I LM-err LM-lag

ln(GSP) 8.88(0.0)

77.58(0.0)

77.58(0.0)

ln(Public) 7.76(0.0)

59.14(0.0)

59.14(0.0)

ln(Private) 9.92(0.0)

96.92(0.0)

96.92(0.0)

Unemp 10.19(0.0)

102.36(0.0)

102.36(0.0)

Hypothesis TestingExample

U. S. Productivity (munnell.3) Pooled Model

Fixed Effects Model

Moran-I LM-err LM-lagRobust LM-err

RobustLM-lag Hetero.

11.85(0.0)

135.89(0.0)

0.1167(0.733)

138.79(0.0)

3.018(0.08)

19.18(0.0)

Moran-I LM-err LM-lagRobust LM-err

RobustLM-lag Hetero.

14.31(0.0)

199.45(0.0)

87.15(0.0)

118.02(0.0)

5.72(0.02)

179.31(0.0)

Hypothesis TestingAnother Example

China Provincial Productivity (china.7) ln(Q) = + ln(L) + ln(K) +

(numbers in parentheses are p-values of the tests)

Moran-I LM-err LM-lag

ln(Q) 5.725(0.0)

31.47(0.0)

31.47(0.0)

ln(L) 5.396(0.0)

27.92(0.0)

27.92(0.0)

ln(K) 11.2(0.0)

122.07(0.0)

122.07(0.0)

Hypothesis TestingAnother Example

China Provincial Productivity (china.7) Pooled Model

Fixed Effects Model

Moran-I LM-err LM-lagRobust LM-err

RobustLM-lag Hetero.

8.211(0.0)

64.094(0.0)

2.101(0.147)

67.21(0.0)

5.217(0.02)

5.546(0.06)

Moran-I LM-err LM-lagRobust LM-err

RobustLM-lag Hetero.

7.781(0.0)

58.44(0.0)

78.15(0.0)

4.302(0.038)

24.01(0.0)

0.017(0.992)

References Anselin L., J. Le Gallo, and H. Jayet. 2008. Spatial Panel Econometrics. In

The Econometrics of Panel Data, Fundamentals and Recent Developments in Theory and Practice (3rd Edition), eds. L. Matyas and P. Sevestre, 627-662. Springer-Verlag.

Baltagi, B. H., S. H. Song, W. Koh, “Test Panel Data Regression Models with Spatial Error Correlation,” Journal of Econometrics 117, 2003: 123-150.

Baltagi, B. H., S. H. Song, B. C. Jung, W. Koh, “Test for Serial Correlation, Spatial Autocorrelation and Random Effects Using Panel Data,” Journal of Econometrics 140, 2007: 5-51.

Baltagi, B., Liu, L. (2008) Testing for Random Effects and Spatial Lag Dependence in Panel Data Models. Statistics and Probability Letters, 78, 3304-3306.

Lee, L. F., and J. Yu, “Some Recent Developments in Spatial Panel Data Models,” Regional Science and Urban Economics,