Spatial Econometric Analysis

2

Kuan-Pin LinPortland 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 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: SARAR(1,1) Allowing spatial weights matrix to be different in

the regression and in the error. The special case is W = M.

W W y y Xβ Xγ ε

M M ε ε υ υ

2

( | , , ) 0

( | , , ) ( ')

E W M

Var W M 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

2'2

2

2 '

ˆ ˆ1~ (1)

ˆ

ˆˆ ˆ ˆ ˆ, /

( ' )

WLM Error

b

n

b trace WW W W

ε ε

y Xβ ε ε

LM-Based Test Statistics

LM Test Statistic for Spatial Lag

2'2

2

'

2

ˆ1~ (1)

ˆ ˆ

ˆ ˆ( ) ( )ˆ

ˆ

( ' )

WLM Lag

b

Wy M Wy

b trace WW W W

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 ˆ 1 1~ (1)

ˆ ˆ ˆ ˆW b W

LM Errorb b

ε ε ε y

2 2' ' '* 2

2 2 2

ˆ ˆ ˆ ˆ ˆ1 1~ (1)

ˆ ˆ ˆ ˆ ˆW W W

LM Lag

ε y ε y ε ε

LM-Based Test Statistics

Joint LM Test for Spatial Correlation(Spatial Lag and Spatial Error)

2 2' '2

2 2

*

*

ˆ ˆ ˆ ˆ1 1~ (2)

ˆ ˆ ˆ

( ) ( )

( ) ( )

W WLM

b

LM Error LM Lag

LM Lag LM Error

ε ε ε y

Hypothesis TestingExample

Crime Equation (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)

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.