Spatial Econometric Analysis

4

Kuan-Pin LinPortland State Univerisity

Model EstimationSpatial Error Model

Spatial AR(1)1( )W

W

y Xβ ε y Xβ I υ

ε ε υ

2

12

2 1

( | , ) 0

( | , )

( | , ) ( ) '( )

( , ) ( ) 0

E W

Var W

Var W W W

Cov W W W

υ X

υ X I

ε X I I

ε υ I

Model EstimationSpatial Error Model

Spatial MA(1)

( )W

W

y Xβ ε y Xβ I υ

ε υ υ

2

2

( | , ) 0

( | , )

( | , ) ( )( ) '

E W

Var W

Var W W W

υ X

υ X I

ε X I I

Model EstimationSpatial Error Model

Spatial ARMA(1,1)1( ) ( )W W

W W

y Xβ ε y Xβ I I υ

ε ε υ υ

2

2 1 1

( | , ) 0

( | , )

( | , )

( ) ( )( ) '( ) '

E W

Var W

Var W

W W W W

υ X

υ X I

ε X

I I I I

Spatial Error AR(1) ModelMaximum Likelihood Estimation

Normal Density Function2( ) ( ) ~ (0, )W W normal iid I y I Xβ υ I

22

( ) ( ) ( )'

1 '( ) exp

22

( )( )

n

f f f W

f

W

υy υ υ I

y

υ υυ

υ I y Xβ

Spatial Error AR(1) ModelMaximum Likelihood Estimation

Log-Likelihood Function

2 2

2

( , , ; , , ) ln(2 ) ln( )2 2

( ) '( ) '( )( )ln

2

n nL W

W WW

β y X

y Xβ I I y XβI

1

1 2

min max

ln ln(1 )

, ,...,

: 1/ 1/ 1

n

ii

n

W

are eigenvalues of W

Stability

I

Spatial Error AR(1) ModelMaximum Likelihood Estimation

Quasi Maximum Likelihood (QML) Estimator

2 2

1 12 2

2

ˆˆ ˆ ˆ( ', , ) ' max arg ( , , ; , , )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )ˆˆ ( )' ' '

ˆ ˆ'ˆˆ ˆ ˆ( )( ),

L W

L L L LVar

Wn

β β y X

υ υυ I y Xβ

Spatial Error AR(1) ModelMaximum Likelihood Estimation

Generalization to consider spatial MA(1) and spatial ARMA(1,1) is straightforward.

22

' 'ln(2 ) ln( ) ln2 2 2

n n J JL J

ε ε

ε y Xβ

J

SPAR(1) (I-W)

SPMA(1) (I+W)-1

SPARMA(1,1) (I+W)-1(I-W)

Crime EquationAnselin (1988)

Spatial Error Model: AR, MA, ARMA(Crime Rate) = + (Family Income) + (Housing Value) + = W + or = W +

SPAR(1) QMLParameter

SPAR(1)QMLs.e

SPMA(1)QMLParameter

SPMA(1)QMLs.e

0.56178 0.14142

0.79909 0.24514

-0.94131 0.43774 -0.92181 0.41823

-0.30225 0.16214 -0.28739 0.14551

59.893 5.0994 59.253 5.4177

L -183.38 -183.07

Crime EquationAnselin (1988)

QML Estimator: SPLAG(1) vs. SPAR(1)

SPAR(1) QMLParameter

SPAR(1)QMLs.e

SPLAG(1)QMLParameter

SPLAG(1)QMLs.e

0.56178 0.14142

0.43101 0.12962

-0.94131 0.43774 -1.0316 0.42108

-0.30225 0.16214 -0.26593 0.17309

59.893 5.0994 45.080 6.4051

L -183.38 -182.39

Spatial Error AR(1) ModelGeneralized Method of Moments

Moment Functions (Kelejian and Prucha, 1998)

W y Xβ ε

ε ε υ

1ˆ ˆˆ ( ' ) '

ˆ ˆW

W

ε y Xβ β X X X y

υ ε ε

υ υ' 2

' ' ' 2

' ' 2

( )

( ) [ ( )] '

( ) [ ( )]

E

E W E W WW

E W E W

υυ I

υυ υυ

υυ υυ

Spatial Error AR(1) ModelGeneralized Method of Moments

Sample Moment Functions

2 2

1

2

1

1( ')

1( ) 0

n

ii

n

i ii

trace WWn

trace Wn

2 2

1

1 n

iin

1

1

n

i i ij jj

n

i ij jj

w

w

Spatial Error AR(1) ModelGeneralized Method of Moments

Nonlinear GMM: 1 Parameter, 2 Equationsˆ min arg ( )

( ) ( ) ' ( )

Q

where Q m m

is positive definite

1

' 2 2

1( ) ( )

ˆ( ) ( ; , , , ) ( ')

1, 2,...,

n

ii

i i i i i i i

m mn

m m y W trace WW

i n

x β

Spatial Error AR(1) ModelGeneralized Method of Moments

Nonlinear GMM: 1 Parameter, 2 Equations

2

ˆ( )ˆ( )

ˆ( )ˆ ˆ2 ( ) ( ) 0

ˆ( )ˆ ˆ2 ( ) ( )

'

mLet G

QG m

QG G positive definite

1 1' ' ' 'ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ( )) ( ) ( ) ( )Var G G G Var m G G G

Spatial Error AR(1) ModelGeneralized Method of Moments

Minimum Distance (MD) Estimator

Efficient GMM Estimator

1

ˆ min arg ( )

( ) ( ) '[ ( ( ))] ( )

Q

where Q m Var m m

ˆ min arg ( )

( ) ( ) ' ( )

Q

where Q m m

1 1' ' 'ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ( )) ( ) ( ) ( )Var G G G Var m G G G

11'ˆ ˆ ˆ ˆ( ) ( ) ( ( )) ( )Var G Var m G

Spatial Error AR(1) ModelGeneralized Method of Moments

Estimation of the variance-covariance matrix of moment functions

'2 1 1

'

max

( ( ))

1( ) ( )

ˆ( ) ( ; , , , )

( / ) ( / )

( 1, 0)

n n

ij i ji j

i i i

ij ij ij ij

ij ij ii ij

Var m can be consistently estimated by

k m mn

where m m y W

k K d d or k K d d

d k k k

x β

( ( )) ( ( ) ( ) ')Var m E m m

Model EstimationSpatial Error Model

Spatial AR(1) Model

Estimate and simultaneously: QML Estimate and iteratively: GMM/GLS

OLS GMM GLS

W y Xβ ε

ε ε υ

Crime EquationAnselin (1988)

Spatial Error AR(1) Model(Crime Rate) = + (Family Income) + (Housing Value) + = W +

GMM vs. QML Estimator

GMM Parameter

GMMs.e

QML Parameter

QMLs.e

0.54904 0.10596 0.56179 0.14142

-0.95537 0.33081 -0.94131 0.43774

-0.30193 0.09017 -0.30225 0.16214

60.096 5.3245 59.893 5.0994

Q 0.06979

References H. Kelejian and I. R. Prucha,1998. A Generalized Spatial Two-stage

Least Squares Procedure for Estimating a Spatial Autoregressive Model with Autoregressive Disturbance. Journal of Real Estate Finance and Economics, 17, 99-121.

L.F.Lee， 2003. Best Spatial Two-stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances. Econometrics Reviews, 22, 307-335.

L.F. Lee, 2007. GMM and 2SLS Estimation of Mixed Regressive Spatial Autoregressive Models. Journal of Econometrics, 137, 489-514.