spatial econometric analysis
DESCRIPTION
Spatial Econometric Analysis. 4 Kuan-Pin Lin Portland State Univerisity. Model Estimation Spatial Error Model. Spatial AR(1). Model Estimation Spatial Error Model. Spatial MA(1). Model Estimation Spatial Error Model. Spatial ARMA(1,1). - PowerPoint PPT PresentationTRANSCRIPT
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.