Transcript Spatial Econometric Analysis Kuan-Pin Lin Portland State University Introduction  Spatial Data   Cross Section Panel Data yi   i   xi   i  yi.

Spatial Econometric Analysis
1
Kuan-Pin Lin
Portland State University
Introduction

Spatial Data


Cross Section
Panel Data
yi   i   xi   i
 yi  xi' βi   i
yit   i   xit   it

Spatial Dependence


Spatial Heterogeneity
Spatial Correlation
 yit  xit' βi   it
Cov ( yi , y j )  0
Cov ( yit , y jt )  0, t
Cov ( i ,  j )  0
Cov ( it ,  jt )  0, t
Spatial Dependence

Least Squares Estimator
y  Xβ  ε
βˆ  (X ' X)1 X ' y
  12  12

2


2
   21


 n1  n 2
E (ε | X)  0
Var (ε | X)  
 1n 

 2n 

2 
 n 
Spatial Dependence
Nonparametric Treatment

Robust Inference
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Spatial Heteroscedasticity Autocorrelation
Variance-Covariance Matrix
Var(βˆ )  (X ' X)1 XE(εε ')X '(X ' X)1
ˆ (βˆ )  ( X ' X) 1 X '[εε
ˆˆ ']X( X ' X) 1 ?
Var
εˆ  y  Xβˆ
Spatial Dependence
Nonparametric Treatment

SHAC Estimator
 kij ˆiˆ j

ˆ

 E (εε ')  

i, j  1, 2,..., n 
1
1
ˆ
ˆ
ˆ
Var (β)  ( X ' X) X ' X( X ' X)


Kernel Function
Normalized Distance
dij / d , d  bandwidth
kij  K (dij / d )
0  kij  1, kii  1, kij  k ji
Spatial Dependence
Parametric Representation

Spatial Weights Matrix
wii 0 , wij  0



j 1
wij  1, i
Spatial Contiguity
Geographical Distance


n
 0
w
W   21


 wn1
w12
0
wn 2
w1n 
w2 n 


0 
First Law of Geography: Everything is related to everything
else, but near things are more related than distant things.
K-Nearest Neighbors
Spatial Dependence
Parametric Representation
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Characteristics of Spatial Weights Matrix
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
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Sparseness
Weights Distribution
Eigenvalues
Higher-Order of Spatial Weights Matrix
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

W2, W3, …
Redundandency
Circularity
Spatial Weights Matrix
An Example
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3x3 Rook Contiguity
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List of 9 Observations with 1-st
Order Contiguity, #NZ=24
1
2
3
1
2,4
4
5
6
2
1,3,5
7
8
9
3
2,6
4
1,5,7
5
2,4,6,8
6
3,5,9
7
4,8
8
5,7,9
9
6,8
W
1st-Order Contiguity (Symmetric)
0
1
0
1
0
0
0
0
0
0
1
0
1
0
0
0
0
0
0
0
1
0
0
0
0
1
0
1
0
0
0
1
0
1
0
0
0
0
1
0
1
0
0
1
0
W
All-Order Contiguity (Symmetric)
0
1
2
1
2
3
2
3
4
0
1
2
1
2
3
2
3
0
3
2
1
4
3
2
0
1
2
1
2
3
0
1
2
1
2
0
3
2
1
0
1
2
0
1
0
An Example of Kernel Weights
K = 1/(ii’ + W)
1
1/2
1/3
1/2
1/3
1/4
1/3
1/4
1/5
1
1/2
1/3
1/2
1/3
1/4
1/3
1/4
1
1/4
1/3
1/2
1/5
1/4
1/3
1
1/2
1/3
1/2
1/3
1/4
1
1/2
1/3
1/2
1/3
1
1/4
1/3
1/2
1
1/2
1/3
1
1/2
1
W1
Non-Symmetric Row-Standardized
0
1/2
0
1/2
0
0
0
0
0
1/3
0
1/3
0
1/3
0
0
0
0
0
1/2
0
0
0
1/2
0
0
0
1/3
0
0
0
1/3
0
1/3
0
0
0
1/4
0
1/4
0
1/4
0
1/4
0
0
0
1/3
0
1/3
0
0
0
1/3
0
0
0
1/2
0
0
0
1/2
0
0
0
0
0
1/3
0
1/3
0
1/3
0
0
0
0
0
1/2
0
1/2
0
W2
Non-Symmetric Row-Standardized
0
0
1/3
0
1/3
0
1/3
0
0
0
0
0
1/3
0
1/3
0
1/3
0
1/3
0
0
0
1/3
0
0
0
1/3
0
1/3
0
0
0
1/3
0
1/3
0
1/4
0
1/4
0
0
0
1/4
0
1/4
0
1/3
0
1/3
0
0
0
1/3
0
1/3
0
0
0
1/3
0
0
0
1/3
0
1/3
0
1/3
0
1/3
0
0
0
0
0
1/3
0
1/3
0
1/3
0
0
Oregon Counties
U. S. States
Spatial Lag Variables



Spatial Independent Variables
  n wij x'j 
 j 1

W
X

Spatial Dependent Variables
i  1, 2,..., n 
Spatial Error Variables
  n wij j 

Wε   j 1
i  1, 2,..., n 
  n wij y j 

Wy   j 1
i  1, 2,..., n 
Spatial Econometric Models
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Linear Regression Model with Spatial
Variables
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
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Spatial Exogenous Model
Spatial Lag Model
Spatial Error Model
Spatial Mixed Model
Examples

Anselin (1988): Crime Equation
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Basic Model
(Crime Rate) =  +  (Family Income) + g (Housing Value) + 

Spatial Lag Model
(Crime Rate) =  +  (Family Income) + g (Housing Value)
+ l W (Crime Rate) + 

Spatial Error Model
(Crime Rate) =  +  (Family Income) + g (Housing Value) + 
 = r W + u

Data (anselin.txt, anselin_w.txt)
Examples

Ertur and Kosh (2007): International
Technological Interdependence and Spatial
Externalities


91 countries, growth convergence in 36 years
(1960-1995)
Spatial Lag Solow Growth Model
ln(y(t)) - ln(y(0)) =  +  ln(y(0)) + g ln(s) + g ln(n+g+d) + l W ln(y(t)) - ln(y(0))) + 

Data (data-ek.txt)
References
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L. Anselin, Spatial Econometrics: Methods and Models. Kluwer Academic Publishers,
Boston, 1988.
L. Anselin. “Spatial Econometrics,” In T.C. Mills and K. Patterson (Eds.), Palgrave
Handbook of Econometrics: Volume 1, Econometric Theory. Basingstoke, Palgrave
Macmillan, 2006: 901-969.
L. Anselin, “Under the Hood: Issues in the Specification and Interpretation of Spatial
Regression Models,” Agricultural Economics 17 (3), 2002: 247-267.
T.G. Conley, “Spatial Econometrics” Entry for New Palgrave Dictionary of
Economics, 2nd Edition, S Durlauf and L Blume, eds. (May 2008).
C. Ertur and W. Kosh, “Growth, Technological Interdependence, Spatial Externalities:
Theory and Evidence,” Journal of Econometrics, 2007.
J. LeSage and R.K. Pace, Introduction to Spatial Econometrics, Chapman & Hall,
CRC Press, 2009.
H. Kelejian and I.R. Prucha, “HAC Estimation in a Spatial Framework,” Journal of
Econometrics, 140: 131-154.