Download Spatial Econometric Analysis Using GAUSS

Spatial Econometric Analysis
Using GAUSS
3
Kuan-Pin Lin
Portland State University
Spatial Weights Matrix
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


Anselin (1988) [anselin.1]
Ertur and Kosh (2007) [ek.1]
China 30 Provinces [china.1, china.2]
Homework


U.S. 48 Lower States [us48_w.txt]
U.S. 3109 Counties [us3109_w.zip]
[us3109_wlist.txt]
Spatial Contiguity Weights Matrix
Anselin (1988): W1, W2, W3
use gpe2;
n=49;
load
wd[n,n]=c:\course10\ec596\SEAUG\data\anselin\anselin_w.txt;
w1=spw(wd);
w2=spwpower(w1,2);
w3=spwpower(w1,3);
w4=spwpower(w1,4);
w5=spwpower(w1,5);
w6=spwpower(w1,6);
call spwplot(w1);
end;
#include gpe\spatial.gpe;
Spatial Contiguity Weights Matrix
China, 30 Provinces and Cities: W1, W2, W3
Distance-Based Spatial Weights
Ertur and Kosh (2007)
proc gcd(xc,yc);
local x,y,d;

x=pi*xc/180; @ convert to radian @
y=pi*yc/180;
 Longitude
(x)
d=3963*arccos(sin(y').*sin(y)+cos(y').*cos(y).*cos(abs(x'-x)));
@ 3963
 Latitude
(y)miles or 6378 km = radius of the earth @
retp(real(d));

endp;
Geographical Location (x,y)
Great Circle Distance



d=gcd(x,y)
(x,y) is in degree decimal units
Distance-Based Spatial Weights Matrix

Using Kernel Weight Function
Distance-Based Spatial Weights
Ertur and Kosh (2007)

Kernel Weight Function
K : R  [1,1]
Either K ( z )  0 if | z | z0 for some z0
Or | z | K ( z )  0 as | z | 
 K ( z )dz  1,  zK ( z )dz  0, | K ( z ) | dz  
 z K ( z )dz  k where k is a constant
2




Parzen Kernel
Bartlett Kernel (Tricubic Kernel)
Turkey-Hanning Kernel
Guassian or Exponenetial Kernel
Kernel Weights Spatial Matrix
An Example
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
Negative Exponential Distance
kij  K (dij / d )  exp  2dij / d 
Negative Gaussian Distance

kij  K (dij / d max )  exp (dij / d max ) 2

 kii  1
wij  
W  K  I
k
i

j
ij

Gaussian Distance Weights Matrix
Ertur and Kosh (2007)
Spatial HAC Estimator

The Classical Model
y  Xβ  ε
βˆ  ( X ' X)1 X ' y
E (ε | X)  0
Var (ε | X)  
 kij ˆiˆ j

ˆ
ˆ
X ' X    
   E (εε ')

i, j  1, 2,..., n 
ˆ X( X ' X) 1
ˆ (βˆ )  ( X ' X) 1 X ' 
Var
Spatial HAC Estimator
General Heteroscedasticity

Huber-White Estimator
n
ˆ
XX '   i 1 ˆi2 xi xi'
1
1
ˆ
ˆ
ˆ
Var (β)  ( X ' X) X ' X( X ' X)
ˆiˆ j
k

1
i

j



ij
ˆX
ˆ 
X'
i, j  1, 2,..., n  , kij  0 i  j



Spatial HAC Estimator
General Heteroscedasticity and Autocorrelation

First Law of Geography dij   kij 
kij  K (dij / d ) or kij  K (dij / dmax )

Kelejian and Prucha (2007)
n
n
ˆ
X ' X   i 1  j 1 kijˆiˆ j xi x'j
1
1
ˆ
ˆ
ˆ
Var (β)  ( X ' X) X ' X( X ' X)
Time Series HAC Estimator
General Heteroscedasticity and Autocorrelation

Newey-West Estimator
n
ˆ
X ' X   i 1 ˆi2 xi xi'

ˆˆ
'
'
  1  i  1 1 


(
x
x

x
x
i i )
 i i i i
 L 1 
ˆ X( X ' X) 1
ˆ (βˆ )  ( X ' X) 1 X ' 
Var
L
n
Crime Equation
Anselin (1988) [anselin.2]

Basic Model
(Crime Rate) = a + b (Family Income) + g (Housing Value) + 

Spatial HAC Estimator
OLS
Parameter
OLS
s.e.
Robust
s.e/hc
Robust
s.e/hac
b
-1.5973
0.33413
0.44664
0.45552
g
-0.27393
0.10320
0.15752
0.15626
a
68.619
4.7355
4.1014
5.3639
R2
0.5520
GDP Output Production
China 2006 [china.3]

Cobb-Douglass Production Function
ln(GDP) = a + b ln(L) + g ln(K) + 

Spatial HAC Estimator
OLS
Parameter
OLS
s.e.
Robust
s.e/hc
Robust
s.e/hac
b
0.76938
0.08054
0.09081
0.11397
g
0.30923
0.09459
0.09463
0.10716
a
-2.6294
0.73630
0.59170
0.46106
R2
0.89137
Spatial Exogeneity
Lagged Explanatory Variables

Spatial Exogenous Model
y  Xβ  WXγ  ε
E (ε | X,W )  0
  n wij x'j 

WX   j 1
i  1, 2,..., n 
 2I
Var (ε | X,W )  E (εε ')  

GDP Output Production
China 2006 [china.4]

Cobb-Douglass Production Function
ln(GDP) = a + b ln(L) + g ln(K) + bw W ln(L) + gw W ln(K) + 
OLS
Parameter
OLS
s.e.
Robust
s.e/hc
Robust
s.e/hac
b
0.77653
0.05892
0.05674
0.05134
g
0.30974
0.07198
0.07891
0.08419
bw
-0.50975
0.11508
0.10334
0.09197
gw
0.56380
0.11626
0.12052
0.07242
a
-3.0745
0.83787
0.68982
0.51367
R2
0.94690
Spatial Endogeneity
Lagged Dependent Variable

Spatial Lag Model
y  Wy  Xβ  ε
E (ε | X,W )  0
  wij y j 

Wy   j 1
i  1, 2,..., n 
 2I
Var (ε | X,W )  E (εε ')  

n
References
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T. Conley, 1999 “GMM estimation with cross sectional dependence,”
Journal of Econometrics 92, 1999, 1–45.
H. Kelejian and I.R. Prucha, “HAC Estimation in a Spatial
Framework,” Journal of Econometrics 140, 2007, 131-154.
W. Newey, and K. West, 1987, “A simple, positive semi-definite,
heteroskedastic and autocorrelated consistent covariance matrix,”
Econometrica, 55, 1987, 703–708.
H. White, “Maximum Likelihood Estimation of Misspecified Models,”
Econometrica, 50, 1982, 1-26.