Spatial Econometric Analysis Using GAUSS

3

Kuan-Pin LinPortland State University

Spatial Weights Matrix 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 MatrixAnselin (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 MatrixChina, 30 Provinces and Cities: W1, W2, W3

Distance-Based Spatial WeightsErtur and Kosh (2007)

Geographical Location (x,y) Longitude (x) Latitude (y)

Great Circle Distance d=gcd(x,y) (x,y) is in degree decimal units

Distance-Based Spatial Weights Matrix Using Kernel Weight Function

proc gcd(xc,yc); local x,y,d; x=pi*xc/180; @ convert to radian @ y=pi*yc/180; d=3963*arccos(sin(y').*sin(y)+cos(y').*cos(y).*cos(abs(x'-x))); @ 3963 miles or 6378 km = radius of the earth @ retp(real(d));endp;

Distance-Based Spatial WeightsErtur and Kosh (2007)

Kernel Weight Function

Parzen Kernel Bartlett Kernel (Tricubic Kernel) Turkey-Hanning Kernel Guassian or Exponenetial Kernel

0 0

: [ 1,1]( ) 0 | |

| | ( ) 0 | |

K REither K z if z z for some zOr z K z as z

2

( ) 1, ( ) 0, | ( ) |

( )

K z dz zK z dz K z dz

z K z dz k where k is a constant

Kernel Weights Spatial MatrixAn Example

Negative Exponential Distance

Negative Gaussian Distance ( / ) exp 2 /ij ij ijk K d d d d

2max max( / ) exp ( / )ij ij ijk K d d d d

1iiij

ij

kw W K

k i j

I

Gaussian Distance Weights MatrixErtur and Kosh (2007)

Spatial HAC Estimator The Classical Model

1 1

ˆ ˆˆ ˆ' ( '), 1,2,...,

ˆ ˆˆ ( ) ( ' ) ' ( ' )

ij i jkE

i j n

Var

X X εε

β X X X X X X

1ˆ ( ' ) '

y Xβ ε

β X X X y( | ) 0( | )

EVar

ε Xε X

Spatial HAC EstimatorGeneral Heteroscedasticity

Huber-White Estimator2 '

1

1 1

ˆ ˆ'

ˆ ˆˆ ( ) ( ' ) ' ( ' )

ni i ii

Var

X X x x

β X X X X X X

ˆ ˆ 1ˆ ˆ' ,, 1, 2,..., 0

ij i jij

k i jk

i j n i j

X X

Spatial HAC EstimatorGeneral Heteroscedasticity and Autocorrelation

First Law of Geography

Kelejian and Prucha (2007)

'1 1

1 1

ˆ ˆ ˆ'

ˆ ˆˆ ( ) ( ' ) ' ( ' )

n nij i j i ji jk

Var

X X x x

β X X X X X X

max( / ) ( / )ij ij ij ijk K d d or k K d d ij ijd k

Time Series HAC EstimatorGeneral Heteroscedasticity and Autocorrelation

Newey-West Estimator

2 '1

' '1 1

1 1

ˆ ˆ'

ˆ ˆ1 ( )1

ˆ ˆˆ ( ) ( ' ) ' ( ' )

ni i ii

L ni i i i i ii L

Var

X X x x

x x x x

β X X X X X X

Crime EquationAnselin (1988) [anselin.2]

Basic Model(Crime Rate) = + (Family Income) + (Housing Value) +

Spatial HAC Estimator

OLSParameter

OLSs.e.

Robusts.e/hc

Robusts.e/hac

-1.5973 0.33413 0.44664 0.45552

-0.27393 0.10320 0.15752 0.15626

68.619 4.7355 4.1014 5.3639

R2 0.5520

GDP Output ProductionChina 2006 [china.3]

Cobb-Douglass Production Function ln(GDP) = + ln(L) + ln(K) +

Spatial HAC EstimatorOLSParameter

OLSs.e.

Robusts.e/hc

Robusts.e/hac

0.76938 0.08054 0.09081 0.11397

0.30923 0.09459 0.09463 0.10716

-2.6294 0.73630 0.59170 0.46106

R2 0.89137

Spatial ExogeneityLagged Explanatory Variables

Spatial Exogenous Model

'1

1,2,...,

nij jjw

Wi n

xX

W y Xβ Xγ ε2

( | , ) 0

( | , ) ( ')

E W

Var W E

ε X

Iε X εε

GDP Output ProductionChina 2006 [china.4]

Cobb-Douglass Production Function ln(GDP) = + ln(L) + ln(K) + w W ln(L) + w W ln(K) +

OLSParameter

OLSs.e.

Robusts.e/hc

Robusts.e/hac

0.77653 0.05892 0.05674 0.05134

0.30974 0.07198 0.07891 0.08419

w -0.50975 0.11508 0.10334 0.09197

w 0.56380 0.11626 0.12052 0.07242

-3.0745 0.83787 0.68982 0.51367

R2 0.94690

Spatial EndogeneityLagged Dependent Variable

Spatial Lag ModelW y y Xβ ε

1

1,2,...,

nij jjw y

Wi n

y

2

( | , ) 0

( | , ) ( ')

E W

Var W E

ε X

Iε X εε

References 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.