Spatial%20Econometric%20Analysis

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Spatial Econometric Analysis

• 4
• Kuan-Pin LinPortland State Univerisity

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Model EstimationSpatial Error Model

• Spatial AR(1)

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Model EstimationSpatial Error Model

• Spatial MA(1)

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Model EstimationSpatial Error Model

• Spatial ARMA(1,1)

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Spatial Error AR(1) ModelMaximum Likelihood
Estimation

• Normal Density Function

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Spatial Error AR(1) ModelMaximum Likelihood
Estimation

• Log-Likelihood Function

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Spatial Error AR(1) ModelMaximum Likelihood
Estimation

• Quasi Maximum Likelihood (QML) Estimator

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Spatial Error AR(1) ModelMaximum Likelihood
Estimation

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

J
SPAR(1) (I-rW)
SPMA(1) (IqW)-1
SPARMA(1,1) (IqW)-1(I-rW)
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Crime EquationAnselin (1988)

• Spatial Error Model AR, MA, ARMA(Crime Rate)
  a b (Family Income) g (Housing Value) ee
  r We u, or e q Wu u

SPAR(1) QML Parameter SPAR(1) QML s.e SPMA(1) QML Parameter SPMA(1) QML s.e
r 0.56178 0.14142
q 0.79909 0.24514
b -0.94131 0.43774 -0.92181 0.41823
g -0.30225 0.16214 -0.28739 0.14551
a 59.893 5.0994 59.253 5.4177
L -183.38 -183.07
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Crime EquationAnselin (1988)

• QML Estimator SPLAG(1) vs. SPAR(1)

SPAR(1) QML Parameter SPAR(1) QML s.e SPLAG(1) QML Parameter SPLAG(1) QML s.e
r 0.56178 0.14142
l 0.43101 0.12962
b -0.94131 0.43774 -1.0316 0.42108
g -0.30225 0.16214 -0.26593 0.17309
a 59.893 5.0994 45.080 6.4051
L -183.38 -182.39
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Spatial Error AR(1) ModelGeneralized Method of
Moments

• Moment Functions (Kelejian and Prucha, 1998)

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Spatial Error AR(1) ModelGeneralized Method of
Moments

• Sample Moment Functions

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Spatial Error AR(1) ModelGeneralized Method of
Moments

• Nonlinear GMM 1 Parameter, 2 Equations

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Spatial Error AR(1) ModelGeneralized Method of
Moments

• Nonlinear GMM 1 Parameter, 2 Equations

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Spatial Error AR(1) ModelGeneralized Method of
Moments

• Minimum Distance (MD) Estimator
• Efficient GMM Estimator

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Spatial Error AR(1) ModelGeneralized Method of
Moments

• Estimation of the variance-covariance matrix of
  moment functions

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Model EstimationSpatial Error Model

• Spatial AR(1) Model
• Estimate b and r simultaneously QML
• Estimate b and r iteratively GMM/GLS
• OLS
• GMM
• GLS

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Crime EquationAnselin (1988)

• Spatial Error AR(1) Model(Crime Rate) a b
  (Family Income) g (Housing Value) ee r We
  u
• GMM vs. QML Estimator

GMM Parameter GMM s.e QML Parameter QML s.e
r 0.54904 0.10596 0.56179 0.14142
b -0.95537 0.33081 -0.94131 0.43774
g -0.30193 0.09017 -0.30225 0.16214
a 60.096 5.3245 59.893 5.0994
Q 0.06979
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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.