15 spatial panel 315_spatial_panel_3.key author: luc created date: 20170522140632z
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Luc Anselin
Spatial Regression15. Spatial Panels (3)
http://spatial.uchicago.edu
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• spatial SUR
• spatial lag SUR
• spatial error SUR
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Spatial SUR
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Specification
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Classic Seemingly Unrelated Regressions
• general cross-sectional covariance
• time series for different (cross-sectional) units
• classic example is investment by firms
• contemporaneous cross-sectional correlation between error terms in time series for different cross-sectional units
• E[eitejt] = σij
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial SUR
• general temporal covariance
• cross-sections for different time periods
• contemporaneous temporal correlation between error terms of cross-sections for different time periods
• E[eiteis] = σts
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial SUR Specification
• cross-sectional regressions, one for each t
• serial (cross-time) covariance is constant across cross-sectional observations
• serial covariance is non-parametric
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial SUR System
• system of T equations
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Motivation
• temporal fixed effects
• different coefficient in each time period t
• efficiency gain
• exploit cross-equation covariance
• only when Xt different in each t
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Estimation
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• FGLS
• special case of non-spherical error variance-covariance matrix
• iterated FGLS is equivalent to ML
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Copyright © 2017 by Luc Anselin, All Rights Reserved
SUR two step FGLS estimation
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Copyright © 2017 by Luc Anselin, All Rights Reserved
SUR iterated FGLS estimation
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Three Stage Least Squares (3SLS)
• allow for endogenous variables on RHS
• in general, stacked form
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• 3SLS Estimation
• need for instruments, similar to 2SLS, but stacked
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Three Step Estimation
• 2SLS on each equation
• estimate σts from 2SLS residuals
• FGLS on full system
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Copyright © 2017 by Luc Anselin, All Rights Reserved
SUR 3SLS estimation
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Specification Tests
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Test on Structure of Σ
• H0: off-diagonal elements are 0
• Likelihood Ratio Test
• Lagrange Multiplier Test R is correlation matrix
χ2 with T(T-1)/2 d.f.
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Illustration - Test on off-diagonal elements
error correlation matrix - 4 equation example
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Test on Coefficient Homogeneity
• H0: coefficients are the same over time, either jointly or individually
• example
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Test on Coefficient Homogeneity (2)
• special case of Chow test, ~ χ2(T-1)
• example
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Spatial Lag SUR
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial SUR - LAG Specification
• different lag model/coefficient in each time period
• general temporal error correlation
• stacked equations
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Estimation Strategies
• special case of S3SLS
• maximum likelihood estimation
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial Lag Spatial 3SLS
• special case with WXt as instruments for Wyt
• all standard results hold
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Copyright © 2017 by Luc Anselin, All Rights Reserved
SUR Lag 3SLS
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Copyright © 2017 by Luc Anselin, All Rights Reserved
SUR Lag 3SLS with endogenous variables
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Maximum Likelihood Estimation
• from the full log-likelihood
• using
• to the concentrated log-likelihood
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• ML Estimation Strategy
• iterative approach
• generalize results from cross-section ML-Lag
• complex coefficient variance-covariance matrix
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• LM Test for Spatial Lag SUR
• apply general principle
• complex expression
• χ2 with T degrees of freedom
• U stacked residual vectors
• requires information matrix
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Spatial Error SUR
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial SUR - Error Specification
• different error model/coefficient in each time period
• cross-equation temporal correlation through remainder error term
• Spatial SUR - Error Specification
• different error model/coefficient in each time period
• cross-equation temporal correlation through remainder error term
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Spatial SUR Error - Covariance
• covariance between t and s
• overall covariance
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Estimation
• special case of FGLS estimation, or spatially weighted least squares
using spatially filtered BX and By, and residuals Be
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Estimation Strategies
• nuisance parameter perspective
• generalized moments estimator (GM)
• maximum likelihood estimation
• full likelihood specification
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• GM Estimator
• generalization of Kelejian-Prucha single-equation GM
• moment equations for residuals from each time period
• solve for λ and construct spatially filtered By, BX and spatially filtered residuals Be
• stack spatially filtered residuals as E (NxT)
• estimate Σ as (1/T)(E’E)
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• GM Moment Equations
uL and uLL spatially lagged residuals
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Copyright © 2017 by Luc Anselin, All Rights Reserved
GM estimation Spatial Error SUR
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• Maximum Likelihood Estimation
• log-likelihood in spatially filtered residuals
• concentrated log-likelihood
• complex coefficient variance matrix
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Copyright © 2017 by Luc Anselin, All Rights Reserved
ML Estimation - Spatial Error SUR Model
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Copyright © 2017 by Luc Anselin, All Rights Reserved
• LM Test for Spatial Error SUR
• apply general principle
• complex expression
• χ2 with T degrees of freedom
• U stacked residual vectors
• T1 = tr(WW), T2 = tr(W’W)
• requires information matrix
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Copyright © 2017 by Luc Anselin, All Rights Reserved
SUR spatial diagnostics (LM tests)
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