15 spatial panel 315_spatial_panel_3.key author: luc created date: 20170522140632z

Luc Anselin

Spatial Regression15. Spatial Panels (3)

http://spatial.uchicago.edu

• spatial SUR

• spatial lag SUR

• spatial error SUR

Spatial SUR

Specification

• 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

• 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

• 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

• Spatial SUR System

• system of T equations

• Motivation

• temporal fixed effects

• different coefficient in each time period t

• efficiency gain

• exploit cross-equation covariance

• only when Xt different in each t

Estimation

• FGLS

• special case of non-spherical error variance-covariance matrix

• iterated FGLS is equivalent to ML

SUR two step FGLS estimation

SUR iterated FGLS estimation

• Three Stage Least Squares (3SLS)

• allow for endogenous variables on RHS

• in general, stacked form

• 3SLS Estimation

• need for instruments, similar to 2SLS, but stacked

• Three Step Estimation

• 2SLS on each equation

• estimate σts from 2SLS residuals

• FGLS on full system

SUR 3SLS estimation

Specification Tests

• 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.

Illustration - Test on off-diagonal elements

error correlation matrix - 4 equation example

• Test on Coefficient Homogeneity

• H0: coefficients are the same over time, either jointly or individually

• example

• Test on Coefficient Homogeneity (2)

• special case of Chow test, ~ χ2(T-1)

• example

Spatial Lag SUR

• Spatial SUR - LAG Specification

• different lag model/coefficient in each time period

• general temporal error correlation

• stacked equations

• Estimation Strategies

• special case of S3SLS

• maximum likelihood estimation

• Spatial Lag Spatial 3SLS

• special case with WXt as instruments for Wyt

• all standard results hold

SUR Lag 3SLS

SUR Lag 3SLS with endogenous variables

• Maximum Likelihood Estimation

• from the full log-likelihood

• using

• to the concentrated log-likelihood

• ML Estimation Strategy

• iterative approach

• generalize results from cross-section ML-Lag

• complex coefficient variance-covariance matrix

• LM Test for Spatial Lag SUR

• apply general principle

• complex expression

• χ2 with T degrees of freedom

• U stacked residual vectors

• requires information matrix

Spatial Error SUR

• 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

• Spatial SUR Error - Covariance

• covariance between t and s

• overall covariance

• Estimation

• special case of FGLS estimation, or spatially weighted least squares

using spatially filtered BX and By, and residuals Be

• Estimation Strategies

• nuisance parameter perspective

• generalized moments estimator (GM)

• maximum likelihood estimation

• full likelihood specification

• 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)

• GM Moment Equations

uL and uLL spatially lagged residuals

GM estimation Spatial Error SUR

• Maximum Likelihood Estimation

• log-likelihood in spatially filtered residuals

• concentrated log-likelihood

• complex coefficient variance matrix

ML Estimation - Spatial Error SUR Model

• 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

SUR spatial diagnostics (LM tests)

15 spatial panel 315_spatial_panel_3.key author: luc created date: 20170522140632z

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