14 spatial panel 22 author: luc created date: 5/21/2017 8:22:02 pm

Copyright © 2017 by Luc Anselin, All Rights Reserved

Luc Anselin

Spatial Regression14. Spatial Panels (2)

http://spatial.uchicago.edu

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• fixed effects models

• random effects models

• ML estimation

• IV/2SLS estimation

• GM estimation

• specification tests

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Fixed Effects Models

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Model Specification

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• Fixed Effects

• unobserved heterogeneity αi

• individual-specific but constant over time

• if included as indicator variables, can be correlated with other explanatory variables

• fixed N approach, for large N or asymptotics with N creates incidental parameter problem

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• Spatial Fixed Effects

• individual heterogeneity

• indicator variable for each location

• yi,t = αi + Xi,tβ + εi,t

• yt = α + Xtβ + εt

• with α’ιN = 0, overall constant and N-1 αi or no constant in β

• y = (ιT ⊗ α) + Xβ + ε

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• Spatial Lag Fixed Effects

• standard pooled lag model, but with additional indicator variables

• yt = ρWyt + α + Xtβ + εt

• y = ρ(IT ⊗ WN)y + (ιT ⊗ α) + Xβ + ε

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• Spatial Error Fixed Effects

• standard pooled error model, but with additional indicator variables

• yt = α + Xtβ + εt

• with error term

• εt = λWN εt + ut

• εt = (IN - λWN)-1ut

• ε = [IT ⊗ (IN - λWN)]-1u

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Estimation Strategies

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• Within Estimator

• wipes out constant and any space-specific effects by taking deviations from group mean (over time) for each i

• for both dependent and explanatory variables

• zit - zim, with zim = Σt zit / T

• demeaning operator Q, a NT x NT matrix

• Qz, applied to a constant yields 0

• consistent estimate only for β not for the αi

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• Demeaning Operator

• Q = INT - (ιTιT’/T ⊗ IN)

• ⊗ is a Kronecker product

• each element in first matrix times second

• ιT is a T x 1 vector of ones

• ιTιT’ is a T x T matrix of ones

• non-standard formulation due to stacking of cross-sections

• standard textbook case = stacking of time series

• Q = INT - (IN ⊗ ιTιT’/T)

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• Properties of Q

• Q is idempotent

• QQ = Q’Q = Q

• Q is singular |Q| = 0

• requires a generalized inverse

• Q- such that QQ-Q = Q

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• Spatial Lag Model in De-Meaned Variables

• apply Q to y, Wy, X and ε

• QWy = WQy

• Qy = ρWQy + QXβ + Qε

• E[Qεε’Q'] = σ2 QQ’ = σ2 Q with Q singular

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• Spatial Error Model in De-Meaned Variables

• apply Q to y, Wy, X and ε

• QWε = WQε

• Qy = QXβ + Qε

• with Qε = λWQε + Qu

• E[Quu’Q'] = σ2 QQ’ = σ2 Q with Q singular

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Random Effects Models

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Model Specification

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• Individual-Level Heterogeneity

• yi,t = μi + Xi,tβ + νi,t

• μi random, becomes part of error term

• μi uncorrelated with X

• εi,t = μi + νit

• for each cross-section t

• εt = μ + νt ,

• μ as a Nx1 random vector

• ε = (ιT ⊗ IN)μ + ν

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• Variance Matrix Non-Spherical

• E[εε’] = E{[(ιT ⊗ IN)μ + ν][(ι’T ⊗ IN)μ’ + ν’]}

• no cross-correlation between μ and ν

• E[εε’] = Σ = σ2μ(ιTι’T ⊗ IN) + σ2ν INT

• NT x NT matrix dimension

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• Simplifying Results

• use matrix properties to simplify expressions for matrix determinant and inverse

• | Σ | = (σ2ν + Tσ2μ)N (σ2ν)T-1

• Σ-1 = (1/T) ιTι’T ⊗ [1/(σ2ν + Tσ2μ)] IN + (IT - (1/T) ιTι’T) ⊗ (1/σ2ν) IN

• no actual matrix inverse required

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ML random effects - Log L = -38077

! = σ2μ / σ2ν

= 0.283 σ2μ = 6.571σ2ν = 23.199

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• Spatial Lag with Random Effects

• special case of lag model with non-spherical error variance

• y = ρ(IT ⊗ WN)y + Xβ + ε

• with ε = (ιT ⊗ IN)μ + ν

• and E[εε’] = Σ = σ2μ(ιTι’T ⊗ IN) + σ2ν INT

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Spatial Error Autocorrelation in Random Effects Models

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• Three Main Specifications

• εt = μ + νt

• SAR in νt (Anselin 88)

• SAR in εt (Kapoor, Kelejian, Prucha 03)

• encompassing (Baltagi et al 06)

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• SAR in Time-Variant Component νt

• εt = μ + νt with νt = θWNνt + ut

• using B = IN - θWN, then νt = B-1ut

• ε = (ιT ⊗ IN)μ + (IT ⊗ B-1)u

• variance matrix

• Σ = σ2μ(ιTιT’ ⊗ IN) + σ2u[IT ⊗ (B’B)-1]

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• SAR in Error εt

• SAR process applies to full error term

• ε = θ(IT ⊗ WN)ε + ν, or, with B = I - θW

• ε = (IT ⊗ B-1)ν

• innovation ν as a one-way error component

• ν = (ιT ⊗ IN)μ + u

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• SAR in Error εt (continued)

• composite error term

• ε = (IT ⊗ B-1)[(ιT ⊗ IN)μ + u]

• variance

• Σ = (IT ⊗ B-1)[σ2uQ0 + σ21Q1](IT ⊗ B-1’)

• with

• σ21 = σ2u + T σ2μ

• Q0 = (IT - JT) ⊗ IN

• Q1 = JT/T ⊗ IN

• JT = ιTιT’

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• Comparison

• SAR in νt

• spatial spillovers only in time variant errors

• SAR in εt

• spatial spillovers in both permanent (individual heterogeneity μ) and time variant error components

• same mechanism in both

• different conceptualizations of spatial effects

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• Encompassing Model

• permanent spatial correlation (random effect)

• μ = θ1WNμ + u1 = (I - θ1WN)-1u1 = A-1u1

• time variant spatial correlation

• νt = θ2WNνt + u2t = (I - θ2WN)-1u2t = B-1u2t

• composite error

• ε = (ιT ⊗ IN)A-1u1 + (IT ⊗ B-1)u2

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• Encompassing Model (continued)

• overall variance

• Σ = σ2u1[ιTιT’ ⊗ (A’A)-1] + σ2u2[IT ⊗ (B’B)-1]

• special cases

• θ1 = 0 → model in νt

• θ1 = θ2 → model in εt

• θ1 = θ2 = 0 → non-spatial random effects

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Estimation Strategies

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• Maximum Likelihood

• special case of model with non-spherical error variance matrix Σ

• complex log-likelihood function

• spatial lag model with error components

• spatial error model with error components

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• IV/2SLS Estimation

• spatial lag model as special case of model with endogenous explanatory variables

• Baltagi (1981) error components 2SLS estimator

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• Generalized Moments Estimator

• only for Kapoor et al spatially correlated error components model

• generalization of Kelejian-Prucha generalized moments estimator in cross-sectional regression

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ML Estimation

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• ML Spatial Lag

• spatial lag model with error components

• complex log-likelihood function

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ML Lag random effects - Log L = -38023

! = σ2μ / σ2ν

= 0.263 σ2μ = 6.095σ2ν = 23.145ρ = 0.132

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• ML Spatial Error with Error Components

• special case of non-spherical error variance-covariance

• likelihood function contains determinant and inverse of the error variance-covariance matrix

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• Variance (Anselin-Baltagi specification)

• slight reparameterization

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• Determinant and Inverse

• determinant

• inverse

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• Likelihood

complex optimization problem

often fails to converge

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• KPP Specification

• variance expression in log-likelihood

• Σ = (IT ⊗ B-1)[σ2uQ0 + σ21Q1](IT ⊗ B-1’)

• with

• σ21 = σ2u + T σ2μ

• Q0 = (IT - JT) ⊗ IN

• Q1 = JT/T ⊗ IN

• JT = ιTιT’

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! = σ2μ / σ2ν

= 0.274 σ2μ = 6.353σ2ν = 23.159λ = 0.103

ML Error KKP - Log L = -38052

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IV/2SLS Estimation

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• Baltagi EC2SLS Estimator

• treat Wy as an endogenous variable

• instruments WX, W2X, etc.

• matrix weighted average of within and between 2SLS estimators

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• Baltagi EC2SLS Estimator (continued)

• three step process

• 2SLS within estimator

• demeaning operator Q applied to all variables and instruments, deviations from temporal mean

• 2SLS between estimator

• operator P, applied to all variables and instruments, temporal mean

• compute σ2μ and σ21 from respective residuals

• EC2SLS as matrix-weighted average

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Lag w Error Components - EC2SLS

σ21 = 45.336 σ2μ = 5.635σ2ν = 22.796ρ = 0.344

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GM Estimation

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• GM Estimation

• extension of GM estimator of Kelejian and Prucha (1998, 1999)

• nuisance parameter approach

• uses Kapoor et al. (2007) error component specification

• actual estimation is FGLS

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• Moment Equations

• notation

• equations

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GM KKP spatial error components

σ21 = 46.954 σ2μ = 5.858σ2ν = 23.522λ = 0.124

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comparison of estimates

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Specification Tests

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• Test Strategies in Error Components Models

• tests are LM, based on estimation under the null

• null can be standard OLS model

• no spatial effects, no error components

• null can be a random effects or a spatial model

• random effects model, no spatial effects

• spatial model, no random effects

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• Classification of Tests

• marginal

• single null hypothesis, irrespective of values for other parameters

• joint

• composite null hypothesis, considering all the parameters

• conditional

• single null hypothesis, conditional on value(s) of other parameter(s)

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• Examples of Null Hypotheses

• marginal

• H0: θ = 0

• joint

• H0: θ = 0 and σ2μ = 0

• conditional

• H0: θ = 0 with σ2μ ≥ 0

• H0: σ2μ = 0 with θ ≠ 0

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• Test Statistics

• complex expressions

• see Anselin et al (2006), Baltagi et al (2003)

• example

• conditional test for H0: θ = 0 with σ2μ ≥ 0 is "2(1)

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14 spatial panel 22 author: luc created date: 5/21/2017 8:22:02 pm

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