5 specificiation dependence · copyright © 2017 by luc anselin, all rights reserved...
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Copyright © 2017 by Luc Anselin, All Rights Reserved
Luc Anselin
Spatial Regression5. Specification of Spatial Dependence
http://spatial.uchicago.edu
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• spatially lagged variables
• spatial lag model
• spatial error model
• other specifications
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Spatially Lagged Variables
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• Specifying Spatial Dependence
• in time series analysis, the concept of a shift yt-k is observation shifted by k periods
• for regular lattices, shift north, south, east, west: yi-1,j , yi+1, j , yi,j-1, yi, j+1 spatial shift of yi,j
• no analog for irregular spatial layouts
• instead, the notion of a spatially lagged variable (Anselin 1988)
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• Spatial Lag
• weighted average of neighboring values
• neighbors defined by spatial weights (wij)
• yiL = wi1.y1 + wi2.y2 + ... + wiN.y = Σj wij.yj
• in practice: very few neighbors (weights are sparse)
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• Spatial Lag in Matrix Notation
• spatial weights matrix times the vector of observations
• yL = Wy
• Wy as such is often used as symbol for a spatially lagged dependent variable
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• Spatial Lag vs. Window Average
• similar to a window average, the spatial lag is a smoother
• lag Wy has smaller variance than original variable y
• spatial lag is NOT a window average since wii = 0 observation at “center” of window is not included
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• Spatially Lagged Variables in a Regression
• spatially lagged dependent variable: Wyspatial (autoregressive) lag model
• spatially lagged explanatory variables: WXspatial cross-regressive model or SLX model
• spatially lagged error terms: Wespatial (autoregressive) error model
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Spatial Lag Model
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Motivation
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• Motivation
• explicit model for spatial interaction = substantive spatial dependence
• peer-effects, etc.
• equilibrium outcome of spatial interaction process, a spatial reaction function (Brueckner 2003)
• non-behavioral motivation = data issue (scale)
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• Spatial Reaction Function (Brueckner 2003)
• yi = R(y-i, xi)
• y decision variable
• x resources
• a linear function for R yields a spatial lag specification
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• Behavioral Motivation (1)
• spillover
• y-i enters into utility function for i
• U(yi, y-i ; xi)
• e.g., yardstick competition, spillovers among state expenditures, race to the bottom
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• Behavioral Motivation (2)
• resource flow
• resources used si enter into utility function
• U(yi, si; xi)
• si is a function of other agents action
• si = H(yi, y-i, xi)
• e.g., tax competition, environmental regulation
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• Identification Issues
• inverse problem
• different processes can yield the same pattern
• reflection problem (Manski 1993)
• parameter identification in spatial/social interaction models
• new economic geography critique (Gibbons and Overman 2012)
• difficulty with interpretation of causal effects
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The Reflection Problem
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• Types of Social Interaction
• interaction effects among individual agents = endogenous effects
• exogenous group characteristics = contextual effects
• observed or unobserved characteristics that agents have in common = correlated effects
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• General Model for Social/Spatial Interaction
• individual as well as group characteristics
• variables: y individual decisions, x group characteristics, z individual observed, u individual unobserved
• y = α + β E[y | x] + E[z | x]! + z’θ + u
• mean group effect E[y | x] = endogenous effect
• E[z | x] = contextual effect
• unobserved individual characteristics correlated across individuals in the group E[u | z, x] = x’δ
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• Social Interaction Regression Model
• conditional expectation
• E[ y | z,x ] = α + β E[y | x] + E[z | x]’! + z’θ + x’δ
• endogenous effects: β ≠ 0
• contextual effects: ! ≠ 0
• correlated effects: δ ≠ 0
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• Identification Problems
• reduced form (move endogenous effects to LHS)
• E [y|x] = α/(1-β) + E[z|x](!+θ)/(1-β) + x’δ/(1-β)
• different social effects cannot be separately identified
• need for parameter constraints, instruments
• note: spatial lag is NOT the group mean
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Specification
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• Mixed Regressive-Spatial Autoregressive
• Wy = spatial autoregressive (spatial lag)
• X = regressive
• y = ρWy + Xβ + u
• ρ = spatial autoregressive coefficient
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• Spatial Filter
• remove effect of spatial autocorrelation
• y - ρWy = Xβ + u
• (I - ρW)y = Xβ + u
• (I - ρW) is spatial filter
• only form of spatial filter within valid DGP
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• Effect of Spatial Filter
• similar to detrending
• deals with scale problems, i.e., non-behavioral motivation for including spatial lag term
• spatial filter still requires estimate of ρ
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• Spatial Multiplier
• derived from reduced form
• what is change in y as a result of change in X
• E[ y | ∆X ] = (I - ρW)-1 (∆X)β = [I + ρW + ρ2W2 + ... ] (∆X)β
• effect is more than (∆X) β ⇒ multiplier
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• Direct and Indirect Effects
• total effect of a change in exogenous variable
• (I - ρW)-1 (∆X)β
• direct effect
• (∆X)β
• indirect effect
• [ (I - ρW)-1 - I ] (∆X)β
• [ ρW + ρ2W2 + ... ] (∆X)β
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• Applications of Spatial Multiplier
• policy analysis
• effect of a change in a policy variable x at i extends beyond i to its neighbors, neighbors of neighbors, etc.
• simulate the spatial imprint of a policy change by solving the reduced form for a change in X
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Misspecification
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• Effects of Ignoring a Spatial Lag
• = ignoring substantive spatial interaction
• omitted variable problem
• OLS biased and inconsistent
• potentially: wrong estimate, wrong sign, wrong standard error, wrong significance, wrong fit
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effect of ignoring spatial lag on OLS estimate
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Spatial Error Model
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• Motivation
• spatial pattern in error term due to omitted random factors = nuisance spatial dependence
• mismatch spatial scale process with spatial scale observations (administrative units as “markets”)
• no substantive interpretation
• problem of efficiency of the estimates
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• Non-Spherical Error Variance
• due to spatial autocorrelation, error covariances are non-zero
• off-diagonal elements are non-zero
• E [ uu’ ] = Σ ≠ σ2I
• spatial structure in the covariance E[ uiuj ] ≠ 0, for i ≠ j
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• Spatial Autoregressive Error Model
• SAR error
• y = Xβ + u with u = λWu + e
• covariance matrix Σ = σ2 [ (I - λW)’(I - λW) ]-1
• but inverse covariance matrix (used in GLS) does not contain inverse terms: Σ-1 = (1/σ2) [ (I - λW)’(I - λW) ]
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• Reduced Form
• y = Xβ + (I - λW)-1e
• no substantive spatial multiplier effect
• effect of spatial autocorrelation is on error variance, used in kriging (spatial prediction)
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• SAR Errors and Heteroskedasticity
• variance Σ = σ2 [ (I - λW)’(I - λW) ]-1 has non-constant diagonal terms - depends on number of neighbors
• this induces heteroskedasticity in u, even with homoskedastic errors e
• difficult to disentangle true heteroskedasticity from induced heteroskedasticity
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• Spatial Moving Average Error Model
• SMA error
• y = Xβ + u with u = λWe + e
• innovation (e) + smoothing of neighbors (We)
• covariance matrix Σ = σ2 (I + λW)’(I + λW)
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• Other Spatial Error Structures
• direct representation: covariance a declining function of distance
• other spatial processes: moving average, CAR (in hierarchical models)
• spatial error components (Kelejian-Robinson)
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Misspecification
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• Effects of Ignoring SAR Errors
• problem of efficiency
• OLS remains unbiased but inefficient
• potentially: correct estimate, wrong standard error, wrong significance, wrong fit
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effect of ignoring SAR errors on OLS estimate
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Other Specifications
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Spatial Durbin Model
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• Classic Spatial Durbin Model (Anselin, Burridge)
• SAR error model as a spatial lag model
• y = Xβ + u with u = λWu + e
• substitute u = (I - λW)-1e
• y = Xβ + (I - λW)-1e
• spatially filtered variable regression (spatial Cochrane-Orcutt)
• (I - λW)y = (I - λW)Xβ + e
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• Classic Spatial Durbin Model (2)
• y = λWy + Xβ - λWXβ + u
• spatial lag (Wy) and cross-regressive term (WX)
• non-linear model in λ and β
• constant term is (1 - λ) β0
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• Common Factor Hypothesis
• k - 1 nonlinear constraints:
• - ( λ. β ) = - λβ
• constant is not separately identified
• negative the coefficient of the spatial lag term (Wy) times each regression coefficient (of X) equals the coefficient of the matching spatially lagged explanatory variable (of WX)
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• Unconstrained Spatial Durbin Model (LeSage-Pace, Elhorst)
• y = γ1Wy + Xγ2 + WXγ3 + u
• common factor hypothesis: H0: γ1.γ2 = -γ3
• H0 not rejected: proper specification is SAR error model
• H0 rejected: different interpretations
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• Reduced Form
• y = (I - γ1W)-1 Xγ2 + (I - γ1W)-1 WXγ3 + v
• using the standard expansion yields complex expression in X, WX, W2X, etc.
• coefficient of WX is γ1.γ2 + γ3 NOT just γ3
(cross-regressive case) or γ1.γ2 (spatial lag case)
• if common factor hypothesis holds, coefficient of WX is 0 since γ1.γ2 = - γ3
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• Hypothesis Tests
• rejection of H0: γ3 = 0 (coefficient of WX) does NOT imply spatial lag model
• if common factor hypothesis holds, then γ3 = 0 implies either γ1 = 0 (standard non-spatial model) or γ2 = 0 (pure error SAR)
• NOT nested hypotheses
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SLX Model
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• Spatial Cross-Regressive Model (Florax and Folmer 1992)
• include spatially lagged exogenous variables (WX) on RHS
• y = Xβ + WXθ + u
• “rediscovered” as SLX model (Vega and Elhorst 2015)
• reaction to mostly pointless critique
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• Motivation for SLX Model
• no spatially lagged dependent variable
• deal with endogeneity in WX if needed
• a model for local spillovers
• no effect from X beyond first order neighbors
• no need for spatial econometric estimators
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• Extensions
• including higher order spatial lag terms
• parameterizing the W matrix
• non/semi-parametric spatial lag term WX
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• Parameterizing the W matrix
• example in Vega and Elhorst (2015)
• wij = 1 / dij!
• regression term W(!)Xθ• iterative non-linear least squares estimation
• identification issues
• note: wij* = log(wij) = -! log(dij)
• equivalent to linear weights specification
• -! [ W*]Xθ
• ! and θ not separately identifiable
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