5 specificiation dependence · Copyright © 2017 by Luc Anselin, All Rights Reserved...

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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5 specificiation dependence · Copyright © 2017 by Luc Anselin, All Rights Reserved...

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