6 specification heterogeneity - Spatial@UChicago · Copyright © 2017 by Luc Anselin, All Rights Reserved •Poisson Regression • P[Y = y] = e-μ μy / y! heterogeneity. Author:

Copyright © 2017 by Luc Anselin, All Rights Reserved

Luc Anselin

Spatial Regression6. Specification Spatial Heterogeneity

http://spatial.uchicago.edu

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• homogeneity and heterogeneity

• spatial regimes

• spatially varying coefficients

• spatial random effects

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Homogeneity and Heterogeneity

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• Global Perspective

• single equilibrium - stationarity

• functional form fixed

• coefficients fixed

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• Local Perspective

• multiple equilibria

• non-stationarity

• functional and/or parameter variability

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• Extreme Homogeneity

• model same everywhere

• parameters same everywhere

• yi = xiβ + εi

• β constant across i

• εi ∼ i.i.d. with Var[εi] = σ2 for all i

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• Extreme Heterogeneity

• every observation is different

• yi = xiβi + εi

• a different parameters βi for each observation i

• εi ∼ i.n.i.d. with Var[εi] = σi2

• possible different functional forms for i

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• Incidental Parameter Problem

• number of unknown parameters increases with sample size

• no consistent estimation of individual parameters βi, σi2

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• Solutions

• imposing structure

• discrete variation - finite subsets of the data

• continuous variation - parameter surface

• heterogeneity parameters

• fixed effects

• random effects

• spatial heterogeneity may be complicated by spatial autocorrelation

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Spatial Regimes

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Discrete Heterogeneity

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• Spatial Regimes

• systematic discrete spatial subsets of the data

• different coefficient values in each subset

• corrects for heterogeneity, but does not explain

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• Spatial Regime Specifications

• varying intercepts

• spatial ANOVA

• spatial fixed effects

• full spatial regimes

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Varying Intercepts

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• ANOVA - Difference in Means

• approach is standard, regimes are spatial

• E[y1] = μ1 ∀ i ∈ R1 (R1 is region 1)

• E[y2] = μ2 ∀ i ∈ R2 (R2 is region 2)

• H0: μ1 = μ2

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• Dummy Variable Regression - Variant 1

• no constant term

• indicator variable for each regime

• yi = β1d1i + β2d2i + εi

• d1(2)i = 1 ∀ i ∈ R1(2), 0 elsewhere

• H0: β1 = β2

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• Dummy Variable Regression - Variant 2

• constant term for overall mean

• yi = α + βdi + εi

• di = 1 ∀ i ∈ R1 , 0 elsewhere

• H0: β = 0, difference from reference mean α

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• Spatial Fixed Effects

• reference mean and difference by regime

• fixed effects multi-level specification

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• Spatial Fixed Effects and Spatial Autocorrelation (Anselin and Arribas-Bel 2013)

• common misconception that spatial fixed effects “fix” spatial autocorrelation

• only in special case of group weights

• each observation has all other observations as neighbors

• so-called “Case” weights (Case 1992)

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Full Spatial Regimes

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• Spatial Regimes - Full Specification

all coefficients (intercept, slope, variance) vary by regimeequivalent to separate regression by regime

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Testing for Spatial Heterogeneity

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• Test on Spatial Homogeneity

• null hypothesis

• equal intercepts, equal slopes

• alternative hypothesis

• different intercepts

• different slopes

• both

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• Chow Test

• test on structural stability

• based on residual sum of squares in constrained (all coefficients equal - R) and unconstrained (coefficients different - U) regressions

• classic form C =

e′ReR − e′

UeU

k/

e′U

eU

N − 2k∼ F (k, N − 2k)

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• General Test on Coefficient Stability

• as a set of linear constraints on the coefficients in a pooled regression

• can be readily extended to spatial models G = (J - 1)K

J regimesK coefficients

V = variance

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Spatial Regimes with Spatial Dependence

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• Spatial Lag and Spatial Error Models

• allow varying coefficients by regime

• fixed spatial coefficient

• same spatial process throughout

• varying spatial coefficient

• different spatial process for each regime

• difficult assumption - needs to be based on a strong foundation

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• Spatial Regimes - Spatial Lag Model

fixed spatial autoregressive coefficient

varying spatial autoregressive coefficient

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• Spatial Regimes - Spatial Error Model

fixed spatial autoregressive coefficient

varying spatial autoregressive coefficient

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• Spatial Weights Specification

• necessary to construct spatially lagged variables

• neighbors spill over across regimes

• neighbors constrained to be within each regime

• weights truncated, possible isolates

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• Spatial Chow Test

• use general form of the test with V as coefficient variance matrix in pooled model

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Spatially Varying Coefficients

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• Spatially Varying Coefficients

• systematic variation with covariates

• coefficient as a function of other variables (including as a trend surface)

• spatial expansion method

• local estimation over space

• coefficients obtained from a subset (kernel) of nearby data points

• geographically weighted regression (GWR)

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Expansion Method

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• Casetti’s Expansion Method

• special case of varying coefficients

• each coefficient is a function of other covariates

• creates interaction effects

• similar in form to multi-level models

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• Sequential Modeling Strategy

• initial model

• yi = α + xiβi + εi

• expansion equation

• βi = γ0 + zi1γ1 + zi2γ2

• final model

• yi = α + xi (γ0 + zi1γ1 + zi2γ2) + εi

• yi = α + xiγ0 + (zi1xi)γ1 + (zi2xi)γ2 + εi

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• Implementation Issues

• multicollinearity

• t-test values unreliable

• various fixes

• principal components (orthogonal expansion)

• danger of overfitting

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• Random Expansion Model

• random error in expansion equation

• βi = γ0 + zi1γ1 + zi2γ2 + ψi

• error term in final model is heteroskedastic

• yi = α + xi (γ0 + zi1γ1 + zi2γ2 + ψi) + εi

• νi = xiψi + εi

• Var[νi] = xi2 σ2ψ + σ2ε

• similar to random coefficient model and multilevel models

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Geographically Weighted Regression

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• Geographically Weighted Regression

• local regression

• a different set of parameter values for each location

• parameter values obtained from a subset of observations using kernel regression

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• Local Regression

• non-parametric specification

• simple bivariate regression

• yi = m(xi) + ui

• functional form of m is unspecified

• m(xi) yields the conditional expectation of y | x

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• Local Average

• what is the expected value of yi given x, E[yi | x]

• special case: for a given x0 with multiple yi

• example: two values for PATIO dummy, house price

• solution:

• take m(x0) as the average of yi for x0 =0 and x0 =1

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local averagepredictor of PRICE for two values of PATIO

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• Locally Weighted Average

• expand the estimate of m(x0) to include values of yi observed for values of x “close” to x0

• compute a locally weighted average

• weights sum to one

• weights larger as x closer to x0 (for h ↓)

• m(x0) = ∑i wi0,h yi

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locally weighted average (lowess) of PRICE for LOTSZ

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• Kernel Regression

• special case of locally weighted average

• use kernel function as the weights

• m(x0) = ∑i K [(xi - x0)/h ]yi

• K is kernel function

• h is bandwidth s.t. K = 0 for xi - x0 > h

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• Kernel Functions with Finite Bandwidth

• Epanechnikov

• K(z) = 1 - z2

• Bisquare

• K(z) = (1 - z2)2

• with z = (xi - x0) / h

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• Gaussian Kernel

• asymptotic bandwidth

• specified in function of standard error or variance

• K(z) = exp(- z2/2)

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• GWR Estimation

• local estimation based on nearby locations

• not just yi but x-y pairs at nearby locations

• kernel regression yields a different coefficient for each location

• specify kernel function and bandwidth

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• GWR Kernel Regression

• location-specific kernel weights

• W(ui, vi) diagonal elements are weights

• b(ui,vi) = [X’W(ui,vi)X]-1X’W(ui,vi)y

• fixed kernel vs adaptive kernel

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fixed bandwidth kernel adaptive kernel

Source: Fotheringham et al (2002)

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• GWR - Practical Issues

• choice of bandwidth

• use cross-validation

• parameter inference

• still several theoretical loose ends

• visualizing parameter heterogeneity

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Spatial Random Effects

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Random Coefficients

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• Random Coefficient Regression

• extreme heterogeneity, but variability in βi driven by a random process - no space

• βi = β + ψi with E[ψi]=0 and Var[ψi]=σ2

• heteroskedastic regression for mean effect

• yi = α + xiβ + νi , var[νi] =σ2ψxi2 + σ2ε

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• Mixed Linear Models

• both fixed and random coefficients

• y = Xβ + Zψ + ε

• Z a “design matrix”, could be same as X

• ψ random coefficients with mean zero and variance Σψ

• ε random error vector with variance Σε

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• Spatial Random Coefficients

• introduce spatial dependence structure in random variation of coefficient

• βi - β = ρ Σj wij (βj - β) + ψi - SAR model

• βi = β + λ Σj wij ψj + ψi - SMA model

• complex covariance structures

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Spatial Random Effects

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• Spatial Random Effects

• βi = β + ψi with spatial effects introduced through random effect ψi

• typically a CAR process

• Bayesian hierarchical model - BYM model

• βi = β + ψi + νi

• spatial dependence in ψi, heterogeneity in νi

• not identified in Gaussian (linear regression) model

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• Example - Poisson Regression

• spatial autocorrelation needs to be introduced indirectly

• auto-Poisson model only allows negative spatial autocorrelation

• random effects model

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• Poisson Regression

• P[Y = y] = e-μ μy / y!

• μ is the mean

• μ as a function of regressors to model heterogeneity

• μi = exp(xi’β) no error term

• random effects

• μi = exp(xi’β + ψi + νi)

• spatial effects through ψi, e.g, CAR model

• non-spatial heterogeneity through νi

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6 specification heterogeneity - Spatial@UChicago · Copyright © 2017 by Luc Anselin, All Rights Reserved •Poisson Regression • P[Y = y] = e-μ μy / y! heterogeneity. Author: