13 spatial panel 113_spatial_panel_1.key author: luc created date: 5/14/2017 9:20:28 am

Copyright © 2017 by Luc Anselin, All Rights Reserved

Luc Anselin

Spatial Regression13. Spatial Panels (1)

http://spatial.uchicago.edu

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• basic concepts

• dynamic panels

• pooled spatial panels

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Basic Concepts

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Data Structures

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• Two-Dimensional Data

• cross-section/space and time

• observations across space: i = 1, … , N

• observations over time: t = 1, … , T

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• Traditional - Focus on Time Dimension

• N time series with T observations each

• short time series

• focus on individual heterogeneity

• long time series

• focus on cross-sectional correlation (SUR, VAR)

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• Stacking of Data

• “vertical” slices - side by side

• yit, with t = 1, ..., T for each i

• y11, y12, ... , y1T | ... | yN1, yN2, ..., yNT

• iteration: for each i over all t

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• Non-Traditional Data Organization

• spatial approach is to consider T cross-sections of size N

• one cross-section for each time period

• large N and small T

• focus on spatial specifications

• large N and large T

• many possibilities, focus on either cross-sectional dependence or time dependence, or both

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• Stacking of Data

• cross-sections stacked on top of each other

• horizontal slices

• yit with i = 1, …, N for each t

• y11,..., yN1 | ... | y1T, ..., yNT

• iteration: for each t over all i

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spatial panel data setup

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• Balanced vs Unbalanced Panel

• balanced

• same i in each cross-section

• Nt = N

• census tracts/counties over time

• unbalanced

• different i in each cross-section (or some of the i different)

• N not constant, different Nt

• house sales over time

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• Space-Time Weights

• no space-time distance metric

• how far how fast

• simplification, constant weights by time period

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• Space-Time Separability

• space-time interaction from separate spatial and serial covariance

• separate models for spatial covariance and for temporal covariance

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Model Specifications

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• Heterogeneity and Dependence

• cross-sectional heterogeneity vs temporal heterogeneity

• cross-sectional dependence vs temporal dependence

• many combinations

• identification problems

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• Homogeneity

• classic pooled cross-section time series

• yi,t = Xi,tβ + εi,t

• same parameters and functional form for all locations and all times

• typically too rigid, but useful point of departure

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• Heterogeneity

• extreme heterogeneity

• yit = Xitβit + εit

• incidental parameter problem

• not operational in classical paradigm

• all coefficients have a distribution in Bayesian paradigm

• hyperparameters

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• Temporal vs Cross-Sectional Heterogeneity

• classic approach focus on individual heterogeneity (and time dependence)

• unobserved heterogeneity

• spatial approach focus on temporal heterogeneity and cross-sectional dependence

• fixed or random effects approach

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• Individual Heterogeneity - Fixed Effects

• separate intercept for each i

• spatial fixed effects

• yi,t = αi + Xi,tβ + εi,t

• matrix notation - for each cross-section t

• yt = α + Xtβ + εt

• y = (ιT ⊗ α) + Xβ + ε

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• Temporal Heterogeneity - Fixed Effects

• separate intercept for each t

• period-specific indicator variables

• yi,t = αt + Xi,tβ + εi,t


• yt = αtιN + Xtβ + εt

• y = (α ⊗ ιN) + Xβ + ε

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• Individual Heterogeneity - Random Effects

• individual effect as a random variable

• yi,t = μi + Xi,tβ + νi,t

• μi random, becomes part of error term

• εi,t = μi + νit


• εt = μ + νt , μ as a Nx1 random vector

• ε = (ιT ⊗ IN)μ

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• Temporal Heterogeneity - Random Effects

• time effect as a random variable

• yi,t = δt + Xi,tβ + νi,t

• δt random, becomes part of error term

• εi,t = δt + νit

• temporal random effect creates cross-sectional equi-correlation

• E[εi,tεj,t] = σ2δ

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Asymptotics

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• Relative Size of N and T

• which of N or T (or both) goes to the limit

• if both go to the limit, what is their ratio

• dimension that goes to the limit creates an incidental parameter problem for fixed effects

• with N → ∞ problem for individual heterogeneity

• with T → ∞ problem for temporal heterogeneity

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• Small (fixed) N, large T

• use T → ∞, time domain asymptotics

• parameterize dependence in time

• non-parametric estimate of cross-sectional covariance (classic SUR)

• incidental parameters indexed by t

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• Small (fixed) T, large N

• use N → ∞, spatial asymptotics

• parameterize dependence in space

• non-parametric estimate of serial covariance (spatial SUR)

• incidental parameters indexed by i

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• Large N and Large T

• use both T →∞ and N →∞

• parameterize space-time dependence

• properties depend on relative growth of N vs. T

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Dynamic Panels

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• Taxonomy of Space-Time Dynamics

• pure space recursive

• time-space recursive

• time-space simultaneous

• time-space dynamic

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• Pure Space Recursive

• neighboring locations in a previous period

• spatial lag at previous time period

• spatial diffusion model

• spatial lag endogenous when there is also space-time error dependence, but not otherwise

• identification problem if Xt-1 is included

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• Time-Space Recursive

• own time lag and neighbors in a previous period

• space-time forecasting model

• both lags exogenous unless there is serial or space-time dependence

• identification problems when time lagged X on RHS

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• Time-Space Simultaneous

• own time lag and contemporaneous neighbors

• spatial lag always endogenous

• space-time multiplier from time lag

• identification problems when including WXt

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• Time-Space Dynamics

• time, spatial and space-time lags

• complex identification issues

• Xt-1 included through yt-1

• WXt included through Wyt

• WXt-1 included through Wyt-1

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Pooled Spatial Panels

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• Pooled Cross-Section and Time Series Model

• simple extension of cross-sectional model over T periods

• constant coefficients over time and across space

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• Pooled Model - Spatial Lag

• same weights matrix in each time period

• constant spatial lag coefficient

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• Pooled Model - Spatial Error

• spatial autoregressive error process in each time period

• overall error variance

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• Specification Tests in Pooled Model

• straightforward extension of cross-sectional LM test statistics

• distributed as !2(1)

• LM-Error

• LM-Lag

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• Estimation of Pooled Models

• straightforward extension of pure cross-sectional case

• block-diagonal NT x NT weights matrix

• IV and ML for lag model

• GMM and ML for error model

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Illustration

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pooled OLS with time fixed effects

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pooled ML lag with time fixed effects

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pooled lag with time fixed effects as 2SLS

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pooled ML error with time fixed effects

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pooled error GMM with time fixed effects

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13 spatial panel 113_spatial_panel_1.key author: luc created date: 5/14/2017 9:20:28 am

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