13 spatial panel 113_spatial_panel_1.key author: luc created date: 5/14/2017 9:20:28 am
Post on 06-Oct-2020
0 Views
Preview:
TRANSCRIPT
Copyright © 2017 by Luc Anselin, All Rights Reserved
Luc Anselin
Spatial Regression13. Spatial Panels (1)
http://spatial.uchicago.edu
1
Copyright © 2017 by Luc Anselin, All Rights Reserved
• basic concepts
• dynamic panels
• pooled spatial panels
2
Copyright © 2017 by Luc Anselin, All Rights Reserved
Basic Concepts
3
Copyright © 2017 by Luc Anselin, All Rights Reserved
Data Structures
4
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Two-Dimensional Data
• cross-section/space and time
• observations across space: i = 1, … , N
• observations over time: t = 1, … , T
5
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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)
6
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
7
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
8
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
9
Copyright © 2017 by Luc Anselin, All Rights Reserved
spatial panel data setup
10
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
11
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Space-Time Weights
• no space-time distance metric
• how far how fast
• simplification, constant weights by time period
12
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Space-Time Separability
• space-time interaction from separate spatial and serial covariance
• separate models for spatial covariance and for temporal covariance
13
Copyright © 2017 by Luc Anselin, All Rights Reserved
Model Specifications
14
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Heterogeneity and Dependence
• cross-sectional heterogeneity vs temporal heterogeneity
• cross-sectional dependence vs temporal dependence
• many combinations
• identification problems
15
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
16
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Heterogeneity
• extreme heterogeneity
• yit = Xitβit + εit
• incidental parameter problem
• not operational in classical paradigm
• all coefficients have a distribution in Bayesian paradigm
• hyperparameters
17
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
18
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
19
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Temporal Heterogeneity - Fixed Effects
• separate intercept for each t
• period-specific indicator variables
• yi,t = αt + Xi,tβ + εi,t
• matrix notation - for each cross-section t
• yt = αtιN + Xtβ + εt
• y = (α ⊗ ιN) + Xβ + ε
20
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
• matrix notation - for each cross-section t
• εt = μ + νt , μ as a Nx1 random vector
• ε = (ιT ⊗ IN)μ
21
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
22
Copyright © 2017 by Luc Anselin, All Rights Reserved
Asymptotics
23
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
24
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
25
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
26
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Large N and Large T
• use both T →∞ and N →∞
• parameterize space-time dependence
• properties depend on relative growth of N vs. T
27
Copyright © 2017 by Luc Anselin, All Rights Reserved
Dynamic Panels
28
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Taxonomy of Space-Time Dynamics
• pure space recursive
• time-space recursive
• time-space simultaneous
• time-space dynamic
29
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
30
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
31
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Time-Space Simultaneous
• own time lag and contemporaneous neighbors
• spatial lag always endogenous
• space-time multiplier from time lag
• identification problems when including WXt
32
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
33
Copyright © 2017 by Luc Anselin, All Rights Reserved
Pooled Spatial Panels
34
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Pooled Cross-Section and Time Series Model
• simple extension of cross-sectional model over T periods
• constant coefficients over time and across space
35
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Pooled Model - Spatial Lag
• same weights matrix in each time period
• constant spatial lag coefficient
36
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Pooled Model - Spatial Error
• spatial autoregressive error process in each time period
• overall error variance
37
Copyright © 2017 by Luc Anselin, All Rights Reserved
• Specification Tests in Pooled Model
• straightforward extension of cross-sectional LM test statistics
• distributed as !2(1)
• LM-Error
• LM-Lag
38
Copyright © 2017 by Luc Anselin, All Rights Reserved
• 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
39
Copyright © 2017 by Luc Anselin, All Rights Reserved
Illustration
40
Copyright © 2017 by Luc Anselin, All Rights Reserved
pooled OLS with time fixed effects
41
Copyright © 2017 by Luc Anselin, All Rights Reserved
pooled ML lag with time fixed effects
42
Copyright © 2017 by Luc Anselin, All Rights Reserved
pooled lag with time fixed effects as 2SLS
43
Copyright © 2017 by Luc Anselin, All Rights Reserved
pooled ML error with time fixed effects
44
Copyright © 2017 by Luc Anselin, All Rights Reserved
pooled error GMM with time fixed effects
45
top related