Spatial Regression. 13. Spatial Panels (1) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved

Transcription

1 Spatial Regression 13. Spatial Panels (1) Luc Anselin 1

2 basic concepts dynamic panels pooled spatial panels 2

3 Basic Concepts 3

4 Data Structures 4

5 Two-Dimensional Data cross-section/space and time observations across space: i = 1,, N observations over time: t = 1,, T 5

6 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

7 Stacking of Data vertical slices - side by side yit, with t = 1,..., T for each i y 11, y12,..., y1t... yn1, yn2,..., ynt iteration: for each i over all t 7

8 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

9 Stacking of Data cross-sections stacked on top of each other horizontal slices y it with i = 1,, N for each t y11,..., yn1... y1t,..., ynt iteration: for each t over all i 9

10 spatial panel data setup 10

11 Balanced vs Unbalanced Panel balanced same i in each cross-section N t = 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

12 Space-Time Weights no space-time distance metric how far how fast simplification, constant weights by time period 12

13 Space-Time Separability space-time interaction from separate spatial and serial covariance separate models for spatial covariance and for temporal covariance 13

14 Model Specifications 14

15 Heterogeneity and Dependence cross-sectional heterogeneity vs temporal heterogeneity cross-sectional dependence vs temporal dependence many combinations identification problems 15

16 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

17 Heterogeneity extreme heterogeneity yit = Xitβit + εit incidental parameter problem not operational in classical paradigm all coefficients have a distribution in Bayesian paradigm hyperparameters 17

18 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

19 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

20 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

21 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

22 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] = σ 2 δ 22

23 Asymptotics 23

24 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

25 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

26 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

27 Large N and Large T use both T and N parameterize space-time dependence properties depend on relative growth of N vs. T 27

28 Dynamic Panels 28

29 Taxonomy of Space-Time Dynamics pure space recursive time-space recursive time-space simultaneous time-space dynamic 29

30 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 spacetime error dependence, but not otherwise identification problem if Xt-1 is included 30

31 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

32 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

33 Time-Space Dynamics time, spatial and space-time lags complex identification issues X t-1 included through yt-1 WXt included through Wyt WX t-1 included through Wyt-1 33

34 Pooled Spatial Panels 34

35 Pooled Cross-Section and Time Series Model simple extension of cross-sectional model over T periods constant coefficients over time and across space 35

36 Pooled Model - Spatial Lag same weights matrix in each time period constant spatial lag coefficient 36

37 Pooled Model - Spatial Error spatial autoregressive error process in each time period overall error variance 37

38 Specification Tests in Pooled Model straightforward extension of cross-sectional LM test statistics distributed as χ2 (1) LM-Error LM-Lag 38

39 Estimation of Pooled Models straightforward extension of pure crosssectional case block-diagonal NT x NT weights matrix IV and ML for lag model GMM and ML for error model 39

40 Illustration 40

41 pooled OLS with time fixed effects 41

42 pooled ML lag with time fixed effects 42

43 pooled lag with time fixed effects as 2SLS 43

44 pooled ML error with time fixed effects 44

45 pooled error GMM with time fixed effects 45