Spatial Regression. 14. Spatial Panels (2) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved

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1 Spatial Regression 14. Spatial Panels (2) Luc Anselin 1

2 fixed effects models random effects models ML estimation IV/2SLS estimation GM estimation specification tests 2

3 Fixed Effects Models 3

4 Model Specification 4

5 Fixed Effects unobserved heterogeneity αi individual-specific but constant over time if included as indicator variables, can be correlated with other explanatory variables fixed N approach, for large N or asymptotics with N creates incidental parameter problem 5

6 Spatial Fixed Effects individual heterogeneity indicator variable for each location yi,t = αi + Xi,tβ + εi,t yt = α + Xtβ + εt with α ιn = 0, overall constant and N-1 αi or no constant in β y = (ιt α) + Xβ + ε 6

7 Spatial Lag Fixed Effects standard pooled lag model, but with additional indicator variables yt = ρwyt + α + Xtβ + εt y = ρ(i T WN)y + (ιt α) + Xβ + ε 7

8 Spatial Error Fixed Effects standard pooled error model, but with additional indicator variables yt = α + Xtβ + εt with error term εt = λwn εt + ut εt = (IN - λwn) -1 ut ε = [IT (IN - λwn)] -1 u 8

9 Estimation Strategies 9

10 Within Estimator wipes out constant and any space-specific effects by taking deviations from group mean (over time) for each i for both dependent and explanatory variables zit - zi m, with zi m = Σt zit / T demeaning operator Q, a NT x NT matrix Qz, applied to a constant yields 0 consistent estimate only for β not for the αi 10

11 Demeaning Operator Q = INT - (ιtιt /T IN) is a Kronecker product each element in first matrix times second ιt is a T x 1 vector of ones ιtιt is a T x T matrix of ones non-standard formulation due to stacking of cross-sections standard textbook case = stacking of time series Q = INT - (IN ιtιt /T) 11

12 Properties of Q Q is idempotent QQ = Q Q = Q Q is singular Q = 0 requires a generalized inverse Q - such that QQ - Q = Q 12

13 Spatial Lag Model in De-Meaned Variables apply Q to y, Wy, X and ε QWy = WQy Qy = ρwqy + QXβ + Qε E[Qεε Q'] = σ 2 QQ = σ 2 Q with Q singular 13

14 Spatial Error Model in De-Meaned Variables apply Q to y, Wy, X and ε QWε = WQε Qy = QXβ + Qε with Qε = λwqε + Qu E[Quu Q'] = σ 2 QQ = σ 2 Q with Q singular 14

15 Random Effects Models 15

16 Model Specification 16

17 Individual-Level Heterogeneity yi,t = μi + Xi,tβ + νi,t μi random, becomes part of error term μi uncorrelated with X εi,t = μi + νit for each cross-section t εt = μ + νt, μ as a Nx1 random vector ε = (ιt IN)μ + ν 17

18 Variance Matrix Non-Spherical E[εε ] = E{[(ι T IN)μ + ν][(ι T IN)μ + ν ]} no cross-correlation between μ and ν E[εε ] = Σ = σ 2 μ(ιtι T IN) + σ 2 ν INT NT x NT matrix dimension 18

19 Simplifying Results use matrix properties to simplify expressions for matrix determinant and inverse Σ = (σ 2 ν + Tσ 2 μ) N (σ 2 ν) T-1 Σ -1 = (1/T) ιtι T [1/(σ 2 ν + Tσ 2 μ)] IN + (IT - (1/T) ιtι T) (1/σ 2 ν) IN no actual matrix inverse required 19

20 φ = σ 2 μ / σ 2 ν = σ 2 μ = σ 2 ν = ML random effects - Log L =

21 Spatial Lag with Random Effects special case of lag model with non-spherical error variance y = ρ(it WN)y + Xβ + ε with ε = (ιt IN)μ + ν and E[εε ] = Σ = σ 2 μ(ιtι T IN) + σ 2 ν INT 21

22 Spatial Error Autocorrelation in Random Effects Models 22

23 Three Main Specifications εt = μ + νt SAR in νt (Anselin 88) SAR in εt (Kapoor, Kelejian, Prucha 03) encompassing (Baltagi et al 06) 23

24 SAR in Time-Variant Component νt εt = μ + νt with νt = θwnνt + ut using B = IN - θwn, then νt = B -1 ut ε = (ιt IN)μ + (IT B -1 )u variance matrix Σ = σ 2 μ(ιtιt IN) + σ 2 u[it (B B) -1 ] 24

25 SAR in Error εt SAR process applies to full error term ε = θ(it WN)ε + ν, or, with B = I - θw ε = (IT B -1 )ν innovation ν as a one-way error component ν = (ιt IN)μ + u 25

26 SAR in Error εt (continued) composite error term ε = (IT B -1 )[(ιt IN)μ + u] variance Σ = (IT B -1 )[σ 2 uq0 + σ 2 1Q1](IT B -1 ) with σ 2 1 = σ 2 u + T σ 2 μ Q 0 = (IT - JT) IN Q 1 = JT/T IN JT = ιtιt 26

27 Comparison SAR in νt spatial spillovers only in time variant errors SAR in εt spatial spillovers in both permanent (individual heterogeneity μ) and time variant error components same mechanism in both different conceptualizations of spatial effects 27

28 Encompassing Model permanent spatial correlation (random effect) μ = θ1wnμ + u1 = (I - θ1wn) -1 u1 = A -1 u1 time variant spatial correlation νt = θ2wnνt + u2t = (I - θ2wn) -1 u2t = B -1 u2t composite error ε = (ιt IN)A -1 u1 + (IT B -1 )u2 28

29 Encompassing Model (continued) overall variance Σ = σ 2 u1[ιtιt (A A) -1 ] + σ 2 u2[it (B B) -1 ] special cases θ1 = 0 model in νt θ1 = θ2 model in εt θ1 = θ2 = 0 non-spatial random effects 29

30 Estimation Strategies 30

31 Maximum Likelihood special case of model with non-spherical error variance matrix Σ complex log-likelihood function spatial lag model with error components spatial error model with error components 31

32 IV/2SLS Estimation spatial lag model as special case of model with endogenous explanatory variables Baltagi (1981) error components 2SLS estimator 32

33 Generalized Moments Estimator only for Kapoor et al spatially correlated error components model generalization of Kelejian-Prucha generalized moments estimator in cross-sectional regression 33

34 ML Estimation 34

35 ML Spatial Lag spatial lag model with error components complex log-likelihood function 35

36 φ = σ 2 μ / σ 2 ν = σ 2 μ = σ 2 ν = ρ = ML Lag random effects - Log L =

37 ML Spatial Error with Error Components special case of non-spherical error variancecovariance likelihood function contains determinant and inverse of the error variance-covariance matrix 37

38 Variance (Anselin-Baltagi specification) slight reparameterization 38

39 Determinant and Inverse determinant inverse 39

40 Likelihood complex optimization problem often fails to converge 40

41 KPP Specification variance expression in log-likelihood Σ = (IT B -1 )[σ 2 uq0 + σ 2 1Q1](IT B -1 ) with σ 2 1 = σ 2 u + T σ 2 μ Q0 = (IT - JT) IN Q 1 = JT/T IN JT = ιtιt 41

42 φ = σ 2 μ / σ 2 ν = σ 2 μ = σ 2 ν = λ = ML Error KKP - Log L =

43 IV/2SLS Estimation 43

44 Baltagi EC2SLS Estimator treat Wy as an endogenous variable instruments WX, W2 X, etc. matrix weighted average of within and between 2SLS estimators 44

45 Baltagi EC2SLS Estimator (continued) three step process 2SLS within estimator demeaning operator Q applied to all variables and instruments, deviations from temporal mean 2SLS between estimator operator P, applied to all variables and instruments, temporal mean compute σ 2 μ and σ 2 1 from respective residuals EC2SLS as matrix-weighted average 45

46 σ 2 1 = σ 2 μ = σ 2 ν = ρ = Lag w Error Components - EC2SLS 46

47 GM Estimation 47

48 GM Estimation extension of GM estimator of Kelejian and Prucha (1998, 1999) nuisance parameter approach uses Kapoor et al. (2007) error component specification actual estimation is FGLS 48

49 Moment Equations notation equations 49

50 σ 2 1 = σ 2 μ = σ 2 ν = λ = GM KKP spatial error components 50

51 comparison of estimates 51

52 Specification Tests 52

53 Test Strategies in Error Components Models tests are LM, based on estimation under the null null can be standard OLS model no spatial effects, no error components null can be a random effects or a spatial model random effects model, no spatial effects spatial model, no random effects 53

54 Classification of Tests marginal single null hypothesis, irrespective of values for other parameters joint composite null hypothesis, considering all the parameters conditional single null hypothesis, conditional on value(s) of other parameter(s) 54

55 Examples of Null Hypotheses marginal H0: θ = 0 joint H0: θ = 0 and σ 2 μ = 0 conditional H0: θ = 0 with σ 2 μ 0 H0: σ 2 μ = 0 with θ 0 55

56 Test Statistics complex expressions see Anselin et al (2006), Baltagi et al (2003) example conditional test for H0: θ = 0 with σ 2 μ 0 is χ 2 (1) 56