Spatial Regression. 15. Spatial Panels (3) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved

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1 Spatial Regression 15. Spatial Panels (3) Luc Anselin 1

2 spatial SUR spatial lag SUR spatial error SUR 2

3 Spatial SUR 3

4 Specification 4

5 Classic Seemingly Unrelated Regressions general cross-sectional covariance time series for different (cross-sectional) units classic example is investment by firms contemporaneous cross-sectional correlation between error terms in time series for different cross-sectional units E[eitejt] = σij 5

6 Spatial SUR general temporal covariance cross-sections for different time periods contemporaneous temporal correlation between error terms of cross-sections for different time periods E[eiteis] = σts 6

7 Spatial SUR Specification cross-sectional regressions, one for each t serial (cross-time) covariance is constant across cross-sectional observations serial covariance is non-parametric 7

8 Spatial SUR System system of T equations 8

9 Motivation temporal fixed effects different coefficient in each time period t efficiency gain exploit cross-equation covariance only when Xt different in each t 9

10 Estimation 10

11 FGLS special case of non-spherical error variancecovariance matrix iterated FGLS is equivalent to ML 11

12 SUR two step FGLS estimation 12

13 SUR iterated FGLS estimation 13

14 Three Stage Least Squares (3SLS) allow for endogenous variables on RHS in general, stacked form 14

15 3SLS Estimation need for instruments, similar to 2SLS, but stacked 15

16 Three Step Estimation 2SLS on each equation estimate σts from 2SLS residuals FGLS on full system 16

17 SUR 3SLS estimation 17

18 Specification Tests 18

19 Test on Structure of Σ H 0: off-diagonal elements are 0 Likelihood Ratio Test Lagrange Multiplier Test χ 2 with T(T-1)/2 d.f. R is correlation matrix 19

20 error correlation matrix - 4 equation example Illustration - Test on off-diagonal elements 20

21 Test on Coefficient Homogeneity H 0: coefficients are the same over time, either jointly or individually example 21

22 Test on Coefficient Homogeneity (2) special case of Chow test, ~ χ 2 (T-1) example 22

23 Spatial Lag SUR 23

24 Spatial SUR - LAG Specification different lag model/coefficient in each time period general temporal error correlation stacked equations 24

25 Estimation Strategies special case of S3SLS maximum likelihood estimation 25

26 Spatial Lag Spatial 3SLS special case with WX t as instruments for Wyt all standard results hold 26

27 SUR Lag 3SLS 27

28 SUR Lag 3SLS with endogenous variables 28

29 Maximum Likelihood Estimation from the full log-likelihood using to the concentrated log-likelihood 29

30 ML Estimation Strategy iterative approach generalize results from cross-section ML-Lag complex coefficient variance-covariance matrix 30

31 LM Test for Spatial Lag SUR apply general principle complex expression χ 2 with T degrees of freedom U stacked residual vectors requires information matrix 31

32 Spatial Error SUR 32

33 Spatial SUR - Error Specification different error model/coefficient in each time period cross-equation temporal correlation through remainder error term 33

34 Spatial SUR Error - Covariance covariance between t and s overall covariance 34

35 Estimation special case of FGLS estimation, or spatially weighted least squares using spatially filtered BX and By, and residuals Be 35

36 Estimation Strategies nuisance parameter perspective generalized moments estimator (GM) maximum likelihood estimation full likelihood specification 36

37 GM Estimator generalization of Kelejian-Prucha single-equation GM moment equations for residuals from each time period solve for λ and construct spatially filtered By, BX and spatially filtered residuals Be stack spatially filtered residuals as E (NxT) estimate Σ as (1/T)(E E) 37

38 GM Moment Equations ul and ull spatially lagged residuals 38

39 GM estimation Spatial Error SUR 39

40 Maximum Likelihood Estimation log-likelihood in spatially filtered residuals concentrated log-likelihood complex coefficient variance matrix 40

41 ML Estimation - Spatial Error SUR Model 41

42 LM Test for Spatial Error SUR apply general principle complex expression χ 2 with T degrees of freedom U stacked residual vectors T 1 = tr(ww), T2 = tr(w W) requires information matrix 42

43 SUR spatial diagnostics (LM tests) 43