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1 Spatial Regression 10. Specification Tests (2) Luc Anselin 1

2 robust LM tests higher order tests 2SLS residuals specification search 2

3 Robust LM Tests 3

4 Recap and Notation LM-Error test LM-Lag test with with 4

5 What are the Robust Tests? Problem: both LM-Error and LM-Lag have power against the other alternative LM-Error rejects null in presence of lag model LM-Lag rejects null in presence of error model robust forms of the test make an asymptotic adjustment to correct for this (Anselin et al 1996) 5

6 Robust LM Tests Robust LM-Error Robust LM-Lag 6

7 Use of Robust Tests only use when BOTH LM-Error and LM-Lag reject the null do NOT use when neither LM-Error nor LM-Lag are significant select model with most significant statistic = specification search 7

8 Robust LM-Lag = 3.0 Robust LM-Error = 5.1 only Robust LM-Error rejects the null hypothesis 8

9 Higher Order Tests 9

10 Test on SARMA test on higher order alternative, BOTH lag and error dependence H0: λ = ρ = 0 Test χ2 (2) two degrees of freedom (one for lag, one for error) 10

11 Non-Standard Result test is not the sum of two one-directional tests Test = LM-Error + Robust LM-Lag = LM-Lag + Robust LM-Error 11

12 > > < < = =

13 Interpretation of SARMA Test caution! SARMA test will be significant when either LM- Error or LM-Lag is highly significant does NOT mean the alternative is a higher order model higher order only makes sense as alternative when evidence of other form of misspecification remains in lower order model (e.g., remaining error autocorrelation in a spatial lag model) 13

14 2SLS Residuals 14

15 Generalized Moran s I Test extend to residuals other than OLS residuals from a 2SLS regression no longer a maximum likelihood framework requires explicit CLT 15

16 Moran s I for 2SLS Residuals Anselin and Kelejian (1997) apply Moran s I requires generalized expression for variance φ2 16

17 Moran s I for 2SLS Residuals (2) expression for I is standard, using 2SLS residuals expression for variance is complex 17

18 Special Case - No Spatial Lag standard endogenous variable case test statistic simplifies to LM-Error like expression using 2SLS residuals 18

19 Specification Search 19

20 Principle 20

21 Two Strategies (Florax, Folmer, Rey 2003) forward step-wise strategy move from simple to complex model backward step-wise strategy move from complex to simple model (Hendry) 21

22 Forward Step-Wise Strategy start from constrained model = non-spatial model (OLS estimation) use LM test statistics to guide model selection problem: pre-testing, due to multiple tests the p-values become suspect corrections for pre-testing are complex 22

23 Backward Step-Wise Strategy start from unconstrained model and test constraints (Hendry approach) proceed from complex to simpler specification problem: requires estimation of complex spatial models first to test parameter constraints (using Wald or LR tests) 23

24 Specification Search - Step 1 use LM tests first (NOT the robust version) none significant: proceed with OLS LM-Error only significant: spatial error model LM-Lag only significant: spatial lag model both significant: proceed to robust LM tests 24

25 Specification Search - Step 2 Robust LM-Error significant, Robust LM-Lag is not: spatial error model Robust LM-Lag significant, Robust LM-Error is not: spatial lag model both Robust LM tests are significant: alternative is the most significant (largest value) possibility of higher order model or alternative specifications 25

26 26

27 Examples 27

28 Case 1: No Evidence of Spatial Autocorrelation Columbus crime example with center-periphery indicator variable (= spatial heterogeneity) columbus.shp with columbus_rk.gal CRIME on INC, HOVAL, CP 28

29 regression results 29

30 < none of the LM statistics are significant but Moran s I is somewhat, why? < possibly power against heteroskedasticity 30

31 Case 2: Clear Indication of One Type of Spatial Autocorrelation - Spatial Error Columbus crime example with DISCBD as one of the explanatory variables columbus.shp with columbus_d.gwt (distancebased weights) CRIME on INC, HOVAL, DISCBD 31

32 regression results 32

33 <<< < Moran s I significant at p = 0.01 LM-Error significant at p = 0.06 Conclusion: Spatial Error Model 33

34 why is Moran s I so significant? non-normality? heteroskedasticity? 34

35 Case 3: Clear Indication of One Type of Spatial Autocorrelation - Spatial Lag Columbus crime example with PLUMB as one of the explanatory variables columbus.shp with columbus_d.gwt (distancebased weights) CRIME on INC, HOVAL, PLUMB 35

36 regression results 36

37 > <<< > LM-Lag significant at p=0.01 LM-Error not significant Conclusion: Spatial Lag Model 37

38 Case 4: LM Tests Not Significant, but one of the Robust LM Tests is Significant columbus.shp with columbus_rk.gal (rook contiguity) CRIME on INC, HOVAL, NSB, DISCBD 38

39 regression results 39

40 < < Moran s I weakly significant LM Tests NOT significant Robust LM Error somewhat significant = IGNORE 40

41 why is Moran s I weakly significant non-normality? heteroskedasticity? 41

42 Case 5: Both LM-Error and LM-Lag significant, Robust LM-Error Significant south.shp with south_rk.gal (rook contiguity) HR90 on RD90, PS90, UE90, DV90, MA90 42

43 regression results 43

44 >>> >>> < <<< 44

45 Case 6: Both LM-Error and LM-Lag significant, Robust LM-Lag Significant south.shp with south_q.gal (rook contiguity) HR70 on RD70, PS70, UE70, DV70, MA70 45

46 regression results 46

47 >>> >>> <<< < 47