Spatial Regression. 10. Specification Tests (2) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved

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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 LMError test LMLag test with with 4
5 What are the Robust Tests? Problem: both LMError and LMLag have power against the other alternative LMError rejects null in presence of lag model LMLag 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 LMError Robust LMLag 6
7 Use of Robust Tests only use when BOTH LMError and LMLag reject the null do NOT use when neither LMError nor LMLag are significant select model with most significant statistic = specification search 7
8 Robust LMLag = 3.0 Robust LMError = 5.1 only Robust LMError 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 NonStandard Result test is not the sum of two onedirectional tests Test = LMError + Robust LMLag = LMLag + Robust LMError 11
12 > > < < = =
13 Interpretation of SARMA Test caution! SARMA test will be significant when either LM Error or LMLag 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 LMError like expression using 2SLS residuals 18
19 Specification Search 19
20 Principle 20
21 Two Strategies (Florax, Folmer, Rey 2003) forward stepwise strategy move from simple to complex model backward stepwise strategy move from complex to simple model (Hendry) 21
22 Forward StepWise Strategy start from constrained model = nonspatial model (OLS estimation) use LM test statistics to guide model selection problem: pretesting, due to multiple tests the pvalues become suspect corrections for pretesting are complex 22
23 Backward StepWise 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 LMError only significant: spatial error model LMLag only significant: spatial lag model both significant: proceed to robust LM tests 24
25 Specification Search  Step 2 Robust LMError significant, Robust LMLag is not: spatial error model Robust LMLag significant, Robust LMError 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 centerperiphery 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 LMError significant at p = 0.06 Conclusion: Spatial Error Model 33
34 why is Moran s I so significant? nonnormality? 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 > <<< > LMLag significant at p=0.01 LMError 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 nonnormality? heteroskedasticity? 41
42 Case 5: Both LMError and LMLag significant, Robust LMError 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 LMError and LMLag significant, Robust LMLag Significant south.shp with south_q.gal (rook contiguity) HR70 on RD70, PS70, UE70, DV70, MA70 45
46 regression results 46
47 >>> >>> <<< < 47