Spatial Regression 10. Specification Tests (2) Luc Anselin http://spatial.uchicago.edu 1
robust LM tests higher order tests 2SLS residuals specification search 2
Robust LM Tests 3
Recap and Notation LM-Error test LM-Lag test with with 4
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
Robust LM Tests Robust LM-Error Robust LM-Lag 6
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
Robust LM-Lag = 3.0 Robust LM-Error = 5.1 only Robust LM-Error rejects the null hypothesis 8
Higher Order Tests 9
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
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
> > < < 56.43 = 51.36 + 5.07 56.43 = 53.45 + 2.98 12
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
2SLS Residuals 14
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
Moran s I for 2SLS Residuals Anselin and Kelejian (1997) apply Moran s I requires generalized expression for variance φ2 16
Moran s I for 2SLS Residuals (2) expression for I is standard, using 2SLS residuals expression for variance is complex 17
Special Case - No Spatial Lag standard endogenous variable case test statistic simplifies to LM-Error like expression using 2SLS residuals 18
Specification Search 19
Principle 20
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
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
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
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
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
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Examples 27
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
regression results 29
< none of the LM statistics are significant but Moran s I is somewhat, why? < possibly power against heteroskedasticity 30
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
regression results 32
<<< < Moran s I significant at p = 0.01 LM-Error significant at p = 0.06 Conclusion: Spatial Error Model 33
why is Moran s I so significant? non-normality? heteroskedasticity? 34
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
regression results 36
> <<< > LM-Lag significant at p=0.01 LM-Error not significant Conclusion: Spatial Lag Model 37
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
regression results 39
< < Moran s I weakly significant LM Tests NOT significant Robust LM Error somewhat significant = IGNORE 40
why is Moran s I weakly significant non-normality? heteroskedasticity? 41
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
regression results 43
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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
regression results 46
>>> >>> <<< < 47