Section 11.7 Beyond the SAR and SEM models
Given the "galaxy" of spatial models, newer approaches to select the one to run have been proposed. [204] propose a top-down approach that involved starting with the spatial Durbin model (SDM) and based on likelihood ratio tests to select the best model. [203], on the other hand, proposes to start with the OLS model, run the Lagrange multipliers, and then run a likelihood ratio test to see if the spatial Durbin model would be more appropriate. Why the Durbin model? Why not Manski or the SAR model? [203] corroborates that the Manski model can be estimated, but the parameters cannot be interpreted in a meaningful way (the endogenous and exogenous effects cannot be distinguished from each other) and the SAR or Kelejian/Prucha model "will suffer from omitted variables bias if the true data-generation process is a spatial Durbin or a spatial Durbin error model" and that similarly the same happens for the spatial Durbin error model (p. 14).
If you paid attention, the summary of the lag model we ran in the previous section included at the bottom a test for remaining residual spatial autocorrelation. This means our model is still not filtering all the spatial dependency that is present here. Perhaps here, and following the workflow proposed by [203], we need to consider if the spatial Durbin model (SDM) would be more appropriate. We can run this model and then contrast it with the SAR model.
fit4_70_sdm <- lagsarlm(eq2_70_S, data=ncovr_s_sf, rwm_s, Durbin = TRUE,
method ="LU")
summary(fit4_70_sdm)
## ## Call: ## lagsarlm(formula = eq2_70_S, data = ncovr_s_sf, listw = rwm_s, ## Durbin = TRUE, method = "LU") ## ## Residuals: ## Min 1Q Median 3Q Max ## -18.09108 -4.13069 -0.88963 2.84179 67.03240 ## ## Type: mixed ## Coefficients: (asymptotic standard errors) ## Estimate Std. Error z value Pr(>|z|) ## (Intercept) 9.996665 2.080609 4.8047 1.550e-06 ## RD70 2.266521 0.350632 6.4641 1.019e-10 ## PS70 0.297629 0.312080 0.9537 0.3402394 ## MA70 -0.066899 0.055596 -1.2033 0.2288603 ## DV70 0.819995 0.244796 3.3497 0.0008090 ## UE70 -0.185133 0.129563 -1.4289 0.1530326 ## lag.RD70 0.262121 0.515226 0.5087 0.6109276 ## lag.PS70 1.071509 0.518284 2.0674 0.0386949 ## lag.MA70 -0.023245 0.088356 -0.2631 0.7924840 ## lag.DV70 -0.414450 0.429808 -0.9643 0.3349117 ## lag.UE70 -0.737893 0.202560 -3.6428 0.0002697 ## ## Rho: 0.415, LR test value: 79.91, p-value: < 2.22e-16 ## Asymptotic standard error: 0.04542 ## z-value: 9.138, p-value: < 2.22e-16 ## Wald statistic: 83.5, p-value: < 2.22e-16 ## ## Log likelihood: -4759 for mixed model ## ML residual variance (sigma squared): 48.66, (sigma: 6.976) ## Number of observations: 1412 ## Number of parameters estimated: 13 ## AIC: 9544, (AIC for lm: 9622) ## LM test for residual autocorrelation ## test value: 13.98, p-value: 0.00018426
We can then compare the nested models using information criteria with
AIC() and BIC(). Using this information criteria, we choose the model with the lowest AIC and BIC.
output_1 <- AIC(fit4_70, fit4_70_sar, fit4_70_sdm)
output_2 <- BIC(fit4_70, fit4_70_sar, fit4_70_sdm)
table_1 <- cbind(output_1, output_2)
table_1
## df AIC df BIC ## fit4_70 7 9668 7 9704 ## fit4_70_sar 8 9555 8 9597 ## fit4_70_sdm 13 9544 13 9613
Our results indicate that AIC favours the SDM model, whereas the BIC favours the simpler SAR model. We can also run a likelihood ratio test to compare the two spatial nested models using
spatialreg::LR.Sarlm() (that deprecates spdep::LR.sarlm() you may still see mentioned in other textbooks). Here we look for the model with the highest log likelihood:
LR.Sarlm(fit4_70_sdm, fit4_70_sar)
## ## Likelihood ratio for spatial linear models ## ## data: ## Likelihood ratio = 21, df = 5, p-value = 0.001 ## sample estimates: ## Log likelihood of fit4_70_sdm ## -4759 ## Log likelihood of fit4_70_sar ## -4769
The LR test is significant, suggesting we should favour the spatial Durbin model. If it had been insignificant, we would have had to choose the more parsimonious model.
As with the spatial lag model, here we need to be careful when interpreting the regression coefficients. We also need to estimate the "impacts" in a similar manner we illustrated above.
fit4_70_sdm.im <- impacts(fit4_70_sdm, tr=trMC, R=100)
summary(fit4_70_sdm.im, zstats=TRUE, short=TRUE)
## Impact measures (mixed, trace): ## Direct Indirect Total ## RD70 2.3255 1.99733 4.3228 ## PS70 0.3562 1.98440 2.3406 ## MA70 -0.0694 -0.08471 -0.1541 ## DV70 0.8164 -0.12307 0.6933 ## UE70 -0.2251 -1.35289 -1.5779 ## ======================================================== ## Simulation results ( variance matrix): ## ======================================================== ## Simulated standard errors ## Direct Indirect Total ## RD70 0.29855 0.6440 0.6312 ## PS70 0.28645 0.7654 0.7383 ## MA70 0.05711 0.1296 0.1137 ## DV70 0.26395 0.6772 0.6877 ## UE70 0.12024 0.2851 0.2694 ## ## Simulated z-values: ## Direct Indirect Total ## RD70 7.681 3.1783 6.875 ## PS70 1.243 2.6410 3.220 ## MA70 -1.267 -0.6544 -1.382 ## DV70 3.269 -0.2063 1.052 ## UE70 -1.856 -4.8483 -5.961 ## ## Simulated p-values: ## Direct Indirect Total ## RD70 1.6e-14 0.0015 6.2e-12 ## PS70 0.2140 0.0083 0.0013 ## MA70 0.2053 0.5129 0.1671 ## DV70 0.0011 0.8366 0.2930 ## UE70 0.0634 1.2e-06 2.5e-09
Notice how in this model only three inputs remain significant, resource deprivation, population structure, and unemployment. The sign of the coefficients remains the same for all the inputs. In most circumstances we see that when the direct effect is positive the indirect effect is also positive, and when the direct effect is negative the indirect effect is negative. Only for the insignificant effect of divorce we see a positive direct effect and a negative indirect effect.
As suggested by [196], we can also examine the proportion of the direct effect in the total effect:
direct <- fit4_70_sdm.im$res$direct
indirect <- fit4_70_sdm.im$res$indirect
total <- fit4_70_sdm.im$res$total
direct/total
## RD70 PS70 MA70 DV70 UE70 ## 0.5380 0.1522 0.4503 1.1775 0.1426
The proportion of the (significant) direct effect ranges from 14% (for unemployment), fairly marginal, to 54% (for resource deprivation), about half of it. Another way of looking at it is by relation of the direct to the indirect effect by looking at its ratio in absolute values:
abs(direct)/abs(indirect)
## RD70 PS70 MA70 DV70 UE70 ## 1.1643 0.1795 0.8193 6.6332 0.1663
When the ratio is greater than 1, this suggests that the direct effect prevails, as we see for resource deprivation. With a score lower than 1, this indicates that the spillover effect is greater than that of internalisation as we see for population structure and unemployment.
