of Census EB from Section 6 using H3 estimation method (labeled “H3-CensusEB”), traditional ELL estimators (labeled “ELL”), estimators based on the CB-EB approach with H3 method (labeled “H3CBEB (old bootstrap)”), the same with PSUs within areas selected in step 1 (labeled “H3-CBEB (old bootstrap with PSU)”) and direct estimators (labeled “Direct”). These full results are quite sobering. While the bias for the traditional ELL estimators is not remarkable, the bias of the estimators obtained by the CB-EB method from Section 4 is substantial, showing upward bias for all areas. In contrast, the bias of the Census-EB estimators obtained from Section 6 seems negligible, ranging between -0.025 and 0.027, similar to the bias range of the original EB estimators from MR (2010).<sup>23</sup> This figure shows a considerable bias reduction of the proposed MC procedure for obtaining the Census EB estimators with respect to the CB-EB method.

In terms of MSE, results are aligned to those in MR (2010), see Figure 4. The traditional ELL yields an empirical MSE above the one of the direct estimator, which uses no model. Even if ELL estimators do not show a large bias when averaging across all the simulated populations (the ELL bootstrap procedure is supposed to approximate the marginal mean _E_ ( _τc_ ; _θ_ ), which is exactly unbiased), as shown in Figure 2, it does not capture the area effect of the true poverty rates for a single population. Note that, under the model-based setup, welfares are generated from the model and in each simulation, new location effects are generated. This means that the true poverty rates vary for one population to another. In fact, a given area might have a positive location effect in one generated population and a negative effect in another. Since the location effects have an expected value of zero, the ELL method is tracking the average zero effect, which is the same for all the areas, but it does not capture the area effects in a given population. What is even more concerning is that the CB-EB method applied in Nguyen et al. (2018) and PovMap described in Section 4 yields an MSE that is more than 3 times greater that of the traditional ELL. This happens because a large amount of the MSE is the squared bias, which has been shown to be substantial for the CB-EB estimator. The proposed MC method for Census EB estimation presented here represents a massive improvement over the traditional ELL and the CB-EB approaches under the setup of this simulation experiment based on a poor model. The next section describes results when increasing the prediction power of the model.

# **7.2 Simulation experiment with improved model**

This experiment addresses one of the concerns of the previous simulation experiment: the poor explanatory power of the model. Hence, in this section a more informed model is considered, yielding an adjusted _R_<sup>2</sup> of about 0.42, which is much closer to that of real-world applications of ELL. The model includes six covariates. The first two, _x_ 1 and _x_ 2, are generated exactly as in the previous section. Four additional covariates are included, with values generated as follows:

1. _x_ 3 is a binary variable, taking value 1 when a random uniform number between 0 and 1 is less than or equal to 0 _._ 1 + 0 _._ 2 80<sup>_<u>c</u>_.</sup>

2. _x_ 4 is a binary variable, taking value 1 when a random uniform value between 0 and 1 is less than or equal to 0 _._ 5 + 0 _._ 3 80<sup>_<u>c</u>_.</sup> 3. _x_ 5 is a discrete variable, simulated as the rounded integer value of the maximum value between 1 and a random Poisson variable with mean _λ_ = 3 �1 _−_ 0 _._ 1 80<sup>_<u>c</u>_</sup> �.

4. _x_ 6 is a binary variable, taking value 1 when a random uniform value between 0 and 1 is less than or equal to 0.4. Note that the values of _x_ 6 are not related to the area’s label.

> 23Results for the Census EB estimators obtained from the original procedure of MR’s (2010) with REML estimation are not shown since they are aligned to the Census EB estimators from Section 6.

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