The welfare vector for each area is created from the model with these covariates as follows ln _ych_ = 3 + 0 _._ 09 _x_ 1 _,ch −_ 0 _._ 04 _x_ 2 _,ch −_ 0 _._ 09 _x_ 3 _,ch_ + 0 _._ 4 _x_ 4 _,ch −_ 0 _._ 25 _x_ 5 _,ch_ + 0 _._ 1 _x_ 6 _,ch_ + _ηc_ + _ech,_ _iid iid_ where _ech ∼ N_ �0 _,_ 0 _._ 5<sup>2�</sup> and _ηc ∼ N_ �0 _,_ 0 _._ 15<sup>2�</sup> . The poverty line in this scenario is fixed at _z_ = 10 _._ 2. All other steps are the same as in the previous simulation.<sup>24</sup> Even if the performance of CB-EB estimators improves when considering a better model, these are still clearly worse than the Census EB estimators obtained with the proposed MC approach of Section 6, see Figure 5. This conclusion remains true even after decreasing the location effect, by simulating the _iid_ populations with _ηc ∼ N_ �0 _,_ 0 _._ 07<sup>2�</sup> .<sup>25</sup> Similarly, despite the improved explanatory power of the model, Figure 6 still portrays a rather flat ELL hovering around the national average poverty rate. Under a single simulated population from the improved model, the traditional ELL method still performs poorly when compared with the Census EB according to MR (2010).

Results from the full simulation with _L_ = 10 _,_ 000 populations for the poverty gap also resemble those from the previous section, see Figures 7 and 8 showing empirical bias and empirical MSE respectively.<sup>26</sup> The CB-EB method implemented as described in Nguyen et al. (2018), which replicates the methods of PovMap, still presents considerable bias in this new simulation exercise. In comparison, the proposed Census EB procedure seems to be nearly unbiased, although the obtained empirical biases actually range between -0.04 and 0.05. Moreover, even under a better regression model, the MSE of the proposed Census EB estimator is still much smaller. What is concerning is that the MSEs of the direct estimators are smaller than those of the traditional ELL estimators and of the CB-EB estimators.

Another criticism of the EB approach of MR (2010), which extends to the Census EB, is its dependence on normality. In order to study the sensitivity of the procedures to the normality assumption, an additional simulation experiment is run under the same setup of this section, but generating the household-specific errors from a Student’s _t_ distribution with 5 degrees of freedom and scaled by 0.5. This distribution is symmetric but has heavier tails than the normal distribution, which may represent the existence of outlying welfares in the data, often encountered in real applications.

Empirical bias and MSE across the _L_ = 10 _,_ 000 simulations are shown in Figures 9 and 10 respectively, for the proposed Census EB, traditional ELL and direct estimators.<sup>27</sup> Note that the ELL method takes household-specific residuals from their empirical distribution, so its performance is not supposed to rely so strongly on normality. However, results suggest that, even under lack of normality, ELL’s performance is still worse than that of the Census EB update. In terms of bias, some areas under the traditional ELL show quite considerable bias, yet on average across areas the bias can be near zero. This suggests that, even if the bias across all areas of ELL can be small, it is not an accurate estimator in each area.<sup>28</sup> In fact, the empirical MSEs of the traditional ELL estimators for many areas are still much greater than those of the proposed Census EB ones in this experiment.

Even after the better performance of the Census EB procedure compared with the traditional ELL method under normality departure in the form of higher tails, model checking and diagnostics are extremely important, as in any model-based procedure. In fact, good performance of EB and Census EB

> 24Another slight modification made in separate simulations is that the location effect is simulated as _ηciid∼ N_<sup>�</sup> 0 _,_ 0 _._ 07<sup>2�</sup> .

> 25Results available upon request. Although it must be noted that under this scenario under many populations _σ_ ˆ _η_ <0, and thus many of the populations need to be re-simulated.

> 26For other FGT indicators see figures: 24, 25, 26, and 27.

> 27For other FGT indicators see figures: 28, 29, 30, and 31.

> 28Note that the purpose in SAE is not obtaining estimators performing well on average across areas since, in that case, the overall sample mean at the population level would be sufficient.

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