One additional simulation is executed with the population of _N_ = 100 _,_ 000, but the sample size is kept as _nc_ = 10 households per area, which makes the total sample size _n_ = 800 . Each area is made up of _Nc_ = 1 _,_ 250 households, which means that sampling fractions are now 0.8 percent. The purpose of the simulation is to assess the performance of the estimators under a scenario more aligned to ELL. ELL (2002) proposed the location effect be at the cluster level, and in some applications this is a cluster of 10 observations. However, in this case the total survey sample size may not be so realistic, since in household surveys from many countries, the overall sample size is of several thousands. As expected, under this scenario, the differences in bias of the different methods is less stark. Note that with zero area sample size ( _nc_ = 0), Census EB and traditional ELL become approximately the same, so, as we approach this scenario, estimators come together (Figure 15). Because of the limited sample, all methods now show considerably more bias and a higher MSE. Concurrently, H3-CensusEB now appears to be more aligned to the ELL and H3-CBEB. However, H3-Census EB still outperforms the older methods in terms of bias and MSE. Note also, that usually ELL is done by setting the location effect at the cluster level but results are presented at higher levels.<sup>32</sup> However, aggregation to higher levels may underestimate the true MSE as noted by Das and Chambers (2017). The inclusion of contextual level variables may help ( _ibid_ ) although this requires being able to link clusters in the survey sample to the census which in practice is not always feasible.

Aggregate results across areas are presented in Table 1 for the simulations discussed.<sup>33</sup> Results from column 1 are those that replicate the simulation experiment of Molina and Rao (2010). Note that the large values of the average relative root mean squared error (ARRMSE) may arise from true values of FGT2 close to 0, but the true values are common to all estimators. Regardless, the better performance of the H3-Census EB update is quite clear in terms of all the performance measures and under all the simulations run. Moreover, the results in the tables corroborate what is observed in the figures.

# **7.3 Examining the bias of the CB-EB PovMap update**

The bias of the CB-EB estimator observed in our simulation experiments is a reason for concern. In the CB-EB method with clusters equal to the areas, one could argue that the bias comes from the fact that not all areas are equally likely to be included in the sample that is selected in step 1 of this procedure. However, in our simulation experiments, bias also appears for the CB-EB method with PSUs selected within the areas in step 1. Thus, the question remains: what is the source of this bias?

Investigating whether part of the bias comes from the regression coefficients used to draw the welfares _β_ ˆ<sup>_∗_</sup> , which vary across the bootstrap replicates, we have plotted a histogram of the simulated regression intercepts _β_<sup>ˆ</sup><sup>_∗_</sup> obtained in the _M_ replicates of the CB-EB bootstrap procedure, with _M_ = 50 _,_ 100 _,_ 300, for a single population. Figure 17 shows the results for _M_ = 50 and _M_ = 300 bootstrap replicates. This figure shows that the distribution of the simulated regression intercepts _β_ 1<sup>_∗_forsmall</sup><sup>_M_isnotactually</sup> centered around the true value _β_ 1 = 3; in fact, it is right-skewed with a mean clearly exceeding the true value. While increasing the bootstrap replicates _M_ yields better results (right plot), the regression intercepts are still not centered around the true intercept.

A second plausible source of bias is through the simulated location effects, _ηc_<sup>_∗_.Sincethemethodselects</sup> PSUs with replacement, some areas might be selected more than once and others might not be selected. As a consequence, var [ˆ� _ηc_ ] might be underestimated in an area selected more than once because there will artificially be more observations in that area. On the other hand, if an area is not selected, then

> 32For other FGT indicators see figures: 36, 37, 38, and 39.

> 33See tables 2 and 3 for FGT0 and FGT2, respectively.

31