From the created “census” in each of the areas, 20 percent of the observations are sampled using simple random sampling without replacement;<sup>18</sup> this yields our “survey” data. Thus, in this experiment, survey weights are all equal. This generation process is repeated _L_ = 10 _,_ 000 times. In each simulation, the following quantities are computed for the poverty rates and gaps in each area:

1. True poverty indicators _τc_ , using the “census”.

2. Direct estimators _τ_ ˆ _c_<sup>_DIR_</sup> using the “survey”, defined as the sample versions of _τc_ .

3. Original EB estimators ˆ _τc_<sup>_EB_</sup> by MR (2010) with REML fitting as implemented in the sae R package, with _M_ = 50 MC replicates.

4. Traditional ELL estimators _τ_ ˆ _c_<sup>_ELL_</sup> and their estimated standard errors, without location means and also including the location means of the considered auxiliary variables, where _M_ = 50.

5. Census EB (CEB) estimators ˆ _τc_<sup>_CEB_</sup> , obtained as in the MC procedure of MR (2010), but generating full census vectors from the conditional distribution and without appending the survey welfares, using REML fitting and with _M_ = 50.

6. Extended CEB estimators using the MC and parametric bootstrap methods described in Section 6, using either H3 or ELL fitting methods with _M_ = 50. Since in this simulation experiment model errors are homoscedastic and sampling weights are constant, here the extended CEB estimators only differ in the above CEB ones in the model fitting method.

7. Estimators based on the CB-EB approach from Section 4, _τ_ ˆ _c_<sup>_CB−EB_</sup> using either H3 or ELL fitting methods.<sup>19</sup> Since clusters are originally supposed to be nested within the areas and the clustered bootstrap method is supposed to sample clusters (PSUs) instead of areas in step 1, we computed additional CB-EB estimators, where every area is divided into 25 PSUs (with 10 households each) and selecting 5 PSUs within every area (instead of areas). Under this scenario _M_ = 50 as well.

Model bias and MSE are approximated empirically as in MR (2010), as the averages across the _L_ = 10 _,_ 000 simulations of the prediction errors _τ_ ˆ _c_<sup>_j−τc_andofthesquaredpredictionerrorsrespectively,where</sup><sup>_j_</sup> stands for one of the methods: DIR, ELL, CEB or CB-EB.<sup>20</sup> Before getting into the results of the full simulation experiment, the methods are initially compared using just a single generated population along with its corresponding sample. For this one population, the different methods are run by setting _M_ = 1000 to get point estimates and _B_ = 1000 for the MSE.<sup>21</sup> For expediency, results that do not conform to the main message of the paper are relegated to the appendix. However, from the figures in the appendix, we can draw the following conclusions: 1) The differences in the EB and Census EB point estimates are negligible even if the area sampling fractions _fc_ = _nc/Nc_ are not so small in this experiment (20 percent), see Figure 19. Although differences appear in their estimated MSEs (see Figure 20), these differences become closer to zero as the sampling fractions _fc_ decrease. 2) Census EB estimates using H3 method as implemented in the sae Stata package are practically identical to the Census EB ones with REML fitting (Figure 21). Similarly, when comparing the Census EB estimators obtained with the proposed MC procedure using either H3 or ELL estimation methods, we can see that they are also aligned (Figure 22). This indicates that the actual method used

> 18The same units are sampled in every simulation and the values of _x_ 1 and _x_ 2 for all the census units are also kept fixed; this means that the _x_ 1 and _x_ 2 values for the sample units are always the same across simulations.

> 19For a detailed description of the difference between the methods readers should refer to Nguyen et al. (2018).

> 20 _E_<sup>�</sup> _τ_ ˆ _c_<sup>_j−τ_</sup> _c_ � for the bias, and _E_<sup>�</sup> _τ_ ˆ _c_<sup>_j−τ_</sup> _c_ �2 for the MSE.

> 21For ELL from section 2 and H3 CBEB from section 4 since the process relies on a single computational algorithm _M_ = 1 _,_ 000

17