to fit the model is not an important factor. In fact, in real applications, the overall survey sample is typically very large so any fitting method, as long as it provides consistent estimators, is expected to deliver estimates that will be very close to the corresponding true values.

The above conclusions support the use of the Census EB, using either REML, H3 or ELL estimation methods, instead of the original EB procedure in real-world applications, where it is seldom possible to link the survey and census data. Moreover, since the large sampling fractions of this experiment favor the original EB method, in the following we show only results for the Census EB instead of the original EB, where welfares are generated in the MC procedure for all the census units and the survey welfares are not appended.

Figure 1 compares, for the poverty rates, Census EB estimators obtained from the proposed MC procedure of Section 6 using H3 estimation method (labeled “H3-CensusEB”) with the analogues obtained using the CB-EB procedure of Section 4 (labeled “H3-CBEB (old bootstrap)”). We include also the results from the CB-EB method with selection of PSUs within the areas instead of selection of areas in step 1 (labeled “H3-CBEB-old with PSU”). Note that the CB-EB estimates presented in Section 4 differ considerably from the true poverty rates whereas the Census EB ones track approximately the true values. In fact, the sum of absolute differences of the CB-EB estimates to the true values is almost three times that of the Census EB estimates.

The main reason for this difference of behavior, even if both estimates are supposed to implement EB prediction is that, under the CB-EB approach in Section 4 with areas equal to clusters, a sample of areas is drawn in each simulation. This leads to a given area benefiting from EB prediction of the area effect only in a portion of the simulations, since it is likely that an area is not selected in every single bootstrap sample. This leads to CB-EB point estimates that are far off from the Census EB estimates obtained with the proposed MC procedure of Section 6. Another reason might be that the parameters used to simulate the census vectors in CB-EB approach are not kept fixed across simulations, making difficult the convergence to the conditional expectation defining the estimator. When using the CB-EB approach with PSUs, there is still a clear difference with the Census EB estimates obtained with the new bootstrap procedure of Section 6.

Turning now to the traditional ELL approach, Figure 2 shows ELL estimates for the poverty rates, based on the model with only _x_ 1 and _x_ 2 (labeled “ELL”) and the same, but including the area means of _x_ 1 and _x_ 2 in the model (labeled “ELL with context”), together with true poverty rates and Census EB estimates obtained using the original MC procedure of MR (2010), without appending the survey data. In this plot, one finds that the traditional ELL estimates face the same troubles as the CB-EB ones. This is in line with what MR (2010) find in their simulations. Given the present simulation set-up with a poor model fit, ELL performs much worse than MR’s Census EB method in terms of point estimates. Also note the rather flat nature of ELL, which is mostly due to the model providing very little information and, given the limited explanatory power of the correlates, the ELL estimates fall close to the average poverty rate of 16 for all the areas.<sup>22</sup> What is somewhat concerning is that the addition of contextual variables to the model for ELL, in the form of the area means, does little to improve the point estimates. Thus, even if the survey data are not preserved in the Census EB estimation method, the benefit of EB is quite considerable and caution should be taken when doing out of sample predictions in case of little explanatory power being brought in by the auxiliary variables.

The complete simulation experiment with _L_ = 10 _,_ 000 simulated populations are now discussed. Results in this case focus on the poverty gap, although the observed tendencies are the same for the poverty rates. Figures 3 and 4 show respectively the empirical bias and empirical MSE across simulated populations

> 22The ELL estimates are not really flat if sorted from smallest to largest.

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