census for all households. Then a unit-context model and a new unit model are selected, once again following the process described in section 4.2 using the first out of 500 samples where 1% SRS by PSU is selected. This is done to remove the issue of outliers from the data and to ensure that the data generating process follows the one assumed in Eq. (2). The simulation removes the potential misspecification due to deviations from normality in the data and allows us to isolate the problem present in the unit-context model ( _UC − CEBa_ ).

Results for the new hybrid simulations are presented in Figures 22 and 23. Note that in this simulation, where we have removed the normality issue, the upward bias that was present in the unit level model ( _CEBa_ ) is no longer evident. On the other hand, the previously suspected downward bias of the unit context models ( _UC − CEBa_ ) is salient, as can be seen in Figure 22 and by municipality deciles sorted by population in Figure 24. Note that under the _UC − CEBa_ model more than 75% of the municipalities present a downward bias (Fig. 22). This finding is aligned to what we observed in Figure 17. However, because there is no deviation from normality in the hybrid simulation, the downward bias of the unit-context models ( _UC − CEBa_ ) is never offset, and is quite considerable and leading to substantially larger empirical MSEs for the unit context models (Fig. 25). Simulations were repeated, where, instead of performing model selection, the selected model for CEB estimators contains exactly the same covariates as those used to generate the welfare, and considering only the area aggregates for U-C models. This was done just to check whether the observed biases could be due to model misspecification, in the sense that the selected covariates are different from those in the true model. Results were very similar to those observed in the previous hybrid simulation with a model selection step. Hence, the results suggest deviations from the assumed model are an issue and the countering of biases is what is driving the seemingly good results for unit-context models in the two-stage sampling scenario, highlighting the importance of proper data transformations and model selection to ensure that model assumptions hold when using Census EB methods.

Given the direction of the bias of unit-context models is not known _a priori_<sup>50</sup> - and that these might present high bias - unit-context models are unlikely to be preferred over traditional FH methods when the census auxiliary data are not aligned to survey microdata, unless the calculation of variances is not possible for various locations as noted before.<sup>51</sup> In this case benchmarking is not a recommended procedure and may not help in reducing the bias. EB estimators are approximately model unbiased and optimal in terms of minimizing the MSE for a given area, thus when adjusted afterwards for benchmarking, that is, so that these match usual estimates at higher aggregation levels, the optimal properties are lost and estimators usually become worse in terms of bias and MSE under the model. When benchmarking adjustments are large, as those likely required by _UC_ variants, it is an indication that the model does not really hold for the data. In the case of _UC_ models we have shown that the model will not hold due to omitted variable bias.

Also, note bias can lead to considerable re-ranking of locations and thus a limit on the acceptable

> 50Notice how under the simulations presented in Figures 1 and 3 the method appears to be upward biased.

> 51This holds for other measures of welfare, and particularly for ELL variants of the unit-context models.

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