MR (2010) presented evidence that their EB approach is superior to ELL in terms of a smaller MSE. This finding has met some criticism, mostly because the simulation experiment in MR (2010) was conducted under scenarios where ELL is seldom applied. First, the population size in MR (2010) was 20,000 uniformly spread across 80 areas. Second, all the areas were sampled and the sample within each area represented 20 percent of the population, something hardly seen in real-world applications. ELL (2002) illustrates how the noise of the estimator falls as the size of the population increases, and advises against estimating below populations of 100 households. In practice, the level of disaggregation of ELL estimates depends on the model quality, and even with an _R_<sup>2</sup> not smaller than 0.6, it is seldom advised to go below 1,000 households (Elbers et al., 2007).

Another point of departure is that the model considered in the simulation experiments of MR (2010) was really poor, yielding an adjusted _R_<sup>2</sup> of less than 0.01. Under the usual applications of the ELL approach, the number of auxiliary variables is much greater, reaching most of the times an explanatory power, as measured by adjusted _R_<sup>2</sup> , of over 0.5. Moreover, to improve precision, ELL (2002; 2003) also advocated for the use of contextual area-level variables<sup>17</sup> on the right-hand side, since these explain (and thus minimize) the variation across locations. These were not used in the simulation experiment conducted in MR (2010). In such a setup, the much noisier ELL estimates obtained in their simulation experiments are not surprising.

A final important difference is the fact that the error measures of estimators and the bootstrap procedures for obtaining them are considerably different in the three procedures. Currently, the error measures of the traditional ELL and of the more recent CB-EB procedure of Van der Weide (2014) are obtained under a very similar framework to that of multiple imputation. As an error measure, these consider the standard deviation of all the simulated point estimates of the indicators in the _M_ bootstrap replicates, and so every simulated indicator is compared to the average of the simulated indicators. Moreover, in the bootstrap approaches used for that average of the simulated indicators, the initial model parameter estimates _β_<sup>ˆ</sup> 0, _σ_ ˆ _η_<sup>2</sup> 0<sup>and</sup><sup>_σ_ˆ</sup> _e_<sup>2</sup> 0<sup>obtainedusingtheoriginalsurveysamplearenotthoseusedtogeneratethe</sup> welfare vectors in the population. In fact, in each bootstrap replicate, model parameters � _β, ση_<sup>2</sup><sup>_, σ_</sup> _e_<sup>2</sup> � are simulated from the estimated distribution of the initial estimators (or an approximation to it). In contrast, in MR’s (2010) parametric bootstrap procedure, a simple MC simulation procedure is used to approximate the actual MSE under the nested error model with the average across bootstrap simulations of the squared prediction errors. In this bootstrap procedure, the parameter estimates used to generate the bootstrap censuses of welfare are kept fixed to the original survey estimates. Consistency of these initial survey estimators to their corresponding true values as the total sample size _n_ tends to infinity ensures that the bootstrap MSE approximates the true MSE (under the model with the true values of the parameters).

# **6 Extended Census EB estimators**

The EB estimator in MR (2010) requires linking the survey and census households, which may not be possible in real applications. In fact, the survey sample is likely not a subset of the available census. Hence, here we consider the Census EB (CEB) estimator, which avoids this step. In fact, in most cases, the number of sample households for a given area is much smaller than the number of census households and, in such cases, the CEB estimator is expected to perform very much like the original EB.

CEB small area estimators are obtained with a similar MC simulation approach as the one used by MR (2010) and described in Section 3, but simulating the vectors of welfare for all the census households

> 17These may come from the census or administrative records (ELL, 2003 p356).

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