1953) as well as in the predicted area effects as in the pseudo empirical best linear unbiased predictor (EBLUP) proposed by You and Rao (2002). In the absence of survey weights and heteroskedasticity, the fitting method yields parameter estimates that are quite similar to the restricted maximum likelihood (REML) fitting method used by MR (2010). The difference remains, however, in how point and noise estimates of the target indicators are obtained.

As further detailed in CMN (2020), however, even following these revisions, the bootstrap procedure, as implemented by `PovMap` software (Zhao, 2006) for EB and later detailed in Nguyen et al. (2018) differed from the original EB procedure from MR (2010). CMN (2020) note the remaining issues rested in the continued reliance on MI methods as the basis of the methodology. Because under the updated fitting methodology, Van der Weide (2014) does not offer an estimate of var � _ση_<sup>2</sup> � bootstrap samples of the data must be taken for each simulation to obtain the fitting parameters � _β_ ˆ _∗,_ ˆ _ση_ 2 _∗_<sup>_,_ˆ</sup><sup>_σ_</sup> _e_<sup>2</sup><sup>_∗_</sup> �.<sup>4</sup> This approach was taken to allow for an algorithm similar to the procedure used in the original implementation of ELL. However, if clusters were equal to areas, then it is unlikely that all areas are included in the sample and thus an area could benefit from EB only in a subset of the simulations, introducing bias into the resulting point and noise estimates. CMN (2020) also present evidence on the fact that, even if the location effect was modeled at the domain level and bootstrap samples of clusters are taken, some bias would still likely remain in the resulting estimates of the original implementation of EB in `PovMap` and the `sae` Stata package.

The Stata package for small area estimation developed by Nguyen et al. (2018) has been updated to integrate the fitting methods from Van der Weide (2014) with the prediction and bootstrap methods from MR (2010). The new method is referred to as the H3-CensusEB (Molina, 2019), and the previous method (i.e. original bootstrap using EB from Van der Weide (2014)) is called H3-CBEB.<sup>5</sup> CMN (2020) performs a model-based validation of the different methods: i) CensusEB, ii) EB, iii) H3-CBEB and iv) ELL. CMN (2020) extend the simulations done by MR (2010) by i) including the area means of the covariates as additional variables in the model; ii) considering a model that has considerably higher explanatory power by adding more covariates; iii) considering larger population sizes and smaller sampling fractions; and iv) generating errors from a Student’s _t_ 5 instead of a normal distribution. However, in all these simulations, population data are generated under model (1).

Currently, the software available for small area estimation of non-linear parameters, such as the `sae` Stata package by Nguyen et al. (2018) as well as the R package `sae` by Molina and Marhuenda (2015), only allow for estimation under the nested error model specified in (1). However, since household surveys often use two stage sampling, it seems appropriate to consider a twofold nested error model. Marhuenda et al. (2017) extend the EB method from MR (2010) to a twofold nested error model, given by:<sup>6</sup>

> 4ELL (2002) provides the derivation of the estimate for var<sup>�</sup> _ση_<sup>2</sup> � in their appendix.

> 5The method is called CensusEB because it does not link survey and census households. The old method is called the clustered bootstrap EB (CBEB). For the updated Stata package, see https://github.com/pcorralrodas/SAEStata-Package.

> 6For simplicity, we omit the heteroskedasticity weights considered by Marhuenda et al. (2017).

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