independent from each other, following Here, _C_ is the number of locations in which the population is divided and _Nc_ is the number of households in location _c_ , for _c_ = 1 _, . . . , C_ . Finally, _β_ is the _K ×_ 1 vector of coefficients.

Under the original ELL methodology, the locations indexed with _c_ are supposed to be the clusters, or primary sampling units (PSUs), of the sampling design and do not necessarily correspond to the level at which the estimates need to be produced. In fact, clusters are typically nested within the areas of interest (e.g. census tracks). Presenting estimates at a higher aggregation level than the clusters (for which random effects are included in the model) may not be appropriate in cases of considerable betweenarea variability, and may underestimate the estimator’s standard errors (Das and Chambers, 2017). A way to alleviate this is to include covariates that explain sufficiently well the between-area heterogeneity in the model ( _ibid_ ). In this regard, ELL (2002) suggests the inclusion of area-level covariates as a way to improve precision, because they can explain the between-area variation in welfare and hence might reduce the contribution of area-specific residuals. Marhuenda et al. (2017) recommend putting the location effects at the same aggregation level where estimation is desired. According to this, in this paper we consider that the clusters _c_ are equal to the areas where estimation is desired.

A key feature of the implemented ELL approach is that it considers heteroscedasticity and fits a preliminary model for that, called the alpha model. Very little research has gone into this aspect of the method. In most applications, the alpha model usually has a small adjusted _R_<sup>2</sup> (which may not reach 0.05) yet tends to play a considerable role in the obtained point estimates, especially for parameters beyond the poverty rate such as the Gini index, Theil index, Poverty Gap, etc. Heteroscedasticity is included in the description of the EB approach given in Rao and Molina (2015), but this is not implemented in the sae R package (Molina and Marhuenda, 2015). For simplicity, below we spell the steps of the traditional ELL approach only in the case of homoscedasticity.

The nested error model in (1) was originally fitted by Generalized Least Squares (GLS) accounting for heteroscedasticity, but Van der Weide (2014) updated this original GLS method to properly account for survey weights.<sup>6</sup> Currently, the implemented method fits the model by Feasible Generalized Least Squares (FGLS). In this procedure, a simple linear regression obtained by considering as model errors _uch_ = _ηh_ + _ech_ is fit using ordinary least squares (OLS), and afterwards the appropriate covariance matrix of regression parameter estimators is estimated.<sup>7</sup> The first stage of the process being an OLS is an important aspect, because most of the model testing and validation done for the ELL approach is in practice undertaken with the OLS fit. Nevertheless, the parameter estimates used for generating welfare in the census (see the steps below) come from the GLS fit accounting for heteroscedasticity and including the survey weights.

The variances of the model parameter estimators are required in the considered ELL bootstrap approach. To estimate these variances, ELL (2002) proposed to use the delta method. The actual implementation of the delta method is through a computationally intensive simulated numerical gradient. An estimate of the gradient vector is obtained by making perturbations of the parameters, and this is used to estimate the variances of the estimators via the delta method. ELL (2002) mentions also the possibility of drawing from the sampling distribution (parametric) as a way to incorporate the estimation error into the total prediction error. This later method has become the de facto approach used under the ELL methodology.<sup>8</sup>

> 6Haslett et al. (2010) presents the problems in the original GLS implementation of ELL.

> 7For a detailed look into the ELL approach, interested readers should refer to the original ELL papers (ELL, 2003; 2002) and Section 3 of Nguyen et al. (2018), which presents the current GLS estimator from Van der Weide (2014)

> 8In a comparison of the simulation methods proposed by ELL (2002), Demombynes et al. (2008) shows that the delta

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