and Chambers, 2017). The recommended approach to mitigate this issue is to include covariates that sufficiently explain the between-area heterogeneity in the model ( _ibid_ ). In this regard, ELL (2002) suggests the inclusion of cluster-level covariates as a way to explain location effects. Nevertheless, this approach is context specific and may not always suffice to ameliorate the issues with between-area heterogeneity. In this regard, Marhuenda et al. (2017) recommend and show that location effects should be at the same aggregation level at which estimation is desired.

If the location effect is specified at the same level where estimation is desired, then the difference between ELL (2003) and MR (2010) reduces to differences in how estimates are obtained and the addition of Empirical Best (EB) prediction by MR (2010). The EB method from MR (2010) conditions on the survey sample data and thus makes more efficient use of the information at hand, whereas ELL does not include this component. In essence, under ELL, for any given area present in the sample the ELL estimator of the census area mean _y_ ̄ _c_ is obtained by averaging across _M_ simulated censuses and is given by _y_ ̄ _c_<sup>_∗_(</sup><sup>_m_)</sup> ≈ _X_<sup> ̄</sup> _c_<sup>_′β_+</sup><sup>_η_</sup> _c_<sup>_∗_(</sup><sup>_m_)</sup> + ̄ _e_<sup>_∗_</sup> _c_<sup>(</sup><sup>_m_)</sup> , _m_ = 1 _, . . . , M_ , where _E_ [ _ηc_<sup>_∗_]=0</sup> <u>1</u> _M_ and _E_ [ _e_<sup>_∗_</sup> _ch_<sup>]=0.</sup> Thus, the ELL estimator _M_ � _m_ =1<sup>_y_ ̄</sup> _c_<sup>_∗_(</sup><sup>_m_)</sup> , which approximates _E_ ( ̄ _yc_ ), reduces to the regression synthetic estimator, _X_<sup> ̄</sup> _c_<sup>_′β_(MR,2010).Ontheotherhand,underMR(2010),</sup> conditioning on the survey sample ensures the estimator includes the random location effect, since _E_ [ ̄ _yc|ηc_ ] ≈ _X_<sup> ̄</sup> _c_<sup>_′β_+</sup><sup>_ηc_.Mechanically,however,conditioningonsurveydatarequiresthelinkingof</sup> areas across surveys and census, something that is not always straightforward.<sup>2</sup> Other differences between ELL (2003) and MR (2010) are the computational algorithms used to obtain point and noise estimates; see Corral, Molina and Nguyen (2020 - CMN henceforth) for further discussion.

ELL’s (2003) approach to obtain estimates builds upon the multiple imputation (MI) literature in that it uses a single algorithm that produces point and noise estimates by varying model parameters across simulations (see Tarozzi and Deaton, 2009 as well as CMN, 2020). The use of MI methods for obtaining point and noise estimates has shortcomings, however. Under multiple imputation, the method that yields the lowest MSE does not necessarily yield valid statistical inference (Van Buuren, 2018), which is contradictory to the goal of small area estimation in improving precision. Accordingly, CMN (2020) show using simulated populations that the noise estimate (referred to as variance by ELL) is not an appropriate estimate of the MSE. In contrast, MR’s (2010) point estimates are obtained through two separate procedures. Point estimates are obtained by a Monte Carlo simulation and noise estimates are obtained by a parametric bootstrap procedure originally proposed by González-Manteiga et al. (2008).<sup>3</sup> In light of the emerging literature, including MR (2010), the World Bank expanded the original ELL design by adding EB predictors and revising its fitting methodology to incorporate heteroskedasticity and survey weights to account for complex sampling designs (Van der Weide, 2014). The survey weights are incorporated in the estimates of the regression coefficients and in the variance of the components following Huang and Hidiroglou (2003) based on Henderson method III (Henderson,

> 2Under ELL, when including area level covariates, linking the survey and the census areas is also required. Note that it is not necessarily the case that enumeration areas for a census and survey will match.

> 3For more details, see Corral, Molina and Nguyen (2020).

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