This model is referred to as the sampling model, and is assumed to have heteroscedastic error variance var( _ωi|yi_ ) = _σω,i_<sup>2.Thesevariancesareassumedtobeknownalthoughinrealitythese</sup> are estimated using the survey data. The best linear unbiased predictor (BLUP) is obtained by predicting _yi_ through the model, _y_ ˆ _i_<sup>_d_=</sup><sup>_xi_ˆ</sup><sup>_β_+</sup><sup>_ε_ˆ</sup><sup>_i_.The</sup><sup>_β_ˆaretheweightedleastsquaresestimator</sup> ˆ ˆ of _β_ under the Fay-Herriot model and, _εi_ = _ψi yi_<sup>_d−xi_ˆ</sup><sup>_β_</sup> is also the BLUP of the area effect _εi._ � � Additionally, _ψi_ = _σu_<sup>2</sup><sup>_/_</sup> � _σu_<sup>2+</sup><sup>_σ_</sup> _ω,i_<sup>2</sup> �. In essence, the final estimates for areas contained in the survey are the weighted average between the direct estimate _y_ ˆ _i_<sup>_d_and the synthetic estimator</sup><sup>_xi_ˆ</sup><sup>_β_, where the</sup> weights, _ψi_ and (1 _− ψi_ ), are determined by the quality of the model and the quality of the direct estimator. For the non-sampled entities, the area-level model simply uses the synthetic estimates to produce predictions. As Molina et al. (2022) notes, normality assumptions are not required for the estimates to be BLUP, but assumptions are needed for estimation of the mean squared error (MSE). Finally, similar to the machine-learning approach, the area-level model can be implemented at the PSU or municipality levels.<sup>15</sup> Regarding unit-level models, the well-known methods of Elbers et al. (2003) and Molina and Rao (2010) are based on the nested error model originally proposed by Battese et al. (1988). This model assumes that (transformed) household income relates linearly to a vector of household-level covariates according the following model:

where _wiv_ denotes the transformed equivalized income of household _v_ in location _i_ , _xiv_ represents a vector of household-specific characteristics, _δ_ is a vector of coefficients on the household characteristics, _μi_ is location-specific random effect such that _μi ∼ N_ (0 _, σμ_<sup>2),and</sup><sup>_ξiv_isahousehold-level</sup> error term such that _ξiv ∼ N_ (0 _, σξ_<sup>2).Thetwoerrorsareassumedtobeindependentfromeach</sup> other.<sup>16</sup> We use a particular version known as the “census empirical best” method (CensusEB), which is a variant of the “empirical best” method of Molina and Rao (2010).<sup>17</sup> It is evident that the dependent variable of the model is not the poverty status of a given household. Instead, the method attempts to replicate the welfare distribution based on an assumed data generating process. The parameter estimates obtained when fitting model (15) to the survey data ( _δ,_<sup>ˆ</sup> ˆ _σμ_<sup>2,and</sup><sup>_σ_ˆ</sup> _ξ_<sup>2)areappliedtothecensushouseholdleveldatatogenerateasimulatedvalueof</sup><sup>_wiv_</sup> for each household in the census. From the simulated census welfare vector it is possible to obtain estimates of any welfare indicator, not just poverty headcount rates.

The parameter estimates are fit through a suitable method such as REML or Henderson’s Method

> 15Estimates at the PSU level may be aggregated to a higher level, see Rao and Molina (2015) for more details. 16Elbers et al. (2003) offer the option of accommodating for non-normally distributed errors – however, despite this flexibility Corral et al. (2021b) illustrate how even under non-normality the method lags Molina and Rao’s (2010) EB approach.

> 17As opposed to empirical best estimates, census empirical best estimates do not include survey observations when calculating area-level poverty rates. Corral et al. (2021b) show that when the sample size is small relative to the population the difference between the two methods is negligible. For additional discussion, see Molina (2019).

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