welfare vector for any area _c_ is defined as _yc_ = ( _yc,s_<sup>_T, y_</sup> _c,r_<sup>_T_)</sup><sup>_T_,whichcontainsthewelfareforsampled</sup> and non-sampled observations in area _c_ , denoted _yc,s_ and _yc,r_ respectively. Typically, the non-sampled population (of size _Nc − nc_ ) is much larger than the sampled population (of size _nc_ ). It may also happen that _nc_ = 0 for some areas, meaning that the area is not sampled in the survey. The empirical best (EB) predictor of _τc_ for an area _c_ that is sampled ( _nc >_ 0) is defined as the conditional expectation _E_ ( _τc|yc,s_ ; _θ_<sup>ˆ</sup> ). For an out-of-sample area ( _nc_ = 0), the empirical best (EB) predictor of _τc_ is _E_ ( _τc_ ; _θ_<sup>ˆ</sup> ), which is similar to the ELL estimator. Here, _θ_<sup>ˆ</sup> is a consistent estimator of _θ_ as the overall sample size _n_ tends to infinity. In contrast with this procedure, the traditional ELL approach does not require to link the census and survey observations.

When the expectation _E_ ( _τc|yc,s_ ; _θ_<sup>ˆ</sup> ) has no explicit form, the empirical best (EB) predictor of _τc_ can be approximated using a Monte Carlo (MC) simulation method. The MC approximation to the EB ˆ predictor _τc_<sup>_EB_</sup> = _E_ ( _τc|yc,s_ ; _θ_<sup>ˆ</sup> ), as described in Rao and Molina (2015), is obtained as follows:

1. Using the survey data, fit model (1) via any method providing consistent estimators. This yields the vector of parameter estimates:

Usual fitting methods under this approach are maximum likelihood (ML) or restricted maximum likelihood (REML), both based on the normal likelihood, and H3 method, which does not require to specify a distribution. In the implemented sae R package, REML is used.

2. Use the parameter estimates obtained in step 1 as true values to simulate a vector of welfare in the census. First, the welfare from each out-of-sample household is generated as follows, ln( _ych,r_<sup>_∗_) =</sup><sup>_xch,rβ_ˆ0+</sup><sup>_η_</sup> _c_<sup>_∗_+</sup><sup>_e∗_</sup> _ch,r_<sup>_,_</sup> where here we add the subscript _r_ to indicate that the household is not sampled. Here, for an area _c_ that is included in the sample, the area effect _ηc_<sup>_∗_isgeneratedas</sup> where _η_ ˆ _c_ 0 and _γ_ ˆ _c_ are respectively given by Here, _y_ ̄ _c,s_ and _x_ ̄ _c,s_ are respectively the sample means of the welfare variable and the correlates in area _c_ . If area _c_ is not sampled, the area effect is generated as _ηc_<sup>_∗∼N_</sup> �0 _,_ ˆ _ση_<sup>2</sup> 0�. Finally, the household error _e_<sup>_∗_</sup> _ch,r_<sup>isgeneratedas:</sup> For an area _c_ that is sampled in the survey, the generated non-sample vector _yc,r_<sup>_∗_is then augmented</sup> by the survey data _yc,s_ . Therefore, the final vector for the whole census in area _c_ is _yc_<sup>_∗_= (</sup><sup>_y_</sup> _c,s_<sup>_T, y_</sup> _c,r_<sup>_∗T_)</sup><sup>_T_,</sup> which is made up of the generated out-of-sample welfares and the survey ones.

3. The previous step yields a census vector of welfare _yc_<sup>_∗_, simulated using the fitted model parameters.</sup> This census vector is then used to calculate the indicator for each area _c_ = 1 _, . . . , C_ , as where _f_ ( _·_ ) can be any indicator function such as the FGT poverty indicators.

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