4. Repeat steps 1 to 3 a large number of times _M_ . If _yc_<sup>_∗_(</sup><sup>_m_)</sup> denotes the _mth_ replicate of the census vector _yc_<sup>_∗_= (</sup><sup>_y_</sup> _c,s_<sup>_T, y_</sup> _c,r_<sup>_∗T_)</sup><sup>_T_,then</sup><sup>_τ ∗_</sup> _c_<sup>(</sup><sup>_m_)</sup> = _f_ ( _yc_<sup>_∗_(</sup><sup>_m_)</sup> ) is the corresponding indicator, _m_ = 1 _, . . . , M_ . The final MC approximation to the EB estimator of _τc_ is just the average across the _M_ simulations of these indicators Molina (2019) comments that the effect of adding the survey data is negligible when the sample is small relative to the census population of the area. Thus, a slight variation of the EB estimator, called Census EB (see e.g. Molina 2019), avoids appending the survey data in step 2 by generating all the census welfares (including the sample ones), similarly as the non-sampled ones are generated. Thus, the welfare for each household _h_ in the census is generated from the model with _ηc_<sup>_∗_and</sup><sup>_e∗_</sup> _ch_<sup>obtained exactly the same as in step 2.The result is the full census vector for area</sup><sup>_c_given</sup> by _yc_<sup>_∗_= (</sup><sup>_y_</sup> _c_<sup>_∗_</sup> 1<sup>_, . . . , y_</sup> _c,N_<sup>_∗_</sup> _c_<sup>)</sup><sup>_T_,whichisusedtocomputethetargetindicator</sup><sup>_τ ∗_</sup> _c_<sup>=</sup><sup>_f_(</sup><sup>_y_</sup> _c_<sup>_∗_).Unfortunately,the</sup> Census EB estimator is not implemented in the sae R package. One possible approach for applying this software when survey units cannot be identified in the census would be to append the survey data to the available census data, which would lead to a different approximation of the EB estimator. A drawback of this approach is that, in this case, the total size of the area is then _Nc_ + _nc ≥ Nc_ .

The EB estimator _τ_ ˆ _c_<sup>_EB_</sup> approximates the so called “best” predictor of _τc_ , which is the optimal predictor in the sense of minimizing the MSE under the model. This MSE of an estimator _τ_ ˆ _c_ is defined as the expectation of the squared prediction error under the considered model, MSE(ˆ _τc_ ) = _E_ (ˆ _τc − τc_ )<sup>2</sup> . ˆ The best predictor is given by the conditional expectation _τc_<sup>_B_</sup> = _E_ ( _τc|yc,s_ ; _θ_ ), where the expectation is with respect to the conditional distribution of the non-sampled data _yc,r_ given the sample data _yc,s_ , determined by the nested error model. Under normality of the random terms in the model, the conditional distribution is also normal. In step 2 of the above procedure, welfare values for non-sampled households are generated from that conditional distribution, but with estimated model parameters. Hence, the above ˆ MC simulation procedure actually approximates the EB estimator _τc_<sup>_EB_</sup> = _E_ ( _τc|yc,s_ ; _θ_<sup>ˆ</sup> ), which is the best predictor with model parameters replaced by the corresponding estimates based on the original sample survey data _θ_<sup>ˆ</sup> 0 = ( _β_<sup>ˆ</sup> 0 _,_ ˆ _ση_<sup>2</sup> 0<sup>_,_ˆ</sup><sup>_σ_</sup> _e_<sup>2</sup> 0<sup>).Notethat,unliketheELLbootstrapprocedure,heretheseestimates</sup> are kept constant across simulations because there is a unique DGP. Hence, this simulation procedure delivers a simple MC approximation for the EB predictor ˆ _τc_<sup>_EB_</sup> = _E_ ( _τc|yc,s_ ; _θ_<sup>ˆ</sup> ) of _τc_ , where the expectation is replaced by the corresponding MC average.

If the target parameter is just the area mean _τc_ = _Nc_<sup>_−_1</sup> � _Nh_ =1 _d_<sup>_ych_,andthereisnolog-transformation</sup> of welfare, then the best predictor reduces to _E_ ( _τc|yc,s_ ; _θ_ ) = _x_ ̄ _cβ_ + _γc_ ( ̄ _yc,s − x_ ̄ _c,sβ_ ), which corrects the regression synthetic estimator _x_ ̄ _cβ_ by accounting for the area effect using the survey data. Moreover, the size of the correction depends on _γc_ , which measures the share of between-area heterogeneity _ση_<sup>2from the</sup> total variance in the area, _ση_<sup>2+</sup><sup>_σ_</sup> _e_<sup>2</sup><sup>_/nc_.Thus, the correction is stronger in the case of highly heterogeneous</sup> areas and it is weak otherwise, becoming null only when all the existing area heterogeneity is completely explained by the available auxiliary variables. In that case, the EB estimator reduces to the “theoretical” ELL estimator _E_ ( _τc_ ; _θ_ ) = _x_ ̄ _cβ_ . Hence, the EB procedure makes more efficient use of the survey data for areas that are sampled and, for those areas, it does not rely so strongly on the assumed linear regression (which could be misspecified).

Concerning error measures of the estimators, the ELL and EB approaches consider different error measures. In the EB approach of MR (2010), the noise is measured by the actual MSE of the EB estimator 8