2. The **model error** is related to the properties of the model parameters and is unrelated to the size of the target population (ELL, 2002). The magnitude of the error is dependent on the precision of the _β_ coefficients of the welfare model and the sensitivity of the indicator to deviations in welfare (ibid). Consequently, in order to minimize this source of error in the final estimates, it is recommended to remove all non-significant independent variables in the modeling stage.

3. The final source of error is the **computation error** , which is not related to the other two sources of error. This is related to the simulation and can be made as small as possible, as computational resources allow, by running a larger number of simulations.

Consider the case that model parameter estimates were kept fixed across simulations and equal to the initial survey estimates from step 1, _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>).In that case, consistency of these estimators as the</sup> total sample size _n_ tends to infinity,<sup>10</sup> and of Monte Carlo averages to the actual expectations under the model as the number of bootstrap replicates _M_ tends to infinity, would ensure that _τ_ ˆ _c_<sup>_ELL_</sup> approximates correctly the “theoretical” ELL estimator, _E_ ( _τc_ ; _θ_ ), which is unbiased by definition. However, there is an additional source of error due to generation of welfare in each simulation from a different set of parameter values in step 3, instead of using the initial estimates from step 1.

Moreover, if our target indicator is the mean welfare in location _c_ , _τc_ = _Nc_<sup>_−_1</sup> � _Nh_ =1 _d_<sup>_ych_,andthewelfare</sup> is not transformed with log, then _E_ ( _τc_ ; _θ_ ) = _E_ ( ̄ _xcβ_ + _ηc_ + ̄ _ec_ ) = _x_ ̄ _cβ_ , where _x_ ̄ _c_ = _Nc_<sup>_−_1</sup> � _Nh_ =1 _d_<sup>_xch_isthe</sup> area mean of the correlates and _e_ ̄ _c_ = _Nc_<sup>_−_1</sup> � _Nh_ =1 _d_<sup>_ech_istheareameanofthehousehold-levelerrors.This</sup> means that the area effect _ηc_ vanishes in the final ELL estimator, which becomes basically the regression ̄ estimator _xcβ_ . This estimator is called “synthetic” in the SAE literature because it does not account for between-area heterogeneity (or idiosyncrasy of the areas) beyond that explained by the correlates. Note that, if the regression parameter _β_ was actually known (best case), _x_ ̄ _cβ_ would not be using at all the actual survey data, only the census auxiliary information. Synthetic estimators rely very strongly on the assumed regression model and might be misleading if the model is incorrectly specified. In fact, standard errors are obtained assuming that the model actually holds and do not account for model misspecification. Hence, under model failure, they might still be small, giving a lot of credibility to the (misleading) estimates. As will be shown, even if the model is correctly specified, the MSE of the ELL estimators can be substantial, especially if the prediction power of the model is weak.

# **3 Molina and Rao’s EB and Census EB estimators**

Molina and Rao (2010) consider the nested error model (1) as in the ELL procedure, but the location effects are originally defined for the areas of interest. A more recent paper by Guadarrama, Molina and Rao (2018) further extends the EB method of MR (2010) to complex sampling designs, but so far it has not been implemented in any software package as a library or command, unlike the traditional ELL (2002) with the subsequent updates by Van der Weide (2014). For simplicity, this section describes the original EB approach proposed by MR (2010).

The main difference with the ELL approach is that the EB method of MR (2010) conditions on the survey sample data and thus makes a more efficient use of the survey data, which contains the only available (and hence precious) information on actual welfare. Nevertheless, conditioning on the survey data requires areas and households across survey and census to be matched. Thus, if _Pc_ denotes the area’s population of size _Nc_ , then the survey sample, _sc_ , is a sample of size _nc ≤ Nc_ drawn from _Pc_ . The complement to the sampled population is referred to as _rc_ , which is just _Pc − sc_ . Thus, the census

> 10Note that the total survey sample size _n_ is typically large.

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