under the nested error model, MSE(ˆ _τc_<sup>_EB_</sup> ) = _E_ (ˆ _τc_<sup>_EB_</sup> _− τc_ )<sup>2</sup> . Under the model-based framework, the target indicators _τc_ are regarded as random quantities (they are generated from the model) and hence incorporate uncertainty as well. Even if an estimator (or predictor) _τ_ ˆ _c_ is unbiased with respect to the model in the sense that the expectation of its prediction error is zero _E_ (ˆ _τc − τc_ ) = 0, it would still leave the variance of the prediction error var(ˆ _τc −τc_ ), which is not exactly equal to the variance of the predictor var(ˆ _τc_ ) because of the randomness of the target indicator _τc_ .

The MSE of the EB estimator is estimated using a parametric bootstrap procedure for finite populations introduced by González-Manteiga et al. (2008), which is computationally more intensive than the ELL bootstrap approach. This is because, in every bootstrap replicate, all the steps for obtaining the EB estimator are reproduced, including model fitting and the MC procedure for approximation of the conditional expectation defining the EB estimator. This means that the number of simulated census vectors is the product of the MC simulations and the number of bootstrap replicates, except when the target indicator has an explicit EB estimator (such as the poverty rate). In this latter case, the MC simulation procedure can be replaced by an explicit formula calculated without simulation, and then the number of simulated censuses will be only the number of bootstrap replicates.

The steps for the parametric bootstrap estimation of the MSE are as follows (MR, 2015):

1. Using the survey data, fit model (1) using a method providing consistent estimators. This yields the set of parameter estimates from the observed sample:

In the implemented sae R package, the REML method is used.

2. Use the estimates in _θ_<sup>ˆ</sup> 0 to simulate census welfares<sup>11</sup> from the fitted model as follows: where the area effects _ηc_<sup>_∗_aregeneratedas</sup> and the household-specific errors are generated as 3. From the simulated census vector _yc_<sup>_∗_,wecalculatethetruevalueoftheindicatorofinterestfor</sup> each area _c_ : _τc_<sup>_∗_=</sup><sup>_f_(</sup><sup>_y_</sup> _c_<sup>_∗_)</sup><sup>_,_</sup> where _f_ ( _yc_<sup>_∗_)canbeanyindicatorfunctionsuchastheFGTindicators.</sup> 4. Since the survey is regarded as a subset of the census, the survey sample _sc_ is now extracted and, using the corresponding sample welfares _yc,s_<sup>_∗_(newlygeneratedinstep2),onefitsthemodel(1).</sup> This yields bootstrap estimates of the model parameters,

- ˆ ˆ _θ_<sup>_∗_</sup> = � _β_<sup>_∗_</sup> _,_ ˆ _ση_<sup>2</sup><sup>_∗,_ˆ</sup><sup>_σ_</sup> _e_<sup>2</sup><sup>_∗_</sup> � _._

5. With the bootstrap vectors of sample welfare _yc,s_<sup>_∗_,</sup><sup>_c_=1</sup><sup>_, . . . , C_,obtaintheEBestimators</sup><sup>_τ_ˆ</sup><sup>_EB_</sup> _c_<sup>_∗_</sup> , using MC simulation if needed. The estimation procedure is exactly the same EB procedure based on the original sample, but using the bootstrap sample data _yc,s_<sup>_∗_,</sup><sup>_c_= 1</sup><sup>_, . . . , C_and the corresponding</sup> bootstrap model parameter estimates _θ_<sup>ˆ</sup><sup>_∗_</sup> from step 4.

> 11Note that here welfares are generated for all households in the census, sampled and non-sampled.

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