As mentioned before, for sampled clusters, EB prediction makes use of the estimated cluster effects from the survey data to improve the point estimates of the poverty indicators and their errors. A crucial assumption under EB prediction is that model errors _ech_ and _ηc_ are normally distributed. This yields ˆ normality for _ηc_ given _ec,s_ , which leads to the proposed predicted cluster effect _ηc_ .

This EB addition implemented in the PovMap software uses an extended version of Henderson’s method III (Henderson, 1953) to obtain estimators _σ_ ˆ _η_<sup>2and</sup><sup>_σ_ˆ</sup> _e_<sup>2ofthevariancecomponents</sup><sup>_σ_</sup> _η_<sup>2and</sup><sup>_σ_</sup> _e_<sup>2thatin-</sup> corporate the sampling weights.<sup>12</sup> In step 2 of section 2 for the ELL, _ση_<sup>2</sup><sup>_∗_</sup> is drawn from a _Gamma_ distribution, however to avoid this assumption and because var [ˆ _η_ c] is unkonwn,<sup>13</sup> it is necessary to rely on bootstrap re-sampling to obtain the noise of the point estimates in similar fashion to ELL.<sup>14</sup> Even if the estimator is based on the EB method of MR (2010), the bootstrap procedure used to generate the census data is completely different. This new bootstrap algorithm tries to avoid the distributional assumptions by generating in the first step samples of clusters from the survey data set. The full procedure for obtaining the estimator of _τc_ , called here clustered bootstrap EB (CB-EB), is as follows:

1. Take a bootstrap sample of clusters (PSUs) with replacement from the survey data.<sup>15</sup>

2. Fit model (1) to the bootstrapped survey data from step 1. This yields the set of parameter estimates from the bootstrap sampled data ( _β_<sup>ˆ</sup><sup>_∗_</sup> _,_ ˆ _ση_<sup>2</sup><sup>_∗,_ˆ</sup><sup>_σ_</sup> _e_<sup>2</sup><sup>_∗_),alongwithpredictedeffectsforthe</sup> clusters that were sampled in step 1, _η_ ˆ _c_<sup>_∗_,andtheircorrespondingestimatedvariance</sup> var<sup>�</sup> _∗_ [ˆ _ηc_<sup>_∗_].In</sup> the implemented version, the model is fit by FGLS as specified in Nguyen et al. (2018).

3. Use the model parameter estimates from step 2 to simulate welfares _ych_<sup>_∗_inthecensusas16</sup> where, for a cluster _c_ that is sampled in step 1, the cluster effect is generated as where _η_ ˆ _c_<sup>_∗_and</sup> var<sup>�</sup> _∗_ [ˆ _ηc_<sup>_∗_]arethebootstrapversionsof</sup><sup>_η_ˆ</sup><sup>_c_anditscorrespondingvariance,givenin(2)</sup> and (3) respectively. For a cluster that is not sampled in step 1, generate Under homoscedasticity, the household specific errors are generated as and, in the case of heteroscedasticity, as Construct the vector _yc_<sup>_∗_= (</sup><sup>_yc_1</sup><sup>_, . . . , ycN_</sup> _c_<sup>)</sup><sup>_T_of simulated welfare for every household within location</sup> _c_ of size _Nc_ , for all the locations _c_ = 1 _, . . . , C_ in the census.

> 12Interested readers should refer to Van der Weide (2014) and/or Nguyen et al. (2018) for an in-depth look at how these are obtained under ELL and Henderson’s method III.

> 13In the traditional ELL method of Section 2, _σ_ ˆ _η_ 2<sup>isassumedtofollowaGammadistribution,whichdoesnotholdin</sup> this case.

> 14Actually, Van der Weide (2014) does not offer a method for the estimation of the standard errors, but the approach described below was the one implemented on PovMap and consequently also in the Stata sae package from Nguyen et al. (2018).

> 15Note that not all clusters are expected to appear

> 16Note that, in each bootstrap replicate, the census is generated from a different model.

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