4. With the simulated welfare vector _yc_<sup>_∗_,calculatetheindicatorofinterestas</sup><sup>_τ ∗_</sup> _c_<sup>=</sup><sup>_f_(</sup><sup>_y_</sup> _c_<sup>_∗_)suchasthe</sup> FGT indicator of order _α ≥_ 0, which is calculated as follows: 5. Repeat steps 1 to 3 _M_ times. Even if traditionally _M_ = 100, a larger number of replicates is recommended. Let _τc_<sup>_∗_(</sup><sup>_m_)</sup> be the indicator obtained in mth replicate, _m_ = 1 _, . . . , M_ . The CB-EB estimator _τ_ ˆ _c_<sup>_CB−EB_</sup> is then obtained as the average across the _M_ bootstrap replicates, and the variance of the estimator is approximated as Unlike the traditional ELL approach, this procedure requires to match the location codes in the survey and the census. However, in the application of the traditional ELL approach, the authors recommended the inclusion of contextual variables at the location level and thus the linking between the two data sources was still necessary.

As in the previous computational procedures, the final estimator of the poverty indicator for a location is obtained as an average across bootstrap replicates. By the Monte Carlo principle, this average is approximating an expectation. However, here it is not clear with respect to which DGP is the expectation taken, since the bootstrap procedure involves generation of several measures: In step 1, the sample of clusters varies, and this would define an expectation with respect to the sampling design, but considering only the first stage of the design. In step 2, model parameters are newly generated in each simulation. Finally, in step 3, location effects are generated from their conditional distribution given the sample residuals (determined by the nested error model under normality), and household-specific errors are generated from their distribution under the assumed nested error model with normality.

Even if the bootstrap procedure is not the same, standard errors are obtained similarly as in the traditional ELL approach and are thus different from MSEs. Note that the traditional ELL approach was initially based on the multiple imputation literature. According to Rubin (1996, and noted in StataCorp, 2019), the objective of multiple imputation is not to re-create the missing data as closely as possible, but to handle it in a manner that allows for statistical inference.

# **5 Comparison of CB-EB, traditional ELL and EB methods**

Because EB prediction uses the survey data to estimate the location effect, its benefits are evident only in locations present both in the census and the survey. In the traditional ELL approach, the location effect was originally at the cluster level and, in most surveys from developing countries where the World Bank focuses its efforts, the sampled clusters represent a small percentage of all the clusters in the country. This may also be the case for locations above clusters. For example, in a recently completed exercise in Moldova, the survey only contained 129 comunas out of a total of 901. Consequently, only a small share of these comunas benefit from EB prediction (Corral and Cojocaru, 2019). Nevertheless, differences also arise due to the computational procedures used to obtain the point estimates and their corresponding measures of noise.

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