for location _c_ ; for example, for the Foster, Greer and Thorbecke (1984 - FGT) class of decomposable poverty measures, the function _f_ ( _yc_<sup>_∗_)isdefinedas</sup> where _z_ is the poverty line, _pch_ is the size of household _h_ from location _c_ in the census and _I_ ( _ych_<sup>_∗< z_) = 1if</sup><sup>_y_</sup> _ch_<sup>_∗< z_andisequalto0otherwise.</sup> 5. Repeat steps 2 to 4 _M_ times. Although traditionally _M_ = 100, a larger number of replicates _M_ is recommended. Let _τc_<sup>_∗_(</sup><sup>_m_)</sup> be the indicator of interest obtained in mth replicate of the bootstrap. The ELL estimator is then given by the average across the _M_ bootstrap replicates, and the ELL estimated variance of the ELL estimator is given by As opposed to the EB method of MR (2010), which uses a Monte Carlo simulation procedure and a separate bootstrap procedure for MSE (see Section 3), in the above ELL procedure, a single computational algorithm tries to capture the noise of the initial model parameter estimates in the ELL standard error by varying the model parameters across simulations, which is aligned to Rubin’s rules (Rubin, 2004). This means that, in each replicate of the bootstrap, different values of the model parameters _β_<sup>_∗_</sup> , _ση_<sup>2</sup><sup>_∗_and</sup> _σe_<sup>2</sup><sup>_∗_</sup> are used to generate the welfare data. However, this step might entail an increase of noise in the ELL estimators.

Note that, if the model parameters were kept fixed to the initial survey estimates from step 1 (skipping step 2), the above ELL estimate _τ_ ˆ _c_<sup>_ELL_</sup> would be a basic Monte Carlo approximation to the marginal mean of the indicator, _E_ ( _τc_ ; _θ_<sup>ˆ</sup> 0), with respect to the distribution of the welfare (given the correlates) induced by the nested error model (1), with _θ_ replaced by the initial estimate _θ_<sup>ˆ</sup> 0. Note that _E_ ( _τc_ ; _θ_ ), evaluated at the true _θ_ = ( _β, ση_<sup>2</sup><sup>_, σ_</sup> _e_<sup>2),isunbiasedbydefinitionsinceitequalstheexpectationofthe</sup> indicator under the true model.<sup>9</sup> According to this, the prediction error of the ELL estimator can be decomposed as described in ELL (2003), where these sources of error are described as follows:

1. The **idiosyncratic error** is related to how the actual value of the expenditure for a given location, _c_ , deviates from its expected value due to unobserved aspects in the expenditure for the location (ELL, 2002). For locations with smaller population sizes, the underlying distribution is not likely approximated when errors are drawn. This, however, is related to the explanatory power of the independent variables in the model, the smaller the unexplained portion of the model, the smaller the idiosyncratic error. With poorly fitting models, estimates for locations with smaller populations are likely to suffer from more variability across simulations. Therefore, in order to minimize this error, one of the goals of the modeling stage is to obtain the highest possible _R_<sup>2</sup> .

> 9Note that under a model-based approach, the true value of the indicator _τc_ is random. Under this setup, an estimator/predictor _τ_ ˆ _c_ of _τc_ is said to be unbiased when _E_ (ˆ _τc − τc_ ) = 0 or, accordingly, when _E_ (ˆ _τc_ ) = _E_ ( _τc_ ).

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