a is a mixed effect specified at the area level, assumed to be distributed normally. Although traditionally ELL has specified the area effect at the cluster or enumeration area level, doing so underestimates the ηη mean squared error by failing to account for correlation in model error across clusters within areas (Marhuendra et al, 2018, Das and Chambers 2018).<sup>14</sup> a is conditioned on the sample values of observed welfare, denoted ys. ηη is a household level idiosyncratic error term, also assumed to be distributed normally. A key feature of the EBP framework is that the random area effect εεii a is conditioned on the sample values of observed welfare ys. This, along with the assumption that σ<sup>2</sup> is normal, allows for the use of ηη the following model for generating simulated welfare: ηη ∗ ∗ (4) 1 2 3 + η� (5) η� ii ssaa] aa rr aa aa ii GG(YY ) = ββ XX + ββ XX + ββ XX + ηη + εε ∗ 2 (6) aa = EE[ηηaa|yyss )� , where aa aa ηη ~NN�0, σ ηη(1 −γγ σ<sup>2</sup> <u>η</u> (7) = . σ<sup>2</sup> ⁄ γa is the shrinkage factor due to conditioning the random effect on the sample mean for area a. Na is the shrinkage factor due to conditioning the random effect on the sample mean for area a. N is the shrinkage factor due to conditioning the random effect on the sample mean for area a. Na is is γγaa ηη+σ<sup>2</sup> εεNNaa a is is

γa is the shrinkage factor due to conditioning the random effect on the sample mean for area a. Na is the shrinkage factor due to conditioning the random effect on the sample mean for area a. N is the shrinkage factor due to conditioning the random effect on the sample mean for area a. Na is is the number of sample observations for area a. The shrinkage in variance due to conditioning the random effect on the sample will be higher when Na is large or when the variance of the residual ε is low, because in these cases the sample contributes more information about the unexplained component of welfare. Shrinkage falls as the estimated within-area correlation in welfare σ<sup>2</sup> , declines, because in those cases a larger share of unobserved welfare is idiosyncratic to households rather than common to areas, so the sample residuals convey less information about the true area mean. Similarly, ηη the shrinkage factor γ declines as the area sample size Na declines, because the sample contributes less information about the welfare in the area.

This basic structure is appealing in part because it is similar to the Elbers, Lanjouw, and Lanjouw (2003) model that has been used extensively for survey-census poverty mapping for decades, with the main difference being the conditioning of the random effect on the mean sample residuals. The concept of shrinkage in the nested error EBP framework is well-established in the small area estimation context (Battese, Harter, and Fuller, 1988, Jiang and Lahiri, 2006, Molina and Rao, 2010, Van der Weide, 2014). In addition, unlike Bayesian spatial autocorrelation models, nested error random effect models do not impose additional smoothness in model errors across neighboring areas. Spatial correlation is instead modeled entirely through the area random effects, which are conditioned on the sample residuals for that area.

As noted above, it is not clear that Empirical Best methods are preferred in the traditional poverty mapping setting where a household census serves as the auxiliary data. While EB methods benefit from incorporating information from the sample, the sample is typically a small share of the census 14 Omitting the cluster effect also can lead to bias by simultaneously overestimating the variance of epsilon and the variance of eta, but the resulting biases act in opposite directions and therefore partially cancel each other out.

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