population, leading to limited benefits from using EB. Meanwhile, an important drawback of EB is the required assumptions that the components of the error term are distributed normally, which can reduce the accuracy of the estimates if the underlying distribution of the residuals is not normal. The choice of whether to use EB in a typical case, when the auxiliary data are a household census, must weigh the cost of assuming normality against the benefit of obtaining additional information from a relatively small sample.

The calculus changes, however, when the auxiliary data are specified at an aggregate geographic level, such as a subarea or area. In this case, EB estimation is essential to effectively combine the survey data with census-based predictions. Intuitively, the auxiliary data contribute less variation when they only vary across sub-areas, compared to the sample, because the sample, unlike the auxiliary data, contains household level information. In the EBP framework described above, the use of predictors aggregated to the sub-area level increases the variance of the area effect σ<sup>2</sup> , which also represents the covariance of unexplained welfare across households within the same area. This covariance rises when using sub-area predictors because all households within each sub-area share the same set of ηη characteristics used to predict welfare in the model. The increase in σ<sup>2</sup> due to the use of regionally aggregated predictors in turn increases the shrinkage factor γ, leading to a greater increase in precision when using an EBP model versus the traditional ELL estimator. ηη The assumption of normality is critical in the EBP framework, as a distributional assumption is required to condition the area level random effect on the average residual of the regression in that area. In the EBP model, this conditioning is equivalent in a Bayesian framework to specifying the average of the residuals as a prior distribution for the random effect under the assumption of normality. The normality assumption is also necessary for the Monte-Carlo simulations of the parameters, which are required because headcount poverty is a non-linear function of welfare. In general, software packages generate small area poverty estimates by repeatedly drawing area effects and the household idiosyncratic error from a normal distribution and aggregating the simulation results. This results in biased estimates of poverty rates if the true error terms implied by the model are not in fact normal. Below, we discuss a monotonic transformation to the welfare measure that makes the assumptions that the area effect and the household error term are distributed normally more palatable.

# D.  Software packages to estimate EBP models

Table 4 summarizes the three software packages that we utilize to estimate household level models. The primary set of results reported below are estimated using a modified version of the R EMDI package (Kreutzmann et al, 2018).<sup>15</sup> We use this package for the main results because it is well-documented, flexible, and fast when running on powerful computers that can exploit parallel processing. In addition, it appends the sample to the census in order to generate slightly more accurate estimates. This package, like others designed for poverty mapping, estimates the mixed model (3) and then repeatedly draws both area effects and household error terms to simulate welfare. Simulated welfare is compared to the

> 15 The R EMDI package is available on CRAN. It is an extension of the R SAE package, and runs substantially faster because it allows the user to parallelize the bootstrap simulations across several cores.

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