and these parametric assumptions are used for the generation of simulated census data sets (see Section 2.3), they also have to hold. Note that these assumptions may be valid even if dated census data are used for predicting survey consumption values. Thus, one crucial part is the diagnosis of the estimated error components from the first-stage regression. If plots or statistical tests on the estimated cluster effects and residuals suggest violations of distributional assumptions, one should adjust the model accordingly. More specifically, heteroscedasticity, serial correlation, and non-normality can be detected and accounted for, for instance by choosing different predictor specifications, transforming the dependent variable, or explicit modeling of heteroscedasticity as discussed in Section 2.1. The variance of the model error also depends fully on the properties of the first-stage estimators. It decreases in survey sample size.

If the assumptions of the ELL method hold and the models are correctly specified, the ELL estimator will usually exhibit a smaller variance of the prediction error than our estimator. The reason is that the latter is a between estimator that ignores variation within clusters. Intuitively, both estimators would only be similarly efficient if the explanatory variables differed distinctly more between clusters than within clusters. In practice, another exception might occur if there are many missing values in the explanatory variables in the survey. Without imputation methods that are subject to estimation uncertainty, the ELL first-stage estimator would be based on a smaller sample than our estimator.

In practice, the variance components of the idiosyncratic and model error are not estimated separately. Rather, the entire variance of the prediction error is obtained from the variation of the simulated poverty estimates in equation (4). Hence, under correct distributional assumptions on the random components, the bootstrap procedure allows to draw valid inferences, i.e., to build confidence intervals which include the true poverty measure with a predetermined probability. For instance, bootstrap percentile intervals, which can be constructed directly from the bootstrap estimates (see Section 2.3), can be used for inference.

Another potential issue in practice is multicollinearity. Note that the fundamental unit of the predictors in the first stage is a cluster, not a household, and that the number of parameters that can be included in (2) is hence restricted to the number of clusters. However, household budget surveys that are used to estimate poverty incidence typically cover 500 clusters or more, with some covering substantially more. Hence, we believe that our estimator could be based on a moderate number of regressors that would be sufficient to accurately predict household consumption which is assumed to differ between clusters.<sup>7</sup>

> 7One commonly used rule-of-thumb is to restrict the number of predictors to the square root of observations. While our results in Sections 4 and 5 are based on 100 clusters and less than ten variables, 500 clusters would allow the analyst to base the first-stage estimation on more than 20 census averages (or other summary statistics computed at the cluster-level).

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