income values. The latter is done due to the non-negligible amount of zero income values.<sup>11</sup> In the second-stage bootstrap procedure, the regression coefficients are sampled from a multivariate normal distribution where the expected values are the first stage estimates and the robust variancecovariance matrix accounts for correlation within the clusters. The error components are generated by a nonparametric bootstrap. In particular, cluster effects are drawn with replacement from the 100 firststage estimates. The household errors are drawn with replacement from the first-stage residuals belonging to this specific cluster. See also Section 2.3.

For computing an overall state-level poverty measure, it is crucial to know at least approximately the distributions of households over municipalities in the population at the time of the recent survey: The proposed approach imputes poverty measures for the municipalities by using the dated census households. Clearly, a composite measure of those single poverty measures has to account for the number of households in the municipalities at the time of the recent survey.<sup>12</sup> We compare the performance of our estimator for the headcount ratio<sup>13</sup> in the state of Minas Gerais with the ELL estimator and a simple (weighted) mean based solely on the recent survey. Note that the sample is, in contrast to many real-world applications, representative and rich at the small-area level such that the weighted survey mean is an unbiased poverty estimator by construction. For the ELL firststage regression, the same explanatory variables are used, yet on the household level and using the 2010 survey data. In a regression based on all households from the 2010 census, this simple model specification already yields an _R_<sup>2</sup> of 0.33. We conduct 300 Monte Carlo simulations with 200 bootstrap census data sets generated in each replication.

For our estimator, the coverage of the confidence intervals is below the nominal one of 95% (Table 4). The estimator is slightly biased which may be because of unmodeled heterogeneity in the error distribution, e.g. between clusters. In a regression based on all households from the 2010 census, variances and skewness of the residuals differ considerably between clusters (Figure 1). However, we found no clear pattern with respect to the fitted values from a first-stage regression or other explanatory variables. As the number of clusters is relatively small, already one cluster with an extreme behavior of its errors can potentially have a large effect on estimates of welfare measures. In practice, it can be challenging to detect and model such peculiarities in the error distribution. Potential remedies are discussed in Section 6.

Due to the bias in the headcount ratio estimator, a comparison with a (weighted) mean purely based

> 11The proportion of all households in the 2010 census data with an income of zero amounts to 3.16 percent.

> 12In fact, this requirement ensures that changes in the distribution of the explanatory variables are accounted for in our approach. While it is not guaranteed to know the distribution of households at the time of the survey, it is arguably much more realistic than assuming the distribution of the explanatory variables on the household level not to change over time, as done by EEL, for instance.

> 13We also estimated the poverty gap in the same simulation setting and obtained qualitatively similar results.

14