falls by a factor of three. The household level model performs well in this stylized setting because the residuals are distributed normally by construction, and a large percentage of variation in simulated welfare is explained by the two predictor variables. It is reassuring, though, that the small area estimator performs well in a favorable controlled environment with populations that vary across simulations.

# b. Design-based simulation

While model-based simulations are useful to explore the performance of the estimator in different populations, the results of model-based simulations inevitably depend on the set of assumptions used to construct the simulated data. To verify that the household-level model works well when estimated in multiple samples, this section describes the results of a design-based simulation. To reduce computation time, the simulation is limited to three regions in Northeast Tanzania near the Kenyan border: Arusha, Kilimanjaro, and Tanga. Between them, these three regions according to the census contain 24 districts, 2,078 villages, 10,964 enumeration areas, and nearly 1.17 million households.<sup>38</sup> The 2018 Household Budget Survey sampled 1,217 households in 97 enumeration areas from these three regions, for an average sample of 12.5 households per enumeration area. The three regions are disproportionately poor, as the non-monetary poverty rate is 46.5 percent when using a poverty line set at the 20<sup>th</sup> percentile of the national distribution.

To conduct the design-based simulations, we repeat the following seven-step procedure 250 times:

1. Draw a simple random sample of 4 EAs for each of the 24 districts from the census, which selects a total of 96 EAs from the population of 10,964 census EAs contained in the three regions.

2. Draw a simple random sample of 13 households for each selected EA to generate a sample of 1,248 households.

3. Construct sample weights for each household equal to the inverse of their selection probability.<sup>39</sup> 4. Use the ordered quantile normalization to construct normalized non-monetary welfare in the sample.

5. Using the household sample from step 2, the sample weights from step 3, and the normalized welfare measure from step 4, estimate a lasso regression of normalized non-monetary welfare on all candidate geospatial indicators.<sup>40</sup> 6. Using the same inputs as the previous step, calculate the 46.5<sup>th</sup> percentile of the sample distribution of normalized non-monetary welfare to use as the poverty line.

> 38 We selected three regions to use for this exercise because the lasso procedure did not select any variables when using only one or two regions, when mimicking the sample size of the budget survey.  We selected these particular three regions because they are disproportionately poor and because they were the first three regions in numeric order following the capital of Dodoma.

> 39 The household sample weights are the product of two ratios: The number of census EAs in each district divided by 4, and the number of households in each sampled EA divided by 13. These are multiplied by household size to obtain population weights.

> 40 As before, we use the plugin method for selecting lambda in the lasso.

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