where _nsc_ denotes the number of households selected in cluster _c_ within state _s_ (10 in this case) and _ρs_ is the intra-cluster correlation of per capita income. Given the above, we can calculate the number of clusters needed to achieve the minimum sample size for each state and then multiply this number by 10 to find the final household sample size. Taking these sample size requirements as given, each of our samples is then drawn in accordance with the two-stage design.

Specifically, the clusters within each state are selected with probability proportional to size (without replacement), where the measure of size is the number of households within each cluster. The households within each cluster are then selected via simple random sampling. According to this design, the inclusion probability for a given household is then approximately: where _n_ ̃ _sc_ is the total number of households in cluster _c_ within state _s_ , _Ns_ is the number of clusters ̃ selected in state _s_ , and _ns_ is the total number of households in state _s_ .<sup>8</sup> The sample size across the 500 samples is roughly 23,500 households. Under this sampling scenario, not all municipalities are included, and the number of municipalities included varies from sample to sample, ranging between 951 to 1,020 municipalities. The median municipality included in a given sample is represented by a single cluster and thus its sample size is 10 households.

In addition to the Intercensal Survey, INEGI has also released geo-referenced data that is similar to the remotely-sensed data often used in machine-learning applications. These geographic variables were calculated by INEGI for the year 2015 (the same year as the Intercensal Survey) and are only available at the municipality level. The geo-referenced data includes 105 different indicators. Specifically, the dataset includes information from 21 different dimensions and each dimension is associated with five different summary measures. For example, we summarize the atmospherically resistant vegetation index for each municipality using the minimum, maximum, mean, sum, and standard deviation of the index. The same procedure is applied to all dimensions.<sup>9</sup> We then have two alternative sets of covariates that we use in our simulations: the geo-referenced data and the standard set of sociodemographic characteristics from the census.

> 8The equation used to calculate inclusion probabilities assumes sampling with replacement, but is used here as an approximation of inclusion probabilities under proportional selection without replacement. This should provide a reasonable approximation in this case since there are a relatively large number of clusters present in the frame. The design weight for each household is simply the inverse of the inclusion probability. In a typical survey, the design weights would be further adjusted for nonresponse and calibrated to known population characteristics. However, since the sampling is only a simulation exercise, there is no nonresponse and thus no nonresponse adjustment is required. Calibration or post-stratification could be performed, but was not implemented to simplify the process.

> 9The 21 dimensions are as follows: enhanced vegetation index, normalized difference vegetation index, normalized difference built-up index, built-up index, simple ratio, atmospherically resistant vegetation index, urban index, index of biological integrity, normalized difference water index, modified normalized difference water index, new builtup index, band ratio for built-up area, normalized built-up area index, built-up area extraction index, normalized difference snow index, visible atmospherically resistant index, soil-adjusted vegetation index, optimized soil-adjusted vegetation index, normalized difference moisture index, digital altitude model, and digital slope model.

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