where _Ns_ is the total number of households in the census for state _s_ , _Nsc_ is the number of households in cluster _c_ from state _s_ , _Cs_ is the number of clusters selected in state _s_ , and _nsc_ is the number of households selected in cluster _c_ within state _s_ , which is fixed at 10.<sup>31</sup> The sample size across the 500 samples is roughly 23,540. Under the proposed 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 sole PSU and thus its sample size is of 10 households.

# **4.2 Model selection**

Model selection is conducted using the first sample drawn from the scenario detailed in the previous section. The target variable is household per capita income. However, this variable is highly skewed and to achieve an approximately normal distribution we test three transformations: i) natural logarithm,<sup>32</sup> ii) log-shift transformation, and iii) Box-Cox transformation of the natural logarithm.<sup>33</sup> As one can see in Figures 6, 7 and 8, the Box-Cox transformation as well as the log shift fix the skewness in the distribution of model residuals that appears after taking the natural logarithm of per capita income.<sup>34</sup> The goal of the model selection process is to arrive at a model that only includes stable covariates. Under each transformation, model selection is done using a least absolute shrinkage and selection operator, commonly known as lasso,<sup>35</sup> where the candidates for covariates include household characteristics and characteristics at the PSU, municipal and state level. Two models are selected: i) a model that includes household characteristics and characteristics at the PSU, municipal, and state levels and ii) another model that only includes characteristics at the PSU, municipal and state levels. The second model is used for the unit-context approach. All household level characteristics that are included also as aggregates at the PSU, municipality, and state levels have been previously standardized to ensure that these have mean 0 and standard deviation 1 for each PSU, municipality and state respectively. Note that aggregated covariates have been obtained from the “census,” thus, the aggregated covariates will be the same within the sample and the census. The lasso model selection process applied here ignores the error structure presented in (1) and (2) and does not ensure that selected covariates will be significant under the assumed models. Consequently, after the initial lasso model selection, model (1) is fit using Henderson’s Method III (with sampling weights) and random effects specified at the municipality level. Because the resulting model may

> 31The 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.

> 32In any given sample, roughly 11 observations have an income of 0, these are assigned an income of 1 prior to transformation.

> 33For more details on transformations, see Tzavidis et al. (2018).

> 34Figures make use of sample from a two-stage clustered design.

> 35Model is selected using 20 fold cross validation and shrinkage parameter ( _λ_ ) that is within 1 standard error of the one that minimizes the cross validated MSE.

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