still include covariates that are not significant under the fitted model, all non-significant (at the 1% level) covariates are removed sequentially.<sup>36</sup> Finally, we check for multicollinearity and remove covariates with a variance inflation factor (VIF) above 3.

Fitted models using the first sample and the described model selection are presented in Tables 3 and 4 in the appendix. Since in small area estimation models are used for prediction, assessing the predictive power of the model is important. This may be measured using the coefficient of determination, _R_<sup>2</sup> (Tzavidis et al., 2018). For the household level model used in this exercise the _R_<sup>2</sup> is close to 0.45 while that of the unit-context model centers around 0.25.

Figures 9, 10, and 11 represent checks on the normality assumptions using normal _Q-Q_ -plots of household level residuals and estimated area effects for the onefold unit level model with random municipality location effects. The resulting plots for the unit-context models can be seen in the appendix figures 26, 27, and 28. The natural logarithm transformation (Fig. 9) presents evidence of deviation from normality. Marhuenda et al. (2017) notes in applications that use real data the exact fit to a distribution is barely met; however the Box-Cox and log-shift transformations provide considerably better normal approximations (see Figures 10 and 11).<sup>37</sup>

# **4.3 Results**

Once the models to be used have been selected and _L_ = 500 samples under two-stage sampling described in section 4.1 have been taken from the “census” created using the Mexican Intra Censal Survey of 2015 we obtain estimates using the different considered models. The target parameters for this simulation are mean welfare and the Foster et al. (1984) - FGT class of decomposable poverty measures for municipalities present in the census. The true values of the target parameters at the municipal level are based on the census data.

As a first step, we compare results using the three transformations discussed in the previous section. Figures 12 and 13 show box plots of empirical absolute relative bias and MSE of CensusEB estimates of poverty rates based on model (1) under each transformation. The Box-Cox and log-shift transformation yield estimates that are not only less biased, but also results with a lower empirical design MSE. As can be seen in the results from the first 3 columns in Table 2 the log-shift transformation yields aggregate results somewhat preferable to the Box-Cox. Additionally, in Table 2 it is quite clear that the natural logarithm may actually result in aggregate results that present considerably smaller gains over direct ones and thus model and residual checks should always be done to ensure adequate transformations are used to avoid such an outcome. The rest of the discussion will focus on estimates obtained from the log-shift transformation applied to the two-stage samples.<sup>38</sup>

> 36The removal is done sequentially starting with the least significant covariate, then the model is fit again without the covariate. The process is repeated until all covariates in the model are significant at the 1% level.

> 37 Box-Cox and log-shift transformations are available under R’s `sae` package as well as under Stata’s `sae` package. In the World Bank’s `PovMap` software, the only available transformation was natural logarithm. However, `PovMap` does allow for drawing residuals from the empirical distribution as well as from a Student’s _t_ distribution.

> 38Figures under other transformations are available upon request.

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