A possible explanation for the good performance of unit-context models under the considered experiments could be due to estimated random effects, where _σ_ ˆ _ac_<sup>2</sup><sup>_>σ_ˆ</sup> _a_<sup>2,whichcoincideswiththe</sup> better performing scenario in subsection 3.1. Although this observation is likely a coincidence since the performance of unit-context models depends on the particular covariates included in the model and on the shape of the target indicator. Under the Mexican data with the covariates used, the unit-context model performs rather well. However, the method considerably lags all others for estimation of the welfare mean by area. A similar result is observed in the results presented in subsection 3.1.<sup>45</sup> Under the simulations of column 3 of Table 2, the gains in MSE for the unit-context methods appear to come in municipalities with larger populations.<sup>46</sup> This is expected because in our sampling scenarios, municipalities with a larger population are more likely to be included in all the samples. For example, in the census there are 16,293 PSUs spread over 1,865 municipalities. The median municipality in the created census has 7 PSUs, and the median municipality in a given two-stage sample is only represented by one PSU. Consequently, it is not surprising the unit-context variants under-perform relative to unit models in terms of bias and MSE in municipalities with smaller populations, as can be observed in Figure 16. For municipalities with larger populations, and hence those likely to consist of more PSUs and more of these PSUs in the sample, direct estimates for FGT0 are more precise and unit-context models begin to catch up to unit models (see Fig. 32 in the appendix).

Under the simulations from a 1% SRS by PSU, shown in Table 2, column 4 and in the appendix Figures 29 and 30, we notice the unit models ( _CEBa_ ) present a slight upward bias that seems to become more pronounced in more populous municipalities (Fig. 17). Notice that some bias is acceptable since gains in MSE are achieved at the expense of bias, however in this case the presence of outliers seems to lead to increased upward bias that affects more municipalities as we move to more populous deciles (box-plots for _CEBa_ in Fig. 17). Unit-context ( _UC − CEBa_ ) models on the other hand, appear to have a downward bias. In the box-plots in Figure 17 for _UC − CEBa_ , the bias in lower deciles is downward and as we move to upper deciles the downward bias is considerably reduced.

As noted in section 3.1, the problem faced by unit-context models is omitted variable bias manifested as a lack of linearity (see Fig. 18).<sup>47</sup> Figure 19 further presents the issue. Under _CEBa_ residuals appear to follow a random pattern, however under _UC − CEBa_ households in municipalities represented by only one PSU in the sample will all have the same linear fit. This manifests itself in the figures for _UC − CEBa_ as a column of vertical dots. The problem can also be observed

> 45See Table 7 in the appendix.

> 46Similar findings are obtained under the simulations for column 1.

> 47The true model includes household size at the household level. Consequently, it is a determinant of the dependent  ̄ variable. Household size at the household level ( _xach_ ) can be broken down into _zach−xac_ , thus the omitted component, _zach_ , is also a determinant of the dependent variable. The unit-context model only includes the PSU average household size ( ̄ _xac_ ) obtained from the census as a covariate. When a given survey sample is taken, the non-included covariate

> ( _zach_ ) is correlated with the PSU average household size and the dependent variable.

29