and there is clear evidence of selection, given the statistical differences in characteristics between refugees and their host communities in Chad. Secondly, the cost of aid to refugees only depends on poor refugees (which is why it is based on the poverty gap indicator), which is a non-linear transformation of the welfare indicator. Using a standard OLS regression with a linear approximation of the welfare indicator may not reliably recover the real cost of hosting refugees. Despite a significant body of literature on the value of small area estimation techniques for estimating poverty headcount ratios from OLS (Rao & Molina, 2015), there is a lack of corresponding evidence for other poverty indicators, such as the poverty gap.

Here, an alternative methodological approach is followed to estimate the cost of hosting Sudanese refugees in Chad. The key objective of our framework involves constructing a counterfactual distribution of the welfare indicator for these refugees under the counterfactual that they are fully integrated into the home country system. Specifically, the welfare indicator is assumed to be a function of some observable characteristics, while letting unspecified the functional form dictating the relationship between the welfare indicator and its assumed correlates. As a result, the framework offers greater flexibility compared to, for example, the OLS modeling approach.

The proposed methodology is especially well-suited for evaluating counterfactual nonlinear transformations of the welfare indicator, such as the poverty gap. For instance, this approach is more ambitious as it seeks to estimate the counterfactual distribution of the welfare indicator for Sudanese refugees. Notably, this methodological framework was initially applied to analyze the impact of changes in labor market institutions on the distribution of wages (DiNardo, Fortin, and Lemieux, 1996).

Let the per capita consumption of the Chadian be determined by: . 1)<sup>llll</sup> In (E . 1) , is the actual per capita consumption of the Chadian, which is expressed as a function of , yy<sup>l</sup> = YY<sup>l</sup> (zz, μμ) (E a vector of observable consumption correlates, and<sup>ll</sup> , a vector of unobserved (all other not in ) factors yy<sup>l</sup> zz that determine per capita consumption. The function is the metric of aggregating the μμ zz components of the observed correlates vector. This function is left unspecified: no specific functional<sup>ll</sup> YY<sup>l</sup> (zz, μμ) restriction or assumption. The metric of aggregation can be thought of as the systems and mechanisms under which the characteristics of local Chadian are rewarded in Chad. The vector of correlates includes important determinants of poverty, such as education and access to assets, etc. (see Table 2). zz Following (E . 1) , by analogy, the per capita consumption, , for the Sudanese refugees can be determined by a similar metric of aggregation, . The key identifying assumption in this<sup>rr</sup> yy<sup>r</sup> framework is that the unobserved correlate, , conditional on the observed correlates,<sup>rr</sup> , has the same YY<sup>r</sup> (zz, μμ) distribution for the Chadian and the Sudanese refugees. Under this identifying assumption, the same μμ zz aggregation metric for the Chadian, , can be used to construct a counterfactual distribution of per capita consumption for the Sudanese refugees if they were fully integrated, that is, if they were<sup>ll</sup> treated as the Chadian. Let � lYY<sup>l</sup> (zz, μμ) be the density function of Chadian’s per capita consumption conditional on .<sup>ll</sup> ff�yy<sup>l</sup> ll , zz) The methodology of DiNardo, Fortin and Lemieux (1996) to obtain zz � r , the conditional density of per capita consumption of refugee participation, is to reweight the Chadian’s distribution of<sup>ll</sup> ff�yy<sup>l</sup> ff , zz) such that the distribution of correlates after reweighting matches that of the Sudanese refugees. More formally, it can be shown that:<sup>ll</sup> �yy<sup>l</sup> , zz� zz 16