To empirically apply these methods, we utilize data and definitions from the Global Multidimensional Poverty Index (MPI) developed by the Oxford Poverty and Human Development Initiative (OPHI) with the United Nations Development Programme (UNDP). The Global MPI is a robust measure that evaluates multidimensional poverty, using 10 different poverty indicators across three dimensions (health, education, and living standards). The 10 indicators include nutrition, child mortality, years of schooling, school attendance, cooking fuel, sanitation, drinking water, electricity, housing, and assets. A household is considered deprived in an indicator if it falls below a defined threshold (see Table A1.1 in Appendix 1 for the deprivation criteria used for each indicator). The underlying survey data for the indicators draws primarily from the Demographic Health Survey (DHS) program and the Multiple Indicator Cluster Surveys (MICS). The Global MPI was chosen for its comprehensive coverage and consistency in measuring poverty indicators.

However, this paper makes an important deviation from the standard Global MPI measure. While the OPHI/UNDP MPI assigns different weights to each of the ten indicators,<sup>6</sup> this paper weighs all of the indicators equally. The reason for this departure lies in the nature of our research question. By assigning equal weight to each indicator, we ensure that no single dimension is prioritized over another, allowing us to investigate the complex relationships in an agnostic manner. It is important to note that this weighting scheme may not necessarily reflect the relative importance of different dimensions of poverty in real-world scenarios, but it is a methodological choice that allows us to delve deeper into the structural relationships among these dimensions. Throughout the analysis, we assume that a household is multidimensionally poor if it is deprived in at least one of the 10 indicators used by the MPI. This means that any household that is deprived is automatically considered multidimensionally poor. To test the robustness of the results, we also replicate the analysis under different thresholds for multidimensional poverty. In Appendix 2 we show the results using a threshold of 3 (a household is considered multidimensionally poor if it is deprived in 3 or more indicators). We find that the results are consistent across thresholds. Note that for case described in the main body of the text (a threshold of 1), the censored headcount ratio (the proportion of the population that is both multidimensionally poor and also deprived in that specific indicator) is the same as the headcount ratio for that indicator.

> 6 The Global MPI's three dimensions (health, education and living standards) are weighted to contribute equally to the index. Thus, the underlying ten indicators are each weighted accordingly to ensure 1/3 contribution at the dimension level. This means that each of the two health indicators has a weight of 1/6, each of the two education indicators has a weight of 1/6, and each of the six living standards indicators has a weight of 1/18. Note that the selection of dimensions and indicators for the Global MPI was prepared following a process of consultation and comparison against the available data. For individual country MPIs, the selection of dimensions and weighting structure may reflect context specific values.

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