where the joint distribution is known. The underlying (strong) assumption is that the joint distribution is the same as in the context in which the model is trained. Another clear drawback is that imputations techniques are relatively complex to implement.

Such imputation methods has significant advantages over the method proposed in Section 3.4.1. First, it artificially generates a database where all outcomes are available for each household. As a result, the multidimensionally poor households can be identified with methods that are more elaborated than the AF identification implicit in Section 3.4.1. Indeed, the monetary dimension need not necessarily be dichotomized when identifying the poor.<sup>51</sup> Hence, imputations allow constructing more sophisticated multidimensional poverty measures. Second, and related, the “comprehensive” database generated by this imputation could potentially also be used for inter-personal well-being comparisons (Objective 1). This is unlike the method proposed in Section 3.2, which only allowed cross-regions well-being comparisons (Objective 2).

Of course, the added-value of these advantages depend on the relevance of the joint distribution assumed. And given the scarcity of contexts where such an imputation model can be trained, it is far from clear that the joint distribution assumed will often decently approximate the unknown joint distribution. Yet, one fundamental point to keep in mind is that this approximation needs not be decent. The key comparison is not how close is the assumed joint distribution from the unobserved joint distribution. Rather, what truly matters is whether the well-being indicators constructed on the “comprehensive” imputed database yield better well-being comparisons than those constructed using the classical approach, which “wastes” available data. This is the message illustrated in Table 2, where even very bad assumptions cˆ on the unknown joint distribution c yield better well-being comparisons than those of H<sup>ˆ</sup> M (or those of H<sup>ˆ</sup> OD).

The most discussed question in this area asks which particular imputation method yields most reliable results given the particular data constraints (Dang, 2021; Dang et al., 2023). Another question, which perhaps deserves more prominence given our purpose, is whether the use of an imputation method with limited reliability is better than no imputation at all. In the case of multidimensional well-being, an imputation method that is only semi-reliable could improve well-being comparisons by allowing to draw information from both a monetary and a non-monetary survey.

It would be interesting to study the reliability of well-being comparisons yielded by indicators constructed on a “comprehensive” database obtained by imputing the monetary dimension into a non-monetary survey (DHS or MICS). Studying this reliability requires at least two contexts for which the joint distribution between monetary and non-monetary outcomes is observed. The more dissimilar these two contexts, the more conservative the assessment. The imputation model is trained on one context and then tested on the second context. Let uˆ and P<sup>ˆ</sup> denote the well-being indicators we would like to

> 51For instance, one could use the PX identification method described in Section 3.2.

28