used by practitioners in preparing welfare aggregates for distributional analysis: the log-linear hedonic model and the self-assessment method. According to a World Bank survey on the main characteristics of the welfare aggregates used for measuring poverty in 70 developing countries, only 28 cases include housing. Among these 28, 12 use a hedonic model, 11 use self-assessment, and other four adopted a combination of hedonic model and self-assessment (World Bank 2015).<sup>9</sup> While we refer the reader to Balcázar et al. (2017) for an extensive review of the literature on rent imputation for welfare analysis, we briefly discuss in the following the log-linear hedonic model and the self-assessment method.

As underlined by Malpezzi (2002), the semi-logarithmic model has several advantages. First, the coefficients have a straightforward interpretation: they show approximately the percentage change in the imputed rent for a given unit-change in the covariates.<sup>10</sup> It also mitigates the heteroskedasticity problem (Diewert 2003). Second, it allows for flexible specifications of the covariates that can be either continuous or binary. Third, and most importantly, it allows the marginal rental value to be a nonlinear function of size and quality of the dwelling, which is a theoretical requirement of the hedonic model (Rosen 1974; Freeman 1993).

The second imputation approach used in this analysis is self-assessment. The self-assessment method relies on the assumption that owners can provide a good estimate of the market value of their dwellings, perhaps with the help of interviewers, as suggested in Garner and Kogan (2007). This information is frequently found in household budget surveys, where home-owners and tenants enjoying subsidized arrangements are asked to estimate the rental price they would pay to live in the current dwelling (Frick et al. 2010). The question may be asked in two ways that differ in whether the homeowner is asked to play the role of the landlord or the tenant. In the first case, the question would be some variant of “ _How much would you receive as a rent if you were to lend your apartment?_ ” In the second case, the question would read like “ _How much would you pay if you were to rent the dwelling you are currently living in?_ ” The extensive theoretical and experimental literature on the difference between willingness to accept (WTA) and willingness to pay (WTP) for the same good (among others, Hanemann 1991; Fehr et al. 2015; Tunçel and Hammit 2014) argues that measures of WTA usually exceed measures of WTP. This is due to the fact that a person’s perceived value of an item is often higher when the person owns the item

> 9 According to World Bank (2015), countries using hedonic models to impute rents are the following: Afghanistan (2008); Timor-Leste (2007); Malawi (2004); Sierra Leone (2003); Kenya (2005); Mozambique (2002); Nigeria (2004); Guatemala (2006); Bangladesh (2005); Peru (2010); Tanzania (2007); Zambia (2006). Among countries using self-assessment to impute rents, there are Nepal (2010); Nicaragua (2009); Zimbabwe (2011); Cambodia (2011); Jordan (2010); Belize (2002); Uganda (2005); Indonesia (2010); Panama (2008); Ecuador (2006); West Bank and Gaza (2007). Countries using a combination of hedonic models and self-assessment: Mongolia (2008); Nicaragua (2001); Peru (2013); Zambia (2010). 10 This is not accurate for dummy variables. In this case, Halvorsen and Palmquist (1980) demonstrate that the percentage change is better approximated by �−1�. ii �100 ∗exp�ββ̂

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