survey data. To ensure that our model works well for data deprived countries, we check if our model selection and performance are robust to using only the countries with at most three poverty estimates, which reflects the 25% most data deprived countries for which we have some data. These countries tend to be more authoritarian and conflict-ridden but also include a couple of wealthy countries which simply do not share much data (Table A.1).

With respect to _coherence_ , we mean that we are willing to tolerate marginal increases in errors if doing so results in a model that is easier to rationalize. This is in part why we are restricting ourselves to predictions consistent with the log-logistic distribution, but it also matters for the covariates we will select. For example, if we find that a model using GDP per capita in 2011 PPPs performs slightly better than a model using GDP per capita in 2017 PPPs, we will favor the latter as the welfare distributions are expressed in 2017 PPPs, and hence makes for a more consistent model.

The use of these three principles in addition to the error of the model means that, at times, value judgments are needed to assess when an increase in the error may be merited by a greater adherence to the three principles. This complicates the model selection somewhat in contrast to a sole reliance on the model error. As a robustness check, we use machine learning to select a model guided exclusively on model performance as a test of how much accuracy we give up by including these three additional principles (section 5.2).

# 4 Results

Our first objective is to select the main variable to use for the tier 1 model. To that end, we run separate quantile regressions following equation (4) using one candidate covariate at a time, replacing ) in the equation. We evaluate the fit in-sample for reasons of efficiency, since evaluating all models out-of-sample is too computationally intensive. Given that all models are llGG(GGl identical and simple, overfitting is unlikely to be an issue. The left panel of Figure 3 plots the error from these regressions against the availability of each indicator from 1991-2020 for the 218 economies considered by the World Bank. We exclude variables with less than 50% availability given that they have too much missingness to be useful for our application.

We are primarily interested in variables for which no other variable has a lower error and lower missingness. When using the full sample (panel a), this concerns national accounts variables and under-five mortality. The national accounts variables on the frontier are GDP, Household Final Consumption Expenditure (HFCE), and Gross National Income (GNI) (the national accounts variables are expressed in logs and per capita terms throughout). We use versions of these variables expressed in 2017 PPP-adjusted USD even though at times non-PPP adjusted versions perform slightly better (see the unlabeled orange dots in Figure 3). These variables have the advantage of being highly correlated with welfare but the drawback of not always being available. For example, GNI per capita is available only for 58% of the country-year observations. Hence, national accounts data can only offer a partial solution to predict the distribution of welfare. To the contrary, under-five mortality is universally available due to modeling done by the UN Population Division.

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