The lasso and stepwise regression did not include interactions. We next test if adding interactions with the income/consumption dummy, or between the variables and <u>p</u> (i.e. letting the ~~�~~ � variables influence the Gini) could lower the error. As we show in the next section, regional llGG 1−pp dummies are most predictive of the Gini index among all 1,000+ variables. Therefore, we also test whether accuracy improves when we allow for variation in the Gini index by region. We try both Gini dummies for all regions and for the three regions that show notable differences in their Gini indices from the rest of the world. These regions are Latin America & the Caribbean and SubSaharan Africa which are regions with high inequality (Haddad et al. 2024), and Europe & Central Asia, which tends to have lower inequality. Finally, we test if the Gini should be a second-order polynomial of log GDP per capita, as the Kuznets curve would suggest (Kuznets 1955).

The error of both tiers would be reduced by letting the Gini differ by the three regions mentioned above (by 1.4 for tier 1 and 1.2 for tier 2). For tier 1, there is additional evidence in favor of adding a dummy for welfare type (income or consumption) and interacting this with GDP per capita. This means that as GDP grows, it has differential impacts on income and consumption vectors, which is in line with existing evidence (Lakner et al 2022, Mahler et al. 2022, Prydz et al 2022). This reduced the error of tier 1 further to 26.7. On the other hand, for tier 2, there is no similar case for adding a welfare type dummy or interacting the income group classifications with welfare type. Finally, we substitute under-60 mortality for the rural population share in the tier 1 model, which increases the error very marginally (from 26.67 to 26.69). This improves the consistency between the two models, and hence reduces revisions to the predictions if a country moves from one tier to another. In sum, we end up with two almost parallel models, where GDP per capita is proxied by income group in tier 2. The final out-of-sample error of the two tiers are 26.7 and 30.1. The regression outputs from the final models are shown in Table 1 (for the full sample).

|**Table 1: Re**<br>Outcome variable: Logwelfare(lG(y))|**gression**<br>|**output**<br>Tier 1|Ti|er 2|
|---|---|---|---|---|
|Intercept|-1.975***|(0.052)|1.837***|(0.047)|
|Log GDP per capita (2017 PPP)|0.393***|(0.004)|||
|Log under-5 mortality (per 1,000 live births)|-0.185***|(0.003)|-0.307***|(0.004)|
|Life expectancy (years)|0.016***|(0.000)|0.017***|(0.000)|
|Rural population share (0-100)|-0.003***|(0.000)|-0.008***|(0.000)|
|Income group|||||
|Low income|||Base||
|Lower-middle income|||0.229***|(0.006)|
|Upper-middle income|||0.450***|(0.008)|
|High income|||1.102***|(0.011)|
|Welfare type (income =1, consumption = 0)<br>|-3.671***|(0.030)|||
|Welfare type * log GDP per capita<br> <br>|0.392***|(0.003)|||
|lG(p/(1 −p))<br> <br>|0.354***|(0.002)|0.350***|(0.002)|
|<br>lG(p/(1 −p))* I[Europe & Central Asia]<br> <br>|-0.045***|(0.002)|-0.034***|(0.003)|
|<br> <br>lG(p/(1 −p))* I[Latin America & Caribbean]<br>|0.159***|(0.002)|0.166***|(0.003)|
|<br> <br>lG(p/(1 −p))* I[Sub-Saharan Africa]|0.060***|(0.002)|0.069***|(0.003)|
|<br><br>Observations|194,626||194,824||
|Pseudo R<sup>2</sup>|0.7512||0.719||

Note: *=0.05, **=0.01, ***=0.001. Robust standard errors in parentheses.

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