l 1 w The error component, (yycch) = ββ , represents a community-level random effect that shifts the intercept of the 0cc + ββ xxcch + ηηcc + εεcch wwh h= 1, . . . , NNcc ; cc= 1, . . . , CC (1) regression up or down for each commune, while the household idiosyncratic error within communes cc ) ηη is assumed to be independent across all households in the sample. Both and are assumed to be cch (εε normally distributed as in ) , and ) . cc cch ηη εε cc ηη<sup>2</sup> cch εε<sup>2</sup> In the poverty mapping literature, multilevel models are often called nested-error models. Among several ηη ~ NN(0, σσ εε ~ NN(0, σσ In the poverty mapping literature, multilevel models are often called nested-error models. Among several estimation strategies, Empirical Best (EB) methods are preferred as they provide accuracy and efficiency gains over previous applications, such as Elbers, Lanjouw, and Lanjouw (2003), allow for the inclusion of random location effects and the recovery of location and household-specific idiosyncratic errors (Corral Rodas, Molina, and Nguyen 2021). In the nested-error model (Battese, Harter, and Fuller 1988), welfare (and poverty) are imputed into census data by using the census empirical best approach of Corral Rodas, Molina, and Nguyen (2021).

In the application here, model fitness is maximized by estimating equation 1 independently for each region in Senegal. For each region, the estimation steps detailed in Corral Rodas et al. (2022) are followed, in particular: (1) using the variance inflation factor (VIF) to remove highly collinear variables, (2) estimating a separate regression for each region, (3) dropping high-leverage observations, and (4) avoiding overfitting by limiting the set of right-hand-side variables using a Lasso regression.<sup>9</sup> The model parameters and error terms allow the consumption and probability of falling into poverty ( �)ı of every household in the 2013 census to be predicted using equation (2). The predicted probability decreases with the vv predicted household consumption and increases on the variance of the errors.<sup>10</sup> As in Günther and Harttgen (2009), households exhibiting a probability of becoming poor above 50 percent (at least once) in the next two years are considered vulnerable, and the group of vulnerable households includes those households predicted to be poor (poverty-induced vulnerability) and those households that are nonpoor, but are at risk of poverty (risk-induced vulnerability).

> 9 Leverage measures the influence on the fitted values of a given observation. Following Corral Rodas et al. 2022, the analysis here controls for high-leverage observations by eliminating observations with leverage above . Because this incurs a loss of information, the predicted poverty rates in the new sample are checked to ensure they are similar to the original data. The census and survey questionnaires had 43 variables in common. The lasso (2kk+ 2)/ll selection process yields a selected set of covariates, although some of the included covariates may be nonsignificant. Reducing the number of variables avoids situations where the number of right-hand-side variables is large relative to the number of observations in a region, especially in regions with smaller populations.

> 10 � See Corral Rodas, Molina, and Nguyen (2021) for the estimation of . cc

> ηη 8

8