_ANDR EE_<sup>_ ́_</sup> _SEPTEMBER 28, 2022_ _12_ covariates.<sup>14</sup> This is an iterative Markov chain Monte Carlo algorithm that converges rather fast. The number of updating iterations is based on the convergence of cross-validated prediction performance as in Andr ́ee (2021a). Since the combined predictor set ( _X, Z_ ) is relatively large, recursive feature elimination is performed by estimating variable importance using a Random Forest, then iteratively discarding the least important predictor. The variable selection that yields the best cross-validation performance for the severe food insecurity prevalence rates is used to run the first 10 imputation iterations, after which the algorithm continues with the fully specified regressions that use all covariates.

To adjust for the fact that the synthetic cases are estimated with uncertainty, with some cases generated in more uncertain regions than others, a total of 10 imputations are generated for each entry of observed covariates. This uses the stochastic properties of the estimated model of Equation 2, to generate multiple likely values. The final model is then trained by creating 10 training data sets _{Y_<sup>1</sup> _, ..., Y_<sup>10</sup> _}_ , each replacing _Ymissing_<sup>_m_with different predictions</sup><sup>_Y_ˆ</sup> _missing_<sup>_m_generated</sup> from models _{g_ ˆ<sup>1</sup> _, ...,_ ˆ _g_<sup>10</sup> _}_ . This results in 10 different models _{f_<sup>ˆ1</sup> _, ..., f_<sup>ˆ10</sup> _}_ which can be ensembled into a single model _f_<sup>ˆ</sup> by aggregating the predictions of the individual base learners _{f_<sup>ˆ1</sup> _, ..., f_<sup>ˆ10</sup> _}_ .<sup>15</sup> The accuracy of the final model can be validated using standard validation techniques.

What this strategy achieves is essentially a form of simulation-based training. The intuition is that the final forecast model has to project a global food insecurity situation for 2020 and after, that is beyond anything seen in the 2015-2019 data but likely comparable to the dynamics in the 2007-2011 data when food prices spiked globally during a time when extreme poverty was more prevalent. Therefore, simulated data for this historical period adds crucial information about conditions outside of the 2015-2019 data perimeter. The synthetic data strategy should therefore not in the foremost place improve the average prediction for the 2015-2019 period (as a standard cross-validation exercise would be indicative of), but act to aid the extrapolation when covariates move out of 2015-2019 country intervals. If the non-parametric extrapolation is based solely on 2015-2019 dynamics, they may become highly unstable when many covariates go outside

> 14This is an adaptation of Multiple Imputation using Chained Equations, see for instance Azur et al. (2011) for details, in which a fully fledged optimized machine learning model is built and cross-validated (an algorithm built around the work of van Buuren and Groothuis-Oudshoorn (2011); van Buuren (2012)).

> 15There are several choices for aggregating base learners. In the current context, the downside of averaging linearly is that it makes the ensemble sensitive to outliers. The arithmetic mean has an upward tendency. Suppose the true value is 0, and 9 out of 10 base learners predict this correctly but one base learner predicts a large value, then this results in a relatively large prediction error of the ensemble. A popular choice to prevent this, is by taking a median. The downside of the median is that it is completely insensitive to changes in the highest or lowest values predicted by the individual base learners. Here, since both outlier robustness and smoothness are valued, the ensemble is created by taking a harmonic average of the median and mean. The harmonic mean is appropriate when combining ratios of different origins and has a downward bias. Therefore, when the mean increases due to a single base learner so that the median remains unchanged, more weight is given to the median relative to the mean. Only when both the median and mean are close to each other, so that there is little influence of outliers, and increase jointly, does the harmonic mean of both move up approximately proportionally. This way of aggregating improved the holdout predictions when compared to using a linear average.