_ESTIMATING FOOD PRICE INFLATION_ _13_ Particularly within the matrix _X_ there can be highly correlated predictors, while the matrix _G_ may contain multiple identical indicators, for instance if there is only one market within an administrative unit. The following hard-coded variable selection rules are used. First, linear combinations are removed to avoid dummy problems. Highly collinear variables with _> ._ 95 correlation are removed by iteratively recalculating the correlation matrix and removing the variable with the highest overall correlation. Finally, variables with near-zero variance are removed.<sup>17</sup> All predictors are centered and scaled.

Note that equation 4 is estimated using an elastic net model that uses _L_ 1 and _L_ 2 penalties to shrink the predictor space, see again (Friedman et al., 2010; Hastie et al., 2021). The cubist regression specification of equation 5 is essentially an observation specific counterpart of the linear model. As such, only the regional dummies are removed. The cubist model can partition the data and fit local regressions. Since the coordinates are supplied, the model can learn spatial fixed effects using adaptive neighborhood sizes as well as spatial interaction effects by partitioning by coordinates rather than relying on explicit spatial dummies, see again (Quinlan, 1992; Kuhn et al., 2012; Witten et al., 2016).

# _D. Validation of imputation accuracy_

There are no countries in the data where the food price data is complete and the true inflation of a basket of food items is fully known. This makes validation against true data difficult.<sup>18</sup> Cross-validation techniques are used to adjust the flexibility and assess the predictive accuracy of the models in each iteration. The elastic net model optimizes over the standard _L_ 1 and _L_ 2 penalties and the mixing parameter.<sup>19</sup> The cubist model tunes over the neighborhood size used for smoothing, and the boosting iterations, using grid of all _Neighborhood × Committees_ = (2 _,_ 4 _,_ 8) _×_ (1 _,_ 25 _,_ 50 _,_ 100) combinations.

The training and out-of-sample validating are as usual on mutually exclusive draws of observations. A 4-fold validation is used to limit the computational burden of the application to manageable levels.<sup>20</sup>

> 17Food price series with zero variance do not occur in the data, but the cross-validation sampling might in theory land a draw that has extremely low price variation, particularly if the code would be applied to other countries.

> 18For instance, if a set of countries where the WFP data are complete would have been available, an interesting exercise would be to construct the _true_ food basket price and accompanying inflation. Then, in this same sample, randomly a subset of the data could be removed to simulate a pattern of missing observations and the final basket price imputation could be imputed and compared to the true data.

> 19 _L_ 1 penalizes likelihood by the absolute sum of coefficients, and _L_ 2 by the sum of squared parameter values, thereby discouraging large parameter estimates but having very different impacts when redundant parameters approach 0. In particular, _L_ 1 penalization can set parameter estimates to 0 as the penalty remains influential when a parameter approaches 0. The mixing parameter determines whether only _L_ 1, _L_ 2 or a mix of penalties, is used, balancing between the well-known Lasso Regression and Ridge regression, or mixing them (elastic net).

> 20Note that for _D_ foods, a total of _countries × M × I × D_  ̄ _country ×_ ( _folds_ + 1) _×_ Θ regressions need to be estimated, with _θ ∈_ Θ indexing the different tuning configurations and _D_<sup> ̄</sup> _country_ being the average number of food items in a country.