_ANDR EE_<sup>_ ́_</sup> _DECEMBER - 2021_ _16_ that from a nutritional perspective, some foods should be weighted more strongly to model a food price index that reflects preferable consumption ratios.<sup>27</sup> The simple approach here is just to ensure that an item measured in larger quantities does not dominate the final index simply because it has a larger price range, or dominate the validation result simply because the unit of measurement is small. This is not ideal, but sensible, given that expenditure shares for specific food items used to construct traditional CPI are not widely available in the countries analyzed by the paper. Recall that the estimation exercise of the paper is in fact motivated by the unavailability of traditional data. Tables A2 and A3 list for all countries the food-specific Index Weights used to scale price levels to correspond to prices for comparable units of measurement.

# _E. The initialization_

A two-step approach is taken to initialize the regression chain. First, univariate imputation methods are used to pre-impute the starting values of missing entries.<sup>28</sup> The iterative modeling then initializes randomly around the pre-imputed values by adding an additive disturbance term to the initialization drawn from a uniform distribution centered around 0, scaled to 10% of the range in the data. The variance of the prices is stabilized with logarithms, so this disturbance term impacts the initialization roughly evenly across levels.

It is important to clean the data from outliers before calculating the preimputations and applying the imputation algorithm. In particular, outliers can lead to explosive predictions or generally bad model fit depending on the types of learning methods used. Since the outliers need to be removed from incomplete time series, standard time series methods such as those relying on non-parametric reasonable and avoids that a food commodity measured in bags of 100 _kg_ dominates the food basket price, but is obviously not smart enough to deal with all situations effectively. A specific rule is specified to deal with eggs, which may be described in some countries as 12 _units_ , while in other countries eggs are measured as 1 _dozen_ . In the case of ‘12 units’, a conversion to 1 dozen is made. All the price estimates from the paper are provided for analysis, and the index weights are provided with the index estimates. If it is suspect that conversion factors may have an impact on the final inflation rate estimates, researchers are encouraged to construct their own indexes relevant to the studies at hand using the individual food price estimates.

> 27For example, 1 kg of salt and 1 kg of sorghum are given equal weight in the simple index generated here, whereas the latter may carry more dietary importance. The reason for using equal weights is that it takes expert judgment to define weights based on dietary needs, which is not easily automated given the heterogeneity of the data. In an ideal world, future price survey programs would be accompanied with surveys on household expenditure shares or food specific trading volumes.

> 28First, minor gaps of up to 3 consecutive months are filled with a univariate Seasonal Kalman smoother using a Basic Structural Model. Then food-market combinations with at least 67% data coverage are completed using the same method. Next, all markets trajectories completed in this way are averaged to construct a preliminary country mean. All other market level series with _>_ 50% data coverage are then imputed using predictions from a Generalized Additive Model with observed price points as dependent variable and the preliminary country mean as the predictor. The parameters are estimated by Restricted Maximum Likelihood. The country average is recalculated, if there are any further markets with missing data, but _>_ 20% data coverage, the same method is again applied. Any market which has _<_ 20% data coverage, is pre-imputed by using an inverse distance weighing interpolation based on the other imputations. The spatial interpolation uses a neighborhood size cutoff which is selected using cross-validation.