_ESTIMATING FOOD PRICE INFLATION_ _15_ mula for MAE, _MAEd|μ_ is the MAE calculated using observed data and the unconditional mean. Since the true data range is not observed, and averaging is known to improve ensemble performance, the quantity from equation 6 is likely a conservative estimate.<sup>24</sup>

� The focus next is on the quantity 1 _− NMAE_ , which is the share of the total absolute variation in the demeaned data explained by the imputation model. The _D_ values are averaged as follows.

where _Z_ is the Fischer Z-transformation and _Z_<sup>_−_1</sup> its inverse, and _w_ are the relative weights of the price component in the final price index. The final score from equation 7 roughly has the interpretation of the average _R_<sup>2</sup> of the food price index, using a robust calculation of out-of-sample errors.<sup>25</sup> The unit of measurements are harmonized so that each food item has equal weight after in the index once scaled to a comparable unit of measurement. Specifically, the food item specific predictions are aggregated into food price indexes by summing the prices of all foods in the basket, after bringing the prices to comparable units of measurement (1 kg for foods, 1 liter for fluids and vegetable oils, 1 dozen for single packaged eggs, 1 unit for foods that come in other units of measurement – such as some fruits that come in bundles). As a simple example, if the index consists of 1 kg of sorghum and 100 Kg of maize, the latter price is multiplied by 0.01 and an equally weighted index is constructed with the result.<sup>26</sup> Note such,24Notethe estimatedthat _MAENMAE|_ � _μ_ only _d_ usesis likely larger than the true value since the denominator is underestimatedthe observed data range while the true data range is likely larger. As in equation 6. There are further related challenges to the validation due to the fact that the general objective of imputing unobserved data can only be validated using observations. Some further rationale for25theNote that (1divergence _−NMAE_ metric� and) _≥_ how(1 _−NRMSE_ it�relates), where the second value equals the out-of-sample calculationto the objective is provided in section B.B3. of the _R_<sup>2</sup> . Keep in mind, however, that the data for validation may still contain outliers, see again section� � B.B3, so that (1 _− NRMSE_ ) _>_ (1 _− NRMSE_ ). Regardless, when _NMAE_ is low, such�as in nearly all countries as the results will show and particularly in the outlier case,�the value (1 _− NMAE_ ) approaches the out-of-sample _R_<sup>2</sup> given by (1 _− NRMSE_ ). Moreover, �when _NMAE_ is small, then model fit is good and average bias must be small, which means that 1 _− NMAE_ approaches the in-sample _R_<sup>2</sup> which is the squared Pearson-correlation between the predictions and the observations. It is well known that the arithmetic mean of multiple correlation coefficients underestimates the total correlation and that the distribution of the correlation coefficient must first be normalized before averaging to obtain a less biased estimate, particularly when the number of coefficients is small. See (Corey et al., 1998) on this matter and the use of the Fischer Z-transformation in this context.

> 26The standardization in the units of measurement is automated and works by parsing the text in the WFP data base that describes the food items and inferring multipliers that standardize the prices to comparable units. In particular, each text string is parsed as _amount×unit_ and a multiplier for food item _d_ is calculated as _wd_ = _amount_ <u>1</u><sup>_c_,where</sup><sup>_c_isaconversionfactorfrom</sup><sup>_unit_toeitherKilogramsorLiters.</sup> For example, the text string 10 _Pounds of Rice_ would result in a multiplier of _wrice_ = 101<sup>_×_</sup> 0 _._ 4535921<sup>=</sup> 0 _._ 2204624, while 1 _kg of wheat_ would simply result in a weight of _wwheat_ = 1. This strategy is usually