_ANDR EE_<sup>_ ́_</sup> _DECEMBER - 2021_ _8_ direction so that old price data quickly loses its relevance.<sup>11</sup> Kalman filtering is a popular method that is known to produce results that are optimal in common settings. These methods are discussed at length by Durbin and Koopman (2013). The state of the methodologies at present, however, is such that the current application brings together too many issues to derive a state space specification that generalizes across all data situations. Essentially, a good approach may be specified for narrow studies of prices within any given country individually, but likely not across all prices and all countries.<sup>12</sup>

# _B. The predictive imputation framework_

The literature on missing data often distinguishes between different mechanisms of missing data: MCAR (missing completely at random), MAR (missing at random), and MNAR (missing not at random). This taxonomy, introduced by Rubin (1976) and well-explained by van Buuren (2012), becomes important when missing data occurs at the covariate side, and different corrections are needed for unbiased inference depending on the type of missing data problem. In the current application, it is required that the data is at least MAR and possibly MNAR and that the value of one food item can be accurately predicted by making reference to prices of other food items and the location in space and time.<sup>13</sup> As such, the

> 11Closely related interpolation methods are plagued by similar issues. For example, linear interpolation takes information from the future to the past. The linear interpolation estimate for _c_ 5 can only be made after _c_ 6 has already been observed. Alternatively, moving average imputation based on past values results in biases when used for real-time monitoring. When prices are rising, a moving average imputation of _a_ 6 would be based on an average of lower prices and a real-time monitoring system using moving average imputation would regularly suggest false trend reversals. These simple interpolation approaches are also highly influenced by outliers. Interpolating linearly from _b_ 2 to _b_ 6, through outlier _b_<sup>_∗_</sup> 4<sup>,replicates</sup> information into two new outliers _b_<sup>_∗_</sup> 3<sup>and</sup><sup>_b∗_</sup> 5<sup>.</sup> More flexible spline methods are known to introduce spurious cycles in time series data and may result in explosive extrapolation due to their quadratic or higher order nature, particularly in the presence of outliers.

> 12Optimality in a Kullback-Leibler or Root-Mean-Square sense is only under restrictive specification assumptions which require tedious design effort and diagnosing. Generally, Kalman filtering estimates time varying states and uses current state estimates and observations to estimate future states. When observations are missing, the imputation will rely fully on the transition equation. This means that the dynamic properties deteriorate rapidly during periods of large data gaps. This may in part be tackled by considering multivariate state space approaches so that realistic predictions for one sequence can be made by utilizing recent information from another sequence. Fast multivariate filtering implementations that could be deployed at scale are for example explored by Koopman and Durbin (2000). Due to outliers, and general non-Gaussian behavior, non-Gaussian outlier models may need to be considered. The univariate context is for example explored by Shephard (1994); Koopman et al. (2019). The above methods all rely on restrictive linearity assumptions. Nonlinear filtering is explored by Wan and Van Der Merwe (2000). Deciding between approaches is complicated by the variability in quality and quantity of data, the patterns of missingness, and the dynamic properties of the data-generating process. The considerations are likely unique to each country-specific study. Finally, predictions with time-series methods are based on historical information and not designed to smooth data that is simultaneously missing in multiple trajectories based on contemporaneous relations between intermittently observed time series with limited overlap.

> 13From the traditional missing data perspective, the MCAR situation is an easy problem as it effectively implies simply dropping incomplete cases will not bias subsequent inference. On the other side of the spectrum are MNAR problems in which the probability of a case being missing systematically varies for reasons that are unknown. In this case, data is needed that explains why certain observations are missing. From an inference perspective, MNAR problems are not even salvaged by predictive regressionbased methods that make predictions for missing data based on other covariates. These methods simply