_GLOBAL FOOD INSECURITY OUTLOOK_ _11_ Any synthetic data case consists of elements describing both the L.H.S. and R.H.S. of the regression model. A drawback from popular non-parametric sampling approaches that mix observed attributes by creating synthetic cases out of dynamically weighted observed cases is that these procedures may result in combinations of covariate values that are internally inconsistent. For instance, nonparametric sampling methods could generate synthetic cases with higher poverty and lower inequality, or simultaneously increase the shares of all types of land cover, ignoring that there is typically a systematic relationship between such variables. This issue can be circumvented when there are a high number of entries in the data for which covariates are observed and only the dependent variable is missing. In this, case synthetic observations can be generated whose covariate values are consistent with actual observations.<sup>13</sup> Here, synthetic cases are generated stochastically using a historical prediction model applied to historical observed covariates. Recall that the restrictions on the use of covariates in the final model are based on the availability of reliable future values of those covariates. For a historical prediction model used only to impute past observations, those restrictions do not apply. This fact is exploited to generate synthetic cases from historical covariates combined with historical proxies for food insecurity. The observations for the prevalence of severe food insecurity start only in 2015, while other food insecurity proxies go back much further in time. Importantly, these include data on undernourishment rates, child mortality rates, life expectancy, poverty at national lines, but also environmental rents and the percentage of urban population in addition to the full set of covariates used in the main model. This allows designing a more accurate model for the sole task of completing historical data which can be used to draw synthetic cases from. This historical imputation model in vector notation is (2) _Y_ = _g_ ( _X, Z_ ) _._ In this model, _Z_ now includes the additional covariates that are historically predictive but cannot be used for projection purposes. Particularly the historical malnutrition rates, life expectancy and child mortality rates are strong predictors for historical food insecurity and allow creating good additional synthetic cases. Imputations are generated using the multiple prediction framework of Andr ́ee (2021a), which iteratively estimates fully optimized machine learning models to replace previous imputations with better predictions generated from all other been trained on data that has artificially been created from generative models (Li and Wen, 2014; Abdul Lateh et al., 2017) or even within fully simulated environments Nikolenko (2021).

> 13As a simple example, Andr ́ee et al. (2020) train a model on monthly food insecurity assessments that are published every 4 months while covariates can be observed every month. They produce simple synthetic cases by assuming that the categorical outcome of the food insecurity assessment would be the same one month prior to or after the assessment. They argue that since the official assessments themselves are not exactly timed measurements but qualitative estimates prone to error, the errors in the synthetic cases would likely be of a similar degree as the measurement error in the observations. This allows them to expand the size of the training data by a factor 3, while ensuring that all covariate values are internally consistent, and this improves prediction on holdout observations.