Extended Data Fig. 1: Structured crop yield model selection for weather and mediating variables.

For each staple crop, cross-validation is used to first select key weather variables that influence yields (rows) and then applied to select mediating variables that capture adaptation or resource access (columns). Each selected mediating variable interacts with each selected weather variable. All observations are residualized before model selection, to remove cross-sectional and time series confounders. All candidates models are tested for each crop; a and b illustrate the process only for maize. a, First step: cross-validation is used to select weather variables that influence maize (see Extended Data Figs. 2 and 3 for all crops). Top panel (blue) shows the unconditional distribution of OOS root-mean-squared error (RMSE) for all candidate models. Subsequent rows show pairs of conditional distributions that partition the sample. Green distributions depict model performance for all model permutations that include the indicated weather variable; grey distributions represent all model permutations that exclude the weather variable. Vertical lines indicate average values for each distribution. Strongly non-overlapping distributions indicate that models including the weather variable systematically tend to outperform those that exclude it. Fully overlapping distributions indicate that the indicated weather variable used to stratify models does not improve model fit. Weather terms with black stars were retained in the maize model. b, Second step: using cross-validation to select mediating variables that capture adaptation to climate (for example, average temperature) or resource access (for example, GDP per capita). Candidate mediating variables are all interacted with weather variables from a. Sub-panels depict the mediating influence of the these factors (columns) on the response to weather variables (rows). Grey panels indicate pairs of variables that are not selected for the final model. Colours indicate a weather response that is empirically recovered for a lower (blue) or upper (red) tercile, although actual model specifications are continuous interactions. All coefficients and interactions are estimated simultaneously from a single joint model. Standard errors robust⁸⁸ to autocorrelation over time within first-level administrative units (for example, states) and across all units within country–year. c, Similar to b but indicating weather-mediating variable pairs selected in the final model (white squares) for each crop.