We develop an indirect inference approach relying on a linear supply and demand model serving as an auxiliary model to provide the first full empirical test of the rational expectations commodity storage model. We build a rich storage model that incorporates a supply response and four structural shocks and show that exploiting information on both prices and quantities is critical for relaxing previous restrictive identifying assumptions and assessing the empirical consistency of the model's features. Finally, we carry out a structural estimation on the aggregate index of the world's most important staple food products. Our estimations show that supply shocks are the main drivers of food market dynamics and that our storage model is consistent with most of the moments in the data, including the high price persistence so far the subject of a long‐standing puzzle.