Understanding the drivers of commodity prices dynamics is crucial. Unfortunately the central economic model for representing commodity prices, the competitive storage model, is not yet empirically validated. In this work, we develop a rich storage model with four demand and supply shocks, elastic supply, and longrun trends and estimate it structurally on a caloric aggregate of the four most important grains. Our estimated model is consistent with most of the moments in the data, validating the empirical relevance of the storage model. The estimated model shows that speculative storage while crucial cannot explain alone the persistence of grain price. It explains 42% of the price autocorrelation, the rest being accounted for by the price trend, the persistence of demand shocks, and the presence of news shocks about production.