Understanding commodity prices dynamics is of crucial importance for assessing the persistence of cost-push costs or for countries dependent on commodity exports. Unfortunately, despite decades of research, the workhorse theoretical model in the field, the rational expectations storage model, is yet to be empirically validated. This paper provides the first full empirical test of the storage model. We first build a new storage model featuring a supply response, long-run demand and cost trends, and four structural shocks. We then develop a flexible empirical approach which relies on the indirect inference method and exploits the joint dynamics of prices and quantities unlike previous estimations which only use price information. The information contained in quantities is essential to relax restrictive identifying assumptions and empirically assess the overall consistency of the model's new features. Finally, we carry out a structural estimation on the aggregate index of the world most important staple food products: maize, rice, soybeans, and wheat. The results show that our extended storage model is consistent with most of the moments in the data, including the high price autocorrelation of which up to 42% can be explained by the transfer of inventories over time. They also show that, although for these commodities supply shocks are the main drivers of market dynamics, over the past 60 years all price spikes have been associated with large positive demand shocks.