  We study a setting where an analyst has access to purely aggregate
information about the consumption choices of a heterogenous population of
individuals. We show that observing the statistical moments of market demand
allows the analyst to test aggregate data for rationality. Interestingly, just
the mean and variance of demand carry observable restrictions. This is in stark
contrast to impossibility result of the Sonnenschein-Mantel-Debreu theorem,
which shows that aggregate demand carries no observable restrictions at all. We
leverage our approach to deliver a characterization of rationality in terms of
moments for the common two-good case. We illustrate the usefulness of
moment-based restrictions through two applications: (i) improving the precision
of demand and welfare estimates; and (ii) testing for the existence of a
welfare-relevant representative consumer.
