Reflecting uncertainty about economic theory when estimating consumer demand

Economic theory provides a great deal of information about demand models. Specifically, theory can dictate many relationships that expenditure and price elasticities should fulfill. Unfortunately, analysts cannot be certain whether these relationships will hold exactly. Many analysts perform hypothesis tests to determine if the theory is correct. If the theory is accepted then the relationships are assumed to hold exactly, but if the theory is rejected they are ignored. In this paper we outline a hierarchical Bayesian formulation that allows us to consider the theoretical restrictions as holding stochastically or approximately. Our estimates are shrunk towards those implied by economic theory. This technique can incorporate information that a theory is approximately right, even when exact hypothesis tests would reject the theory and ignore all information from it. We illustrate our model with an application of this data to a store-level system of demand equations using supermarket scanner data.