We propose a new approach to estimating and testing asset pricing models in the context of a bilinear paradigm introduced by Kruskal [18]. This approach is both simple and at the same time quite general. As an illustration we apply it to the special case of the arbitrage pricing model where the number of factors is pre‐specified. The data appear to be generally in conflict with a five or seven factor representation of the model used by Roll and Ross [30]. When we consider the number of replications of our test and the large number of observations on which it is performed, the frequency with which we reject the three factor APM does not lead us to conclude that this model is unrepresentative of security returns. Further, the rejection of the five and seven factor versions is to be expected if the three factor version is correct. The paradigm gives insight into the appropriate specification of the model and suggests that there may be a small number of economy wide factors that affect security returns.