Bartlett Identities Tests

In this note we propose a general testing procedure for parametric models based on Bartlett Identities. A well-know example is the Information Matrix test, which is based on the Bartlett Identity of order 1. The Identities are shown to induce a sequence of testable restrictions on the data generating process. When all the restrictions are considered jointly, they are often complete, in the sense that they are satisfied if and only if the model is correctly specified. We show that this is the case for normal, exponential and Poisson models. A test of the joint validity of an arbitrarily chosen subset of restrictions is proposed, and its first order asymptotic properties are presented. An interpretation of the test as a score test for neglected parameter heterogeneity is also given.