The primary concern of this article is the provision of definitions and tests for exogeneity appropriate for models defined through sets of conditional moment restrictions. These forms of exogeneity are expressed as additional conditional moment constraints and may be equivalently formulated as a countably infinite number of unconditional restrictions. Consequently, tests of exogeneity may be seen as tests for an additional set of infinite moment conditions. A number of test statistics are suggested based on GMM and generalised empirical likelihood. The asymptotic properties of the statistics are described under both null hypothesis and a suitable sequence of local alternatives. An extensive set of simulation experiments explores the relative practical efficacy of the various test statistics in terms of empirical size and size-adjusted power.

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