September 25, 2025

• Working Papers
• Publication
• Macroeconomy

Working Paper 2025-9a

Abstract: This paper studies the limiting behavior of the test for instrument exogeneity in linear models when there is uncertainty about the strength of the identification signal. We consider the test for conditional moment restrictions with an expanding set of constructed instruments. We establish the uniform validity of the standard normal asymptotic approximation, under the null, of this specification test over all possible degrees of model identification. As a result, this allows the researcher to use standard inference for testing instrument exogeneity without the need of any prior knowledge if the instruments are strong, semi-strong, weak, or completely irrelevant. Furthermore, we show that the test is consistent regardless of the instrument strength; i.e., even in cases (weak and completely irrelevant instruments) where the standard tests fail to exhibit asymptotic power. To obtain these results, we characterize the rate of the estimator under a drifting sequence for the identification signal. We illustrate the appealing properties of the test in simulations and an empirical application.

JEL classification: C12, C14, C26, C52

Key words: linear instrumental variables (IV) model, conditional test for instrument exogeneity, uniform inference, instrument strength, generalized method of moments (GMM) estimator, drifting sequences, expanding set of basis functions

Digital Object Identifier (DOI): https://doi.org/10.29338/wp2025-09

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We would like to thank the editor, an associate editor, two referees, Bertille Antoine, Marine Carrasco, Jean-Marie Dufour, Sílvia Gonçalves, and participants at the 2024 EC² meeting (Amsterdam), the 2024 meeting of the Canadian Econometric Study Group (Toronto), the Spring 2025 Virtual Time Series Seminar, the 2025 meeting of the Société Canadienne de Science Économique (Orford, Québec) and the 2026 CSW Asia Meeting of the Econometric Society (Abu Dhabi) for very helpful comments and suggestions. Prosper Dovonon acknowledges financial support from the Social Sciences and Humanities Research Council of Canada. The views expressed here are those of the authors and do not necessarily reflect those of the Federal Reserve Bank of Atlanta or the Federal Reserve System.

Prosper Dovonon (corresponding author) is in the Economics Department, Concordia University, 1455 de Maisonneuve Blvd. West, H 1155, Montreal, Quebec, Canada, H3G 1M8. Nikolay Gospodinov is in the Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309-4470.

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