Nikolay Gospodinov, Raymond Kan, and Cesare Robotti
Working Paper 2014-12
August 2014

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This paper studies some seemingly anomalous results that arise in possibly misspecified and unidentified linear asset-pricing models estimated by maximum likelihood and one-step generalized method of moments (GMM). Strikingly, when useless factors (that is, factors that are independent of the returns on the test assets) are present, the models exhibit perfect fit, as measured by the squared correlation between the model's fitted expected returns and the average realized returns, and the tests for correct model specification have asymptotic power that is equal to the nominal size. In other words, applied researchers will erroneously conclude that the model is correctly specified even when the degree of misspecification is arbitrarily large. We also derive the highly nonstandard limiting behavior of these invariant estimators and their t-tests in the presence of identification failure. These results reveal the spurious nature of inference as useless factors are selected with high probability, while useful factors are driven out from the model. The practical relevance of our findings is demonstrated using simulations and an empirical application.

JEL classification: C12; C13; G12

Key words: asset pricing, irrelevant risk factors, unidentified models, model misspecification, continuously updated GMM, maximum likelihood, rank test, test for overidentifying restrictions

The authors thank Seung Ahn, Alex Horenstein, Lei Jiang, Robert Kimmel, Francisco Peñaranda, Enrique Sentana, Chu Zhang, seminar participants at EDHEC Business School, the National University of Singapore, the University of Cantabria, the University of Geneva, and Western University as well as conference participants at the 2013 All-Georgia Finance Conference, the 2013 Metro-Atlanta Econometric Study Group Meeting, the 2013 Seventh International Conference on Computational and Financial Econometrics, the 2014 Society for Financial Economics conference, the 2014 Tsinghua Finance Workshop, and the 2014 China International Conference in Finance for helpful discussions and comments. Kan gratefully acknowledges financial support from the Social Sciences and Humanities Research Council of Canada and the National Bank Financial of Canada. The views expressed here are the authors' and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors' responsibility.

Please address questions regarding content to Nikolay Gospodinov (corresponding author), Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309, 404-498-7892,; Raymond Kan, Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6,; or Cesare Robotti, Imperial College Business School, Tanaka Building, South Kensington Campus, London, SW7 2AZ, United Kingdom, 44 (0)20 7594 2682,

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