Too Good to Be True? Fallacies in Evaluating Risk Factor Models

Nikolay Gospodinov, Raymond Kan, and Cesare Robotti

Working Paper 2017-9
November 2017

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This paper is concerned with statistical inference and model evaluation in possibly misspecified and unidentified linear asset-pricing models estimated by maximum likelihood and one-step generalized method of moments. Strikingly, when spurious factors (that is, factors that are uncorrelated with 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. Furthermore, factors that are spurious are selected with high probability, while factors that are useful are driven out of the model. Although ignoring potential misspecification and lack of identification can be very problematic for models with macroeconomic factors, empirical specifications with traded factors (e.g., Fama and French, 1993, and Hou, Xue, and Zhang, 2015) do not suffer of the identification problems documented in this study.

JEL classification: G12, C12, C13

Key words: asset pricing, spurious risk factors, unidentified models, model misspecification, continuously updated GMM, maximum likelihood, goodness-of-fit, rank test


A part of this paper was circulated under the title "Spurious Inference in Unidentified Asset-Pricing Models." The authors thank Seung Ahn, Alex Horenstein, Lei Jiang, Robert Kimmel, Frank Kleibergen, Francisco Peñaranda, Enrique Sentana, Chu Zhang, seminar participants at Bocconi University, EDHEC, ESSEC, Federal Reserve Bank of New York, Boston University, Hong Kong University of Science and Technology, National University of Singapore, Queen Mary University of London, University of California-San Diego, University of Cantabria, University of Exeter, University of Geneva, University of Hong Kong, University of Lugano, University of Reading, University of Rome Tor Vergata, University of Southampton, Vanderbilt University, 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 China International Conference in Finance, the 2014 NBER-NSF Time-Series Conference, the 2014 Northern Finance Association Conference, the 2014 SoFiE Conference, the 2014 Tsinghua Finance Workshop, the 2015 Brunel Workshop in Empirical Finance, the 2015 Toulouse Financial Econometrics Conference, and the 2015 York Conference on Macroeconomic, Financial, and International Linkages for helpful discussions and comments. 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 corresponding author: Nikolay Gospodinov, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street NE, Atlanta, GA 30309, 404-498-7892, nikolay.gospodinov@atl.frb.org; Raymond Kan, Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario M5S 3E6, Canada, kan@chass.utoronto.ca; or Cesare Robotti, Department of Finance, Terry College of Business, University of Georgia, B328 Amos Hall, 620 South Lumpkin Street, Athens, GA 30602, robotti@uga.edu.
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