Reference Variables, Factor Structure, and the Approximate Multibeta Representation



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    • Faculty of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa 32000, Israel. I received helpful comments from seminar participants at the University of Maryland, the University of Rochester, Washington University, and Yale University. I am especially grateful to J. Shanken for suggestions and comments that improved the exposition of the paper.


The Arbitrage Pricing Theory (APT) implies that if asset returns have a factor structure, then an approximate multibeta representation holds with respect to the factors as reference variables. This paper assumes that asset returns satisfy a factor structure and derives a condition under which the approximate multibeta representation holds with respect to a set of reference variables which may not be the factors. This condition is that the regression matrix of the reference variables on the factors is nonsingular. Implications for the testability of the APT are also discussed.