Approximate Factor Structures: Interpretations and Implications for Empirical Tests

Abstract

ABSTRACT This paper provides some new insights about approximate factor structures, as defined by Chamberlain and Rothschild [2], and their implications for empirical tests. First, we show that any economy that satisfies an approximate factor structure can be transformed, in a manner that does not alter the characteristics of investor portfolios, into an economy that satisfies an exact factor structure, as defined by Ross [9]. Second, we show that principal components analysis represents just one of many methods of forming groups of well‐diversified portfolios with no idiosyncratic risk in large samples. Correct factor loadings will be obtained by regressing security returns on any group of these portfolios. Our interpretations of the Chamberlain and Rothschild results also provide additional insights into the testability of the Arbitrage Pricing Theory. We show that securities cannot be repackaged to hide factors in the manner suggested by Shanken [10] without the variance of some of the repackaged securities approaching infinity in large economies.

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