Empirical Tests of Arbitrage Pricing any given set of assets, Shanken demonstrates that factor analysis can produce many different factor structures from the manipulated portfolios In the extreme case where the constructed portfolios are mutually uncorrelated factor analysis produces no common factor. Of course, forming uncorrelated portfolios by longing and shorting securities merely repackages the risk bearing and potential reward associated with the original securities and does not alter the fundamental forces and characteristics inherent in the economy. However, as a statistical tool, factor analysis can no longer detect those pervasive forces from such manipulated portfolios. This, of course, should not be construed as a criticism of the theory or of the testability of the aPt, but rather should serve as a reminder of the potential problems involved in doing statistical analysis on unrepresentative samples. In this paper, we select 180 securities for each of the initial factor analyses. If we miss an important factor because of unrepresentativeness, all the tests that follow will be biased against the apt C. Testable Hypothesis of the APT We regard Equation(2)as the main result of the apt that explains the cross- sectional differences in asset returns, and it is(2)that will be tested in the following sections A logical first step in testing(2)would be to look for priced factors. However, the task of finding priced factors turns out to be not particularly straightforward If we have determined that k factors exist (in the sense of Connor [13 ])in the generating process of asset returns in the economy, then the number of priced factors-as long as there is at least one-is not well defined. Intuitively this can be most easily seen by noting that a k factor pricing equation can always be collapsed into a single beta equation via mean-variance efficient set mathematics (and with an additional orthogonal transformation similar to that described below, the number of priced factors can be arbitrary) Mathematically, let u be the risk premium vector for a particular set of risk factors and let u be any vector of the same dimension with u'u=U'u. Then there exists an orthogonal trans formation that will transform the original set of factors to a new set of orthogonal factors whose associated risk premium vector is u. In other words, if it has been established that k factors are present, the number of priced factors can be any number between 1 and k. It should be emphasized here however, that those factors that are not priced are just as important as those that are priced in an individuals investment decision(see Breeden [4], Constantinides [12] and roll [32] for related issues). They are irrelevant only in predicting expected return This should be borne in mind when interpreting the cross-sectional results in Section Ill a question that naturally arises in this investigation is how the aPt fa against other asset pricing models. It is immediately apparent from Equations The term "pervasive forces"was made popular by Connor [13]. See Shanken [37 for his interpretation of Connor's result and its implications to the aPt, Contrary to some beliefs, Shanken's results were not driven by the idiosync erms in a finite sample or the approximate nature of Ross'original formulation. The no common factor result can be obtained in an economy with exactly k factors and no idiosyncratic risks