American Finance Association Yes, The APT is Testable Author(s): Philip H. Dybvig and Stephen A. Ross Source: The Journal of Finance, Vol. 40, No. 4 (Sep., 1985), pp. 1173-1188 Published by: Blackwell Publishing for the American Finance Association Stable URL: http://www.jstor.org/stable/2328401 Accessed: 15/11/2010 00:35 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. Blackwell Publishing and American Finance Association are collaborating with JSTOR to digitize, preserve and extend access to The Journal of Finance. http://www.jstor.org

THE JOURNAL OF FINANCE * VOL. XL, NO. 4 * SEPTEMBER 1985 Yes, The APT Is Testable PHILIP H. DYBVIG and STEPHEN A. ROSS* ABSTRACT The Arbitrage Pricing Theory (APT) has been proposed as an alternative to the mean- variance Capital Asset Pricing Model (CAPM). This paper considers the testability of the APT and points out the irrelevance for testing of the approximation error. We refute Shanken's objections, including his assertion that Roll's critique of the CAPM is applicable to the APT. We also explain the testability of the APT on subsets, and we explore the relationship between the APT and the CAPM. THE ARBITRAGE PRICING THEORY (APT) has been the subject of extensive research efforts since it was introduced by Ross [36, 37]. The research has included theoretical papers' and a growing empirical literature.2 Nonetheless, there seems to be some confusion in the literature about the relation between the theoretical underpinnings of the APT and its empirical tests. This paper is an attempt to alleviate the confusion. To illustrate the issues and to provide a focus, Section I of this paper is devoted to examining and refuting some points raised in Shanken [40] which considers the testability of the APT. The underlying premise of the APT is that asset returns, Ri, are generated by a factor model of the form Ri = it + j^=1 bijbj + si, (1) where a is a vector of systematic factors which commonly influence assets returns, * Both authors are from the Yale School of Management, Yale University. The authors are grateful for valuable comments from Steve Brown, Jon Ingersoll, Dick Roll, Jay Shanken, and Mark Weinstein. We would like to recommend related papers by Pfleiderer [28] and Pfleiderer and Reiss [29]. Dybvig is grateful for support under the Batterymarch fellowship program, and Ross is grateful for support from the National Science Foundation. 1 On the theoretical front, several separate but related strands can be identified. Chamberlain and Rothschild [5] Huberman [22], Ingersoll [24], and Stambaugh [42] have derived the APT using a definition of no arbitrage in an economy with infinitely many assets. Connor [11], Dybvig [17], and Grinblatt and Titman [21] derive the APT in an equilibrium setting and have refined the degree of approximation. Chen and Ingersoll [8], Cragg and Malkiel [14], and Dybvig [16] have derived the APT by assuming that some agent holds a well-diversified portfolio. Brock [1, 2], Constantinides [12], Cox, Ingersoll, and Ross [13], Garman and Ohlsen [18], Prescott [30], Roll and Ross [35], and Schmalensee [39] have examined the intertemporal consistency of the theory in dynamic equilibrium models. Shanken [41] futher examines equilibrium derivations of the APT. 2 APT has been examined by Brown and Weinstein [3], Chen [7], Chan, Chen, and Hsieh [6], Chen, Roll, and Ross [9], Dhrymes, Friend, and Gultenkin [15], Cho, Elton, and Gruber [10], Gehr [19], Gibbons [20], Hughes [23], Jobson [25], Oldfield and Rogalski [27], Reinganum [31], and Roll and Ross [35]. (Without judging its other merits, e.g., relative power, the Brown and Weinstein test conclusively demonstrates that the APT is testable, since their test is a maximum likelihood test.) 1173

1174 The Journal of Finance fi is the vector of factor loadings on asset i, Ei is the expected return on asset i, and ei is the idiosyncratic noise term associated with asset i. Although weaker assumptions suffice, for concreteness we can assume that ii is independent across assets, is independent of the factors, and has a mean of zero. We will also assume that the factors, &j, have mean zero and that variances exist.3 The APT says simply that the new assets' expected returns are linear in the factor loadings: = r + E=1 Xj, (2) where r is a constant (which equals the riskless rate of return if there is a riskless asset), and X is a vector of risk premia. The "Arbitrage" in the "Arbitrage Pricing Theory" comes from the simple proof that (1) implies (2), using absence of arbitrage alone, for the case in which all the si's are identically zero.4 More generally, deriving the APT involves proving that the si's can be neglected when pricing assets. Of the many existing derivations of the APT, most are based on the intuition that the idiosyncratic error terms represent diversifiable risk and therefore should not affect pricing. Because diversification is a notion that holds only approximately in a finite economy, theoretical motivations of the APT have generally dealt in approximations. The theoretical derivations of the APT differ both in the sense of approximation and in the additional assumptions that are made.5 Some theoretical models conclude that the APT is a good approxi- mation in a sequence economy, when there are "sufficiently many" assets (Cham- berlain [4], Chamberlain and Rothschild [5], Huberman [22], Ingersoll [24], and Ross [37]) or in a finer sense (Connor [11]).6 Other models show that the APT should be a good approximation in a finite economy (Cragg and Malkiel [14], Dybvig [17], Grinblatt and Titman [21], and in continuous time, Ross [38]), with 'Chamberlain [4], Chamberlain and Rothschild [5], Connor [11], Ingersoll [24], and Stambaugh [42], have significantly weakened the requirement of independent idiosyncratic error terms. One message that emerges from these papers is that a factor model is essentially one in which the covariance matrix has k large eigenvalues. Another message is that what is important is that the factors be separately identifiable from asset returns, i.e. that each factor be approximated closely by some linear combination of asset returns. In Connor's model, this assumption is combined with an assumption that no asset's idiosyncratic noise plays a significant role in aggregated wealth. 4Suppose that all the Zi's are identically zero. If the linearity condition in (2) did not hold, there would be two portfolios with the same factor loadings but different expected returns. Consequently, going long in the portfolio with higher expected return, financed by shorting the other portfolio, would be an arbitrage opportunity. 5 This has led some researchers to conclude that empirical investigations of the APT should account explicitly for the error term arising from one or another theoretical model. However, including the error term in an empirical test defeats the whole purpose of the theoretical models, which are intended to demonstrate that the error term is negligible. Furthermore, including the error term would require a new empirical test for each such version of the model. Fortunately, the notion of including the error term in the test appears in methodological discussions but not in the actual empirical tests. 6 Specifically, the first papers show that for whatever factor structure we extract, then (E - re - Xj) ' -1(E - re - XA) is uniformly bounded as more assets are added, where z is the covariance matrix for the residual "idiosyncratic" noise and e is a vector of ones. Note that z could even be the original covariance matrix. Connor shows that the sum across assets of squared deviations, (, - re - Xf) ' (, - re - Xf3), goes to zero as assets are added. Connor also admits a general residual idiosyncratic noise matrix, but makes assumptions to ensure that the APT is valid. (One of Connor's assumptions is that 11 (( '-11)'I -* 0 as the number of assets increases.)