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An important issue in applications of multifactor models of asset returns is the appropriate number of factors. Most extant tests for the number of factors are valid only for strict factor models, in which diversifiable returns are uncorrelated across assets. In this paper we develop a test statistic to determine the number of factors in an approximate factor model of asset returns, which does not require that diversifiable components of returns be uncorrelated across assets. We find evidence for one to six pervasive factors in the cross-section of New York Stock Exchange and American Stock Exchange stock returns. (1976) has generated an increased interest in the application of linear factor models in the study of capital asset pricing. The APT has the attractive feature that it makes a minimal number of assumptions about the nature of the economy (a factor structure for the returns generating process, a large number of assets, and frictionless trading). The costs of these minimalist assumptions include certain ambiguities such as an approximate pricing relation and an unknown number of pervasive factors. THE ARBITRAGE PRICING THEORY (APT) of RossIn order to estimate and test the APT, one must specify the number of pervasive factors in asset returns. The issue of the appropriate number of factors has been the subject of some controversy (see, for example, Roll and Ross (1980, 1984); Dhrymes, Friend, and Gultekin (1984); Luedecke (1984); Trzcinka (1986); Conway and Reinganum (1988); and Brown (1989)). In this paper we propose a new approach to estimating the number of pervasive economic factors generating asset returns. An important feature of our approach is that it is valid when asset returns follow an approximate, rather than a strict, factor model.A strict factor structure is one in which the idiosyncratic, or diversifiable, components of asset returns have zero correlation across assets. Ross (1976) assumes a strict factor structure in his original development of the APT. However, he notes that this assumption can be weakened. The key requirement for the APT is that nonfactor risk can be diversified away in many-asset

This paper derives Ross's mutual fund separation theory and a new, equilibrium version of Ross's arbitrage pricing theory as special cases of a general theory. The paper also reveals that the two theories are identical in their predictions of asset prices and portfolio returns. The capital asset pricing model (a restricted case of the mutual fund separation theory) receives special treatment.

With some blurring at the boundaries, multifactor models of asset returns can be divided into three types: macroeconomic, statistical, and fundamental. Our empirical findings confirm the conventional wisdom that statistical factor models and fundamental factor models outperform macroeconomic factor models in tenns of explanatory power. The findings also indicate that the fundamental factor model slightly outperforms the statistical factor model. This result is at first surprising, because statistical factor models are estimated by maximizing explanatory power. So, how can an alternative outperform them by this criterion? The explanation lies in the much larger number of external data sources used in fundamental factor models, particularly the large set of industry dummies. Another empirical finding is that the marginal explanatory power of a macroeconomic factor model is zero when it is added to the fundamental factor model. This result may indicate that the fundamental factors {in some unknown combination) capture the same risk characteristics as the macroeconomic factors. TYPES OF FACTOR MODELSMacroeconomic factor models are the simplest and most intuitive type. They use observable economic time series as measures of the pervasive factors in security returns.^ Some of the macroeconomic variables typically used as factors are inflation^ the Gregory Connor is director of research, Europe, for BARRA international. percentage change in industrial production, the excess retum to long-term government bonds, and the realized return premium of low-grade corporate bonds relative to high-grade bonds. The random retum of each security is assumed to respond linearly to the macroeconomic shocks. As in all factor models, each security also has an assetspecific retum unrelated to the factors. A security's linear sensitivities to the factors are called the factor betas of the security. A drawback to macroeconomic factor models is that they require identification and measurement of all the pervasive shocks affecting security returns. A small number of pervasive sources of risk may exist, but without knowing exactly what they are, or lacking data to measure them, they are of little use in explaining returns.Statistical factor models use various maximumlikelihood and principal-components-based factor analysis procedures on cross-sectional/time-series samples of security returns to identify the pervasive factors in returns.Macroeconomic and statistical factor models both estimate a firm's factor beta by time-series regression. Given the nature of security returns data, this limitation is substantial. Time-series regression requires a long and stable history of returns for a security to estimate the factor betas accurately.Fundamental factor models do not require timeseries regression. They rely on the empirical finding that company attributes such as firm size, dividend yield, book-to-market ratio, and industry 42

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