On the empirical identification of risk factors in arbitrage pricing models

In the last ten years multiple factor models were widely used both in the academic and the real world as the leading paradigm describing the behaviour of stock returns. Unfortunately, so far academic research has not succeeded in theoretically deriving which factors influence stock returns. An important class of arbitrage pricing theory (APT) models are the fundamental factor models. They rely on the empirical finding that company attributes such as market capitalization, book to market ratio, dividend yield, etc. are posited to have an impact on average stock returns. As a result of a multiple cross-sectional regression one determines which variables have discriminatory power, i.e. have significant factor returns. Another important class of APT models are the macroeconomic factor models. These models assume that the addition of macroeconomic variables leads to an improvement in the explanation of the cross-section of expected returns. A security's sensitivities to the factors are called the factor betas of the security. The macroeconomic factor models estimate a firm's factor betas by time-series regression. In the present paper it is shown that empirical tests of multiple factor models based on the cross-section of sample mean returns may be misleading. In general, the empirical studies are based on a pricing relation that contains idiosyncratic risk. Then, the correct econometric specification results in a cross-sectional regression model that can no longer be estimated because there are asset specific intercepts. Estimating a traditional cross-sectional relationship between sample mean returns and factor betas (including a single intercept) renders the least squares estimates of the regression coefficients biased. As a consequence conclusions about the significance of the included risk factors are wrong. Therefore, some of the ‘anomalies’ found in empirical studies may be due to this bias. Moreover, it is shown that even in the case of an exact pricing relation the underlying factors of the return generating process must be known. However, this contradicts the attempt to identify the risk factors empirically using a cross-sectional regression model.