Spurious Factors in Linear Asset Pricing Models

When a risk factor has small covariance with asset returns, risk premia in the linear asset pricing models are no longer identified. Weak factors, similar to weak instruments, make the usual estimation techniques unreliable. When included in the model, they generate spuriously high significance levels of their own risk premia estimates, overall measures of fit and may crowd out the impact of the true sources of risk. I develop a new approach to the estimation of cross-sectional asset pricing models that : a) provides simultaneous model diagnostics and parameter estimates; b) automatically removes the effect of spurious factors; c) restores consistency and asymptotic normality of the parameter estimates, as well as the accuracy of standard measures of fit; d) performs well in both small and large samples. I provide new insights on the pricing ability of various factors proposed in the literature. In particular, I identify a set of robust factors (e.g. Fama-French ones, but not only), and those that suffer from severe identification problems that render the standard assessment of their pricing performance unreliable (e.g. consumption growth, human capital proxies and others). JEL classification: G12, C13, C58. Sharpe (1964) and Lintner (1965) CAPM pioneered the class of linear factor models in asset pricing. Now, decades later, what started as an elegant framework has turned into a well-established and successful tradition in finance. Linear models, thanks to their inherent simplicity and ease of interpretation, are widely used as a reference point in much of the empirical work, having been applied to nearly all kinds of financial assets1. In retrospect, however, such heavy use produced a rather puzzling outcome: Harvey, Liu, and Zhu (2013) list over 300 factors proposed in the literature, all of which have been claimed as important (and significant) drivers of the cross-sectional variation in stock returns2. One of the reasons for such a wide range of apparently significant risk factors is perhaps a simple lack of model identification, and consequently, an invalid inference about risk premia parameters. As pointed out in a growing number of papers (see e.g. Jagannathan and Wang (1998), Kan and Zhang (1999b), Kleibergen (2009), Kleibergen and Zhan (2013), Burnside (2010), Gospodinov, Kan, and Robotti (2014a)), in the presence of factors that only weakly correlate with assets (or do not correlate at all), all the risk premia parameters are no longer strongly identified and standard estimation and inference techniques become unreliable. As a result, identification failure often leads to the erroneous conclusion that such factors are important, although they are totally spurious by nature. The impact of the true factors could, in turn, be crowded out from the model. The shrinkage-based estimators that I propose (Pen-FM and Pen-GMM, from the penalised version of the Fama-MacBeth procedure or GMM, accordingly), not only allow to detect the overall problem of rank-deficiency caused by irrelevant factors, but also indicate which particular variables are causing it, and recover the impact of strong risk factors without compromising any of its properties (e.g. consistency, asymptotic normality, etc). My estimator can bypass the identification problem because, in the case of useless (or weak) factors, we know that it stems from the low correlation between these variables and asset returns. This, in turn, is reflected in the regression-based estimates of betas, asset exposures to the corresponding sources of risk. Therefore, one can use the L1−norm of the vector of β̂’s (or related quantities) to assess the overall factor strength for a given cross-section of returns, and successfully isolate the cases when it is close to zero. Consequently, I modify the second stage of the Fama-MacBeth procedure3 (or the GMM objective function) to include a penalty that is inversely proportional to the factor strength, measured by the L1−norm of the vector β̂. Notable examples are the 3-factor model of Fama and French (1992), Fama and French (1993); the conditional CAPM of Jagannathan and Wang (1996); the conditional CCAPM of Lettau and Ludvigson (2001b), the Q1-Q4 consumption growth of Jagannathan and Wang (2007), the durable/nondurable consumption CAPM of Yogo (2006); the ultimate consumption risk of Parker and Julliard (2005); the pricing of currency portfolios in Lustig and Verdelhan (2007) and Lustig, Roussanov, and Verdelhan (2011); and the regression-based approach to the term structure of interest rates in Adrian, Crump, and Moench (2013) In the context of predictive regressions, Novy-Marx (2014) recently demonstrated that many unconventional factors, such as the party of the U.S. President, sunspots, the weather in Manhattan, plannet location and the El-Nino phenomenon have a statistically significant power for the performance of many popular trading strategies, such as those based on market capitalisation, momentum, gross profitability, earnings surprises and others. The problem of identification is not a consequence of having several stages in the estimation. It is well known that the two-pass procedure gives exactly the same point estimates as GMM with the identity weight matrix under a particular moment normalisation.