Weighing asset pricing factors: a least squares model averaging approach

Abstract

Empirical evidence has demonstrated that certain factors in asset pricing models are more important than others for explaining specific portfolio returns. We propose a technique that evaluates the factors included in popular linear asset pricing models. Our method has the advantage of simultaneously ranking the relative importance of those pricing factors through comparing their model weights. As an empirical verification, we apply our method to portfolios formed following Fama and French [A five-factor asset pricing model. J. Financ. Econ., 2015, 116, 1–22] and demonstrate that models accommodated to our factor rankings do improve their explanatory power in both in-sample and out-of-sample analyses.

Keywords:

• Asset pricing
• HJ-distance
• Model averaging
• Model screening

JEL classification:

• C1
• G12

Acknowledgements

We wish to thank Kee-Hong Bae, Bruno Solnik, Jun Yu, Lu Zhang, Xinyu Zhang, seminar participants at the European Financial Management Association (EFMA) 2017 Xiamen annual meeting, Young Econometricians around Pacific (YEAP) 2017 annual conference, Chinese Academy of Sciences, Renmin University, and Xiamen University for helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

† Examples include Campbell (1996), Jagannathan and Wang (1996), Lettau and Ludvigson (2001), Cohen (2012), Cohen and Lou (2012), Kim (2012), Moskowitz (2012), Palazzo (2012), Eiling (2013), Hirshleifer (2013), Fama and French (2015). See Campbell (2000) and Goyal (2012) for a literature review.

‡ Their finding is supported by Kan and Robotti (2009) and Barillas and Shanken (2015) that the empirical performance of an asset pricing model can vary vastly across different samples.

§ In a pioneering study, Hansen (2007) proposed the Mallows model average (MMA) estimator which calculates weights based on the original Mallows' $C_p$ proposed by Mallows (1973). In the past two decades, many least squares model averaging methods have been proposed, for example, Wan (2010), Hansen and Racine (2012), Liu and Okui (2013), Xie (2015), and Zhang (2015), among others.

† In their seminal book, Burnham and Anderson (2002) demonstrated the advantage of using model averaging relative to other methods (model selection, statistical tests, etc.) for model inference. Specifically, Burnham and Anderson (2002) recommended using cumulative weights to assess the relative importance of factors.

‡ See, for example, Grueber et al. (2011), Mundry (2011), and Symonds and Moussalli (2011). Burnham (2011) provide an extensive literature survey.

† Note that the sample HJ distance in equation (2(2) ${\mathrm{HJ}}_T^s\left({\mathit{\delta }}^s\right)\equiv \sqrt{{\mathit{h}}^s\left({\mathit{\delta }}^s{)}^{\mathrm{\top }}{\mathit{G}}^{-1}{\mathit{h}}^s\right({\mathit{\delta }}^s)},$(2) ) is also well defined for other choices of ${\mathit{\delta }}^s$. We choose equation (4(4) ${\hat{\mathit{\delta }}}^s={\left(({\mathit{D}}^s{)}^{\mathrm{\top }}{\mathit{G}}^{-1}{\mathit{D}}^s\right)}^{-1}({\mathit{D}}^s{)}^{\mathrm{\top }}{\mathit{G}}^{-1}{\mathbf{1}}_N,$(4) ) because it is the most popular way to calculate the HJ distance in the empirical literature.

‡ In our experiment, the estimation bias of ${\hat{\mathit{\delta }}}^s$ by equation (4(4) ${\hat{\mathit{\delta }}}^s={\left(({\mathit{D}}^s{)}^{\mathrm{\top }}{\mathit{G}}^{-1}{\mathit{D}}^s\right)}^{-1}({\mathit{D}}^s{)}^{\mathrm{\top }}{\mathit{G}}^{-1}{\mathbf{1}}_N,$(4) ) can be quite large when N and $k^s$ are close. However, if we fix N and $k^s$, then this bias vanishes quickly as T increases to infinity. Such incidents are well interpreted by the panel data literature, such as Bai and Ng (2002).

§ The vector of the pricing errors ${\mathit{h}}^s\left({\mathit{\delta }}^s\right)$ measures the errors of the pricing model s. Therefore, it is viable to construct a risk function based on ${\mathit{h}}^s\left({\mathit{\delta }}^s\right)$ such that $L\left({\mathit{h}}^s\right({\mathit{\delta }}^s\left)\right):{\mathbb{R}}^N\to {\mathbb{R}}^1$. By searching through various possible choices of $\mathit{w}$, we can always find a specific $\hat{\mathit{w}}$ such that $L\left(\mathit{h}\right(\hat{\mathit{w}}\left)\right)$ is smaller than any $L\left({\mathit{h}}^s\right({\mathit{\delta }}^s\left)\right)$. Even in the extreme case, where there exists a dominating model among the approximation models that yields the lowest possible value of $infL$, the optimal weighting vector ${\mathit{w}}_{opt}$ should assign a weight of 1 to this model and 0 to the other models such that $L\left(\mathit{h}\right({\mathit{w}}_{opt}\left)\right)=infL$.

¶ The performance of an asset pricing model is measured by the term $\mathit{h}\left(\mathit{w}{)}^{\mathrm{\top }}{\mathit{G}}^{-1}\mathit{h}\right(\mathit{w})$, which equals the square of the HJ distance defined in (2(2) ${\mathrm{HJ}}_T^s\left({\mathit{\delta }}^s\right)\equiv \sqrt{{\mathit{h}}^s\left({\mathit{\delta }}^s{)}^{\mathrm{\top }}{\mathit{G}}^{-1}{\mathit{h}}^s\right({\mathit{\delta }}^s)},$(2) ). The complexity of a model is measured by the penalty term $(N+k(\mathit{w}\left)\right)/(N-k(\mathit{w}\left)\right)$. Note that a linear asset pricing model's performance can be improved by adding extra factors that are proved to be appropriate. This decreases the value of $\mathit{h}\left(\mathit{w}{)}^{\mathrm{\top }}{\mathit{G}}^{-1}\mathit{h}\right(\mathit{w})$, which is at the cost of simultaneously increasing the value of the penalty term $(N+k(\mathit{w}\left)\right)/(N-k(\mathit{w}\left)\right)$ through a higher effective number of parameters $k\left(\mathit{w}\right)$.

† Due to the nonlinearity, the distribution of the averaged estimator $h\left(\mathit{w}\right)$ is not a conventional distribution. In spirit of Liu (2015), the limiting distribution of $h\left(\mathit{w}\right)$ can be derived through a plug-in type estimator in a local asymptotic framework. We leave this for future research.

† In fact, as shown in many theoretical works, for example, Wan (2010), Xie (2015), and Zhang (2015), it is necessary to keep the total number of candidate models small or have it converge to the infinity slowly enough, so as to preserve the asymptotic optimality of the model averaging estimators.

‡ As long as the values of c and L are tolerant enough, the results are shown to be insensitive to c and L. In the extreme case, c=0.99 and $L=2^{\tilde{k}}-1$. In our empirical application, we set c=0.5 and L=100.

§ Note that L is an arbitrary parameter chosen by the researcher. How to choose or estimate the optimal number of L is beyond the scope of this paper, and we leave it for future research.

† For example, see Campbell (1996), Lettau and Ludvigson (2001), Kan and Robotti (2009), Moskowitz (2012), Hirshleifer (2013), Cohen (2012), and Cohen and Lou (2012), among others.

‡ In fact, we compare our estimation results based on both the screened and unscreened model sets. The results are similar.

† Note that we only include part of the factor ranking results for presentation convenience. Additional results are available on request.

1. We suppose the dimension of $\mathcal{F}$ to be K and $i=1,\dots ,K$.

Additional information

Funding

This paper is partially supported by the Natural Science Foundation of China (71701175), Ministry of Education of the People's Republic of China (17YJC790174), under Project of Humanities and Social Sciences. Ren's research is supported by the Natural Science Foundation of China (71771192, 71631004). The usual caveat applies.