Chapter 4 Factor investing and asset pricing anomalies

Asset pricing anomalies are the foundations of factor investing. In this chapter our aim is twofold:

  • present simple ideas and concepts: basic factor models and common empirical facts (time-varying nature of returns and risk premia);
  • provide the reader with lists of articles that go much deeper to stimulate and satisfy curiosity.

The purpose of this chapter is not to provide a full treatment of the many topics related to factor investing. Rather, it is intended to give a broad overview and cover the essential themes so that the reader is guided towards the relevant references. As such, it can serve as a short, non-exhaustive, review of the literature. The subject of factor modelling in finance is incredibly vast and the number of papers dedicated to it is substantial and still rapidly increasing.

The universe of peer-reviewed financial journals can be split in two. The first kind is the academic journals. Their articles are mostly written by professors, and the audience consists mostly of scholars. The articles are long and often technical. Prominent examples are the Journal of Finance, the Review of Financial Studies and the Journal of Financial Economics. The second type is more practitioner-orientated. The papers are shorter, easier to read, and target finance professionals predominantly. Two emblematic examples are the Journal of Portfolio Management and the Financial Analysts Journal. This chapter reviews and mentions articles published essentially in the first family of journals.

Beyond academic articles, several monographs are already dedicated to the topic of style allocation (a synonym of factor investing used for instance in theoretical articles (Barberis and Shleifer (2003)) or practitioner papers (Asness et al. (2015))). To cite but a few, we mention:

  • Ilmanen (2011): an exhaustive excursion into risk premia, across many asset classes, with a large spectrum of descriptive statistics (across factors and periods),
  • Ang (2014): covers factor investing with a strong focus on the money management industry,
  • Bali, Engle, and Murray (2016): very complete book on the cross-section of signals with statistical analyses (univariate metrics, correlations, persistence, etc.),
  • Jurczenko (2017): a tour on various topics given by field experts (factor purity, predictability, selection versus weighting, factor timing, etc.).

Finally, we mention a few wide-scope papers on this topic: Goyal (2012), Cazalet and Roncalli (2014) and Baz et al. (2015).

4.1 Introduction

The topic of factor investing, though a decades-old academic theme, has gained traction concurrently with the rise of equity traded funds (ETFs) as vectors of investment. Both have gathered momentum in the 2010 decade. Not so surprisingly, the feedback loop between practical financial engineering and academic research has stimulated both sides in a mutually beneficial manner. Practitioners rely on key scholarly findings (e.g., asset pricing anomalies) while researchers dig deeper into pragmatic topics (e.g., factor exposure or transaction costs). Recently, researchers have also tried to quantify and qualify the impact of factor indices on financial markets. For instance, Krkoska and Schenk-Hoppé (2019) analyze herding behaviors while Cong and Xu (2019) show that the introduction of composite securities increases volatility and cross-asset correlations.

The core aim of factor models is to understand the drivers of asset prices. Broadly speaking, the rationale behind factor investing is that the financial performance of firms depends on factors, whether they be latent and unobservable, or related to intrinsic characteristics (like accounting ratios for instance). Indeed, as Cochrane (2011) frames it, the first essential question is which characteristics really provide independent information about average returns?. Answering this question helps understand the cross-section of returns and may open the door to their prediction.

Theoretically, linear factor models can be viewed as special cases of the arbitrage pricing theory (APT) of Ross (1976), which assumes that the return of an asset \(n\) can be modelled as a linear combination of underlying factors \(f_k\): \[\begin{equation} \tag{4.1} r_{t,n}= \alpha_n+\sum_{k=1}^K\beta_{n,k}f_{t,k}+\epsilon_{t,n}, \end{equation}\]

where the usual econometric constraints on linear models hold: \(\mathbb{E}[\epsilon_{t,n}]=0\), \(\text{cov}(\epsilon_{t,n},\epsilon_{t,m})=0\) for \(n\neq m\) and \(\text{cov}(\textbf{f}_n,\boldsymbol{\epsilon}_n)=0\). If such factors do exist, then they are in contradiction with the cornerstone model in asset pricing: the capital asset pricing model (CAPM) of Sharpe (1964), Lintner (1965) and Mossin (1966). Indeed, according to the CAPM, the only driver of returns is the market portfolio. This explains why factors are also called ‘anomalies’.

Empirical evidence of asset pricing anomalies has accumulated since the dual publication of Fama and French (1992) and Fama and French (1993). This seminal work has paved the way for a blossoming stream of literature that has its meta-studies (e.g., Green, Hand, and Zhang (2013), Harvey, Liu, and Zhu (2016) and McLean and Pontiff (2016)). The regression (4.1) can be evaluated once (unconditionally) or sequentially over different time frames. In the latter case, the parameters (coefficient estimates) change and the models are thus called conditional (we refer to Ang and Kristensen (2012) and to Cooper and Maio (2019) for recent results on this topic as well as for a detailed review on the related research). Conditional models are more flexible because they acknowledge that the drivers of asset prices may not be constant, which seems like a reasonable postulate.

4.2 Detecting anomalies

4.2.1 Challenges

Obviously, a crucial step is to be able to identify an anomaly and the complexity of this task should not be underestimated. Given the publication bias towards positive results (see, e.g., Harvey (2017) in financial economics), researchers are often tempted to report partial results that are sometimes invalidated by further studies. The need for replication is therefore high and many findings have no tomorrow (Linnainmaa and Roberts (2018)), especially if transation costs are taken into account (Patton and Weller (2020), A. Y. Chen and Velikov (2020)). Nevertheless, as is demonstrate by Chen (2019), \(p\)-hacking alone cannot account for all the anomalies documented in the literature. One way to reduce the risk of spurious detection is to increase the hurdles (often, the \(t\)-statistics) but the debate is still ongoing (Harvey, Liu, and Zhu (2016), A. Y. Chen (2020)), or to resort to multiple testing (Harvey, Liu, and Saretto (2020)).

Some researchers document fading anomalies because of publication: once the anomaly becomes public, agents invest in it, which pushes prices up and the anomaly disappears. McLean and Pontiff (2016) document this effect in the US but Jacobs and Müller (2020) find that all other countries experience sustained post-publication factor returns. With a different methodology, A. Y. Chen and Zimmermann (2020) introduce a publication bias adjustment for returns and the authors note that this (negative) adjustment is in fact rather small. Penasse (2019) recommends the notion of alpha decay to study the persistence or attenuation of anomalies.

The destruction of factor premia may be due to herding (Krkoska and Schenk-Hoppé (2019), Volpati et al. (2020)) and could be accelerated by the democratization of so-called smart-beta products (ETFs notably) that allow investors to directly invest in particular styles (value, low volatility, etc.). For a theoretical perspective on the attractivity of factor investing, we refer to Jin (2019). On the other hand, DeMiguel, Martin Utrera, and Uppal (2019) argue that the price impact of crowding in the smart-beta universe is mitigated by trading diversification stemming from external institutions that trade according to strategies outside this space (e.g., high frequency traders betting via order-book algorithms).

The remainder of this subsection was inspired from Baker, Luo, and Taliaferro (2017) and C. Harvey and Liu (2019).

4.2.2 Simple portfolio sorts

This is the most common procedure and the one used in Fama and French (1992). The idea is simple. On one date,

  1. rank firms according to a particular criterion (e.g., size, book-to-market ratio);
  2. form \(J\ge 2\) portfolios (i.e., homogeneous groups) consisting of the same number of stocks according to the ranking (usually, \(J=2\), \(J=3\), \(J=5\) or \(J=10\) portfolios are built, based on the median, terciles, quintiles or deciles of the criterion);
  3. the weight of stocks inside the portfolio is either uniform (equal weights), or proportional to market capitalization;
  4. at a future date (usually one month), report the returns of the portfolios.
    Then, iterate the procedure until the chronological end of the sample is reached.

The outcome is a time series of portfolio returns \(r_t^j\) for each grouping \(j\). An anomaly is identified if the \(t\)-test between the first (\(j=1\)) and the last group (\(j=J\)) unveils a significant difference in average returns. More robust tests are described in Cattaneo et al. (2020). A strong limitation of this approach is that the sorting criterion could have a non-monotonic impact on returns and a test based on the two extreme portfolios would not detect it. Several articles address this concern: Patton and Timmermann (2010) and Romano and Wolf (2013) for instance. Another concern is that these sorted portfolios may capture not only the priced risk associated to the characteristic, but also some unpriced risk. K. Daniel et al. (2020) show that it is possible to disentangle the two and make the most of altered sorted portfolios.

Instead of focusing on only one criterion, it is possible to group asset according to more characteristics. The original paper Fama and French (1992) also combines market capitalization with book-to-market ratios. Each characteristic is divided into 10 buckets, which makes 100 portfolios in total. Beyond data availability, there is no upper bound on the number of features that can be included in the sorting process. In fact, some authors investigate more complex sorting algorithms that can manage a potentially large number of characteristics (see e.g., Feng, Polson, and Xu (2019) and Bryzgalova, Pelger, and Zhu (2019)).

Finally, we refer to Ledoit, Wolf, and Zhao (2020) for refinements that take into account the covariance structure of asset returns and to Cattaneo et al. (2020) for a theoretical study on the statistical properties of the sorting procedure (including theoretical links with regression-based approaches). Notably, the latter paper discusses the optimal number of portfolios and suggests that it is probably larger than the usual 10 often used in the literature.

In the code and Figure 4.1 below, we compute size portfolios (equally weighted: above versus below the median capitalization). According to the size anomaly, the firms with below median market cap should earn higher returns on average. This is verified whenever the orange bar in the plot is above the blue one (it happens most of the time).

The size factor: average returns of small versus large firms.

FIGURE 4.1: The size factor: average returns of small versus large firms.

4.2.3 Factors

The construction of so-called factors follows the same lines as above. Portfolios are based on one characteristic and the factor is a long-short ensemble of one extreme portfolio minus the opposite extreme (small minus large for the size factor or high book-to-market ratio minus low book-to-market ratio for the value factor). Sometimes, subtleties include forming bivariate sorts and aggregating several portfolios together, as in the original contribution of Fama and French (1993). The most common factors are listed below, along with a few references. We refer to the books listed at the beginning of the chapter for a more exhaustive treatment of factor idiosyncrasies. For most anomalies, theoretical justifications have been brought forward, whether risk-based or behavioral. We list the most frequently cited factors below:

  • Size (SMB = small firms minus large firms): Banz (1981), Fama and French (1992), Fama and French (1993), Van Dijk (2011), Asness et al. (2018) and Astakhov, Havranek, and Novak (2019).
  • Value (HM = high minus low: undervalued minus `growth’ firms): Fama and French (1992), Fama and French (1993), C. S. Asness, Moskowitz, and Pedersen (2013).
  • Momentum (WML = winners minus losers): Jegadeesh and Titman (1993), Carhart (1997) and C. S. Asness, Moskowitz, and Pedersen (2013). The winners are the assets that have experienced the highest returns over the last year (sometimes the computation of the return is truncated to omit the last month). Cross-sectional momentum is linked, but not equivalent, to time series momentum (trend following), see e.g., Moskowitz, Ooi, and Pedersen (2012) and Lempérière et al. (2014). Momentum is also related to contrarian movements that occur both at higher and lower frequencies (short-term and long-term reversals), see Luo, Subrahmanyam, and Titman (2020).
  • Profitability (RMW = robust minus weak profits): Fama and French (2015), Bouchaud et al. (2019). In the former reference, profitability is measured as (revenues - (cost and expenses))/equity.
  • Investment (CMA = conservative minus aggressive): Fama and French (2015), Hou, Xue, and Zhang (2015). Investment is measured via the growth of total assets (divided by total assets). Aggressive firms are those that experience the largest growth in assets.
  • Low `risk’ (sometimes, BAB = betting against beta): Ang et al. (2006), Baker, Bradley, and Wurgler (2011), Frazzini and Pedersen (2014), Boloorforoosh et al. (2020), Baker, Hoeyer, and Wurgler (2020) and Asness et al. (2020). In this case, the computation of risk changes from one article to the other (simple volatility, market beta, idiosyncratic volatility, etc.).

With the notable exception of the low risk premium, the most mainstream anomalies are kept and updated in the data library of Kenneth French (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). Of course, the computation of the factors follows a particular set of rules, but they are generally accepted in the academic sphere. Another source of data is the AQR repository: https://www.aqr.com/Insights/Datasets.

In the dataset we use for the book, we proxy the value anomaly not with the book-to-market ratio but with the price-to-book ratio (the book value is located in the denominator). As is shown in Clifford Asness and Frazzini (2013), the choice of the variable for value can have sizable effects.

Below, we import data from Ken French’s data library. We will use it later on in the chapter.

TABLE 4.1: Sample of monthly factor returns.
date MKT_RF SMB HML RMW CMA RF
1963-07-31 -0.0039 -0.0047 -0.0083 0.0066 -0.0115 0.0027
1963-08-31 0.0507 -0.0079 0.0167 0.0040 -0.0040 0.0025
1963-09-30 -0.0157 -0.0048 0.0018 -0.0076 0.0024 0.0027
1963-10-31 0.0253 -0.0129 -0.0010 0.0275 -0.0224 0.0029
1963-11-30 -0.0085 -0.0084 0.0171 -0.0045 0.0222 0.0027
1963-12-31 0.0183 -0.0189 -0.0012 0.0007 -0.0030 0.0029

Posterior to the discovery of these stylized facts, some contributions have aimed at building theoretical models that capture these properties. We cite a handful below:

  • size and value: Berk, Green, and Naik (1999), K. D. Daniel, Hirshleifer, and Subrahmanyam (2001), Barberis and Shleifer (2003), Gomes, Kogan, and Zhang (2003), Carlson, Fisher, and Giammarino (2004), Arnott et al. (2014);
  • momentum: Johnson (2002), Grinblatt and Han (2005), Vayanos and Woolley (2013), Choi and Kim (2014).

In addition, recent bridges have been built between risk-based factor representations and behavioural theories. We refer essentially to Barberis, Mukherjee, and Wang (2016) and K. Daniel, Hirshleifer, and Sun (2020) and the references therein.

While these factors (i.e., long-short portfolios) exhibit time-varying risk premia and are magnified by corporate news and announcements (Engelberg, McLean, and Pontiff (2018)), it is well-documented (and accepted) that they deliver positive returns over long horizons. We refer to Gagliardini, Ossola, and Scaillet (2016) and to the survey Gagliardini, Ossola, and Scaillet (2019), as well as to the related bibliography for technical details on estimation procedures of risk premia and the corresponding empirical results. A large sample study that documents regime changes in factor premia was also carried out by Ilmanen et al. (2019). Moreover, the predictability of returns is also time-varying (as documented in Farmer, Schmidt, and Timmermann (2019), Tsiakas, Li, and Zhang (2020) and Liu, Pan, and Wang (2020)), and estimation methods can be improved (Johnson (2019)).

In Figure 4.2, we plot the average monthly return aggregated over each calendar year for five common factors. The risk free rate (which is not a factor per se) is the most stable, while the market factor (aggregate market returns minus the risk-free rate) is the most volatile. This makes sense because it is the only long equity factor among the five series.

Average returns of common anomalies (1963-2020). Source: Ken French library.

FIGURE 4.2: Average returns of common anomalies (1963-2020). Source: Ken French library.

The individual attributes of investors who allocate towards particular factors is a blossoming topic. We list a few references below, even though, they somewhat lie out of the scope of this book. Betermier, Calvet, and Sodini (2017) show that value investors are older, wealthier and face lower income risk compared to growth investors who are those in the best position to take financial risks. The study Cronqvist, Siegel, and Yu (2015) leads to different conclusions: it finds that the propensity to invest in value versus growth assets has roots in genetics and in life events (the latter effect being confirmed in Cocco, Gomes, and Lopes (2020), and the former being further detailed in a more general context in Cronqvist et al. (2015)). Psychological traits can also explain some factors: when agents extrapolate, they are likely to fuel momentum (this topic is thoroughly reviewed in Barberis (2018)). Micro- and macro-economic consequences of these preferences are detailed in Bhamra and Uppal (2019). To conclude this paragraph, we mention that theoretical models have also been proposed that link agents’ preferences and beliefs (via prospect theory) to market anomalies (see for instance Barberis, Jin, and Wang (2020)).

Finally, we highlight the need of replicability of factor premia and echo the recent editorial by Harvey (2020). As is shown by Linnainmaa and Roberts (2018) and Hou, Xue, and Zhang (2020), many proclaimed factors are in fact very much data-dependent and often fail to deliver sustained profitability when the investment universe is altered or when the definition of variable changes (Clifford Asness and Frazzini (2013)).

Campbell Harvey and his co-authors, in a series of papers, tried to synthesize the research on factors in Harvey, Liu, and Zhu (2016), C. Harvey and Liu (2019) and Harvey and Liu (2019). His work underlines the need to set high bars for an anomaly to be called a ‘true’ factor. Increasing thresholds for \(p\)-values is only a partial answer, as it is always possible to resort to data snooping in order to find an optimized strategy that will fail out-of-sample but that will deliver a \(t\)-statistic larger than three (or even four). Harvey (2017) recommends to resort to a Bayesian approach which blends data-based significance with a prior into a so-called Bayesanized p-value (see subsection below).

Following this work, researchers have continued to explore the richness of this zoo. Bryzgalova, Huang, and Julliard (2019) propose a tractable Bayesian estimation of large-dimensional factor models and evaluate all possible combinations of more than 50 factors, yielding an incredibly large number of coefficients. This combined with a Bayesianized Fama and MacBeth (1973) procedure allows to distinguish between pervasive and superfluous factors. Chordia, Goyal, and Saretto (2020) use simulations of 2 million trading strategies to estimate the rate of false discoveries, that is, when a spurious factor is detected (type I error). They also advise to use thresholds for t-statistics that are well above three. In a similar vein, Harvey and Liu (2020) also underline that sometimes true anomalies may be missed because of a one time \(t\)-statistic that is too low (type II error).

The propensity of journals to publish positive results has led researchers to estimate the difference between reported returns and true returns. A. Y. Chen and Zimmermann (2020) call this difference the publication bias and estimate it as roughly 12%. That is, if a published average return is 8%, the actual value may in fact be closer to (1-12%)8%=7%. Qualitatively, this estimation of 12% is smaller than the out-of-sample reduction in returns found in McLean and Pontiff (2016). ### Predictive regressions, sorts, and p-value issues For simplicity, we assume a simple form: \[\begin{equation} \tag{4.2} \textbf{r} = a+b\textbf{x}+\textbf{e}, \end{equation}\] where the vector \(\textbf{r}\) stacks all returns of all stocks and \(\textbf{x}\) is a lagged variable so that the regression is indeed predictive. If the estimate \(\hat{b}\) is significant given a specified threshold, then it can be tempting to conclude that \(\textbf{x}\) does a good job at predicting returns. Hence, long-short portfolios related to extreme values of \(\textbf{x}\) (mind the sign of \(\hat{b}\)) are expected to generate profits. This is unfortunately often false because \(\hat{b}\) gives information on the past* ability of \(\textbf{x}\) to forecast returns. What happens in the future may be another story.

Statistical tests are also used for portfolio sorts. Assume two extreme portfolios are expected to yield very different average returns (like very small cap versus very large cap, or strong winners versus bad losers). The portfolio returns are written \(r_t^+\) and \(r_t^-\). The simplest test for the mean is \(t=\sqrt{T}\frac{m_{r_+}-m_{r_-}}{\sigma_{r_+-r_-}}\), where \(T\) is the number of points and \(m_{r_\pm}\) denotes the means of returns and \(\sigma_{r_+-r_-}\) is the standard deviation of the difference between the two series, i.e., the volatility of the long-short portfolio. In short, the statistic can be viewed as a scaled Sharpe ratio (though usually these ratios are computed for long-only portfolios) and can in turn be used to compute \(p\)-values to assess the robustness of an anomaly. As is shown in Linnainmaa and Roberts (2018) and Hou, Xue, and Zhang (2020), many factors discovered by reasearchers fail to survive in out-of-sample tests.

One reason why people are overly optimistic about anomalies they detect is the widespread reverse interpretation of the p-value. Often, it is thought of as the probability of one hypothesis (e.g., my anomaly exists) given the data. In fact, it’s the opposite; it’s the likelihood of your data sample, knowing that the anomaly holds. \[\begin{align*} p-\text{value} &= P[D|H] \\ \text{target prob.}& = P[H|D]=\frac{P[D|H]}{P[D]}\times P[H], \end{align*}\] where \(H\) stands for hypothesis and \(D\) for data. The equality in the second row is a plain application of Bayes’ identity: the interesting probability is in fact a transform of the \(p\)-value.

Two articles (at least) discuss this idea. Harvey (2017) introduces Bayesianized \(p\)-values: \[\begin{equation} \tag{4.3} \text{Bayesianized } p-\text{value}=\text{Bpv}= e^{-t^2/2}\times\frac{\text{prior}}{1+e^{-t^2/2}\times \text{prior}} , \end{equation}\] where \(t\) is the \(t\)-statistic obtained from the regression (i.e., the one that defines the p-value) and prior is the analyst’s estimation of the odds that the hypothesis (anomaly) is true. The prior is coded as follows. Suppose there is a p% chance that the null holds (i.e., (1-p)% for the anomaly). The odds are coded as \(p/(1-p)\). Thus, if the t-statistic is equal to 2 (corresponding to a p-value of 5% roughly) and the prior odds are equal to 6, then the Bpv is equal to \(e^{-2}\times 6 \times(1+e^{-2}\times 6)^{-1}\approx 0.448\) and there is a 44.8% chance that the null is true. This interpretation stands in sharp contrast with the original \(p\)-value which cannot be viewed as a probability that the null holds. Of course, one drawback is that the level of the prior is crucial and solely user-specified.

The work of Chinco, Neuhierl, and Weber (2020) is very different but shares some key concepts, like the introduction of Bayesian priors in regression outputs. They show that coercing the predictive regression with an \(L^2\) constraint (see the ridge regression in Chapter 6) amounts to introducing views on what the true distribution of \(b\) is. The stronger the constraint, the more the estimate \(\hat{b}\) will be shrunk towards zero. One key idea in their work is the assumption of a distribution for the true \(b\) across many anomalies. It is assumed to be Gaussian and centered. The interesting parameter is the standard deviation: the larger it is, the more frequently significant anomalies are discovered. Notably, the authors show that this parameter changes through time and we refer to the original paper for more details on this subject.

4.2.4 Fama-Macbeth regressions

Another detection method was proposed by Fama and MacBeth (1973) through a two-stage regression analysis of risk premia. The first stage is a simple estimation of the relationship (): the regressions are run on a stock-by-stock basis over the corresponding time series. The resulting estimates \(\hat{\beta}_{i,k}\) are then plugged into a second series of regressions: \[\begin{equation} r_{t,n}= \gamma_{t,0} + \sum_{k=1}^K\gamma_{t,k}\hat{\beta}_{n,k} + \varepsilon_{t,n}, \end{equation}\] which are run date-by-date on the cross-section of assets.6 Theoretically, the betas would be known and the regression would be run on the \(\beta_{n,k}\) instead of their estimated values. The \(\hat{\gamma}_{t,k}\) estimate the premia of factor \(k\) at time \(t\). Under suitable distributional assumptions on the \(\varepsilon_{t,n}\), statistical tests can be performed to determine whether these premia are significant or not. Typically, the statistic on the time-aggregated (average) premia \(\hat{\gamma}_k=\frac{1}{T}\sum_{t=1}^T\hat{\gamma}_{t,k}\): \[t_k=\frac{\hat{\gamma}_k}{\hat{\sigma_k}/\sqrt{T}}\] is often used in pure Gaussian contexts to assess whether or not the factor is significant (\(\hat{\sigma}_k\) is the standard deviation of the \(\hat{\gamma}_{t,k}\)).

We refer to Jagannathan and Wang (1998) and Petersen (2009) for technical discussions on the biases and losses in accuracy that can be induced by standard ordinary least squares (OLS) estimations. Moreover, as the \(\hat{\beta}_{i,k}\) in the second-pass regression are estimates, a second level of errors can arise (the so-called errors in variables). The interested reader will find some extensions and solutions in Shanken (1992), Ang, Liu, and Schwarz (2018) and Jegadeesh et al. (2019).

Below, we perform Fama and MacBeth (1973) regressions on our sample. We start by the first pass: individual estimation of betas. We build a dedicated function below and use some functional programming to automate the process.

TABLE 4.2: Sample of beta values (row numbers are stock IDs).
Constant MKT_RF SMB HML RMW CMA
1 0.008 1.438 0.525 0.567 1.006 -0.301
3 -0.002 0.791 1.084 0.854 0.220 -0.423
4 0.005 0.364 0.305 -0.072 0.589 0.267
7 0.006 0.414 0.682 0.284 0.294 0.119
9 0.005 0.810 0.682 1.047 0.009 0.147
11 0.000 0.937 0.120 0.494 -0.247 0.085

In the table, MKT_RF is the market return minus the risk free rate. The corresponding coefficient is often referred to as the beta, especially in univariate regressions. We then reformat these betas from Table 4.2 to prepare the second pass. Each line corresponds to one asset: the first 5 columns are the estimated factor loadings and the remaining ones are the asset returns (date by date).

TABLE 4.3: Sample of reformatted beta values (ready for regression).
MKT_RF SMB HML RMW CMA 2000-01-31 2000-02-29 2000-03-31
1 1.4384642 0.5251567 0.5670364 1.0063901 -0.3013492 -0.036 0.263 0.031
3 0.7912419 1.0843206 0.8536687 0.2200451 -0.4227554 0.077 -0.024 0.018
4 0.3640251 0.3047825 -0.0718367 0.5885632 0.2666427 -0.016 0.000 0.153
7 0.4143588 0.6817543 0.2839718 0.2941460 0.1193298 -0.009 0.027 0.000
9 0.8104622 0.6823502 1.0465634 0.0085424 0.1468137 0.032 0.076 -0.025
11 0.9368310 0.1197819 0.4939755 -0.2465353 0.0849845 0.144 0.258 0.049

We observe that the values of the first column (market betas) revolve around one, which is what we would expect. Finally, we are ready for the second round of regressions.

TABLE 4.4: Sample of gamma (premia) values.
Constant MKT_RF SMB HML RMW CMA
2000-01-31 -0.031 0.037 0.227 -0.157 -0.276 0.044
2000-02-29 0.020 0.081 -0.134 0.050 0.089 -0.027
2000-03-31 0.007 -0.011 -0.016 0.054 0.036 0.039
2000-04-30 0.127 -0.132 -0.104 0.088 0.117 -0.002
2000-05-31 0.042 -0.005 0.075 -0.113 -0.080 -0.045
2000-06-30 0.028 -0.029 -0.019 0.054 0.045 0.017

Visually, the estimated premia are also very volatile. We plot their estimated values for the market, SMB and HML factors.

Time series plot of gammas (premia) in Fama-Macbeth regressions.

FIGURE 4.3: Time series plot of gammas (premia) in Fama-Macbeth regressions.

The two spikes at the end of the sample signal potential colinearity issues; two factors seem to compensate in an unclear aggregate effect. This underlines the usefulness of penalized estimates (see Chapter 6).

4.2.5 Factor competition

The core purpose of factors is to explain the cross-section of stock returns. For theoretical and practical reasons, it is preferable if redundancies within factors are avoided. Indeed, redundancies imply collinearity which is known to perturb estimates (Belsley, Kuh, and Welsch (2005)). In addition, when asset managers decompose the performance of their returns into factors, overlaps (high absolute correlations) between factors yield exposures that are less interpretable; positive and negative exposures compensate each other spuriously.

A simple protocol to sort out redundant factors is to run regressions of each factor against all others: \[\begin{equation} \tag{4.4} f_{t,k} = a_k +\sum_{j\neq k} \delta_{k,j} f_{t,j} + \epsilon_{t,k}. \end{equation}\] The interesting metric is then the test statistic associated to the estimation of \(a_k\). If \(a_k\) is significantly different from zero, then the cross-section of (other) factors fails to explain exhaustively the average return of factor \(k\). Otherwise, the return of the factor can be captured by exposures to the other factors and is thus redundant.

One mainstream application of this technique was performed in Fama and French (2015), in which the authors show that the HML factor is redundant when taking into account four other factors (Market, SMB, RMW and CMA). Below, we reproduce their analysis on an updated sample. We start our analysis directly with the database maintained by Kenneth French.

We can run the regressions that determine the redundancy of factors via the procedure defined in Equation (4.4).

We obtain the vector of \(\alpha\) values from Equation (). Below, we format these figures along with \(p\)-value thresholds and export them in a summary table. The significance levels of coefficients is coded as follows: \(0<(***)<0.001<(**)<0.01<(*)<0.05\).

TABLE 4.5: Factor competition among the Fama and French (2015) five factors.
Dep. Variable Intercept MKT_RF SMB HML RMW CMA
MKT_RF 0.008 (***) NA 0.258 (***) 0.104 -0.368 (***) -0.911 (***)
SMB 0.003 (*) 0.131 (***) NA 0.072 -0.43 (***) -0.127
HML 0 0.028 0.038 NA 0.148 (***) 1.02 (***)
RMW 0.004 (***) -0.096 (***) -0.219 (***) 0.143 (***) NA -0.287 (***)
CMA 0.002 (***) -0.11 (***) -0.03 0.456 (***) -0.133 (***) NA

We confirm that the HML factor remains redundant when the four others are present in the asset pricing model. The figures we obtain are very close to the ones in the original paper (Fama and French (2015)), which makes sense, since we only add 5 years to their initial sample.

At a more macro-level, researchers also try to figure out which models (i.e., combinations of factors) are the most likely, given the data empirically observed (and possibly given priors formulated by the econometrician). For instance, this stream of literature seeks to quantify to which extent the 3-factor model of Fama and French (1993) outperforms the 5 factors in Fama and French (2015). In this direction, De Moor, Dhaene, and Sercu (2015) introduce a novel computation for p-values that compare the relative likelihood that two models pass a zero-alpha test. More generally, the Bayesian method of Barillas and Shanken (2018) was subsequently improved by Chib, Zeng, and Zhao (2020).

Lastly, even the optimal number of factors is a subject of disagreement among conclusions of recent work. While the traditional literature focuses on a limited number (3-5) of factors, more recent research by DeMiguel et al. (2020), He, Huang, and Zhou (2020), Kozak, Nagel, and Santosh (2019) and Freyberger, Neuhierl, and Weber (2020) advocates the need to use at least 15 or more. Green, Hand, and Zhang (2017) even find that the number of characteristics that help explain the cross-section of returns varies in time.

4.2.6 Advanced techniques

The ever increasing number of factors combined to their importance in asset management has led researchers to craft more subtle methods in order to ``organize’’ the so-called factor zoo and, more importantly, to detect spurious anomalies and compare different asset pricing model specifications. We list a few of them below. - Feng, Giglio, and Xiu (2020) combine LASSO selection with Fama-MacBeth regressions to test if new factor models are worth it. They quantify the gain of adding one new factor to a set of predefined factors and show that many factors reported in papers published in the 2010 decade do not add much incremental value;
- C. Harvey and Liu (2019) (in a similar vein) use bootstrap on orthogonalized factors. They make the case that correlations among predictors is a major issue and their method aims at solving this problem. Their lengthy procedure seeks to test if maximal additional contribution of a candidate variable is significant;
- Fama and French (2018) compare asset pricing models through squared maximum Sharpe ratios;
- Giglio and Xiu (2019) estimate factor risk premia using a three-pass method based on principal component analysis;
- Pukthuanthong, Roll, and Subrahmanyam (2018) disentangle priced and non-priced factors via a combination of principal component analysis and Fama and MacBeth (1973) regressions;
- Gospodinov, Kan, and Robotti (2019) warn against factor misspecification (when spurious factors are included in the list of regressors). Traded factors (\(resp.\) macro-economic factors) seem more likely (\(resp.\) less likely) to yield robust identifications (see also Bryzgalova (2019)).

There is obviously no infallible method, but the number of contributions in the field highlights the need for robustness. This is evidently a major concern when crafting investment decisions based on factor intuitions. One major hurdle for short-term strategies is the likely time-varying feature of factors. We refer for instance to Ang and Kristensen (2012) and Cooper and Maio (2019) for practical results and to Gagliardini, Ossola, and Scaillet (2016) and Ma et al. (2020) for more theoretical treatments (with additional empirical results).

4.3 Factors or characteristics?

The decomposition of returns into linear factor models is convenient because of its simple interpretation. There is nonetheless a debate in the academic literature about whether firm returns are indeed explained by exposure to macro-economic factors or simply by the characteristics of firms. In their early study, Lakonishok, Shleifer, and Vishny (1994) argue that one explanation of the value premium comes from incorrect extrapolation of past earning growth rates. Investors are overly optimistic about firms subject to recent profitability. Consequently, future returns are (also) driven by the core (accounting) features of the firm. The question is then to disentangle which effect is the most pronounced when explaining returns: characteristics versus exposures to macro-economic factors.

In their seminal contribution on this topic, Daniel and Titman (1997) provide evidence in favour of the former (two follow-up papers are K. Daniel, Titman, and Wei (2001) and Daniel and Titman (2012)). They show that firms with high book-to-market ratios or small capitalizations display higher average returns, even if they are negatively loaded on the HML or SMB factors. Therefore, it seems that it is indeed the intrinsic characteristics that matter, and not the factor exposure. For further material on characteristics’ role in return explanation or prediction, we refer to the following contributions: - Section 2.5.2. in Goyal (2012) surveys pre-2010 results on this topic;
- Chordia, Goyal, and Shanken (2019) find that characteristics explain a larger proportion of variation in estimated expected returns than factor loadings;
- Kozak, Nagel, and Santosh (2018) reconcile factor-based explanations of premia to a theoretical model in which some agents’ demands are sentiment driven;
- Han et al. (2019) show with penalized regressions that 20 to 30 characteristics (out of 94) are useful for the prediction of monthly returns of US stocks. Their methodology is interesting: they regress returns against characteristics to build forecasts and then regress the returns on the forecast to assess if they are reliable. The latter regression uses a LASSO-type penalization (see Chapter 6) so that useless characteristics are excluded from the model. The penalization is extended to elasticnet in Rapach and Zhou (2019).
- Kelly, Pruitt, and Su (2019) and Kim, Korajczyk, and Neuhierl (2019) both estimate models in which factors are latent but loadings (betas) and possibly alphas depend on characteristics. Kirby (2020) generalizes the first approach by introducing regime-switching. In contrast, Lettau and Pelger (2020a) and Lettau and Pelger (2020b) estimate latent factors without any link to particular characteristics (and provide large sample asymptotic properties of their methods).
- In the same vein as Hoechle, Schmid, and Zimmermann (2018), Gospodinov, Kan, and Robotti (2019) and Bryzgalova (2019) and discuss potential errors that arise when working with portfolio sorts that yield long-short returns. The authors show that in some cases, tests based on this procedure may be deceitful. This happens when the characteristic chosen to perform the sort is correlated with an external (unobservable) factor. They propose a novel regression-based approach aimed at bypassing this problem.

More recently and in a separate stream of literature, Ralph S.J. Koijen and Yogo (2019) have introduced a demand model in which investors form their portfolios according to their preferences towards particular firm characteristics. They show that this allows them to mimic the portfolios of large institutional investors. In their model, aggregate demands (and hence, prices) are directly linked to characteristics, not to factors. In a follow-up paper, Ralph SJ Koijen, Richmond, and Yogo (2019) show that a few sets of characteristics suffice to predict future returns. They also show that, based on institutional holdings from the UK and the US, the largest investors are those who are the most influencial in the formation of prices. In a similar vein, Betermier, Calvet, and Jo (2019) derive an elegant (theoretical) general equilibrium model that generates some well-documented anomalies (size, book-to-market). The models of Arnott et al. (2014) and Alti and Titman (2019) are also able to theoretically generate known anomalies. Finally, in I. Martin and Nagel (2019), characteristics influence returns via the role they play in the predictability of dividend growth. This paper discussed the asymptotic case when the number of assets and the number of characteristics are proportional and both increase to infinity.

4.4 Hot topics: momentum, timing and ESG

4.4.1 Factor momentum

A recent body of literature unveils a time series momentum property of factor returns. For instance, Gupta and Kelly (2019) report that autocorrelation patterns within these returns is statistically significant.7 In the same vein, Arnott et al. (2020) make the case that the industry momentum found in Moskowitz and Grinblatt (1999) can in fact be explained by this factor momentum. Going even further, Ehsani and Linnainmaa (2019) conclude that the original momentum factor is in fact the aggregation of the autocorrelation that can be found in all other factors. Given the data obtained on Ken French’s website, we compute the autocorrelation function (ACF) of factors. We recall that \[\text{ACF}_k(\textbf{x}_t)=\mathbb{E}[(\textbf{x}_t-\bar{\textbf{x}})(\textbf{x}_{t+k}-\bar{\textbf{x}})].\]

Autocorrelograms of common factor portfolios.

FIGURE 4.4: Autocorrelograms of common factor portfolios.

Of the four chosen series, only the size factor is not significantly autocorrelated at the first order.

4.4.2 Factor timing

Given the abundance of evidence of the time-varying nature of factor premia, it is legitimate to wonder if it is possible to predict when factor will perform well or badly. The evidence on the effectiveness of timing is diverse: positive for Greenwood and Hanson (2012), Hodges et al. (2017), Haddad, Kozak, and Santosh (2020) and Lioui and Tarelli (2020), negative for Asness et al. (2017) and mixed for Dichtl et al. (2019). There is no consensus on which predictors to use (general macroeconomic indicators in Hodges et al. (2017), stock issuances versus repurchases in Greenwood and Hanson (2012), and aggregate fundamental data in Dichtl et al. (2019)). In ML-based factor investing, it is possible to resort to more granularity by combining firm-specific attributes to large-scale economic data as we explain in Section 5.7.2.

4.4.3 The green factors

The demand for ethical financial products has sharply risen during the 2010 decade, leading to the creation of funds dedicated to socially responsible investing (SRI - see Camilleri (2020)). Though this phenomenon is not really new (Schueth (2003), Hill et al. (2007)), its acceleration has prompted research about whether or not characteristics related to ESG criteria (environment, social, governance) are priced. Dozens and even possibly hundreds of papers have been devoted to this question, but no consensus has been reached. More and more, researchers study the financial impact of climate change (see Bernstein, Gustafson, and Lewis (2019), Hong, Li, and Xu (2019) and Hong, Karolyi, and Scheinkman (2020)) and the societal push for responsible corporate behavior (Fabozzi (2020), Kurtz (2020)). We gather below a very short list of papers that suggests conflicting results:

  • favorable: ESG investing works (Kempf and Osthoff (2007), Cheema-Fox et al. (2020)), can work (Nagy, Kassam, and Lee (2016), Alessandrini and Jondeau (2020)), or can at least be rendered efficient (Branch and Cai (2012)). A large meta-study reports overwhelming favorable results (Friede, Busch, and Bassen (2015)), but of course, they could well stem from the publication bias towards positive results.
  • unfavorable: Ethical investing is not profitable according to Adler and Kritzman (2008) and Blitz and Swinkels (2020). An ESG factor should be long unethical firms and short ethical ones (Lioui (2018)).
  • mixed: ESG investing may be beneficial globally but not locally (Chakrabarti and Sen (2020)). Portfolios relying on ESG screening do not significantly outperform those with no screening but are subject to lower levels of volatility (Gibson et al. (2020), Gougler and Utz (2020)). As is often the case, the devil is in the details, and results depend on whether to use E, S or G (Bruder et al. (2019)).

On top of these contradicting results, several articles point towards complexities in the measurement of ESG. Depending on the chosen criteria and on the data provider, results can change drastically (see Galema, Plantinga, and Scholtens (2008), Berg, Koelbel, and Rigobon (2019) and Atta-Darkua et al. (2020)).

We end this short section by noting that of course ESG criteria can directly be integrated into ML model, as is for instance done in Franco et al. (2020).

4.6 Coding exercises

  1. Compute annual returns of the growth versus value portfolios, that is, the average return of firms with above median price-to-book ratio (the variable is called `Pb’ in the dataset).
  2. Same exercise, but compute the monthly returns and plot the value (through time) of the corresponding portfolios.
  3. Instead of a unique threshold, compute simply sorted portfolios based on quartiles of market capitalization. Compute their annual returns and plot them.

References

Adler, Timothy, and Mark Kritzman. 2008. “The Cost of Socially Responsible Investing.” Journal of Portfolio Management 35 (1): 52–56.

Alessandrini, Fabio, and Eric Jondeau. 2020. “Optimal Strategies for Esg Portfolios.” SSRN Working Paper 3578830.

Alti, Aydoğan, and Sheridan Titman. 2019. “A Dynamic Model of Characteristic-Based Return Predictability.” Journal of Finance 74 (6): 3187–3216.

Ammann, Manuel, Guillaume Coqueret, and Jan-Philip Schade. 2016. “Characteristics-Based Portfolio Choice with Leverage Constraints.” Journal of Banking & Finance 70: 23–37.

Ang, Andrew. 2014. Asset Management: A Systematic Approach to Factor Investing. Oxford University Press.

Ang, Andrew, Robert J Hodrick, Yuhang Xing, and Xiaoyan Zhang. 2006. “The Cross-Section of Volatility and Expected Returns.” Journal of Finance 61 (1): 259–99.

Ang, Andrew, and Dennis Kristensen. 2012. “Testing Conditional Factor Models.” Journal of Financial Economics 106 (1): 132–56.

Ang, Andrew, Jun Liu, and Krista Schwarz. 2018. “Using Individual Stocks or Portfolios in Tests of Factor Models.” SSRN Working Paper 1106463.

Arnott, Robert D, Mark Clements, Vitali Kalesnik, and Juhani T Linnainmaa. 2020. “Factor Momentum.” Journal of the American Statistical Association 3116974.

Arnott, Robert D, Jason C Hsu, Jun Liu, and Harry Markowitz. 2014. “Can Noise Create the Size and Value Effects?” Management Science 61 (11): 2569–79.

Asness, Cliff, Andrea Frazzini, Niels Joachim Gormsen, and Lasse Heje Pedersen. 2020. “Betting Against Correlation: Testing Theories of the Low-Risk Effect.” Journal of Financial Economics 135 (3): 629–52.

Asness, Clifford, Swati Chandra, Antti Ilmanen, and Ronen Israel. 2017. “Contrarian Factor Timing Is Deceptively Difficult.” Journal of Portfolio Management 43 (5): 72–87.

Asness, Clifford, and Andrea Frazzini. 2013. “The Devil in Hml’s Details.” Journal of Portfolio Management 39 (4): 49–68.

Asness, Clifford, Andrea Frazzini, Ronen Israel, Tobias J Moskowitz, and Lasse H Pedersen. 2018. “Size Matters, If You Control Your Junk.” Journal of Financial Economics 129 (3): 479–509.

Asness, Clifford, Antti Ilmanen, Ronen Israel, and Tobias Moskowitz. 2015. “Investing with Style.” Journal of Investment Management 13 (1): 27–63.

Asness, Clifford S, Tobias J Moskowitz, and Lasse Heje Pedersen. 2013. “Value and Momentum Everywhere.” Journal of Finance 68 (3): 929–85.

Astakhov, Anton, Tomas Havranek, and Jiri Novak. 2019. “Firm Size and Stock Returns: A Quantitative Survey.” Journal of Economic Surveys 33 (5): 1463–92.

Atta-Darkua, Vaska, David Chambers, Elroy Dimson, Zhenkai Ran, and Ting Yu. 2020. “Strategies for Responsible Investing: Emerging Academic Evidence.” Journal of Portfolio Management 46 (3): 26–35.

Back, Kerry. 2010. Asset Pricing and Portfolio Choice Theory. Oxford University Press.

Baker, Malcolm, Brendan Bradley, and Jeffrey Wurgler. 2011. “Benchmarks as Limits to Arbitrage: Understanding the Low-Volatility Anomaly.” Financial Analysts Journal 67 (1): 40–54.

Baker, Malcolm, Mathias F Hoeyer, and Jeffrey Wurgler. 2020. “Leverage and the Beta Anomaly.” Journal of Financial and Quantitative Analysis Forthcoming: 1–24.

Baker, Malcolm, Patrick Luo, and Ryan Taliaferro. 2017. “Detecting Anomalies: The Relevance and Power of Standard Asset Pricing Tests.”

Bali, Turan G, Robert F Engle, and Scott Murray. 2016. Empirical Asset Pricing: The Cross Section of Stock Returns. John Wiley & Sons.

Ballings, Michel, Dirk Van den Poel, Nathalie Hespeels, and Ruben Gryp. 2015. “Evaluating Multiple Classifiers for Stock Price Direction Prediction.” Expert Systems with Applications 42 (20): 7046–56.

Ban, Gah-Yi, Noureddine El Karoui, and Andrew EB Lim. 2016. “Machine Learning and Portfolio Optimization.” Management Science 64 (3): 1136–54.

Bansal, Ravi, and Salim Viswanathan. 1993. “No Arbitrage and Arbitrage Pricing: A New Approach.” Journal of Finance 48 (4): 1231–62.

Banz, Rolf W. 1981. “The Relationship Between Return and Market Value of Common Stocks.” Journal of Financial Economics 9 (1): 3–18.

Barberis, Nicholas. 2018. “Psychology-Based Models of Asset Prices and Trading Volume.” In Handbook of Behavioral Economics-Foundations and Applications.

Barberis, Nicholas, Lawrence J Jin, and Baolian Wang. 2020. “Prospect Theory and Stock Market Anomalies.” SSRN Working Paper 3477463.

Barberis, Nicholas, Abhiroop Mukherjee, and Baolian Wang. 2016. “Prospect Theory and Stock Returns: An Empirical Test.” Review of Financial Studies 29 (11): 3068–3107.

Barberis, Nicholas, and Andrei Shleifer. 2003. “Style Investing.” Journal of Financial Economics 68 (2): 161–99.

Barillas, Francisco, and Jay Shanken. 2018. “Comparing Asset Pricing Models.” Journal of Finance 73 (2): 715–54.

Baz, Jamil, Nicolas Granger, Campbell R Harvey, Nicolas Le Roux, and Sandy Rattray. 2015. “Dissecting Investment Strategies in the Cross Section and Time Series.” SSRN Working Paper 2695101.

Belsley, David A, Edwin Kuh, and Roy E Welsch. 2005. Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. Vol. 571. John Wiley & Sons.

Berg, Florian, Julian F Koelbel, and Roberto Rigobon. 2019. “Aggregate Confusion: The Divergence of ESG Ratings.” SSRN Working Paper 3438533.

Berk, Jonathan B, Richard C Green, and Vasant Naik. 1999. “Optimal Investment, Growth Options, and Security Returns.” Journal of Finance 54 (5): 1553–1607.

Bernstein, Asaf, Matthew T Gustafson, and Ryan Lewis. 2019. “Disaster on the Horizon: The Price Effect of Sea Level Rise.” Journal of Financial Economics 134 (2): 253–72.

Betermier, Sebastien, Laurent E Calvet, and Evan Jo. 2019. “A Supply and Demand Approach to Equity Pricing.” SSRN Working Paper 3440147.

Betermier, Sebastien, Laurent E Calvet, and Paolo Sodini. 2017. “Who Are the Value and Growth Investors?” Journal of Finance 72 (1): 5–46.

Bhamra, Harjoat S, and Raman Uppal. 2019. “Does Household Finance Matter? Small Financial Errors with Large Social Costs.” American Economic Review 109 (3): 1116–54.

Blitz, David, and Laurens Swinkels. 2020. “Is Exclusion Effective?” Journal of Portfolio Management 46 (3): 42–48.

Boloorforoosh, Ali, Peter Christoffersen, Christian Gourieroux, and Mathieu Fournier. 2020. “Beta Risk in the Cross-Section of Equities.” Review of Financial Studies Forthcoming.

Bouchaud, Jean-philippe, Philipp Krueger, Augustin Landier, and David Thesmar. 2019. “Sticky Expectations and the Profitability Anomaly.” Journal of Finance 74 (2): 639–74.

Branch, Ben, and Li Cai. 2012. “Do Socially Responsible Index Investors Incur an Opportunity Cost?” Financial Review 47 (3): 617–30.

Brandt, Michael W, Pedro Santa-Clara, and Rossen Valkanov. 2009. “Parametric Portfolio Policies: Exploiting Characteristics in the Cross-Section of Equity Returns.” Review of Financial Studies 22 (9): 3411–47.

Bruder, Benjamin, Yazid Cheikh, Florent Deixonne, and Ban Zheng. 2019. “Integration of ESG in Asset Allocation.” SSRN Working Paper 3473874.

Bryzgalova, Svetlana. 2019. “Spurious Factors in Linear Asset Pricing Models.”

Bryzgalova, Svetlana, Jiantao Huang, and Christian Julliard. 2019. “Bayesian Solutions for the Factor Zoo: We Just Ran Two Quadrillion Models.” SSRN Working Paper 3481736.

Bryzgalova, Svetlana, Markus Pelger, and Jason Zhu. 2019. “Forest Through the Trees: Building Cross-Sections of Stock Returns.” SSRN Working Paper 3493458.

Bustos, O, and A Pomares-Quimbaya. 2020. “Stock Market Movement Forecast: A Systematic Review.” Expert Systems with Applications Forthcoming.

Camilleri, Mark Anthony. 2020. “The Market for Socially Responsible Investing: A Review of the Developments.” Social Responsibility Journal Forthcoming.

Cao, Li-Juan, and Francis Eng Hock Tay. 2003. “Support Vector Machine with Adaptive Parameters in Financial Time Series Forecasting.” IEEE Transactions on Neural Networks 14 (6): 1506–18.

Carhart, Mark M. 1997. “On Persistence in Mutual Fund Performance.” Journal of Finance 52 (1): 57–82.

Carlson, Murray, Adlai Fisher, and Ron Giammarino. 2004. “Corporate Investment and Asset Price Dynamics: Implications for the Cross-Section of Returns.” Journal of Finance 59 (6): 2577–2603.

Castaneda, Pablo, and Jorge Sabat. 2019. “Microfounding the Fama-Macbeth Regression.” SSRN Working Paper 3435141.

Cattaneo, Matias D, Richard K Crump, Max Farrell, and Ernst Schaumburg. 2020. “Characteristic-Sorted Portfolios: Estimation and Inference” Forthcoming. Review of Economics; Statistics: 1–47.

Cazalet, Zélia, and Thierry Roncalli. 2014. “Facts and Fantasies About Factor Investing.” SSRN Working Paper 2524547.

Chakrabarti, Gagari, and Chitrakalpa Sen. 2020. “Time Series Momentum Trading in Green Stocks.” Studies in Economics and Finance.

Cheema-Fox, Alexander, Bridget Realmuto LaPerla, George Serafeim, David Turkington, and Hui Stacie Wang. 2020. “Decarbonization Factors.” SSRN Working Paper 3448637.

Chen, Andrew Y. 2019. “The Limits of P-Hacking: A Thought Experiment.” SSRN Working Paper 3272572.

Chen, Andrew Y. 2020. “Do T-Stat Hurdles Need to Be Raised?” SSRN Working Paper 3254995.

Chen, Andrew Y, and Mihail Velikov. 2020. “Zeroing in on the Expected Returns of Anomalies.” SSRN Working Paper 3073681.

Chen, Andrew Y, and Tom Zimmermann. 2020. “Publication Bias and the Cross-Section of Stock Returns.” Review of Asset Pricing Studies Forthcoming.

Chen, Luyang, Markus Pelger, and Jason Zhu. 2020. “Deep Learning in Asset Pricing.” SSRN Working Paper 3350138.

Chib, Siddhartha, Xiaming Zeng, and Lingxiao Zhao. 2020. “On Comparing Asset Pricing Models.” Journal of Finance 75 (1): 551–77.

Chinco, Alexander, Adam D Clark-Joseph, and Mao Ye. 2019. “Sparse Signals in the Cross-Section of Returns.” Journal of Finance 74 (1): 449–92.

Chinco, Alexander, Andreas Neuhierl, and Michael Weber. 2020. “Estimating the Anomaly Baserate.” Journal of Financial Economics Forthcoming.

Chinco, Alex, Samuel M Hartzmark, and Abigail B Sussman. 2019. “Necessary Evidence for a Risk Factor’s Relevance.” SSRN Working Paper 3487624.

Choi, Seung Mo, and Hwagyun Kim. 2014. “Momentum Effect as Part of a Market Equilibrium.” Journal of Financial and Quantitative Analysis 49 (1): 107–30.

Chordia, Tarun, Amit Goyal, and Alessio Saretto. 2020. “Anomalies and False Rejections.” Review of Financial Studies 33 (5): 2134–79.

Chordia, Tarun, Amit Goyal, and Jay Shanken. 2019. “Cross-Sectional Asset Pricing with Individual Stocks: Betas Versus Characteristics.” SSRN Working Paper 2549578.

Cocco, Joao F, Francisco Gomes, and Paula Lopes. 2020. “Evidence on Expectations of Household Finances.” SSRN Working Paper 3362495.

Cochrane, John H. 2009. Asset Pricing: Revised Edition. Princeton University Press.

Cochrane, John H. 2011. “Presidential Address: Discount Rates.” Journal of Finance 66 (4): 1047–1108.

Cong, Lin William, and Douglas Xu. 2019. “Rise of Factor Investing: Asset Prices, Informational Efficiency, and Security Design.” SSRN Working Paper 2800590.

Cooper, Ilan, and Paulo F Maio. 2019. “New Evidence on Conditional Factor Models.” Journal of Financial and Quantitative Analysis 54 (5): 1975–2016.

Cronqvist, Henrik, Alessandro Previtero, Stephan Siegel, and Roderick E White. 2015. “The Fetal Origins Hypothesis in Finance: Prenatal Environment, the Gender Gap, and Investor Behavior.” Review of Financial Studies 29 (3): 739–86.

Cronqvist, Henrik, Stephan Siegel, and Frank Yu. 2015. “Value Versus Growth Investing: Why Do Different Investors Have Different Styles?” Journal of Financial Economics 117 (2): 333–49.

Daniel, Kent D, David Hirshleifer, and Avanidhar Subrahmanyam. 2001. “Overconfidence, Arbitrage, and Equilibrium Asset Pricing.” Journal of Finance 56 (3): 921–65.

Daniel, Kent, David Hirshleifer, and Lin Sun. 2020. “Short and Long Horizon Behavioral Factors.” Review of Financial Studies 33 (4): 1673–1736.

Daniel, Kent, Lira Mota, Simon Rottke, and Tano Santos. 2020. “The Cross-Section of Risk and Return.” Review of Financial Studies 33 (5): 1927–79.

Daniel, Kent, and Sheridan Titman. 1997. “Evidence on the Characteristics of Cross Sectional Variation in Stock Returns.” Journal of Finance 52 (1): 1–33.

Daniel, Kent, and Sheridan Titman. 2012. “Testing Factor-Model Explanations of Market Anomalies.” Critical Finance Review 1 (1): 103–39.

Daniel, Kent, Sheridan Titman, and KC John Wei. 2001. “Explaining the Cross-Section of Stock Returns in Japan: Factors or Characteristics?” Journal of Finance 56 (2): 743–66.

DeMiguel, Victor, Alberto Martin Utrera, and Raman Uppal. 2019. “What Alleviates Crowding in Factor Investing?” SSRN Working Paper 3392875.

DeMiguel, Victor, Alberto Martin Utrera, Raman Uppal, and Francisco J Nogales. 2020. “A Transaction-Cost Perspective on the Multitude of Firm Characteristics.” Review of Financial Studies 33 (5): 2180–2222.

De Moor, Lieven, Geert Dhaene, and Piet Sercu. 2015. “On Comparing Zero-Alpha Tests Across Multifactor Asset Pricing Models.” Journal of Banking & Finance 61: S235–S240.

Dichtl, Hubert, Wolfgang Drobetz, Harald Lohre, Carsten Rother, and Patrick Vosskamp. 2019. “Optimal Timing and Tilting of Equity Factors.” Financial Analysts Journal 75 (4): 84–102.

Dunis, Christian L, Spiros D Likothanassis, Andreas S Karathanasopoulos, Georgios S Sermpinis, and Konstantinos A Theofilatos. 2013. “A Hybrid Genetic Algorithm–Support Vector Machine Approach in the Task of Forecasting and Trading.” Journal of Asset Management 14 (1): 52–71.

Ehsani, Sina, and Juhani T Linnainmaa. 2019. “Factor Momentum and the Momentum Factor.” SSRN Working Paper 3014521.

Engelberg, Joseph, R David McLean, and Jeffrey Pontiff. 2018. “Anomalies and News.” Journal of Finance 73 (5): 1971–2001.

Fabozzi, Frank J. 2020. “Introduction: Special Issue on Ethical Investing.” Journal of Portfolio Management 46 (3): 1–4.

Fama, Eugene F, and Kenneth R French. 1992. “The Cross-Section of Expected Stock Returns.” Journal of Finance 47 (2): 427–65.

Fama, Eugene F, and Kenneth R French. 1993. “Common Risk Factors in the Returns on Stocks and Bonds.” Journal of Financial Economics 33 (1): 3–56.

Fama, Eugene F, and Kenneth R French. 2015. “A Five-Factor Asset Pricing Model.” Journal of Financial Economics 116 (1): 1–22.

Fama, Eugene F, and Kenneth R French. 2018. “Choosing Factors.” Journal of Financial Economics 128 (2): 234–52.

Fama, Eugene F, and James D MacBeth. 1973. “Risk, Return, and Equilibrium: Empirical Tests.” Journal of Political Economy 81 (3): 607–36.

Farmer, Leland, Lawrence Schmidt, and Allan Timmermann. 2019. “Pockets of Predictability.” SSRN Working Paper 3152386.

Feng, Guanhao, Stefano Giglio, and Dacheng Xiu. 2020. “Taming the Factor Zoo: A Test of New Factors.” Journal of Finance 75 (3): 1327–70.

Feng, Guanhao, Nicholas G Polson, and Jianeng Xu. 2019. “Deep Learning in Characteristics-Sorted Factor Models.” SSRN Working Paper 3243683.

Franco, Carmine de, Christophe Geissler, Vincent Margot, and Bruno Monnier. 2020. “ESG Investments: Filtering Versus Machine Learning Approaches.” arXiv Preprint, no. 2002.07477.

Frazzini, Andrea, and Lasse Heje Pedersen. 2014. “Betting Against Beta.” Journal of Financial Economics 111 (1): 1–25.

Freyberger, Joachim, Andreas Neuhierl, and Michael Weber. 2020. “Dissecting Characteristics Nonparametrically.” Review of Financial Studies 33 (5): 2326–77.

Friede, Gunnar, Timo Busch, and Alexander Bassen. 2015. “ESG and Financial Performance: Aggregated Evidence from More Than 2000 Empirical Studies.” Journal of Sustainable Finance & Investment 5 (4): 210–33.

Gagliardini, Patrick, Elisa Ossola, and Olivier Scaillet. 2016. “Time-Varying Risk Premium in Large Cross-Sectional Equity Data Sets.” Econometrica 84 (3): 985–1046.

Gagliardini, Patrick, Elisa Ossola, and Olivier Scaillet. 2019. “Estimation of Large Dimensional Conditional Factor Models in Finance.” SSRN Working Paper 3443426.

Galema, Rients, Auke Plantinga, and Bert Scholtens. 2008. “The Stocks at Stake: Return and Risk in Socially Responsible Investment.” Journal of Banking & Finance 32 (12): 2646–54.

Gibson, Rajna, Simon Glossner, Philipp Krueger, Pedro Matos, and Tom Steffen. 2020. “Responsible Institutional Investing Around the World.” SSRN Working Paper 3525530.

Giglio, Stefano, and Dacheng Xiu. 2019. “Asset Pricing with Omitted Factors.” SSRN Working Paper 2865922.

Gomes, Joao, Leonid Kogan, and Lu Zhang. 2003. “Equilibrium Cross Section of Returns.” Journal of Political Economy 111 (4): 693–732.

Gospodinov, Nikolay, Raymond Kan, and Cesare Robotti. 2019. “Too Good to Be True? Fallacies in Evaluating Risk Factor Models.” Journal of Financial Economics 132 (2): 451–71.

Goto, Shingo, and Yan Xu. 2015. “Improving Mean Variance Optimization Through Sparse Hedging Restrictions.” Journal of Financial and Quantitative Analysis 50 (6): 1415–41.

Gougler, Arnaud, and Sebastian Utz. 2020. “Factor Exposures and Diversification: Are Sustainably-Screened Portfolios Any Different?” Financial Markets and Portfolio Management Forthcoming.

Goyal, Amit. 2012. “Empirical Cross-Sectional Asset Pricing: A Survey.” Financial Markets and Portfolio Management 26 (1): 3–38.

Green, Jeremiah, John RM Hand, and X Frank Zhang. 2013. “The Supraview of Return Predictive Signals.” Review of Accounting Studies 18 (3): 692–730.

Green, Jeremiah, John RM Hand, and X Frank Zhang. 2017. “The Characteristics That Provide Independent Information About Average Us Monthly Stock Returns.” Review of Financial Studies 30 (12): 4389–4436.

Greenwood, Robin, and Samuel G Hanson. 2012. “Share Issuance and Factor Timing.” Journal of Finance 67 (2): 761–98.

Grinblatt, Mark, and Bing Han. 2005. “Prospect Theory, Mental Accounting, and Momentum.” Journal of Financial Economics 78 (2). Elsevier: 311–39.

Gu, Shihao, Bryan T Kelly, and Dacheng Xiu. 2020a. “Autoencoder Asset Pricing Models.” Journal of Econometrics Forthcoming.

Gu, Shihao, Bryan T Kelly, and Dacheng Xiu. 2020b. “Empirical Asset Pricing via Machine Learning.” Review of Financial Studies 33 (5): 2223–73.

Guida, Tony, and Guillaume Coqueret. 2018b. “Machine Learning in Systematic Equity Allocation: A Model Comparison.” Wilmott 2018 (98): 24–33.

Gupta, Tarun, and Bryan Kelly. 2019. “Factor Momentum Everywhere.” Journal of Portfolio Management 45 (3): 13–36.

Haddad, Valentin, Serhiy Kozak, and Shrihari Santosh. 2020. “Factor Timing.” Review of Financial Studies 33 (5): 1980–2018.

Han, Yufeng, Ai He, D Rapach, and Guofu Zhou. 2019. “Firm Characteristics and Expected Stock Returns.” SSRN Working Paper 3185335.

Hansen, Lars Peter. 1982. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica, 1029–54.

Harvey, Campbell, and Yan Liu. 2019. “Lucky Factors.” SSRN Working Paper 2528780.

Harvey, Campbell R. 2017. “Presidential Address: The Scientific Outlook in Financial Economics.” Journal of Finance 72 (4): 1399–1440.

Harvey, Campbell R. 2020. “Replication in Financial Economics.” Critical Finance Review, 1–9.

Harvey, Campbell R, and Yan Liu. 2019. “A Census of the Factor Zoo.” SSRN Working Paper 3341728.

Harvey, Campbell R, and Yan Liu. 2020. “False (and Missed) Discoveries in Financial Economics.” Journal of Finance Forthcoming.

Harvey, Campbell R, Yan Liu, and Alessio Saretto. 2020. “An Evaluation of Alternative Multiple Testing Methods for Finance Applications.” Review of Asset Pricing Studies 10 (2): 199–248.

Harvey, Campbell R, Yan Liu, and Heqing Zhu. 2016. “… And the Cross-Section of Expected Returns.” Review of Financial Studies 29 (1): 5–68.

He, Ai, Dashan Huang, and Guofu Zhou. 2020. “New Factors Wanted: Evidence from a Simple Specification Test.” SSRN Working Paper 3143752.

Henrique, Bruno Miranda, Vinicius Amorim Sobreiro, and Herbert Kimura. 2019. “Literature Review: Machine Learning Techniques Applied to Financial Market Prediction.” Expert Systems with Applications 124: 226–51.

Hill, Ronald Paul, Thomas Ainscough, Todd Shank, and Daryl Manullang. 2007. “Corporate Social Responsibility and Socially Responsible Investing: A Global Perspective.” Journal of Business Ethics 70 (2): 165–74.

Hjalmarsson, Erik, and Petar Manchev. 2012. “Characteristic-Based Mean-Variance Portfolio Choice.” Journal of Banking & Finance 36 (5): 1392–1401.

Hodges, Philip, Ked Hogan, Justin R Peterson, and Andrew Ang. 2017. “Factor Timing with Cross-Sectional and Time-Series Predictors.” Journal of Portfolio Management 44 (1): 30–43.

Hoechle, Daniel, Markus Schmid, and Heinz Zimmermann. 2018. “Correcting Alpha Misattribution in Portfolio Sorts.” SSRN Working Paper 3190310.

Hong, Harrison, G Andrew Karolyi, and José A Scheinkman. 2020. “Climate Finance.” Review of Financial Studies 33 (3): 1011–23.

Hong, Harrison, Frank Weikai Li, and Jiangmin Xu. 2019. “Climate Risks and Market Efficiency.” Journal of Econometrics 208 (1): 265–81.

Hou, Kewei, Chen Xue, and Lu Zhang. 2015. “Digesting Anomalies: An Investment Approach.” Review of Financial Studies 28 (3): 650–705.

Hou, Kewei, Chen Xue, and Lu Zhang. 2020. “Replicating Anomalies.” Review of Financial Studies 33 (5): 2019–2133.

Huang, Wei, Yoshiteru Nakamori, and Shou-Yang Wang. 2005. “Forecasting Stock Market Movement Direction with Support Vector Machine.” Computers & Operations Research 32 (10): 2513–22.

Ilmanen, Antti. 2011. Expected Returns: An Investor’s Guide to Harvesting Market Rewards. John Wiley & Sons.

Ilmanen, Antti, Ronen Israel, Tobias J Moskowitz, Ashwin K Thapar, and Franklin Wang. 2019. “Factor Premia and Factor Timing: A Century of Evidence.” SSRN Working Paper 3400998.

Jacobs, Heiko, and Sebastian Müller. 2020. “Anomalies Across the Globe: Once Public, No Longer Existent?” Journal of Financial Economics 135 (1): 213–30.

Jagannathan, Ravi, and Zhenyu Wang. 1998. “An Asymptotic Theory for Estimating Beta-Pricing Models Using Cross-Sectional Regression.” Journal of Finance 53 (4): 1285–1309.

Jegadeesh, Narasimhan, Joonki Noh, Kuntara Pukthuanthong, Richard Roll, and Junbo L Wang. 2019. “Empirical Tests of Asset Pricing Models with Individual Assets: Resolving the Errors-in-Variables Bias in Risk Premium Estimation.” Journal of Financial Economics 133 (2): 273–98.

Jegadeesh, Narasimhan, and Sheridan Titman. 1993. “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency.” Journal of Finance 48 (1): 65–91.

Jin, Dunhong. 2019. “The Drivers and Inhibitors of Factor Investing.” SSRN Working Paper, no. 3492142.

Johnson, Timothy C. 2002. “Rational Momentum Effects.” Journal of Finance 57 (2): 585–608.

Johnson, Travis L. 2019. “A Fresh Look at Return Predictability Using a More Efficient Estimator.” Review of Asset Pricing Studies 9 (1): 1–46.

Jurczenko, Emmanuel. 2017. Factor Investing: From Traditional to Alternative Risk Premia. Elsevier.

Kelly, Bryan T, Seth Pruitt, and Yinan Su. 2019. “Characteristics Are Covariances: A Unified Model of Risk and Return.” Journal of Financial Economics 134 (3): 501–24.

Kempf, Alexander, and Peer Osthoff. 2007. “The Effect of Socially Responsible Investing on Portfolio Performance.” European Financial Management 13 (5): 908–22.

Kim, Kyoung-jae. 2003. “Financial Time Series Forecasting Using Support Vector Machines.” Neurocomputing 55 (1-2). Elsevier: 307–19.

Kim, Soohun, Robert A Korajczyk, and Andreas Neuhierl. 2019. “Arbitrage Portfolios.” SSRN Working Paper 3263001.

Kirby, Chris. 2020. “Firm Characteristics, Stock Market Regimes, and the Cross-Section of Expected Returns.” SSRN Working Paper 3520131.

Koijen, Ralph SJ, Robert J Richmond, and Motohiro Yogo. 2019. “Which Investors Matter for Global Equity Valuations and Expected Returns?” SSRN Working Paper 3378340.

Koijen, Ralph S.J., and Motohiro Yogo. 2019. “A Demand System Approach to Asset Pricing.” Journal of Political Economy 127 (4): 1475–1515.

Kozak, Serhiy, Stefan Nagel, and Shrihari Santosh. 2018. “Interpreting Factor Models.” Journal of Finance 73 (3): 1183–1223.

Kozak, Serhiy, Stefan Nagel, and Shrihari Santosh. 2019. “Shrinking the Cross-Section.” Journal of Financial Economics 135: 271–92.

Krkoska, Eduard, and Klaus Reiner Schenk-Hoppé. 2019. “Herding in Smart-Beta Investment Products.” Journal of Risk and Financial Management 12 (1): 47.

Kurtz, Lloyd. 2020. “Three Pillars of Modern Responsible Investment.” Journal of Investing 29 (2): 21–32.

Lakonishok, Josef, Andrei Shleifer, and Robert W Vishny. 1994. “Contrarian Investment, Extrapolation, and Risk.” Journal of Finance 49 (5): 1541–78.

Ledoit, Olivier, Michael Wolf, and Zhao Zhao. 2020. “Efficient Sorting: A More Powerful Test for Cross-Sectional Anomalies.” Journal of Financial Econometrics 17 (4): 645–86.

Lempérière, Yves, Cyril Deremble, Philip Seager, Marc Potters, and Jean-Philippe Bouchaud. 2014. “Two Centuries of Trend Following.” arXiv Preprint, no. 1404.3274.

Lettau, Martin, and Markus Pelger. 2020a. “Estimating Latent Asset-Pricing Factors.” Journal of Econometrics Forthcoming.

Lettau, Martin, and Markus Pelger. 2020b. “Factors That Fit the Time Series and Cross-Section of Stock Returns.” Review of Financial Studies 33 (5): 2274–2325.

Linnainmaa, Juhani T, and Michael R Roberts. 2018. “The History of the Cross-Section of Stock Returns.” Review of Financial Studies 31 (7): 2606–49.

Lintner, John. 1965. “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets.” Review of Economics and Statistics 47 (1): 13–37.

Lioui, Abraham. 2018. “ESG Factor Investing: Myth or Reality?” SSRN Working Paper 3272090.

Lioui, Abraham, and Andrea Tarelli. 2020. “Factor Investing for the Long Run.” SSRN Working Paper 3531946.

Liu, Li, Zhiyuan Pan, and Yudong Wang. 2020. “What Can We Learn from the Return Predictability over Business Cycle?” Journal of Forecasting Forthcoming.

Luo, Jiang, Avanidhar Subrahmanyam, and Sheridan Titman. 2020. “Momentum and Reversals When Overconfident Investors Underestimate Their Competition.” Review of Financial Studies Forthcoming.

Ma, Shujie, Wei Lan, Liangjun Su, and Chih-Ling Tsai. 2020. “Testing Alphas in Conditional Time-Varying Factor Models with High Dimensional Assets.” Journal of Business & Economic Statistics 38 (1). Taylor & Francis: 214–27.

Martin, Ian, and Stefan Nagel. 2019. “Market Efficiency in the Age of Big Data.” SSRN Working Paper 3511296.

Matı'as, José M, and Juan C Reboredo. 2012. “Forecasting Performance of Nonlinear Models for Intraday Stock Returns.” Journal of Forecasting 31 (2). Wiley Online Library: 172–88.

McLean, R David, and Jeffrey Pontiff. 2016. “Does Academic Research Destroy Stock Return Predictability?” Journal of Finance 71 (1). Wiley Online Library: 5–32.

Moskowitz, Tobias J, and Mark Grinblatt. 1999. “Do Industries Explain Momentum?” Journal of Finance 54 (4): 1249–90.

Moskowitz, Tobias J, Yao Hua Ooi, and Lasse Heje Pedersen. 2012. “Time Series Momentum.” Journal of Financial Economics 104 (2): 228–50.

Mossin, Jan. 1966. “Equilibrium in a Capital Asset Market.” Econometrica: Journal of the Econometric Society 34 (4): 768–83.

Nagy, Zoltán, Altaf Kassam, and Linda-Eling Lee. 2016. “Can ESG Add Alpha? An Analysis of ESG Tilt and Momentum Strategies.” The Journal of Investing 25 (2): 113–24.

Patton, Andrew J, and Allan Timmermann. 2010. “Monotonicity in Asset Returns: New Tests with Applications to the Term Structure, the CAPM, and Portfolio Sorts.” Journal of Financial Economics 98 (3): 605–25.

Patton, Andrew J, and Brian M Weller. 2020. “What You See Is Not What You Get: The Costs of Trading Market Anomalies.” Journal of Financial Economics Forthcoming.

Penasse, Julien. 2019. “Understanding Alpha Decay.” SSRN Working Paper 2953614.

Perrin, Sarah, and Thierry Roncalli. 2019. “Machine Learning Optimization Algorithms & Portfolio Allocation.” SSRN Working Paper 3425827.

Petersen, Mitchell A. 2009. “Estimating Standard Errors in Finance Panel Data Sets: Comparing Approaches.” Review of Financial Studies 22 (1): 435–80.

Pukthuanthong, Kuntara, Richard Roll, and Avanidhar Subrahmanyam. 2018. “A Protocol for Factor Identification.” Review of Financial Studies 32 (4): 1573–1607.

Rapach, David E, Jack K Strauss, and Guofu Zhou. 2013. “International Stock Return Predictability: What Is the Role of the United States?” Journal of Finance 68 (4): 1633–62.

Rapach, David, and Guofu Zhou. 2019. “Time-Series and Cross-Sectional Stock Return Forecasting: New Machine Learning Methods.” SSRN Working Paper 3428095.

Reboredo, Juan C, José M Matı'as, and Raquel Garcia-Rubio. 2012. “Nonlinearity in Forecasting of High-Frequency Stock Returns.” Computational Economics 40 (3): 245–64.

Romano, Joseph P, and Michael Wolf. 2013. “Testing for Monotonicity in Expected Asset Returns.” Journal of Empirical Finance 23: 93–116.

Ross, Stephen A. 1976. “The Arbitrage Theory of Capital Asset Pricing.” Journal of Economic Theory 13 (3): 341–60.

Schueth, Steve. 2003. “Socially Responsible Investing in the United States.” Journal of Business Ethics 43 (3): 189–94.

Shanken, Jay. 1992. “On the Estimation of Beta-Pricing Models.” Review of Financial Studies 5 (1): 1–33.

Sharpe, William F. 1964. “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk.” Journal of Finance 19 (3): 425–42.

Tsiakas, Ilias, Jiahan Li, and Haibin Zhang. 2020. “Equity Premium Prediction and the State of the Economy.” Journal of Empirical Finance Forthcoming.

Van Dijk, Mathijs A. 2011. “Is Size Dead? A Review of the Size Effect in Equity Returns.” Journal of Banking & Finance 35 (12): 3263–74.

Vayanos, Dimitri, and Paul Woolley. 2013. “An Institutional Theory of Momentum and Reversal.” Review of Financial Studies 26 (5): 1087–1145.

Volpati, Valerio, Michael Benzaquen, Zoltan Eisler, Iacopo Mastromatteo, Bence Toth, and Jean-Philippe Bouchaud. 2020. “Zooming in on Equity Factor Crowding.” arXiv Preprint, no. 2001.04185.


  1. Originally, Fama and MacBeth (1973) work with the market beta only: \(r_{t,n}=\alpha_n+\beta_nr_{t,M}+\epsilon_{t,n}\) and the second pass included nonlinear terms: \(r_{t,n}=\gamma_{n,0}+\gamma_{t,1}\hat{\beta}_{n}+\gamma_{t,2}\hat{\beta}^2_n+\gamma_{t,3}\hat{s}_n+\eta_{t,n}\), where the \(\hat{s}_n\) are risk estimates for the assets that are not related to the betas. It is then possible to perform asset pricing tests to infer some properties. For instance, test whether betas have a linear influence on returns or not (\(\mathbb{E}[\gamma_{t,2}]=0\)), or test the validity of the CAPM (which implies \(\mathbb{E}[\gamma_{t,0}]=0\)).

  2. Autocorrelation in aggregate/portfolio returns is a widely documented effect since the seminal paper Lo and MacKinlay (1990) (see also Moskowitz, Ooi, and Pedersen (2012)).

  3. In the same spirit, see also Lettau and Pelger (2020a) and Lettau and Pelger (2020b).