## Common Factor Structure in a Cross-Section of Stocks

**An amazing academic paper about multiple equity factor models and about the way how to pick the best one:**

**Authors:** Cooper, Maio, Philip

**Title: **Multifactor Models and the APT: Evidence from a Broad Cross-Section of Stock Returns

**Link:** https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2883765

**Abstract:**

We seek to describe the broad cross-section of average stock returns. We follow the APT literature and estimate the common factor structure among a large cross-section containing 278 decile portfolios (associated with 28 market anomalies). Our statistical model contains seven common factors (with an economic meaning) and prices well both the original portfolio returns and an efficient combination of these portfolios. This model clearly outperforms the empirical workhorses in the literature when it comes to pricing this broad cross-section. Augmenting the empirical models with new factor-mimicking portfolios, based on APT principles, significantly improves their performance.

**Notable quotations from the academic research paper:**

"The traditional workhorse in the empirical asset pricing literature the three-factor model of Fama and French (1993, 1996) (FF3 henceforth) fails to explain the new market anomalies. Moreover, the four-factor model of Carhart (1997) (C4) does a good job in capturing price momentum, but also struggles in terms of explaining some of the profitability- and investment-based anomalies. In response to this gap, we have witnessed the emergence of new multifactor models containing (different versions of) investment and profitability factors, in particular the five-factor model of Fama and French (2015, 2016b) (FF5) and the four-factor model of Hou, Xue, and Zhang (2015, 2016) (HXZ4). However, several dimensions of the broad cross-section of stock returns are still not explained by the new factor models. In particular, the five-factor model does not account for momentum (including industry momentum), while both of these models do not capture several profitability and investment-based (in particular, several forms of accruals) anomalies.

Following such evidence, several questions naturally emerge in the empirical asset pricing literature: __ How many factors do we need, and what are these factors, to describe well the broad cross-section of stock returns? To which dimensions of the cross-section of stock returns are these factors more correlated? To what extent (and how) can we improve the current multifactor models proposed in the literature in order to achieve a better description of large-scale cross-sectional risk premia?__ This paper attempts at providing answers to these questions. In order to achieve this goal, we adopt the general framework of the Arbitrage Pricing Theory (APT).

We follow part of the relatively small empirical APT literature in terms of estimating common stock return factors by applying asymptotical principal components analysis (APCA) to a large cross-section of stock returns. We employ a total of 28 anomalies or portfolio sorts for a total of 278 decile portfolios. __ The estimation results show that there are seven common factors that are statistically significant over our sample period (1972 to 2013). These seven factors cumulatively explain around 91% of the cross-sectional variations in the 278 portfolio returns.__ The first common factor basically captures the average anomaly and thus resembles a market factor. The other six factors capture different dimensions of the large cross-section of market anomalies. In particular, the second, third, and four factors are strongly correlated with value-growth, investment, profitability, and momentum-based anomalies. This is consistent with the role of the seven-factor model in terms of describing well this cross-section of 278 equity portfolios. This statistical model is thus a benchmark for this specific cross-section of stock returns, against which the existent models are compared.

We conduct cross-sectional asset pricing tests of our APT model by using the 278 equity portfolios as testing assets. The results confirm that the seven-factor model explains about 60% of the cross-sectional variation in the risk premia associated with the 278 portfolios. Moreover, most factor risk price estimates are statistically significant. Across categories of anomalies, the APT does a better job in pricing value-growth and intangibles, compared to the group of investment-based anomalies. Moreover, the model prices perfectly an efficient combination of the original portfolios as indicated by the GLS cross-sectional R2 estimates around 100%. This result confirms that the statistical model is a successful APT.

Next, we compare our APT model to some of most popular multifactor models existent in the literature in terms of pricing the 278 portfolios. The models include the already mentioned FF3, C4, HXZ4, FF5, in addition to a restricted version of FF5 that excludes HML (FF4), and the four-factor model of Pastor and Stambaugh (2003) (which includes a stock liquidity factor). The results show that only C4 and HXZ4 offer an economically significant explanatory power for the broad cross-section of stock returns, while the fit of both FF5 and FF4 is quite small. Moreover, the performance of all the six empirical factor models clearly lags behind the fit of the seven-factor APT, suggesting that these models have a large room for improvement in terms of describing large-scale cross-sectional risk premia.

In light of such evidence, we define and estimate new empirical multifactor models to better describe the broad cross-section of anomalies. All these models contain seven factors, to be consistent with our benchmark APT, and represent augmented versions of C4, HXZ4, FF5, and FF4, the best performing empirical models. The new factors in each of these models represent factor-mimicking portfolios (spreads among extreme portfolio deciles) associated with selected anomalies. These anomalies are those for which the original factors in each model do a worse job in terms of describing the time-series variation in the corresponding decile portfolio returns. Thus, our criteria for selecting the new factors relies on the APT restriction that the risk factors should explain well the time-series variation in the returns of the testing assets. The results show that adding the new factors improves all four empirical models, and helps especially the performance of both FF5 and FF4 in terms of explaining the large cross-section of stock returns. Moreover, the augmented models do a very good job in explaining an efficient combination of the original portfolios, thus, showing that they represent valid APTs. Therefore, the performance of the augmented empirical models is quite similar to that of our benchmark APT. Overall, our results indicate that there is a significant room for improving the existing empirical multifactor models in terms of explaining the large cross-section of stock returns in a way that is consistent with the APT."

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