Empirical Cross-Sectional Asset Pricing

Stefan Nagel. “Empirical Cross-Sectional Asset Pricing” The Annual Review of Financial Economics (2013) 5:167-199.

The author reviews research in several aspects of empirical cross-sectional asset pricing. Other posts here have and will highlight papers which touch on some of them, so I summarize only a couple of general themes.

There are approaches to cross-sectional asset pricing which distinguish explanations for why investors price assets the way they do and approaches which do not. “The economic content of pricing models is … in the restrictions that they impose on the [stochastic discount factor].” Ad hoc factor models and production-based models, while useful in cross-sectional tests, do not rely on investor preferences for their restrictions on the SDF and, so, do not explain the “why”. Rational (consumption-based, long-run risk) as well as sentiment-based tests based on theoretical restrictions on investor preferences give them a higher degree of economic content. Frictions and liquidity risk can cause rational models to fail, but studies in these areas provide evidence for “why” assets are priced the way they are. That is, additional risks exist for which investors rationally require additional return in order to bear the risk.

A significant amount of cross-sectional asset pricing research is centered around finding deviations from accepted models (anomalies). Recent studies have urged caution in this pursuit, however. The possibility for data snooping, where some spurious anomalies are randomly found to be significant, is an ever-present problem. Adjusting critical values and performing out-of-sample testing are ways suggested in the literature to help attenuate the problem. However, the author notes that pseudo out-of-sample testing is not necessarily a solution because data can be mined to find significant result here as well. True out-of-sample testing is necessary to lend validation to results. A true out-of-sample test occurs when data is used that was not available or used for the original study. Some ways this has been done in the literature is the use of data from different markets, post-publication data of the same asset(s) used in the original study, and in some cases even pre-sample data that became available after publication. It has additionally been found that use of r-squared statistics as an indication of how well a model or factor explain the cross-section of asset returns can be misleading when the test assets have a low-dimensional factor structure.

Investor sentiment happens when asset prices are evaluated under a subjective probability distribution. Its effect on asset prices is persistent when this sentiment is correlated across enough investors and there are significant enough risks to dissuade or prevent sophisticated investors from taking opposite positions and driving prices back to their fundamental values. Investor sentiment and rational explanations for return predictability need not be mutually exclusive. Research in this area focuses both on finding and testing empirical proxies for investor sentiment and on investigating the limits to arbitrage preventing sophisticated investors from correcting mispricing. Analyst forecasts, style fund returns, and mutual fund inflows are a few examples of proxies used with some success as proxies for investor sentiment.

Real world investors must learn about the return generating process in real time. Therefore, return predictability obvious to researchers in retrospect may not have been known or obvious to investors. Out-of-sample testing on variables shown to have ability to predict returns is one area of research into investor learning. For example, studies have shown that there is a decay of predictability after the publication of research about return predictors. The implication is that something not known to real-world investors becomes known upon publication and they then move to exploit it to the extent possible. Additional research has been done using pseudo out-of-sample testing where early periods in the data are used as a training sample to predict future returns. The author notes that investor learning does not fall under rational expectations or sentiment.

Empirical cross-sectional asset pricing research has come a long way and there is much yet to be explored. Many avenues of research are fruitful, but the ones with the greatest economic content are those which have something to say about the preferences of investors because they offer insight into understanding not just what is happening but “why” investors are pricing assets the way they are.