In my role as an academic, I play down the importance of stationarity to get on with research efforts. When I have to make investment decisions, it is the elephant in the room. In fact, the question of stationarity is so important that it often dominates my investment decision-making and as a result renders much academic research of little practical value. The point of this commentary is to argue that finance research needs to take the question of stationarity more seriously to be more useful to investors.
Formally a stationary stochastic process is a stochastic process whose joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance, if they are relevant, also do not change over time. Non-stationarity should not be confused with unpredictability. All random processes are unpredictable. If the process is non-stationary, even the parameters of the random distribution cannot be estimated with confidence. Putting aside formal definitions, I find the example of drawing colored balls from jugs with replacement to be a great way to explain how the problem of stationarity impacts investment decision making.
If there is one jug and the balls are drawn from it with replacement, the process describing the sequence of balls drawn is stationary even though the actual color of the ball to be drawn is random. If suddenly a new jug is introduced with a different mix of balls and the next series of draws is from a mixture of the two jugs, the process is non-stationary. However, this is what can be called a limited degree of non-stationarity. By simply redefining the procedure for drawing balls, a new stationary process emerges that involves two steps. At the first step, one of the two jugs is randomly selected. At the second step, a ball is drawn from the chosen jug. As long as this procedure is followed the new process, though more complicated, is stationary. In fact, the new process can be interpreted as an example of a regime switching model in which first the regime is chosen and then a random ball draw occurs.
The balls and jugs analogy is useful for conceptualizing differing degrees of non-stationarity. The important questions include: How many jugs are there? Can the number of jugs even be enumerated? What is the distribution of balls within each of the jugs? In the limit, think of the case where there are an immense number of jugs, the contents of which are unknown, and where the probabilities of selecting a given jug are also unknown and may be changing over time. This limiting case I refer to as fundamental non-stationarity. Although this may seem like an extreme case, I argue that it is a problem that investors face on a daily basis. Fundamental non-stationarity is not a rarity, but the normal state of affairs. To explore the issue further, I consider examples of four investment decisions.
The surprising behavior of the VIX index
The VIX index, calculated by the Chicago Board Options Exchange measures the market's expectation of 30-day volatility. It is constructed from the implied volatilities of a wide range of S&P 500 index options with approximately 30 days to maturity. As of October 2017, the VIX had been near record lows for more than a year. The average was about 11% compared to a long-run historical average of 15% or more depending on the sample period. The investment question is whether this abnormal behavior suggests taking a position in VIX derivatives.
One way to approach the question is to turn to the academic literature on fitting stochastic models to the VIX index. It turns out that the literature is both large and highly sophisticated mathematically. A few recent examples among the many papers include Goard and Mazur (2013), “Stochastic volatility models and the pricing of fix options,” Zang, Ni, Huang, and Wu (2016), “Double-jump stochastic volatility model for VIX: Evidence from VVIX,”, and Kaeck and Alexander (2013) “Continuous time VIX dynamics: On the role of stochastic volatility of volatility.” In their defense, these papers, and others like them, do allow for some non-stationarity along the lines of the two jug analogy. They do so by incorporating the possibility of random jumps or stochastic volatility. The problem I have as an investor is that I fear the process during the current quiescent period is not just a result of a random failure of jumps to materialize or a random drop in volatility in a stochastic volatility model, but a fundamentally different process.
Of course, if a model is fit with enough flexibility in its parameters, it will appear to account for the non-stationarity during the sample period but in doing so it will misstate the true nature of the process. From an investment standpoint this is critical because if the true process is fundamentally non-stationary, at some point it will change in a manner unanticipated by investors. If the change involves drawing from an entirely new jug among a vast number of jugs, a complex process fit to historical data will simply be misleading. This is, in effect, the argument Taleb (2007) makes with regard to the financial crisis. But the observation is not limited to the dramatic, “black swan” events that Taleb describes. If the world is fundamentally non-stationary, it is a problem that investors face continually to varying degrees as the social, political and economic environments evolve.
In particular, the stochastic process for the VIX will change when the social, political and economic factors, which are yet to be delineated, that led to its historical low mean value, are transformed. One such factor that could have altered market volatility was the election of Donald Trump. However, the fact that such an hypothesis is speculative is precisely the problem. As Ross (2005) observes, even after the fact it is difficult to identify events that may have altered the stochastic process of asset returns.
The cross section of expected returns
Following the lead of Fama and French (2002), intense interest in factor models designed to explain the cross section of expected returns has led to extensive research in the area. As Harvey, Liu and Zhu (2016) document, that research effort has produced a veritable zoo of allegedly significant factors. Based on their review of the 313 articles, the authors report the identification of 316 priced factors. This factor zoo led Harvey, Liu and Zhu to argue for the use higher cut-offs for statistical significance in order to overcome the impact of apparent data mining.
Data mining and non-stationarity are different issues, but they can have a similar impact from a practical investment standpoint. Data mining refers to the problems that arise when there is repeated sampling from the same historical data set. The most common problem that results from data mining is the “discovery” of idiosyncratic quirks that are unique to the sample, but are not actual true relations. As a result of data mining, spurious relations uncovered in the sample period will fail to hold post sample. When the data are non-stationary, a relation may be found that does, in fact, hold for the historical sample period but that is no longer true. Once again, the relation fails to hold in the post sample period but for a different reason.
The failure of factor models estimated in one period to hold in another may be due to either data mining, non-stationarity, or some combination. Either way, given the vast zoo of factors that have been uncovered, we (the research profession) are almost assured of finding a factor model that explains the cross section of expected returns in any chosen historical sample period. However, it remains unclear what practical value this has for investors who cannot be confident that the relations will hold going forward.
With regard to individual stocks, language is an impediment to appreciating the full extent of potential non-stationarity. Throughout its corporate life, Apple has always been called Apple but the company has reinvented itself numerous times. In the process, it transformed itself from a start-up maker of personal computers into a global consumer product and services powerhouse despite having several brushes with insolvency. Of course, it is possible that the process for stock returns remained stationary while the company was continually transformed because stock returns depend on investor expectations. But it would be foolhardy for an investor to assume that the dramatic evolution of the firm did not have a major impact on investor perceptions, including investor estimates of risk, and thereby on stock returns.
It is worth noting that applied investment research, by that I mean the work of security analysts, appears to take the problem of stationarity for granted. If the stochastic process generating key metrics of financial performance, such as revenue, earnings and free cash flow, were stationary then presumably the best way to project future financial performance would be to fit statistical models much like those used to analyze the VIX index. This is not what analysts do. Instead, they examine the details of the company’s business with the hope that the understanding they achieve will help them predict future financial performance. This can be interpreted as an effort to overcome non-stationarity by attempting to predict how future business conditions will generate revenues, earnings and free cash flow given currently available information. In the context of the balls and jugs analogy, security analysts are using fundamental analysis to select the jug.
Smart beta and factor premiums
As a final example, there has been an active debate recently regarding so called “smart beta” and associated factor premiums. As Asness (2016) notes, smart beta and factor-based strategies have become increasingly popular in recent years. The goal of these strategies is to identify factors, of which Fama and French’s SML is an early example, and then to harvest the factor premium by investing in long-short portfolios.
As Arnott, Beck, Kalesnik and West (ABKW, 2017) repeatedly state, though they do not couch their argument explicitly in terms of stationarity, this investment strategy is based on the assumption that the stochastic process governing factor returns is sufficiently stationary that past average premiums are reasonable estimates of future expected premiums. ABKW argue that the assumption is false. They claim that research identifying historical factor premiums has failed to adequately account for the extent to which rising valuations contributed to the lofty historical returns. Based on their empirical research, ABKW conclude that valuation increases have been the primary driver of smart beta returns over the short term, and even long term, and as a result past excess returns are not likely to be sustainable in the future. In fact, ABKW suggest that factor portfolios that have markedly appreciated could “go horribly wrong” and potentially crash. The point here is not to evaluate whether ABKW are correct, and there are many authors including Asness (2016) who argue their conclusions are exaggerated, but to note that the entire debate is basically a dispute over stationarity.
In the context of the jugs and balls analogy, valuation increases can be thought of as drawing from a jug without replacement. Every time say a red ball is drawn, the probability of drawing another red ball declines. For this reason, the distribution is non-stationary. The probability of drawing a red ball can be interpreted as the probability that a factor portfolio will earn excess returns. The more the valuation increases, the more red balls are drawn, and the less likely it will be that valuations will rise in the future.
Perhaps the most controversial factor premium in this regard is momentum. Early papers such as Jegadeesh and Titman (1993) found significant premiums associated with momentum. Then later papers including Dolvin and Foltice (2017) argued that the anomaly had disappeared. Simultaneously, Moskovitz and Daniel (2016) reported significant crash risk associated with momentum, but Barroso and Santa-Clara (2015) claimed that this risk could be ameliorated by varying leverage of the momentum portfolio. And this is just a sliver of an immense and internally contradictory literature on momentum. From the standpoint of a practical investor, the safe conclusion is that if there is a momentum effect, it is far from stationary.
The four examples offered here are by no means unique. Similar arguments apply to most every investment strategy based on estimates of statistical parameters derived from historical data. All such strategies assume, explicitly or implicitly, that world is sufficiently stationary that such estimates are of practical value to investors.
Conclusions and implications
The basic conclusion is straightforward. Non-stationarity is not a minor statistical annoyance but a fundamental and unavoidable issue that investors face each time they make an investment decision. I argue that there is generally insufficient evidence to support the assumption that the processes underlying social institutions (including financial markets), unlike those underlying many physical systems, are stationary. Such non-stationarity includes not only the possibility of large, unexpected breaks from the past as occurred during the financial crisis, but daily changes in the stochastic processes governing asset returns. It is not surprising, therefore, that fundamental security analysis, which takes non-stationarity for granted, remains the basis for most practitioner-based investment research.
 My favorite example of data mining involves Richard Feynman and the expansion of Pi. Feynman would reel off the first 768 digits of the expansion, the last six of which are 9-9-9-9-9-9, and then say “and so on” before breaking into laughter. The 763rd digit of Pi has now become known as the Feynman point, but the six 9s have no meaning.
 To be fair, the original name of the company was Apple Computer which was shortened to Apple as other devices (which are actually computers) became the predominant source of the company’s revenue. However, throughout its life the company has generally been referred to as Apple.