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.[1] 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.

*Individual Stocks*

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.[2] 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.

[1] 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 763

^{rd}digit of Pi has now become known as the Feynman point, but the six 9s have no meaning.
[2] 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.