Stationarity

Simply put, stationarity is the property of things that do not change over time.

Quant Investment Managers Need Stationarity

At the core of every quantitative investment management endeavor is the assumption that there are patterns in markets that prevailed in the past, that will prevail in the future, and that one can use to make money in financial markets.

A successful search for those patterns, often referred to as *alphas*, and the expectation that they will persist over time, is typically required prior to deploying capital. Thus stationarity is a wishful assumption inherent to quantitative investment management.

Stationarity In Financial Markets Is Self-Destructive

However, alphas are often victim of their own success. The better an alpha, the more likely it will be copied by competitors over time, and therefore the more likely it is to fade over time. Hence, every predictive pattern is bound to be a temporary or transient regime. How long the regime will last depends on the rigor used in the alpha search, and the secrecy around its exploitation.

The ephemerality of alphas is well documented; see for instance Igor Tulchinsky’s latest book, The Unrules: Man, Machines and the Quest to Master Markets, which I highly recommend.

In regards to the widespread perception that financial markets are highly non-stationary though, non-stationarity is often meant in a mathematical sense and usually refers to financial time series.

Time Series Stationarity Can’t Be Disproved With One Finite Sample

In the case of time series (a.k.a. stochastic processes), stationarity has a precise meaning (as expected); in fact two.

A time series is said to be *strongly stationary* when all its properties are invariant by change of the origin of time, or time translation. A time series is said to be *second-order stationary*, or *weakly stationary* when its mean and auto-covariance functions are invariant by change of the origin of time, or time translation.

Intuitively, a stationary time series is a time series whose local properties are preserved over time. It is therefore not surprising that it has been a pivotal assumption in econometrics over the past few decades, so much so that it is often thought that practitioners ought to first make a time series stationary before doing any modeling, at least in the Box-Jenkins school of thought.

This is absurd for the simple reason that, (second order) stationarity, as a property, cannot be disproved from a single finite sample path. Yes, you read that right! Read on to understand why.

But before delving into an almost philosophical argument, let’s take a concrete example.

Let’s consider the plot above. Is this the plot of a stationary time series? If you were to answer simply based on this plot, you would probably conclude that it is not. But I’m sure you see the trick coming, so you would probably want to run a so-called ‘stationarity test’, perhaps one of the most widely used, the Augmented-Dickey-Fuller test. Here’s what you’d get if you were to do so (source code at the end):

ADF Statistic: 4.264155

p-Value: 1.000000

Critical Values:

1%: -3.4370

5%: -2.8645

10%: -2.5683

As you can see, the ADF test can’t reject the null hypothesis that the time series is an AR that has a unit root, which would (kind of) confirm your original intuition.

Now, if I told you that the plot above is a draw from a Gaussian process with mean 100 and auto-covariance function

then I am sure you’d agree that it is indeed a draw from a (strongly) stationary time series. After all, both its mean and auto-covariance functions are invariant by time translation.

If you’re still confused, here’s the same draw over a much longer time horizon:

I’m sure you must be thinking that it looks more like what you’d expect from a stationary time series (e.g. it is visually mean-reverting). Let’s confirm that with our ADF test:

ADF Statistic: -4.2702

p-Value: 0.0005

Critical Values:

1%: -3.4440

5%: -2.8676

10%: -2.5700

Indeed, we can reject the null hypothesis that the time series is non-stationarity at a 0.05% p-Value, which gives us strong confidence.

However, the process hasn’t changed between the two experiments. In fact even the random path used is the same, and both experiments have enough points (at least a thousand each). So what’s wrong?

Intuitively, although the first experiment had a large enough *sample size*, it didn’t span long enough a *time interval* to be characteristic of the underlying process, and there is no way we could have known that beforehand!

*The takeaway is that it is simply impossible to test whether a time series is stationary from a single path observed over a finite time interval, without making any additional assumption.*

Two assumptions are often made but routinely overlooked by practitioners and researchers alike, to an extent that results in misinformed conclusions; an implicit assumption and an explicit assumption.

**The Implicit Assumption**

Stationarity is a property of a stochastic process, not of a path. Attempting to test stationarity from a single path ought to implicitly rely on the assumption that the path at hand is sufficiently informative about the nature of the underlying process. As we saw above, this might not be the case and, more importantly, one has no way of ruling out this hypothesis. Because a path does not look mean-reverting does not mean that the underlying process is not stationary. You might not have observed enough data to characterize the whole process.

Along this line, any financial time series, whether it passes the ADF test or not, can always be extended into a time series that passes the ADF test (hint: there exist stationary stochastic processes whose space of paths are universal). Because we do not know what the future holds, strictly speaking, saying that financial time series are non-stationary is slightly abusive, at least as much so as saying that financial time series are stationary.

In the absence of evidence of stationarity, a time series should not be assumed to be non-stationary — we simply can’t favor one property over the other statistically. This works similarly to any logical reasoning about a binary proposition A: no evidence that A holds is never evidence that A does not hold.

Assuming that financial markets are non-stationarity might make more ** practical sense** as an

**than assuming that markets are stationary for structural reasons. For instance, it wouldn’t be far fetch to expect productivity, global population, and global output, all of which are related to stock markets, to increase over time. However, would not make more**

*axiom***, and it is a working hypothesis that we simply cannot invalidate (in insolation) in light of data.**

*statistical sense*2. **The Explicit Assumption**

Every statistical test of stationarity relies on an assumption on the class of diffusions in which the underlying process’ diffusion must lie. Without this, we simply cannot construct the statistic to use for the test.

Commonly used (unit root) tests typically assume that the true diffusion is an Autoregressive or AR process, and test the absence of a unit root as a proxy for stationarity.

*The implication is that such tests do not have as null hypothesis that the underlying process is non-stationary, but instead that the underlying process is a non-stationary AR process!*

Hence, empirical evidence leading to reject the null hypothesis could point to either the fact that the underlying process is not an AR, or that it is not stationary, or both! Unit root tests by themselves are not enough to rule out the possibility that the underlying process might not be an AR process.

The same holds for other tests of stationarity that place different assumptions on the underlying diffusion. Without a model there is no statistical hypothesis test, and no statistical hypothesis test can validate the model assumption on which it is based.

Seek Stationary Alphas, Not Stationary Time Series

Given that we cannot test whether a time series is stationary without making an assumption on its diffusion, we are faced with two options:

- Make an assumption on the diffusion and test stationarity
- Learn a predictive model, with or without assuming stationarity

The former approach is the most commonly used in the econometrics literature because of the influence of the Box-Jenkins method, whereas the latter is more consistent with the machine learning spirit consisting of flexibly learning the data generating distribution from observations.

Modeling financial markets is hard, very hard, as markets are complex, almost chaotic systems with very low signal-to-noise ratios. Any attempt to properly characterize market dynamics — for instance by attempting to construct stationary transformations — as a requirement for constructing alphas, is brave, counterintuitive, and inefficient.

Alphas are functions of market features that can somewhat anticipate market moves in absolute or relative terms. To be trusted, an alpha should be expected to be preserved over time (i.e. be stationary in a loose sense). However, whether the underlying process itself is stationary or not (in the mathematical sense) is completely irrelevant. Value, size, momentum and carry are some examples of well documented trading ideas that worked for decades, and are unrelated to the stationarity of price or returns series.

But enough with stationarity, let’s move on to the nature of memory in markets.