This is the second in a three-part series on the growing likelihood of a transition in major markets. Part 1 focused on Macro Event Risk and Cycles. This post discusses the application of popular complex systems Early Warning Signals (EWS) to assess a possible transition in the S&P 500.

## Complex Systems Theory

Most people associate complexity science with natural systems such as animal populations, climate ecosystems, and earthquake prediction.

For some years now, there has been expanding research into man-made systems that exhibit similar characteristics, including social networks and financial markets. In markets, the application of complexity science is still nascent and is likely best applied in combination with other analysis toolkits.

### Bifurcations and Critical Transitions

Complex systems can have tipping points where they undergo a sudden transition to a new state or regime. These points are called bifurcations^{1} and the shifts that occur when a system passes bifurcations are called critical transitions^{2}. There are many types of bifurcations, including

- Catastrophic bifurcations where a system transitions from one stable state to a new stable state
- Bifurcations in cyclic and chaotic systems where a system transitions from a stable state to an unstable state

### Early Warning Signals for Critical Transitions

Researchers have come to believe that there exist standard Early Warning Signals (EWS) to identify when critical transitions are approaching.

Various categories of EWS have been produced. Two popular categories are (1) signals based on time series analysis and (2) trait-based signals based on trends in the visual representation of the systems. To date, I have focused on time series signals, but believe there is potential in visual trait-based signals for financial markets.

Discussing every time-series based EWS is beyond the scope of this post. I focus on the most widely researched EWS to date, Critical Slowing Down (CSD), a phenomenon whereby a slight disturbance of the system away from a perceived equilibrium takes a long time to revert back to equilibrium.

The figure below depicts CSD in a fold bifurcation system. In (a), the system quickly recovers from the perturbation (depicted by a clear circle) to its equilibrium (the dark circle) as the slope of recovery is steep. As the system moves toward bifurcation to a new phase (b), the recovery slope is flatter and recovery takes longer. Finally, at bifurcation/phase transition, there is no recovery and the system moves to a new state.

### Measuring Critical Slowing Down

CSD, and thus a system’s proximity to a phase transition, can be evaluated using two statistical signals: rising correlation and increasing variance.

#### Rising Correlation

Rising correlation is best measured in a time-series based system using autocorrelation, which per Wikipedia is:

“The correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations as a function of the time lag between them.”

Source: https://en.wikipedia.org/wiki/Autocorrelation

What this means is that a given data point in a time series is impacted by a “copy of itself”, i.e. a previous data point in the time series. So, the price of the S&P 500 today can be impacted by a different point in the time series, which despite the time lag, is the same actual variable.

To identify CSD, researchers have found that an increase in the lag-1 autocorrelation (AR(1) increases as the system approaches a critical state. AR(1) can be estimated using the following autoregression model:

The above should look familiar as a linear function. What makes it autoregressive is that you are using a time series value at t-1 to predict the time series value at t.

**?**_{t}is the price at time*t***?**_{t-1}is the price at time*t-1***?**^{−?Δ?}is the AR(1) coefficient or λ**??**is random noise

The key element is the AR(1) coefficient, also represented as λ. ? is the magnitude of the recovery rate to equilibrium in the system and Δ? is the change between t-1 and t, which is 1. Thus, as recovery rates move toward zero, i.e. the system is not recovering as fast to equilibrium, the AR(1) coefficient moves toward 1, because ? raised to a number approaching 0, approaches 1.^{1}

#### Increased Variance

Rising variance is also indicative of CSD.

“Again, this can be…intuitively understood: as the eigenvalue [the recovery rate of the system, ?

Scheffer, M. Critical Transitions in Nature and Society (Princeton Univ. Press, 2009).^{−?Δ?}or λ] approaches zero, the impacts of shocks do not decay, and their accumulating effect increases the variance of the state variable”

Scheffer offers a mathematical proof of this, explained briefly below:

The expectation of an AR(1) process (the equation above) is that it will move toward the mean values of ?_{t} and ?_{t-1}, which means you can transform the equation above as follows (with **E** standing for expected value and **c** is a constant):

E(?_{t}) = c + λE(?_{t-1}) + E(??) >> mean (?) = c + λ*mean(?) + 0 >>

mean(y) = c / 1 – λ

With λ (the recovery rate of the system) approaching zero and c = 0, the mean(y) equals zero and Scheffer proves that variance progresses toward infinity when autocorrelation tends to one:

“Close to the critical point, the return speed to equilibrium decreases, implying that [?] approaches zero and the autocorrelation tends to one. Thus, the variance tends to infinity.”

Scheffer, M. Critical Transitions in Nature and Society (Princeton Univ. Press, 2009).

## CSD in the S&P 500

Leveraging the open source ewstools Python library, I calculated core Early Warnings Signals (including some that are not covered in this post) for the S&P 500 over the last two years. I used daily closing price data for this analysis which I adjusted to achieve mean-stationarity of the data.

The analysis shows rising variance and autocorrelation since Fall 2018 in the S&P 500. Taken in isolation, this would not be sufficient to predict a transition. However, in conjunction with the convergence of macroeconomic cycles and shifts in price momentum, it becomes a more relevant data point. The full code for this analysis can be found here.

In the next post in this series, I will discuss price action and momentum in major markets, notably crude oil and equities.

- Kuznetsov, Y. A. Elements of Applied Bifurcation Theory (Springer, 1995).
- Scheffer, M. Critical Transitions in Nature and Society (Princeton Univ. Press, 2009).

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*Any opinions or forecasts contained herein reflect the personal and subjective judgments and assumptions of the author only. There can be no assurance that developments will transpire as forecasted and actual results will be different. The accuracy of data is not guaranteed but represents the author’s best judgment and can be derived from a variety of sources. The information is subject to change at any time without notice.*