# Linear and nonlinear causality in financial markets

Haochun Ma  Ludwig-Maximilians-Universität, Department of Physics,
Schellingstraße 4, 80799 Munich, Germany  Allianz Global Investors, risklab,
Seidlstraße 24, 80335, Munich, Germany  Davide Prosperino  Ludwig-Maximilians-
Universität, Department of Physics, Schellingstraße 4, 80799 Munich, Germany
Allianz Global Investors, risklab, Seidlstraße 24, 80335, Munich, Germany
Alexander Haluszczynski  Allianz Global Investors, risklab, Seidlstraße 24,
80335, Munich, Germany  Christoph Räth<EMAIL_ADDRESS>Deutsches
Zentrum für Luft- und Raumfahrt (DLR), Institut für KI Sicherheit, Wilhelm-
Runge-Straße 10, 89081 Ulm, Germany

###### Abstract

Identifying and quantifying co-dependence between financial instruments is a
key challenge for researchers and practitioners in the financial industry.
Linear measures such as the Pearson correlation are still widely used today,
although their limited explanatory power is well known. In this paper we
present a much more general framework for assessing co-dependencies by
identifying and interpreting linear and nonlinear causalities in the complex
system of financial markets. To do so, we use two different causal inference
methods, transfer entropy and convergent cross-mapping, and employ Fourier
transform surrogates to separate their linear and nonlinear contributions. We
find that stock indices in Germany and the U.S. exhibit a significant degree
of nonlinear causality and that correlation, while a very good proxy for
linear causality, disregards nonlinear effects and hence underestimates
causality itself. The presented framework enables the measurement of nonlinear
causality, the correlation-causality fallacy, and motivates how causality can
be used for inferring market signals, pair trading, and risk management of
portfolios. Our results suggest that linear and nonlinear causality can be
used as early warning indicators of abnormal market behavior, allowing for
better trading strategies and risk management.

††preprint: AIP/123-QED

> Within the complex system of financial markets, understanding the intricate
> ties between assets is crucial. Although the Pearson correlation has been a
> standard measure for these relationships, its linear approach might not
> fully represent the entire spectrum of causality. This study employs
> sophisticated causal inference algorithms and methods to differentiate
> between linear and nonlinear causal contributions. By examining major stock
> indices from Germany and the U.S., we uncover profound and possibly
> nonlinear linkages. More than presenting a new approach, this research
> indicates a significant shift in our perception and quantification of
> financial market behaviors. Such insights hold promise for refining market
> predictions, optimizing trading strategies, and improving portfolio risk
> management.

##  I Introduction

The field of econophysics is garnering heightened attention in the physics
domain, offering a novel lens to conventional financial methodologies
Jovanovic, Mantegna, and Schinckus (2018). This emerging perspective draws
from statistical physics tools, spanning signal processing, agent-based market
frameworks, and random matrix theory Wang  _et al._ (2013). Understanding the
co-dependence of financial assets is paramount across various finance sectors,
especially when quantifying portfolio-associated risks Mantegna and Stanley
(1999). This development has seen industry practitioners keenly monitor the
evolution of co-dependence metrics. Predominantly, mutual dependencies of
financial instruments are characterized via the Pearson correlation of their
return time series. However, there is increasing research underscoring the
nonlinear characteristics of these series Wang, Xie, and Stanley (2018).
Notably, Mantegna and Stanley (1995) showed the power law scaling dynamics of
financial indices’ probability distributions, while Ghashghaie  _et al._
(1996) pinpointed turbulent cascades in foreign exchange markets. Such
insights challenge the adequacy of linear dependency metrics. Addressing this,
Haluszczynski  _et al._ (2017) segregated linear from nonlinear mutual
information contributions using Fourier transform surrogates, aiming to
quantify nonlinear correlations among financial assets. The authors
demonstrated that the integration of nonlinear correlations into portfolio
construction led to an increase in investment performance. A pressing query is
the continued reliance on the Pearson correlation Benesty  _et al._ (2009) as
a causality proxy, given the intricate nature of causality measurement within
dynamic systems. Granger’s initial study in the 1960s Granger (2001) addressed
the difference between causality and correlation, leading to the development
of more advanced causal inference tools. This ranged from information-
theoretic tools Schreiber (2000) to state-space reconstruction models Sugihara
_et al._ (2012). While causal inference has mainly focused on determining
causality Ge and Lin (2021), the study of its linear versus nonlinear
characteristics has not been performed in detail. Beginning work has been
performed by Paluš and Vejmelka (2007) and Hlinka  _et al._ (2014), who
focused on mutual information to detect nonlinear dynamics in time series and
evaluated nonlinearity contributions in climate connectivity.

In this paper, we analyze causality in financial markets by separating linear
and nonlinear contributions to causality using Fourier transform surrogates.
To do so, we use two different causal inference techniques and apply them to
historical stock data of the German DAX and the U.S. Dow-Jones index. We also
identify causality-based statistical properties of financial data and motivate
how linear and nonlinear causality can be separated and measured. We find that
while correlation is a good proxy for linear causality, nonlinear effects are
disregarded and thus, significant amount of nonlinear causality is neglected.
This is potentially dangerous when practitioners evaluate the risk of a
portfolio only using correlation. Therefore, we propose a simple integration
of causality measures into market signal inference, pair trading, and
portfolio construction routines and show that they yield superior results.

Figure 1: Mirage correlations and causality. The top row shows different
regimes of the coupled difference system defined in Equation 1. It appears
that the variables are correlated in the first regime, anti-correlated in the
second, and lose all coherence in the third. The bottom row shows the rolling
correlation (left), causality (center), and linear causality (right). The
causality is measured using Convergent Cross Mapping (CCM). While the
correlation alternates between periods of positive, negative, and zero
correlation, the causality in both directions stays stable over time. This
also holds true for the linear causality. When comparing the measurements to
the governing equations, we see that causality offers a more stable and
accurate representation of the co-dependence between the two variables than
correlation does.

##  II Methods

We structure our methods section in three different parts: data, causality
measures, linear and nonlinear decomposition, and financial frameworks.

###  II.1 Data

In this section, we describe the data used for this study. Before we apply our
framework to real-world data, we demonstrate it on a synthetic example.
Additionally, we use rolling windows in order to evaluate our analysis
dynamically.

####  II.1.1 Coupled Difference

A simple example of a system that displays chaotic behavior is the coupled
difference as introduced in Lloyd (1995). This system was also employed by
Sugihara  _et al._ (2012) to illustrate Convergent Cross Mapping (CCM), a
causality inference method integral to this study. It is defined by the
following two equations:

$\displaystyle\begin{split}x(t+1)&=x(t)\cdot\left[r_{x}-r_{x}\cdot
x(t)-\beta_{y\rightarrow x}\cdot y(t)\right]\\\
y(t+1)&=y(t)\cdot\left[r_{y}-r_{y}\cdot x(t)-\beta_{x\rightarrow y}\cdot
x(t)\right]\,,\end{split}$ (1)

where the standard parameters are: $r_{x}=3.8$, $r_{y}=3.5$,
$\beta_{y\rightarrow x}=0.02$, and $\beta_{x\rightarrow y}=0.1$. We selected
this system due to its exhibition of so-called mirage correlations, which
means that variables may be positively coupled for long periods but can
spontaneously become uncorrelated or decoupled. This can lead to problems when
fitting models or inferring causality from observational data Sugihara  _et
al._ (2012).

Figure 2: Historical Stock Returns and Correlation. The top row shows the
logarithmic returns of the historical stock data of the German DAX (left) and
the U.S. Dow-Jones (right) index, respectively. Each line represents the
logarithmic return of one stock over time. The bottom row shows the pairwise
correlations between the stocks. Each line represents the correlation between
two stocks over time. The black line shows the average correlation inside the
index. The vertical lines represent important economic or political events.

####  II.1.2 Financial Data

For our real-world analysis, we select a subset of stocks from the DAX and
Dow-Jones indices that represent the $30$ highest capitalized and thus most
influential companies in Germany and the U.S., respectively. Beginning on
January 19, 1973, our data consists of the daily closing prices of all stocks
that were in the index through April 20, 2022, to provide a consistent
universe of stocks over the entire period. This yields a total of $N_{DAX}=11$
and $N_{DJ}=17$ time series with $12785$ data points. We would like to note
that the survival bias Brown  _et al._ (1992) is negligible for our analysis.

To ensure stationary time series, we convert the stock prices $p_{t}$ to
logarithmic returns:

$x_{t}=\log{p_{t}}-\log{p_{t-1}}\,.$

The time horizon of our data is long enough to examine a number of important
market events—starting with the global recession of the early 1980s, it also
includes Black Monday (October 19, 1987), when stock markets around the world
collapsed for the first time since World War II. From 1997 to 2001, markets
were characterized by excessive speculation and the overvaluation of many
technology companies, which led to the dot-com bubble DeLong and Magin (2006).
The bubble burst in 2002 with substantial price declines in July and
September. Finally, our data includes the 2007/2008 subprime mortgage crisis,
when the market declined from its all-time high in October 2007 and crashed
after the collapse of Lehman Brothers on September 15, 2008. As a result of
slowing growth of the GDP of China and the Greek debt default, investors sold
shares globally between 2015 and 2016. The data further includes the so-called
Volmageddon on February 5, 2018, where a large sell-off in the U.S. stock
market lead to a spike in implied market volatility Krishnan  _et al._ (2021).
Finally, the data includes the impact of the COVID-19 pandemic, which, among
other events, triggered a sudden global stock market crash on February 20,
2020. In addition, our period under review also includes a number of important
global political events. These include the fall of the Berlin Wall on November
9, 1989, which triggered the collapse of the Soviet Union, the attacks of
September 11, 2001, and the Russian invasion of the Ukraine on February 24,
2022.

####  II.1.3 Rolling Windows

To obtain dynamically evolving results, we divide the data into overlapping
rolling windows Zivot  _et al._ (2003) and compute our measures for each
interval following the approach by Haluszczynski  _et al._ (2017). We use a
sliding window of $T_{w}=1000$ trading days, which corresponds to roughly four
years of data. The gap or stride between successive intervals is set to
$\delta T=20$ trading days, roughly amounting to a month. As such, the $w$-th
interval is represented as:

$\mathbf{x}^{(w)}=\left(x_{1+(w-1)\cdot\delta
T},\dots,x_{T_{w}+(w-1)\cdot\delta T}\right)\,,$ (2)

which yields a total of $w=594$ overlapping windows. A (causality) measure
$\psi\big{(}\mathbf{x},\mathbf{y}\big{)}\mapsto\mathbb{R}$, which maps two
time series to a real number, is thus transformed into a vector
$\Psi\in\mathbb{R}^{w}$.

Figure 3: Transfer Entropy. The first row shows the historical TE of stocks
within the German DAX (left) and the U.S. Dow-Jones (right) indices,
respectively. Each line represents one direction of the TE between two stocks
over time. The bottom row illustrates the corresponding surrogate TE. The
vertical lines represent important economic or political events.

###  II.2 Causality Measures

We select two techniques that represent prominent categories currently used in
causal inference Runge (2018)—however, it is important to note that our
framework is applicable to any method capable of detecting nonlinear
causality.

####  II.2.1 Pearson Correlation

Before describing the causal inference methods, we introduce the Pearson
correlation Pearson (1895). We use it as a benchmark since it is still widely
popular in the financial industry due to its simple calculation and
interpretability Benesty  _et al._ (2009). It quantifies the strength and
direction of the linear relationship between two variables. It is computed as
follows:

$\rho\left(\mathbf{x},\mathbf{y}\right)\equiv\frac{\sum_{i=t}^{T}(x_{t}-\bar{x})(y_{t}-\bar{y})}{\sqrt{\sum_{t=1}^{T}(x_{t}-\bar{x})^{2}}\sqrt{\sum_{t=1}^{T}(y_{t}-\bar{y})^{2}}}\,,$
(3)

where $x_{t}$ denotes the stock returns at time $t$ and
$\bar{x}=\frac{1}{T}\sum_{t=1}^{T}x_{t}$ signifies their expected value. The
correlation is normalized and bounded to the interval $[-1,1]$ and thus allows
direct comparisons across pairwise correlations between different stocks. As
shown by Bonett and Wright (2000) a sample size of $T\leq 56$ is sufficient to
estimate the measure reliably.

####  II.2.2 Transfer Entropy

Transfer Entropy (TE) is a powerful information-theoretic measure introduced
by Schreiber (2000) which has gained popularity in the field of causal
inference, particularly in the analysis of time series data. TE provides a way
to quantify the directed flow of information between variables, which allows
assessing causal relationships in a probabilistic framework. The TE from $X$
to $Y$ is defined as:

$\textit{TE}_{X\rightarrow
Y}=H(Y_{t+1},Y_{t})+H(Y_{t},X_{t})-H(Y_{t+1},Y_{t},X_{t})-H(Y_{t})\,,$

where $H(Y_{t+1},Y_{t})$, $H(Y_{t},X_{t})$, $H(Y_{t+1},Y_{t},X_{t})$, and
$H(Y_{t})$ are the joint and marginal entropies of the respective variables.
To facilitate comparison between different estimations of TE, we apply the
subsequent normalization:

$\textit{TE}_{X\rightarrow
Y}=\frac{H(Y_{t+1},Y_{t})+H(Y_{t},X_{t})-H(Y_{t+1},Y_{t},X_{t})-H(Y_{t})}{\sqrt{H(Y_{t+1},Y_{t})\cdot
H(X_{t+1},X_{t})}}\,.$ (4)

The normalization to $\left[0,1\right]$ stems from our understanding of TE as
an asymmetric causal measure. This interpretation aligns with the concept of
covariance, which, when rescaled, results in the normalized form, the
aforementioned Pearson correlation Pearson (1895).

Figure 4: Convergent Cross Mapping. The setup of this figure is analogous to
Fig. 3.

We would like to point out that the calculation of empirical probability
densities $p$ and hence information-theoretic measures raise unexpected
difficulties exceeding the scope of this work. While it is common to use
histograms with equally distributed bins to estimate densities, Mynter (2021)
showed that this method potentially leads to biases since the estimation is
dependent on the partition details—hence, finding a robust estimator is non-
trivial. However, for the purpose of our research, we find that equally
distributed bins perform reasonably well. Furthermore, it is worth mentioning
that TE might capture false causalities depending on the dimension of
conditioning Paluš and Vejmelka (2007).

####  II.2.3 Convergent Cross Mapping

Convergent Cross Mapping (CCM) is an influential technique utilized for causal
inference within the realm of complex dynamical systems Sugihara  _et al._
(2012). It aims to reveal causal connections between variables by
reconstructing the dynamics that underlie them. CCM operates on the premise
that variables with causal links will exhibit similar dynamical behavior,
leading to a notion referred to as shadowing.

The underlying idea is based on Takens’ theorem, which states that the entire
state space can be reconstructed from a single embedded coordinate of the
system, also called a shadow manifold Takens (2006). Due to transitivity, two
coordinates within a system can then be mapped to each other by neighboring
states in their respective shadow manifolds—this allows for cross prediction.
The quality of the prediction, evaluated using the Pearson correlation,
quantifies the strength of the causal relationship. The algorithm of CCM can
be outlined as follows:

  1. 1.

Time Delay Embedding Embed the time series data of $X$ and $Y$ into higher-
dimensional spaces using the embedding dimension $\kappa$ and time delay
$\tau$.

  2. 2.

Library Construction Create a library of vectors from the reconstructed state
space $\mathbf{X}$, denoted as $\mathcal{L}_{\mathbf{X}}$, and a library of
vectors from the reconstructed state space $\mathbf{Y}$, denoted as
$\mathcal{L}_{\mathbf{Y}}$.

  3. 3.

Nearest Neighbor Selection For each vector $\mathbf{X}(i)$ in the shadow
manifold $\mathcal{M}_{\mathbf{X}}$, find its nearest neighbor in
$\mathcal{M}_{\mathbf{Y}}$, denoted as $\mathbf{Y}(j)$. Similarly, for each
vector $\mathbf{Y}(k)$ in $\mathcal{M}_{\mathbf{Y}}$, find its nearest
neighbor in $\mathcal{M}_{\mathbf{X}}$, denoted as $\mathbf{X}(l)$.

  4. 4.

Cross Mapping Assess the predictability of $X$ based on $Y$ by comparing the
distances between the vector pairs $\mathbf{X}(i)$ and $\mathbf{Y}(j)$, and
the vector pairs $\mathbf{Y}(k)$ and $\mathbf{X}(l)$. A statistical measure,
such as the correlation coefficient $\rho$, can be used to quantify the
predictability.

  5. 5.

Convergence Analysis Repeat the cross mapping procedure for different library
lengths. Evaluate the correlation as a function of the number of points used
and assess the convergence of the results. The convergence of the cross
mapping indicates the presence of a causal relationship between $X$ and $Y$.

In the original application of CCM, convergence typically requires visual
inspection. However, we’ve implemented a more systematic approach using
expanding windows. For a given vector of correlations $\mathbf{\rho}$ of size
$n$, we calculate the standard deviation within each window. Convergence is
determined if the standard deviation consistently decreases, eventually
falling below a predefined threshold $\theta$. If convergence is achieved, the
mean of the last $s$ values is calculated to smooth any outliers. Conversely,
if convergence is not reached, the causality measure in CCM is set to zero.
This process is mathematically expressed as:

$\textit{CCM}_{X\rightarrow
Y}\equiv\begin{cases}\frac{1}{n}\sum_{i=1}^{s}\mathbf{\rho}_{n-s+i}\,&\text{if
$\mathbf{\rho}$ converges}\\\
0\,&\text{otherwise}\end{cases}\in\left[-1,1\right]\,.$ (5)

This process automates the evaluation of CCM causality for various connections
within a system at a reasonable speed. To standardize the measure and render
it comparable with other non-directional causal inference methods, the
correlation distance, denoted as $d=\sqrt{2\left(1-\rho\right)}$, can be
employed.

CCM’s effectiveness in identifying causal relationships within time series
data is affected by multiple aspects. The presence of noise or missing values
in the data can alter the outcomes Mønster  _et al._ (2017), and the choice of
appropriate embedding dimensions $\kappa$ and time delays $\tau$ is subject to
the characteristics of the specific dataset Wallot and Mønster (2018). For
example, the optimal value for $\tau$ can be determined by finding the first
local minimum in the Mutual Information (MI) respective to $\tau$.
Additionally, the False Nearest Neighbor (FNN) algorithm can help finding the
smallest embedding dimension that maintains the attractor’s structure,
ensuring that neighboring points in the original time series stay neighbors in
the embedded version Kennel, Brown, and Abarbanel (1992).

Figure 5: Fallacy Transfer Entropy. The first row shows the historical fallacy
TE of stocks within the German DAX (left) and the U.S. Dow-Jones (right)
indices, respectively. Each line represents one direction of the fallacy TE
between two stocks over time. The bottom row illustrates the corresponding
surrogate TE. The setup of this figure is analogous to Fig. 3.

####  II.2.4 Limits of Causality Measures

We would like to emphasize our recognition of the limitations associated with
the causal inference techniques we are presenting, as well as the broader
challenges inherent in causal inference. Nonetheless, the purpose of this
paper is to utilize these methods as a means to demonstrate a framework for
dissecting causality into linear and nonlinear components within the context
of finance. It is important to note that this paper does not delve into
assessing the accuracy of these methods in capturing genuine causal
relationships, nor does it explore the robustness of the methods themselves.
Despite their drawbacks, these two methodologies have shown successful
applications across various real-world scenarios McCracken and Weigel (2014).
For in-depth analyses on TE and CCM, we recommend referring to Overbey and
Todd (2009) and Krishna and Tangirala (2019), respectively.

Additionally, we acknowledge that TE and CCM operate with reconstructed spaces
and have theoretical vulnerabilities when applied to variables within an
attractor McCracken and Weigel (2014). Nevertheless, the analysis conducted in
this paper relies on simulated data rather than a purely theoretical
foundation. For a comprehensive discussion on the efficacy of state-space
reconstruction methods in establishing causality, we direct interested readers
to Cummins, Gedeon, and Spendlove (2015).

Figure 6: Fallacy Convergent Cross Mapping. The setup of this figure is
analogous to Fig. 5.

###  II.3 Linear and Nonlinear Decomposition

To decompose the causal relationships within time series systems into
components originating from linear and nonlinear drivers, we employ surrogate
techniques based on the Fourier Transform (FT). Employing these surrogates on
(causality) measures, we devise methodologies to systematically capture the
quantitative breakdown of linear and nonlinear influences.

####  II.3.1 Fourier Transform Surrogates

FT surrogates destroy the nonlinear characteristics of a time series $\bm{x}$
while keeping the linear ones unaffected Räth and Monetti (2009). The
algorithm to generate FT surrogates is described as follows Räth  _et al._
(2012):

  1. 1.

Fourier Transform: Given a real-valued time series
$\mathbf{x}=\\{x_{1},x_{2},\ldots,x_{N}\\}$, compute its Fourier transform
$\mathbf{F}(\mathbf{x})$ using the Fast Fourier Transform (FFT) algorithm
Brigham (1988).

$\mathbf{F}(\mathbf{x})=\textit{FFT}(\mathbf{x})$

  2. 2.

Phase Randomization: Preserve the amplitudes but randomize the phases of the
Fourier coefficients. This can be done by multiplying the complex Fourier
coefficients by a random phase factor $e^{i\phi}$, where $\phi$ is uniformly
distributed over the interval $[0,2\pi]$. The phase-randomized Fourier
Transform $\mathbf{F}^{\prime}(\mathbf{x})$ is given by:

$F^{\prime}_{k}=|F_{k}|\cdot e^{i\phi_{k}},\quad\phi_{k}\in[0,2\pi]$

  3. 3.

Inverse Fourier Transform: Compute the inverse FT of the phase-randomized
coefficients to obtain the surrogate time series $\mathbf{\tilde{x}}$:

$\mathbf{\tilde{x}}=\textit{IFFT}(\mathbf{F}^{\prime}(\mathbf{x}))$

By keeping the amplitudes of the original data and only randomizing the
phases, the resulting surrogates maintain the power spectral density of the
original time series but break the higher-order statistical dependencies.

To enhance the reliability of our findings, we average metrics derived from
surrogate time series over various instances $K$ of random phases. The
surrogate of time series $\mathbf{x}$, when subjected to the random phases of
realization $k$, is denoted as $\mathbf{\tilde{x}}^{(k)}$.

Figure 7: Nonlinear causality. The first row shows the historical nonlinear TE
of stocks within the German DAX (left) and the U.S. Dow-Jones (right) indices,
respectively. Each line represents one direction of the TE between two stocks
over time. The bottom row illustrates the nonlinear CCM. The vertical lines
represent important economic or political events.

####  II.3.2 Linear and Nonlinear Measures

In order to evaluate how much of a (causal) measure is attributed to linear or
nonlinear effects, we adopt a specific approach that involves the calculation
of measures on surrogate time series. Within the context of this research, we
focus on a bivariate measure, denoted as
$\psi\big{(}\mathbf{x},\mathbf{y}\big{)}$, which is a function mapping two
time series to a real number. This function’s purpose is to capture the
relationship between the two time series in numerical terms. The corresponding
surrogate or linear measure is defined as the average over $K$ surrogate
realizations of both time series:

$\tilde{\psi}(\mathbf{x},\mathbf{y})\equiv\frac{1}{K}\sum_{k=1}^{K}\psi\big{(}\tilde{\mathbf{x}}^{(k)},\tilde{\mathbf{y}}^{(k)}\big{)}\,.$
(6)

Here, the superscript $k$ indicates that we add the same random phases to both
time series within a single realization. This choice ensures that phase
differences remain unaffected, preserving specific properties such as the
Pearson correlation Prichard and Theiler (1994a). To ensure robustness we
repeat the calculation for $K=50$ surrogate realizations.

####  II.3.3 Nested Measures

As aforementioned, employing rolling windows transforms the measure $\psi$
into a vector. This transition allows for the investigation of interrelations
between two measures through a third expression:

$\psi_{ir}\equiv\rho(\mathbf{\psi}_{1},\mathbf{\psi}_{2})\,.$ (7)

Particularly, we can utilize the Pearson correlation $\rho$ to study the
relationship between the original measure and its corresponding surrogate,
expressed as:

$\rho(\mathbf{\psi},\mathbf{\tilde{\psi}})\,.$ (8)

This method also allows for expressing the coefficient of determination using
the Pearson correlation, as mentioned in Kasuya (2019):

$R^{2}=\rho^{2}\in\left[0,1\right]\,,$ (9)

This enables us to quantify the extent of the measure attributable to linear
influences, more precisely, the fraction of the variability in the measure
$\psi$ that can be explained from the surrogate measure $\tilde{\psi}$. What
remains then emanates from nonlinear characteristics:

$\psi_{nl}\equiv 1-\rho^{2}(\mathbf{\psi},\mathbf{\tilde{\psi}})\,.$ (10)

Furthermore, there’s an application to the exploration of the correlation-
causality fallacy Maziarz (2015). This involves determining how much of the
causality is explained by correlation:

$\psi_{fall}\equiv\rho^{2}(\mathbf{\psi},\mathbf{\rho})\,,$ (11)

serving as a gauge of the causal relationship that can be explained by
correlation. Specifically, this measure for the fallacy can be applied to the
surrogate measure in order to evaluate how much of the linear causality is
captured by correlation:

$\psi_{fall,lin}\equiv\rho^{2}(\tilde{\mathbf{\psi}},\mathbf{\rho})\,.$ (12)

Figure 8: Pair Trading. The stock prices of two companies from the DAX (Bayer
and BASF) are displayed in the top left figure. The top right figure presents
the co-dependence measures over time, with each color corresponding to a
specific co-dependence measure that is included in the legend on the right-
hand side. The bottom left chart illustrates the strategy positions over time,
with long position in Bayer and short position in BASF indicated by $1$, the
opposite indicated by $-1$, and no investment indicated by $0$. The graph in
the lower right corner illustrates the cumulative return achieved by the
strategy over time. The dotted horizontal lines mark the strategy’s most
recent cumulative return value. The vertical lines indicate notable economic
or political events.

###  II.4 Financial Frameworks

Here, we introduce two financial frameworks and demonstrate how causality can
be easily integrated, while simultaneously enhancing performance.

####  II.4.1 Pair Trading

Pair trading is a popular and widely utilized strategy in quantitative finance
that aims to capitalize on relative price movements between two closely
related assets Vidyamurthy (2004). This strategy is grounded in the concept of
mean reversion, which assumes that over time, the prices of assets that are
historically correlated tend to revert to their historical average
relationship. The basic premise is to find two stocks that are highly
correlated. When they deviate from this correlation (i.e., one stock moves up
while the other moves down or vice versa), we take a long position in the
underperforming stock and a short position in the outperforming stock,
expecting them to revert to their historical correlation Hull (2019). Thus, a
basic form of the strategy involves the following steps:

  1. 1.

Correlation Calculation: We calculate the rolling historical and the short-
term correlation between two stocks

  2. 2.

Signal Generation: When the current correlation $\rho_{t}$ deviates from its
historical mean by a certain threshold, a trading signal is generated. A
common approach is to use the z-score $z$ of the spread, which measures the
number of standard deviations by which the current correlation deviates from
its historical mean:

$z_{t}=\frac{\rho_{t}-\bar{\rho}_{hist}}{\sigma_{\rho_{hist}}}\,,$ (13)

where $\bar{\rho}_{hist}$ and $\sigma_{\rho_{hist}}$ denote the mean and
standard deviation of the historical correlation, respectively.

  3. 3.

Trade Execution: When the z-score crosses a predefined threshold (e.g., above
a positive threshold for a long trade or below a negative threshold for a
short trade), a trade is initiated. A long trade involves buying the
underperforming asset and simultaneously shorting the overperforming asset. We
set the threshold at $z_{t}\pm 1.5$.

  4. 4.

Profit Taking: The strategy aims to profit from the mean reversion process. As
the spread narrows and returns to its historical mean, the positions are
unwound, resulting in a profit.

We would like to note that we are aware of the simplifications of the strategy
and that for practical use more fine-tuning is necessary. However, we find the
parametrization of the strategy to be sufficient for illustrative purposes.
For our purposes we exchange the historical Pearson correlation with the TE
and CCM respectively.

Figure 9: Minimum Risk Portfolio Optimization. The top row displays the
optimized Minimum Risk Portfolio weights over time using both the correlation
(on the left) and CCM (on the right) as co-dependence measures. Each colored
area represents a stock from the Dow-Jones, which is mapped in the legend to
the right. The dotted vertical lines depict significant economic or political
events. In the bottom row, the left figure illustrates the distributions of
the downside returns when using different co-dependence measures. The vertical
lines depict the VaR at $\alpha=1\%$ level. The plot to the right displays the
portfolio’s value over time. The vertical lines denote significant economic or
political occurrences. The dotted horizontal lines denote the portfolio’s most
recent value. Each color corresponds to a particular codependence measure,
which is mapped in the right-hand side legend.

####  II.4.2 Portfolio Optimization

In the world of finance, Markowitz Portfolio Theory (MPT), developed by Harry
Markowitz in 1952, is a cornerstone concept for investors and financial
analysts Markowitz (1991). This theory revolutionized the way investors think
about constructing portfolios. It is based on a fundamental premise: rational
investors seek to maximize their portfolio’s expected return while minimizing
its risk. The key insight here is that an asset’s risk and return should not
be evaluated in isolation but rather in the context of the entire portfolio.

The expected return of a portfolio is calculated as a weighted sum of the
expected returns of its individual assets:

$E(R_{p})=\sum_{i=1}^{n}w_{i}\cdot E(R_{i})\,,$ (14)

where $E(R_{p})$ is the expected return of the portfolio, $w_{i}$ is the
weight of asset $i$ in the portfolio, and $E(R_{i})$ is the expected return of
asset $i$. Even though historic returns do not indicate future performance, it
is common to use the historical mean as a proxy for the expected returns Hull
(2019).

The portfolio’s variance is a measure of its risk. It considers not only the
individual asset variances but also the correlation between assets. The
formula for portfolio variance is:

$\sigma^{2}_{p}=\sum_{i=1}^{n}\sum_{j=1}^{n}w_{i}\cdot
w_{j}\cdot\sigma_{i}\cdot\sigma_{j}\cdot\rho_{ij}\,,$ (15)

where $\sigma^{2}_{p}$ is the variance of the portfolio, $w_{i}$ and $w_{j}$
are the weights of assets $i$ and $j$ in the portfolio, and $\sigma_{ij}$ is
the covariance between assets $i$ and $j$. We can replace the correlation with
a causality measure $\psi$ or use the sign of the correlation if the measure
$\psi$ is normalized to $\left[0,1\right]$:

$\sigma^{2}_{p}=\sum_{i=1}^{n}\sum_{j=1}^{n}w_{i}\cdot
w_{j}\cdot\sigma_{i}\cdot\sigma_{j}\cdot\psi_{ij}\cdot\textit{sgn}\left(\rho_{ij}\right)\,,$
(16)

where $\textit{sgn}(\cdot)$ denotes the Sign function.

A popular measure of the riskiness of historical portfolio performance is
Value-at-Risk (VaR), which quantifies the potential loss in value of an
investment or portfolio over a specified time horizon at a $\alpha$ Duffie and
Pan (1997) confidence level. A $1-\alpha$ $VaR=x$ means that there is a
$\alpha$ chance that the portfolio will lose more than $x$. Unlike standard
deviation, VaR measures tail risk and does not assume a normal distribution,
which is particularly important for risk management purposes. We use the
default value of $\alpha=1\%$.

Two portfolios of great importance within MPT are the Minimum Risk Portfolio
and theMaximum Sharpe Ratio Portfolio. These portfolios play a crucial role in
portfolio analysis and optimization:

  * •

Minimum Risk: The Minimum Risk Portfolio represents the portfolio with the
lowest possible risk for a given set of assets. Mathematically, it can be
formulated as an optimization problem. The solution to this problem provides
the weights of assets in the Minimum Risk Portfolio:

Minimize $\displaystyle\quad\sigma^{2}_{p}$ Subject to $\displaystyle\quad
E(R_{p})=\text{target return}$ $\displaystyle\quad\sum_{i=1}^{n}w_{i}=1$
$\displaystyle\quad w_{i}\geq 0\quad\text{for all }i$

  * •

Maximum Sharpe Ratio: The Maximum Sharpe Ratio Portfolio represents the
portfolio that offers the highest risk-adjusted return. The Sharpe Ratio ($S$)
measures this risk-adjusted performance:

$S=\frac{E(R_{p}-R_{f})}{\sigma_{p}}$ (17)

To find the Maximum Sharpe Ratio Portfolio, we maximize the Sharpe Ratio by
adjusting the asset weights. Mathematically:

Maximize $\displaystyle\quad S$ Subject to
$\displaystyle\quad\sum_{i=1}^{n}w_{i}=1$ $\displaystyle\quad w_{i}\geq
0\quad\text{for all }i$

We illustrate a simple way to incorporate causality measures into portfolio
construction through the utilization of these two portfolios. As a result, we
regularly adjust the portfolio by optimizing its weightings with the mentioned
algorithms to align it with the prevailing market conditions. To achieve this,
we apply the rolling causality measures as previously demonstrated in this
paper. Consequently, we can assess the advantages of using causality measures
as the co-dependency metric for the portfolio, examining both its performance
and risk management implications.

Figure 10: Maximum Sharpe Ratio Portfolio Optimization. The setup of this
figure is analogous to Fig. 9. The top row displays the optimized Maximum
Sharpe Ratio Portfolio weights over time using both the correlation (on the
left) and CCM (on the right) as co-dependence measures. Each colored area
represents a stock from the DAX, which is mapped in the legend to the right.
The dotted vertical lines depict significant economic or political events. In
the bottom row, the left figure illustrates the distributions of the returns
when using different co-dependence measures. The vertical lines depict the
standard deviations of the returns. The plot to the right displays the
portfolio’s value over time. The vertical lines denote significant economic or
political occurrences. The dotted horizontal lines denote the portfolio’s most
recent value. Each color corresponds to a particular co-dependence measure,
which is mapped in the right-hand side legend.

##  III Results

In the following we present the results of our analyses, which we structure
into three subsections. As motivated by Figure 1, we observe that for complex
and chaotic systems it is difficult to measure the co-dependence of variables
through correlations as they can exhibit different regimes of positive,
negative, and no correlation even though they are guided by exactly the same
governing equations. This is illustrated by the rolling window analysis of the
correlation, which is unrobust and changes significantly over time. Hence in
order to measure their co-dependence reliably, another measure is needed.
Causality measures, such as CCM, are a valuable technique to measure the
causality of two variables in both directions and provide stable results over
time. Furthermore, by using FT surrogates, we can separate the causality in
linear and nonlinear contributions which helps to understand the intricate
nature of the co-dependence. The Figure shows that the separation of causality
is stable over different windows and also plausible when compared to the
governing Equations 1.

###  III.1 Historical Causality

To demonstrate the practical applicability of our framework, we have employed
it in an analysis of major German and U.S. stock indices. The data and the
dynamic correlation patterns are visually depicted in Figure 2. Notably, these
correlations undergo significant shifts during and after pivotal economic and
political events. This phenomenon can be attributed to the changing behavior
of investors and other market participants in response to these impactful
occurrences. Furthermore, this effect extends to our investigation of
causality measures, as demonstrated in Figures 3 and 5. These figures reveal
that linear and nonlinear causality measures, such as Transfer Entropy (TE)
and Convergent Cross Mapping (CCM), exhibit analogous responses to these
events.

Specifically, when examining TE, it becomes apparent that TE is highly
responsive to these events, displaying sharp fluctuations. In contrast,
surrogate TE remains relatively stable and does not react as drastically.
Conversely, surrogate CCM appears to respond more strongly than regular CCM,
displaying significant jumps similar to the observed patterns in correlation.
One of the most striking examples of this behavior is observed during Black
Monday in 1987, where we witness substantial increases in correlation, TE, and
surrogate CCM, particularly in the context of U.S. stocks. Two other
significant events that exhibit similar patterns are the global financial
crisis in 2009 and the COVID-19 pandemic in 2020. These observations suggest
that these events triggered structural shifts in the market, which is
reasonable given their profound impacts on the global economy. An intriguing
observation is that TE experiences more pronounced fluctuations compared to
surrogate TE during these events, while the opposite is observed for CCM. This
suggests that the linear dynamics in the stock markets were more profoundly
influenced, possibly due to investors simultaneously adjusting their stock
positions in response to the market crashes.

###  III.2 Correlation-Causality Fallacy and Nonlinear Causality

Upon examination of Figure 5, it becomes evident that both the original and
surrogate Transfer Entropy (TE) exhibit a moderate correlation. Notably, there
is an intriguing exception during the period spanning from approximately 1990
to 2002 in the U.S. stock market, where a substantial portion, approximately
$75\%$, of TE can be attributed to correlation. This spike coincided with the
rise and eventual burst of the dotcom bubble, suggesting that it might have
served as an indicator of abnormal market behavior during this period.

One of the most significant findings from this analysis is the observation
that fallacy of surrogate Convergent Cross Mapping (CCM) is remarkably high,
around $90\%$, in both the German and U.S. stock indices, as depicted in
Figure 6. This suggests that correlation effectively acted as a suitable proxy
for linear causality for the majority of the past few decades. However, in
periods where this fallacy diminishes, such as the aftermath of the dotcom
bubble in 2002 and the onset of the global financial crisis in 2008, relying
solely on correlation as a measure of co-dependence significantly
underestimates portfolio risk, as nonlinear effects cannot be disregarded.
This effect is even more pronounced when examining the fallacy of the original
CCM, where we also observe a substantial drop during these phases.

To gauge the extent of nonlinear contributions to our causality measures, we
delve into the analysis of how much of the causality can be accounted for by
its surrogate. In Figure 7, we observe the evolution of nonlinear causality
over time, noting that nonlinear TE and CCM exhibit similar but not identical
behaviors. Both measures reveal heightened levels of nonlinearity during the
period between the dotcom bubble burst and the commencement of the global
financial crisis. In contrast, before and after this period, we observe phases
with less nonlinearity. This indicates that these two major economic events
should be assessed differently, as the dotcom bubble led to increased
nonlinearity in its aftermath, while the global financial crisis, precipitated
by the U.S. housing market crisis, ushered in a phase of more linear market
behavior. Particularly for CCM, this behavior is quite drastic, with jumps
exceeding $20\%$. In conclusion, our analysis suggests that nonlinear
causality can be a valuable tool for anticipating and evaluating financial
impacts, provided it is continually monitored and assessed in the context of
evolving market dynamics.

###  III.3 Pair Trading and Portfolio Optimization

To effectively apply causality measures in practical financial scenarios, we
present two common financial frameworks where the interdependence between
assets plays a pivotal role. The first concept we explore is pair trading, a
logical choice given its reliance on the idea that two assets tend to revert
to a default correlation, and deviations from this norm can be profitably
exploited. In Figure 8, we use two German stocks from the chemical industry,
Bayer and BASF, to illustrate how causality measures can be seamlessly
integrated. It’s noteworthy that even though the differences in the evolution
of co-dependence measures are relatively similar, over time, these subtle
distinctions significantly impact trading performance. Of particular interest
is the fact that the trading strategy employing surrogate Convergent Cross
Mapping (CCM) outperforms the one utilizing correlation by a substantial
margin, approximately six times, despite the measures’ apparent similarity.
Additionally, we observe that Transfer Entropy (TE) and CCM perform better
than correlation, while surrogate TE lags behind and even delivers negative
returns. This straightforward example underscores the potential of a
causality-based pair trading strategy.

As previously highlighted, relying solely on correlation can potentially lead
to an underestimation of risk, a perilous scenario when managing a portfolio.
In Figure 9, we employ stocks from the U.S. Dow-Jones index and minimize risk
by dynamically optimizing the portfolio weights on a monthly basis. It becomes
evident that the allocations of a portfolio using correlation and CCM exhibit
visible disparities over time. This divergence is reflected in the portfolio’s
downside returns and overall performance. Notably, we observe that a portfolio
employing surrogate Transfer Entropy (TE), CCM, and surrogate CCM achieves a
superior $1\%$ Value at Risk (VaR) while slightly enhancing portfolio
performance.

Similarly, in the context of optimizing the Sharpe Ratio, as depicted in
Figure 10, the inclusion of causality measures results in a more favorable
risk-return profile. When optimizing the stocks of the German DAX index, we
note a reduction in portfolio standard deviation and an increase in portfolio
value over time, particularly when employing original and surrogate CCM.

##  IV Conclusion and Outlook

The present study has addressed the issue of identifying and quantifying co-
dependence among financial instruments, which continues to be a paramount
challenge for both researchers and practitioners in the financial industry.
While traditional linear measures like the Pearson correlation have maintained
their prominence, this paper has introduced a novel framework aimed at
analyzing both linear and nonlinear causal relationships within financial
markets. To achieve this, we have employed two distinct causal inference
methodologies, namely Transfer Entropy and Convergent Cross-Mapping, and have
utilized Fourier transform surrogates to disentangle their respective linear
and nonlinear contributions.

Our findings have unveiled that stock indices in Germany and the U.S. exhibit
a substantial degree of nonlinear causality, a phenomenon that has largely
eluded previous investigations. It is important to recognize that while
correlation, exemplified by the Pearson correlation coefficient, serves as an
excellent proxy for linear causality, it falls short in capturing the
intricate nonlinear dynamics that underlie financial markets. Consequently,
relying solely on correlation can lead to an underestimation of causality
itself.

The framework introduced in this study not only facilitates the quantification
of nonlinear causality but also sheds light on the perilous "correlation-
causality fallacy." By delving into the nuances of causality, we have
motivated how these insights can be harnessed for practical applications,
including inferring market signals, implementing pair trading strategies, and
enhancing the management of portfolio risk.

One of the insights derived from our findings underscores the role that both
linear and nonlinear causality can play as early warning indicators for
unusual market dynamics. Furthermore, our results suggest that a
straightforward incorporation of these causality measures into strategies,
such as pair trading and portfolio optimization, can yield better outcomes
compared to a reliance solely on Pearson correlation. This understanding can
significantly empower traders and risk managers, enabling them to craft more
effective trading strategies and to adopt a more proactive approach to risk
mitigation.

Looking ahead, the implications of our findings extend to various facets of
financial research and practice. Further exploration of nonlinear causality
may uncover new dimensions of financial market interactions, potentially
leading to the development of innovative trading algorithms and risk
management tools. Additionally, the integration of causality measures into
existing financial models and frameworks holds the promise of enhancing their
predictive accuracy and robustness.

In conclusion, this paper has introduced a comprehensive framework for
disentangling linear and nonlinear causality within financial markets. The
revelation of substantial nonlinear causality and the recognition of the
limitations of traditional correlation measures underline the importance of
taking a more nuanced approach to co-dependency analysis. The insights gained
from this study have the potential to enhance the way we perceive and navigate
the intricacies of financial markets, contributing to more informed decision-
making, better risk management practices, and more financial stability.

###### Acknowledgements.

We would like to thank the DLR and AllianzGI for providing data and
computational resources.

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