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Enhancing Causal Discovery in Financial Networks with Piecewise Quantile Regression
\mainmatter
\authorrunning{Cornell, Mitchell, Roughan}
Understanding the dynamics that lead to price fluctuations in speculative is critical for a variety of decision-makers. It allows both investors and active market participants (traders) to make more informed transactions --- further providing both participants and policymakers with a deeper comprehension of the risks associated with market participation. A crucial aspect of market dynamics is the concept of price causality, which delves into the temporal dependencies between assets. However, achieving a complete understanding of these causal relationships remains challenging, as existing methodologies often provide only a narrow view of the true interactions. In this paper, we develop probabilistic forecasting methodologies to infer a network representation of the causal dynamics of a system, and we apply our methodology to a cryptocurrency dataset.
The recent emergence of cryptocurrency markets provides a fertile environment for modelling large-scale causal interactions. These assets rapidly rose to public prominence following their remarkable growth . This growth, however, is often viewed as highly unstable, with prices showing extreme levels of volatility and responsivity to social and economic factors --- such as market sentiment, news events, technological advancements, regulatory changes, and the erratic behaviour of market participants. Many speculate that these emerging markets are less efficient than traditional, regulated financial markets --- a view supported by several studies ALYAHYAEE2018228. While the consequences of such inefficiency remain unclear, it stands to incentivize the use of causal analysis.
A common way to investigate causal dynamics in financial markets involves using Vector Autoregression (VAR) to model linear mean-response effects cornell2023vector, rank2, conf. The method is a multivariate expansion of the standard univariate autoregression to account for cross-dependencies, allowing us to construct networks that indicate cross-causal effects. Another frequent approach is to use Generalised Autoregressive Conditional Heteroskedastic (GARCH) techniques to model the causal relationships in return volatility, which can similarly be cast into a network structure rank2,conf. This approach can be extended to higher-order moments (such as skew), with the combined results providing insight into causal dynamics across the target distribution.
However, in the context of distributional representations, these linear-moment models exhibit several inherent limitations. Modelling the moments (mean, volatility, skew, and kurtosis, etc.) independently hinders the discernment of net variable effects, and higher order moments compound outlier effects, leading to estimation difficulties. These limitations are most pronounced in the distributional tails, which are of great importance in volatile financial markets, and tend to be the key region of interest.
Our proposed method, Piecewise Quantile Vector Autoregression (P-QVAR) captures mean-response, volatility and higher-moment causality in a single model, revealing asymmetries and tail-event interactions. The emphasis on these effects, bolstered by the robust estimates produced from probabilistic prediction renders our model particularly suitable for representing financial market data.
The primary contributions of this paper include:
Networks based on correlational structures in financial asset time series data have been explored in many papers. Often, researchers investigate simultaneous correlations between prices, seeking to model the joint structure of these observations as a network stat_analysis_of_network. Several studies analyze the temporal evolution of these networks using sliding-window methods partial_corr,structural_entropy, with others investigating evolution by incrementally adding nodes based on their correlations, developing an asset graph clustering_and_info.
Causal networks can be generated from cross-correlational effects. These types of networks have been constructed before billio2011 and are frequently used to model the joint causal dynamics of price and sentiment connecting_emotions,Combiningsocial_and_financial. Cross-correlational effects are typically modelled in a partial-effects framework to control for confounding variables. The VAR model is a common approach for this purpose, and is equivalent to partial effects of the cross-correlation matrix. Variations to this model have been developed to incorporate specific features, such as long range dependency long_range_VAR and restrictions to acyclic graphs ahelegebey.
Cryptocurrency causality network research has generally focused on the interactions between cryptocurrencies and other data (sentiment, traditional financial assets, {\em etc.}) connecting_emotions,crypto1, amirzadeh_nazari_thiruvady_ee, Elsayed2020CausalityAD, AZQUETAGAVALDON2020122574, Ahelegbey2021NetworkBE. Beyond a small number of studies Ahelegbey2021NetworkBE, cornell2023vector the majority of research in this space employs bivariate analysis, rather than the full partial-effects found in VAR models.
Compared to standard VAR, there is a more substantial body of research on modeling cryptocurrency markets with GARCH models (which can be considered a loose VAR equivalent for volatility). However, these studies are generally not framed in terms of network analysis, and usually involve a limited number of variables. A common theme in these studies is the identification of assymetric effects chu, CHEIKH2020101293, FAKHFEKH2020101075, GARCHN, such as treating positive and negative returns separately.
Quantile regression has been employed to model the autoregressive components of economic data across different target quantiles QAR. Several econometric studies have analyzed `quantile-causality' by assessing covariate effects across a dense set of targets, fitting 100 regressions and observing how coefficients vary across the target range BAUR2013786,CHUANG20091351, with the latter incorporating some nonlinearity by including both raw and squared return covariates. However, this type of dense analysis is not be suitable for our objectives due to its poor scaling in relation to our desired network size. Other studies focus on a select set of representative quantiles, often emphasizing asymmetric effects by separating variable effects into positive and negative regions assymetric1, assymetric2. Extreme value asymmetries in the context of quantile-causality have been explored by fitting independent linear regions covid_indicator, GUO2018251. This specification deviates from the contiguous construction traditionally linked with piecewise linear regression, which we employ in this paper. More generalized nonlinear causal effects for quantile causality have been explored using kernel-based estimation techniques, with several applications to financial data BALCILAR201674, JENA2019615.
While quantile regression's application to network modelling has been somewhat limited, one significant study extended the recently developed “Network Autoregression" NAR into a quantile regression framework ZHU2019345. However, these approaches employ spatial modelling techniques, without a heavy emphasis on causality. This study bridges the gap between the recent advances in causality analysis with complex network based representations. We introduce a model that can identify the type of complex causal patterns that have seen recent academic focus, while also retaining a high degree of interpretability, allowing for a direct representation in a network structure. Our model allows for highly intuitive visual representations, which we demonstrate in our empirical study.
The aim of causal network analysis is to construct a directed graph (digraph) denoted by $G=(V,E)$, where $V$ is a set of nodes and $E$ is a set of edges representing causal dependencies. Each edge $e_{i,j} \in E$ indicates that the next observation of asset $j$ depends on the previous values of $i$. The set of edges $E$ may be represented as an $N \times N$ adjacency matrix $W$ (where $N$=$|V|$ is the number of price series under study), with elements $W_{i,j}=1$ if there is a link $e_{i,j}$, and $W_{i,j}=0$ otherwise.
The primary methodological choice when developing these networks from empirical data is the selection of a suitable model to test the interrelations within the time series, {\em i.e.,} to infer $E$. This section provides an overview of the common methodology for this estimation problem (VAR), as well our proposed extension, P-QVAR.
Vector autoregression (VAR) is a popular statistical model introduced by the macroeconometrician Christopher Sims sims_1980 to model the joint dynamics and causal relations among a collection of time series. It is the natural multivariate extension of the univariate autoregression (AR) model frequently used to analyse the inter-temporal dependency of a sequence of observations. Under the VAR(p) formulation the expectation of the data vector $\boldsymbol{y}_t$ at the next observation is a linear function of $p$ previous observations. Equations 1 and 2 below show the relationship for order-1 and order-p lagged variants:
where $\boldsymbol{y}_t$ is a $N \times 1$ vector of observations at time $t$, $\boldsymbol{c}$ is a constant vector, the $A_k$ are $N \times N$ coefficient matrices for lags $k=1,...,p$, and $\boldsymbol{\epsilon}_t$ is a $N \times 1$ vector of error terms with zero mean and covariance matrix $\Sigma_\epsilon$. The vector $\boldsymbol{\epsilon}_t$ is often assumed to be Gaussian, however this is not a strict requirement. The VAR model assumes that the current value of each variable depends on its past values as well as the past values of potentially all other variables in the system (full conditioning).
The estimation of a VAR model comprises estimating the coefficient matrices $A_k$ and the error covariance matrix $\Sigma_\epsilon$. As we are interested in the causal influence structure within our dataset, we primarily require estimates of $A_k$, as they fully characterise the causal relations. This is often accomplished by the multivariate least squares (MLS) approach under which estimating the VAR is viewed as a general multivariate regression problem, with closed-form solutions generated via orthogonal projection luet.
We may conduct hypothesis tests for the statistical significance of the elements of the coefficient matrices by noting that our estimates $\hat{A_k}$ are asymptoptically normally distributed under finite variance assumptions, {\em i.e.,}
where $\Gamma=\ YY'/N$, $\otimes$ indicates the Kronecker product and $\mbox{Vec}(\cdot)$ denotes casting a matrix into vector form. For the case of a VAR(1) model the term $Y$ is the matrix representation of our response data $\boldsymbol{y}_t$, implying that $\Gamma$ is an estimate of the covariance matrix of returns. For generalised VAR(p) the complexity of this matrix increases, however Equation (ref) is still valid. To establish the existence of link $e_{i,j}$ we construct $t$ values associated with the null hypothesis $A_{i,j}=0$ as $t_{i,j}=\hat{A}_{i,j}/\hat{s}_{i,j}$, where $\hat{s}_{i,j}$ is the relevant term from $\Gamma^{-1}\otimes\Sigma_\epsilon$. These $t$ vales can be used to generate a binary link classification with a false positive probability $\alpha$ as $e_{i,j}=|t_{i,j}|>\Phi(1-\alpha/2)$, where $\Phi$ is the inverse normal CDF. For the empirical networks discussed in (ref) , we select $t$ values corresponding to $\alpha=0.001$, providing a relatively high detection threshold.
To streamline our discussion and estimation of causal networks, we limit our analysis to VAR(1) processes and omit the index $p$ from our discussion, with $A=A_1$ unless otherwise specified. For the empirical networks in Section (ref) we operate under the assumption that any causal link $i\rightarrow j$ will first manifest in order-1 effects, and that lag $p>1$ effects will not occur independently of a $p=1$ dependence. This assumption is intuitive and the scenarios where it doesn't hold are expected to be relatively rare.
Quantile regression (QR) is a probabilistic approach widely used to model linear variable effects across distinct regions of the conditional target density, $F_{Y|X}(y)$. This technique was originally developed in the context of the Least Absolute Deviation (LAD) regression, where linear models trained by minimizing the \(\mathcal{L}_1\) residual loss function: $|y_t-\hat{y}_t|$ produce estimates of the conditional median (50th quantile). Full QR, targeting some arbitrary distributional quantile $q$, is achieved by extending the \(\mathcal{L}_1\) residual loss to the following pinball loss format:
A regression model trained to minimise the expected pinball loss \(\mathcal{L}_q\) for a given quantile $ q\in (0, 1) $ can be shown to result in predictions of the conditional quantile $Q_{Y|X}(q)$. Specifically, these predictions are generated as a linear combination of the $j$ predictor variables:
where the fitted values $\boldsymbol{\hat{\alpha}}$ are determined from the following empirical minimisation on the $T$ training observations:
By training a model ($\boldsymbol{\hat{\alpha}}_i,c_i$ pair) for each time series we can stack our fitted models to produce a functional representation analogous to Equation (ref):
Validating individual links in our estimated recurrence matrices $\hat{A}_1$ is relatively challenging for quantile regression, especially when compared to the conventional asymptotic results available for least squares estimators, such as those used in standard VAR. Our approach to this problem is to once against construct $t$ values associated with a Wald test, $t_{i,j}=\hat{A}_{i,j}/\hat{s}_{i,j}$. However, this necessitates an alternative estimate for the standard error $\hat{s}_{i,j}$. The distributional properties of quantile regression coefficients has been explored in numerous papers, and here we follow the approach outlined in greene2012econometric, which summarises the works from several studies tvals0, tvals1, rogers1993calculation and puts forward the following solution:
where $D_i$ is a diagonal matrix containing elements:
and $f(0)$ is the true probability density function of the disturbances evaluated at $0$. Typically, this value is estimated using a single-parameter kernel density estimator greene2012econometric:
where $\epsilon_{t,q}=y_t-\hat{y}_{t,q}$. In our study we employ a Gaussian kernel for density estimation, which is defined as:
and select the bandwidth function, $h_q$, specified in hseather. For each of the $i$ series, we then calculate $\hat{s}_{i,j}$ as the root of the $j$'th diagonal element of the Covariance matrix found in (ref).
The kernel and bandwidth we select is available in numerous standard software packages. For our analysis, we make use of the Python package Statsmodels seabold2010statsmodels. This package is particularly efficient for our needs, as it solves ((ref)) using recursive least squares --- an approach that scales effectively with the high number of covariates in our study. In addition to the median target $q=0.5$, we select a lower, $q=0.1$, and an upper quantile, $q=0.9$, as targets for this study.
To investigate whether there are unique causal effects in the tail regions of our input variables we extend our analysis beyond basic linear functions. A straightforward method to introduce nonlinear effects into a linear setup involves transforming the covariates, a process commonly known as basis expansion. One popular variant is the polynomial basis expansion, where both the raw covariate values and their higher-order transformations, such as squared returns, are taken into account. In our context, these transformations can be instrumental in capturing autoregression in the higher moments of the return series. For example, the effect of a covariate on the 90th percentile might not be linear with respect to $y_t$, but could be linear in $y^2_t$, suggesting a volatility shift akin to GARCH dynamics.
Yet, applying these methods to our data presents challenges:
To circumvent these challenges, our solution utilises piecewise linear embeddings. This approach enables us to asymmetrically capture higher-order causal effects without compounding outlier estimation issues. The inherent linearity of this method, combined with the pinball loss function facilitates a relatively robust estimation of causal nonlinearities. The specific embedding we consider is a replacement of all instances of $X$ in the covariate equation with a 3-knot piecewise linear embedding:
where:
The breakpoints for our piecewise embeddings ($X^-_{j},X_j,X^+_{j}$) are selected as the unconditional quantiles of the input variables, $Q_{X_j}(0.1)$ & $Q_{X_j}(0.9)$ to ensure that we consistently target tail effects across all coins. From this basis expansion, alongside the representation shown in (ref) we can present our final model in the following format:
This technique allows us to produce 9 separate networks, each associated with one of the $A^*_q$ and designed to locally capture the causal structure in some sub-region of the input and output densities. The interpretation of our sub-networks are relatively intuitive, for instance, a link $i\rightarrow j$ in $A^+_{0.9}$ denotes the effect that large observations of asset $i$ have on the upper quantiles of $j$. Similarly, a link in $A^-_{0.1}$ indicates the low observations of asset $i$ effects the lower quantiles of asset $j$.
To obtain coefficients that match our intended interpretation of links we make several minor structural modifications, such that the model space and parameter interpretation are changed without effecting the actual fitted curves. First, we shift the model space for the $q=0.1$ and $q=0.9$ regression from modelling the raw target quantiles, to fitting the quantiles of the residual from the median regression: $\epsilon_t=y_t-\hat{y}_{t,0.5}$. This changes (ref) to the following:
This allows us to separate the scale and location effects of our target variables, changing the focus of the $Q(0.1)$ and $Q(0.9)$ targets towards describing the risk and volatility effects in the influence structure. For instance, a link in $A^+_{0.9}$ now denotes that the distance of the right tail from the median is being effected by $\boldsymbol{y}_{t-1}$. In the prior specification, we would have a link in $A^+_{0.9}$ if the entire density shifted, even if there was no tail-specific changes.
Second, when evaluating our final networks we change the tail-input subnetworks to a format that represents the net effect, rather than the partial effects originally present in Equations (ref), (ref) and (ref). To do so we analyse the `net slope' coefficient matrices, $\mathbb{A}^\pm_{q}=(A^\pm_{q}+A_{q})$, with the covariances adjusted to reflect this sum:
While (ref), (ref) and (ref) may appear mathematically esoteric, the fitted relationships are quite visually intuitive. (ref) provides a visual indication of the relationships that P-QVAR can fit, showing variable effects estimated from an example network containing only Bitcoin (BTC) and Ethereum (ETH). This simple example demonstrates several archetypal relationships that motivate our design:
The nonlinear profile for all targets motivates our piece-wise embeddings, ${X^+_i, X^-_i}$, while the differing dependence across our target regions drives us to asses multiple quantiles.
In order to visually asses the types of causal effects identified across the entire set of sub-networks, $A^\pm_i$, we create a Quantile Influence Graph (QIG). This visualisation contains nodes for all input and target regions that we consider, with the edges corresponding to the different sub networks, $A^\pm_i$. To demonstrate our QIG, (ref) displays the type of structures that would be identified by running our method on several archetypal models. Specifically, we visualise autoregression in increasing moments, starting with a negative standard VAR, then GARCH effects ($\sim$ARMA of volatility) and lastly show autoregression of the skew, followed by an assymetric effect.
Our dataset comprises hourly prices for 260 cryptocurrencies spanning from January 1, 2021, to January 1, 2022, along with their market capitalizations as of June 2022. The selected currencies were drawn from the top 750 cryptocurrencies by market capitalization at the time of data collection (June 2022). However, a substantial portion of these, primarily those with lower capitalizations, were excluded due to incomplete or missing price histories. The final dataset represents 79.6% of the total market capitalization (836B out of 1.05T). The returns, $y_t$, are computed as the logged ratio of consecutive observations in the original price series, $p_t$, expressed in terms of the Coin/USD pair: {\em i.e.,} $y_t = 100\log\left(p_t/p_{t-1}\right)$. For comprehensive details on this dataset see cornell2023vector. To minimise the effects of data artifacts, such as coordinated pump and dump schemes, we take a stabilising transformation $\text{sign}(y_t)\times \log(|y_t|+1)$. Almost all hourly returns are in the low single digit $\%$ region, where this transform is near-linear, however it flattens the infrequent events where we may see some of the low valuation coins have high double, or even triple digit returns before reverting in the next period.
(ref) presents a QIG of the self-edge effects ({\em i.e.,} the diagonal elements of $A^*_i$) in our overall causal network. In general, there is a high proportion of potential self-links that are statistically significant (65.8% of possible links), exhibiting pronounced directional trends. More specifically:
Overall, we conclude that our cryptocurrency market data exhibits high levels of self-causality. The observed causal structures tend to be highly prototypical, with nearly all links displaying the same directional trends. This relationship is characterized by mean reversion at the density center and mutual excitation between the tails. Additionally, we observe notably weaker dependence for negative tail events. This combination of effects suggests erratic price behavior, implying generally reverting, self-exciting swings with a tendency to enter short-lived bull runs.
Figure (ref) presents the same graphical representation as in Figure (ref), but now for cross-causal effects ({\em i.e.,} off-diagonal elements of $A^*_i$). Generally, there is a greatly reduced number of statistically significant effects (4.5% of possible links), with a comparatively high degree of symmetry. Further:
We conclude that there is comparatively limited, but still present causality between coins in the cryptocurrency market. These relationships generally occur between the tails and often reflect excitatory patterns. The effects are quite symmetric, indicating that for cross-causal effects, negative and positive tail events have similar levels of influence. In the financial context, this implies that while there may be limited return spillover (linear-median links), the volatility and risk of coins may rapidly propagate through complex causal chains. For market participants, this indicates that cross-causal effects are relevant for risk management, but might have less application to statistical arbitrage style trading.
Figure (ref) displays the log-log Complementary Cumulative Distribution Functions (CCDF) for the degree distributions of our 9 sub-networks. We see that across all forms of sub network, the out-degree distributions display comparatively heavier tails, indicating that out-degree, the measure of external influence is more concentrated than incoming influence. This is particularly pronounced for the median response targets, where we see quite slow decay in the survival function. Across all targets, there appears to be higher scaling for the tail-effect distributions, with the largest amount of scaling coming from the left-tail to median response type link.
Table (ref) presents the summary statistics for our individual causal subnetworks, displaying the mean, standard deviation, and correlation to market capitalization for several attributes.
The majority of significant correlations are positive, indicating generally elevated levels of activity for higher capitalisation regions of the networks. The median response networks show relatively high interactions with capitalisation, with many links pointing downstream, and tail-events having larger causal impact for highly capitalised coins. The combination of downstream propensity and negative in-degree correlation for the linear-median network is relatively intuitive, as we would not expect standard return behaviour from low-capitalisation coins to be effecting the largest cryptocurrencies. For a detailed examination of the correlations between degree types across our subnetworks, refer to Additional Figures item A (ref). While summarizing the 324 correlation pairs is challenging, a general pattern emerges: matching degree types tend to correlate positively. This suggests that coins are more likely to be broadly influential or influenced across the networks, rather than confined to specific regions. In contrast, the relatively sparse correlations between non-matching degree types indicate that the strength of outgoing and incoming influence is largely independent for each coin.
Overall, we make the following qualitative interpretations: (1) Tail-events in high capitalisation coins tend to have the most outgoing influence. (2). Median-response information has a slight tendency to flow downstream from large capitalisation influences, while scale/quantile-response information flows across.
We can also construct a single all-effects network by summing all subnetworks, resulting in a multigraph where multiple edges between nodes are possible. For comparison against this general causality network we also construct a `baseline' linear causality network by fitting a standard linear-VAR model. For consistency with our existing sub-networks, this standard VAR model is fit against the median target, which is a small departure from the traditional description shown in Section (ref). We also compare against a Piecewise VAR network, which is the sum of the three median response networks.
Figure (ref) showcases the CCDF for these networks, while Table (ref) provides summary statistics (we divide the multigraph degrees by 9, and the piecewise network by 3 to be on an equivalent scale). The most stark difference is the number of identified links. The multigraph, which considers causality across a variety of variable scenarios identifies around 6 times the number of causal links compared to the standard VAR. Even restricting our attention to the median response, we see that splitting the linearity into our piecewise structure leads to 3 times as many identified relationships. Both the multigraph and standard VAR networks lack a statistically significant relationship between out-degree and capitalisation, with the standard model having a significant negative relationship for in-degree, and the piecewise network having a significant relationship for out degree. While the correlation structure for the standard VAR model resembles the linear-median response sub-network we previously identified, it surprisingly contains even fewer links than this sub-network. It would appear that rather than the tail-median effects producing false positives for linear-median effects, they actually obscure/cancel the linear-median effects. A visual example of this could be seen in the Figure 1b, where the significant right tail-median positive effect could obscure the linear-median negative effect. Comparing the multigraph against the results in Table (ref) underscores the significance of stratified results, emphasizing that combining the results can mask the absence of a market capitalisation effect. Similarly, the non-stratified CCDF analysis under-emphasises the out-degree scaling we see in the individual sub-networks.
It is unsurprising that within the all-effects multigraph we observe Bitcoin to have the highest total out-degree. This intuitive finding indicates that Bitcoin has statistically significant causal effects at 10.5% of all possible outgoing links, and 64.6% of all coins have some causal dependency on Bitcoin. This effect is largely driven by a very high out-degree for the negative-tail to median response subnetwork, where 104 (40%) coins displayed negative causal effect (100% negative), indicating that negative outliers for bitcoin tended to be followed by elevated returns for most other assets in the network.
This study introduces a technique for constructing nonlinear distributional causality networks, enabling a direct investigation of tail-tail causal interactions in financial data. By combining piecewise linear embeddings with quantile regressions, we robustly estimate a set of subnetworks representing distinct, locally linear causal effects across key regions of the input and output densities. In an empirical study on 260 cryptocurrencies, our P-QVAR model identifies intricate self-causality structures, with cryptocurrencies displaying an asymmetric blend of mean-reversion, GARCH-like behaviour, and autoregression in the residual skew. We find that large positive events generally exert more causal influence, showcasing heightened mean reversion and skew transference compared to negative events. Cross-causal effects are more symmetrical, with GARCH like behaviour being the most pronounced influence structure. There is comparatively limited cross-causality for the median response target, indicating that most cross-asset coupling occurs in the higher distributional moments. Our findings have implications for both investors and policymakers, as it highlights the relatively independent action of raw returns may not imply isolated behaviour during tail events and extreme market behaviour. This study serves as a foundation for further research into distributional causality networks, prompting a shift in the focus of financial networks from mean-response causality to more risk-inclusive measures.