A causal interactions indicator between two time series using extreme variations in the first eigenvalue of lagged correlation matrices

1.   Introduction

Time series correlations are widely used dependence metrics in finance and economics. The problem is that correlations are usually spurious, do not represent dependence for non-normal distributions, and heavily depend on the scale and frequency of the time series. Correlations are statistical measures that depend on the chosen time-window as point estimates, making them useless for dynamic dependencies and change point detection (Janse et al., 2021; Rodriguez Dominguez and Stynes, 2022). To avoid these pitfalls, statistical techniques such as copulas, entropy-based measures, network graphs, and parametric distance metrics have been applied to model dependence (Marti et al., 2021). These statistical methods, along with their flaws, are involved in finance and insurance for risk management, portfolio optimization, alpha generation, factor investing, credit lending, and more, with trillions of dollars depending on them every year. Causality could be one solution for measuring true dependencies, but at the cost of being difficult to measure and identify. Granger's economics Nobel Prize-winning causality test (Granger, 1969) detects probabilistic causal relationships between time series using control variables. However, although it provides information about forecasting capabilities, it provides nothing about true causal interactions. The fact that it still remains one of the benchmarks for time series causal identification evidences the need for a statistical method using time series data that could detect causal interactions.

Causal relationships are based on interactions, which if strong enough, can modify the structure of the time series model. This work focuses on physical experiments with a clear definition for causal interactions and mapping statistical properties to time series causal identification. The Coulomb-Dyson gas of charges is a log-gas (or a two-dimensional Coulomb gas) consisting of repelling particles on the line in a global confining potential. An interaction can be defined as compressing the whole gas, which is equivalent to forcing the largest eigenvalue of a random matrix representing the gas inside the allowed band given by the Tracy-Widom distribution (Tracy and Harold, 2002). This is the distribution for the large deviations of the first eigenvalue in a random matrix. The large deviation function is the excess free energy of the gas of eigenvalues forced to stay between two hard walls (Cunden et al., 2018). Greater values in the first eigenvalue in a variogram can explain a third-phase gas transition pulled-to-pushed-type interaction. The first premise of this work is that a greater number of interactions implies greater probability of a true interaction. Finally, this system of a Coulomb-Dyson gas being pulled and compressed by two walls can be equivalent to a system of two pairs of cause-and-effect time series candidates. In this case, the large deviations of the first eigenvalue are analyzed for different lagged $2\times 2$ correlation matrices, with the lag applied to the causal candidate time series. The second premise of this work is that the interaction, detected in nature and by homology to the Coulomb-Dyson gas compression, must be causal due to the direction of time imposed by the lagged correlation matrices. Therefore, the interaction that is true with highest probability and in the direction of time must be causal. The standard deviation of the explanatory power of the first eigenvalue for all lags is computed as a causal interactions indicator. Large values of this indicator in a particular point in time means a high probability of a causal interaction, the interaction being measured by the extreme deviations of the first eigenvalue and the causality enforced by the direction of time in the lagged correlation matrices.

Based on these previous aspects, the main contributions proposed in this paper follow:

● A new method to detect causal interactions between two time series using random matrix theory (RMT) and eigenvalues' distributions being defined from physical experiments is presented, in the form of an indicator.

● To the best of the authors' knowledge, there is no causal identification method that focuses on random matrix theory (RMT) and eigenvalues' distributions. Some work uses RMT to search for structural changes in financial markets (Potters et al., 2005; Bouchaud and Potters, 2009). It differs from this paper in that they do not mention the variability of the variogram of the largest eigenvalue, but focus on its tendency and with no link to causal interactions. Whereas this work uses a different indicator consisting of the standard deviation of the variogram measured, in this case, by the explanatory power of the largest eigenvalue and not the eigenvalue itself to capture extreme variations, which maximize the probability of having a causal interaction.

● The method is able to capture structural changes in time series more dynamically than traditional econometric methods. It is also able to detect causal interactions in a pair of time series. In the experiments, the method is able to identify the causal variable from a pool of more than one thousand time series candidates. The same output is found with synthetic generated data.

● The indicator is relatively simple, easy for model validation of financial institutions and regulators, and can be used, as shown in the experiments, for risk management in flow business, including but not limited to liquidity risk in banks and brokerage units, foreign exchange flows in the global economy (Mensi et al., 2021; Arkol and Azimli, 2024), order-book management (Brown et al., 1997) and high frequency trading (Samsul et al., 2019). It is also useful for market risk analysis, regime-switch detection (Balcilar et al., 2015), structural breaks in the market (Fenghua et al., 2017) and forecasting (Li et al., 2021).

● Finally, to the best of the authors' knowledge, there is no method that connects causality from an experimental physics point of view with causal interactions between time series (Edinburgh et al., 2021).

The paper is organized as follows: in Section 2, a revision of the state-of-the-art method is presented; in Section 3, the Coulomb-Gas experimental setup analogy for causal interactions is described; in Section 4, the methodology for the indicator is shown; in Section 5, experiments are presented; and in Section 6, some conclusions are made and future work suggested.

2.   Literature review

The main issue with applying causality to finance and economics is that it has many varied interpretations by authors in different fields (Pearl, 2009; Karimi and Hamilton, 2011; Yarlagadda and Hershey, 2003; Kosuke et al., 2010). A Nobel Prize-winning causal time series identification approach is the Granger causality test (Granger, 1969), which is not focused on empirical causality, and just considers linear prediction and time-lagged dependence between two time series. Despite being one of the most applied methods in history, it cannot identify true causal relationships as it does not take unknown confounders into account (Cochrane, 1997). The Granger test has been designed for the time and frequency domains, but the same issues persist (Breitung and Candelon, 2006). Causality has been studied with different frameworks, which are sometimes difficult to connect in quantum physics, statistical physics, and relativity theory (Brukner, 2014; Sklar, 2009; Havas, 1969). This work focuses on the concept of causality from statistical physics.

Causality is defined as the requirement that two points in space-time cannot communicate with each other if they are separated by a spacelike distance (Rosenfelder, 1989). Therefore causality implies interactions. Random matrix theory has been applied for detecting structural changes in financial market dynamics, in particular the variogram of the largest eigenvalue (Potters et al., 2005; Bouchaud and Potters, 2009). An expression for the variogram of the top eigenvalue is obtained in (Potters et al., 2005):

This result is expected when a "true" dynamical evolution of the structure of the matrix is absent. These results assume that the correlation matrix is time independent. Any significant deviation from (1) would indicate a genuine evolution of the market structure. Figure 1 shows a real evolution of the strength of the average correlation with time. Finally, the non-stationary effects in financial markets are investigated by studying the dynamics of the top eigenvalue and eigenvector. If the true top eigenvector (or eigenvalue) did not evolve in time, the variograms in (1) should converge to their asymptotic limits. If the correlations' structure changes over extended periods of time, which would lead to an increased asymptotic value for the variograms including a long time tail. The asymptotic value of these variograms is furthermore much larger than the theoretical one. This non-stationarity of financial markets is expected and dangerous for out-of-sample risk management (Potters et al., 2005). In Figure 1, the variograms display the existence of genuine fluctuations of the market mode on a time scale of 100 days, superimposed to the initial noise-dominated regime. These fluctuations, when becoming extreme, can be captured by the indicator presented in this paper and related to causal interactions.

The Tracy-Widom distribution of the largest eigenvalue for extreme values has been studied for random dynamical systems (Majumdar and Schehr, 2014). In particular, the Coulomb gas analogy, which is used in this work for causal interactions in pairs of time series, has been applied before to derive the deviation functions describing the tails of the distribution of the largest eigenvalue (Hastings and McLeod, 1980; Nadal and Majumdar, 2011).

3.   Coulomb gas analogy for detecting causal interactions

Random matrix theory analyzes the eigenvalue spectra of random matrices, and in particular the Tracy-Widom region is focused on the edges. The distribution of eigenvalues of a $N\times N$ Hermitian matrix, with independent and identically distributed (i.i.d.) upper triangular entries, converges in probability as $N\rightarrow \infty$ to the Wigner semicircle distribution. The probability distribution of the largest eigenvalue, $\lambda_{max}$, is more difficult to estimate. It has been shown that the standard deviation of this distribution scales to $N^{-1/6}$. Tracy-Widom revealed that for large $N$, the distribution of a random variable $\chi$ is independent of $N$: $P(\chi \leq x) = F_{\beta}(x)$, which is called the Tracy-Widom distribution and depends on $\beta$. For $\beta = 2$:

Here, $q(x)$ satisfies the special case of $\alpha = 0$ of the Painleve II equation:

For $\alpha = 0$, the solution only requires the right-tail boundary condition for its unique specification: $q(z)\sim Ai(x)$ as $z\rightarrow \infty$. $Ai$ is the Airy function

which satisfies the differential equation $Ai^{\prime\prime}(z)-z Ai(z) = 0$ and vanishes like the following expression: $Ai(z) \sim (1/2\sqrt{\pi}z^{1/4})\exp^{-(2/3)z^{3/2}}$ as $z\rightarrow \infty$ (Nadal and Majumdar, 2011; Hastings and McLeod, 1980). For large deviations of the largest eigenvalue, the probability of $\lambda_{max}$ for large $N$ is (Nadal and Majumdar, 2011):

$F_\beta^{\prime}$ is the Tracy–Widom distribution, $\psi_{-}$ and $\psi_{+}$ are the left and right large deviation functions describing the tails of the distribution of $\lambda_{max}$, and both were computed by simple physical methods exploiting the Coulomb gas analogy (Hastings and McLeod, 1980).

The physics of the left and right tail are very different. In the case of the left tail, the semi-circular charge density of the Coulomb gas is pushed by the hard wall, resulting in a reorganisation of all of the $N$ charges, which creates an energy difference. In contrast, for the right tail, the dominant fluctuations are caused by pulling a single charge away (to the right) from the Wigner sea (the blue area from Figure 2), leading to an energy difference of $O(N)$. The different behavior of the probability distribution leads to a phase transition. In Figure 3, the effect of the presence of a wall on the Wigner semi-circle is shown. If $N$ is large but finite, the Tracy-Widom distribution describes the crossover function between stable and unstable regimes of a random dynamical system (Majumdar and Schehr, 2014), and moreover, there is a third-order phase transition between these regimes, which also takes place in the Coulomb gas, two-dimensional Quickest Change Detection, and several other systems of interest.

4.   Causal interactions time series indicator

Given $\boldsymbol{X}, \boldsymbol{Y}\in\mathbb{R}^{L\times 1}$, a pair of cause-and-effect variables in the form of time series of length $L$. The lagged correlation matrix is computed with the returns of $\boldsymbol{X}$ and $\boldsymbol{Y}$, $\boldsymbol{r_X}, \boldsymbol{r_Y}\in\mathbb{R}^{L\times 1}$, and lagging one of them by $\tau$ time-steps of time $t$. For the relationship to be causal, the direction of time is such that the lagged variable must be the causal candidate time series $\boldsymbol{X}$. The lagged correlation coefficient is:

such that $\mathcal{C}_{XY}(\tau = 0) = \mathcal{C}_{XY}$ is the standard correlation coefficient. Whereas the correlation matrix of the standard case is symmetric, the lagged case $\boldsymbol{R_{XY}} = \left[\begin{matrix}1&C_{YX}(\tau)\\C_{XY}(\tau) & 1\\\end{matrix}\right]$ is non-symmetric.

The indicator for causal interactions, $\sigma_{\Gamma}$, is defined as the standard deviation of the explanatory power for the first eigenvalue, $\Gamma = \frac{\lambda_1}{\lambda_1+\lambda_2}$, in $2\times 2$ lagged correlation matrices for different values of lag $\tau = 1, \dots \max\tau$. The indicator at time $t$ is computed as follows:

● For each lag $\tau$, the lagged correlation matrix is computed as $\mathcal{C}_{XY}(\tau) = \langle \boldsymbol{r_X}^{t-\tau}, \boldsymbol{r_Y}^{t} \rangle$, with the lag applied to the candidate causal time series returns $\boldsymbol{r_X}$, while the effect $\boldsymbol{r_Y}$ remains fixed for all lags.

● A principal component analysis is performed on each lagged $\mathcal{C}_{XY}(\tau)$, and the explanatory power of the first eigenvalue is recorded as $\Gamma(\tau) = \frac{\lambda_1}{\lambda_1 + \lambda_2}$.

● The same process is repeated for all lags, $\tau = 1, \dots \max\tau$, with $\max\tau$ being a model hyperparameter. Finally, the standard deviation of all $\Gamma(\tau)$ values, for $\tau = 1, \dots \max\tau$ gives the indicator at time $t$, $\sigma_{\Gamma}(t)$:

Higher values of $\sigma_{\Gamma}(t)$, $\forall t$, means a higher probability of having a causal interaction from $\boldsymbol{X}$ to $\boldsymbol{Y}$. In Algorithm 1, the method is presented.

5.   Experiments and discussion

5.1.   Datasets description

Two sets of experiments were performed. The first one uses financial market daily time series data sourced from Bloomberg, spanning the years 2013 to 2023. The second set of experiments uses daily time series data from end-of-day files of a brokerage business containing all clients' cash and trading activities, flows, and positions. The anonymous dataset was provided by Miralta Finance Bank S.A. A description of the datasets is given next:

● For the first set of experiments in Section 5.3, a financial market dataset consisting of 110 daily time series of stocks' causal drivers candidates was used. These are enumerated in Table 1, and consist of daily time series of pricing, macroeconomic, products and market indicators data from the following asset classes: credit, rates, commodities and equities indexes, FX spot rates (X-rate and spot is the same), commodities, equities and government bonds futures' indexes, credit default swap (CDS) indexes and implied volatility data. A second dataset consists of effect time series candidates, daily prices for twelve SP500 stocks from different sectors. Notable examples include technology sector stocks such as Apple, consumer discretionary sector stocks like McDonald's Corporation, energy sector stocks such as Schlumberger Limited, and financial sector stocks like Goldman Sachs Group, Inc.

● For the second set of experiments in Section 5.4, brokerage business variables are included, which comprise data for approximately 15,000 clients (as detailed in Table 3). This dataset provides information on various aspects such as trading volume data, cash-flow data, account statuses, and trade executions. This offers valuable insights into brokerage operations, client behavior and liquidity analysis for an institution like a bank, which must be managed optimally for regulatory purposes.

For the first dataset, the goal is to use the indicator, $\sigma_{\Gamma}$, to measure causal interactions and relationships between causal drivers' candidates and stocks from the SP500. The second dataset is focused on finding causal relationships via interactions between a brokerage business and its liquidity levels. But first, in the next section, a randomized experiment is performed.

5.2.   A Randomized experiment for structural change detection

Figure 4 shows the average of $\sigma_{\Gamma}$ for 10 i.i.d. time series generated from a normal distribution. For this, Algorithm 1 is applied to sets of 2 different time series. In Figure 5, the average values of $\sigma_{\Gamma}$ from the i.i.d. series' previous case are compared with the average values of Algorithm 1 applied to time series variables from the brokerage dataset, which was described in the previous section, and includes true causal interactions and structural changes, as prior knowledge for the experiments. The hypothesis that the indicator captures structural changes can be simply verified from this experiment, as for i.i.d. generated time series in Figure 4, and the values of the cumulative sum of the indicator $\sigma_{\Gamma}$ are much smaller than the values for time series with structural relationships and changes in Figure 5.

5.3.   Causal interactions market risk indicator

The first set of experiments tests the ability of the indicator in detecting structural changes in the effect candidate variable. For that, the change is synthetically forced. For the different stocks, autoregressive moving average ARMA models are calibrated to historical time series returns data. At a specific and predefined point in time (03/10/2018), the ARMA parameters are forced to change in different ways, and the ARMA models with this new set of parameters are calibrated with the rest of the historical time series data, from that point onwards.

In Figure 6, a pair of figures are shown for four stocks: the figures on the left are the cases where the ARMA parameters do not change on that particular test day, and the right figures are the ones with changes in the ARMA parameters. In these figures, all of the cases using the four stocks as the effect variable and the 110 causal drivers' candidates from Table 1 as causal variables are shown. It is clearly seen how the indicator $\sigma_{\Gamma}$, computed for all pairs of stock/drivers' candidates, dramatically jumps. This is what was expected for this indicator, the ability to accurately capture structural changes in time series. In the left figures in Figure 6, indicators have higher values for the candidates with more chances to be related by causal interactions; and in the right figures, at the change point, this distance can have a reduced appearance. However, looking at the scale of the axis, it is clear that the gap between the indicators for different drivers is increased, and this is caused by the jump of all of the indicators which changes the scale. Even though all indicators jump at the change point, the causal drivers of the change have higher indicator values post-change. The structural changes have been identified, but there is still the need to test the indicator for identifying causal interactions with the extreme values of the indicator.

Next, the same experiment shown in Figure 6 for the case of the causal indicator $\sigma_{\Gamma}$ is performed for the Granger causality test. In this case, a rolling test is computed for the same time period than in Figure 6, for all causal drivers' candidates and the same four SP500 stocks. The forced change in ARMA structure is done at 01/10/2018, for all cases in these Granger causal test experiments. Results are shown in Figure 7, where the time series of the f-statistic for the Granger causality test (rolling-window-based) are shown for all cases. It can be seen that the Granger causality test captures the date of the structural changes, but with less precision than the proposed causal interactions' indicator $\sigma_{\Gamma}$. Also, the Granger test has an overlap for all the causal candidates' f-statistic time series values, in contrast to the $\sigma_{\Gamma}$ indicator, which can differentiate between ARMAs adjusted to different causal candidates' variables. This means that the $\sigma_{\Gamma}$ indicator has more precision detecting the date of the structural change for all cases than the Granger test. Also, the indicator $\sigma_{\Gamma}$ contains more information about the structural changes and the underlying time series processes, as it shows different values depending on the causal candidates' time series variables used to fit the ARMA models.

In Table 2, the KS and Anderson-Darling statistics, p-values, and the Wasserstein distance are shown for the four stocks presented in the experiments, in order to validate that all come from different distributions, enriching the generalization of the results. Due to space constraints, results are shown for these four stocks only, although original experiments were performed with twelve of them in all cases.

In Figure 8, the top four causal drivers candidates' $\sigma_{\Gamma}$ values for Goldman Sachs during the peak of the COVID crisis are shown. The ranking changes for the causal variables candidates during the period, but the top four values are recorded and shown in the plots. These indicators' values are compared with the Goldman Sachs stock price, volatility, and average returns time series. In Figure 8, the predictability of the 1-month average return and realized volatility from the top indicators for the causal drivers candidates' sample, can be seen. Also, a strong inverse relationship between these indicators and the stock price occurs. All of these experiments show that the indicators with a causal driver having the highest values are strong risk analysis measures indicators, which include prediction (causal) capabilities for the stock. In Figure 9, the distribution of all causal drivers' $\sigma_{\Gamma}$ candidates' values is analyzed for the cases in which Goldman Sachs' stock time series returns are the effect variable from 2014 to 2023. The historical distribution in Figure 9b follows an inverse Wishart distribution, which can be contrasted with Figure 9a.

At this stage, the capabilities of the indicator in detecting structural changes has been verified, with the ability to monitor them at each timestamp, instead of using estimation windows such as in the Granger test, the Chow test, or other likelihood ratio tests. It has shown strong predictability capabilities with respect to the effect variable, with the method itself not based on predictability between the cause-and-effect variable, as in the Granger test. It instead focuses on the distribution properties of the explanatory power of the spectra from lagged correlation matrices. Therefore, the predictability is a consequence and not an assumption of the method, which makes this predictability closer to true causal interactions. To confirm the hypothesis of causal interactions, another experiment is shown in which the causal relationship is known in advance and tested with the indicator.

Experiments are conducted consisting of selecting a known causal relationship in the financial markets, and checking if the rule for the maximum $\sigma_{\Gamma}$ indicator is able to select the same pair of cause-and-effect variables, given a particular effect variable, and from more than a hundred causal drivers' candidates. For the effect variable, the credit default swap (CDS) index for high yield US corporates (CDX), the Stoxx 600 equity index, the Stoxx volatility index, and the CDS investment grade for the Europe corporate sector (iTraxx EU) were used. The indicator is given in the form of a daily time series, and in order to measure the level of causal interactions, all of these values are added up and displayed in a bar graph. Figure 10 shows how the cumulative sum of all of the indicator's values during the period from 2014 to 2023 for the CDX index as effect candidate identifies the US high yield corporate credit as the top causal driver candidate couple. This is the one among all of the 110 candidates which has the highest prior probability to be a causal variable for the CDX. The reason is that the insurance contract on the high yield US corporates credit is the CDX index, so in consequence it is causally connected with the dynamics of that type of credit itself.

Next, Figure 11 shows, for the Stoxx 600 index, which consists of the largest companies' stocks in Europe by market capitalization, the strongest indicators' cumulative sum values for the main equity indexes of the European countries. This is obvious, as both sets of indexes share the same constituents. In the case of the Stoxx volatility index, the same rationale applies as for the Stoxx 600 index (Figure 12). Being an implied volatility index, the constituents are not shared this time by both sets of indexes, however the stocks and their indexes must be causally connected with their implied volatility, and as a consequence, the one in the index. In the case of the iTraxx Europe CDS index, displayed in Figure 13, the main European equity indexes are shown as the causal counterparts. Not having a European credit investment grade index in the sample demands the choice of using the equity component of these corporates instead. This equity component has strong connections with the credit component and their CDS index, and from the sample, are known to be the most causally connected, as they are the stock prices of the corporate names in which the default insurances are written (all of the CDS contracts that make up the iTraxx index).

In Figure 14, the distributions of the average values of the Jensen-Shannon distance are shown, between the historical distribution of the indicator $\sigma_{\Gamma}$ and the inverse Wishart distribution, for 1, 3, and 6 degrees of freedom, fitted to the historical data of the indicators. Values of the three distances for the three degrees of freedom are averaged. The distribution of the average is shown for the cases in which the effect variable is the CDX index and the causes variables consist of all of the candidates from the sample of more than one hundred candidates. It can be seen in Figure 14 that the top causal variable candidate for the CDX index, from experiments and from common knowledge, is the U.S Corporate high yield, as previously stated in the paper, which has the indicator with the shortest distance between the historical distribution and the fitted inverse Wishart for different degrees of freedom. This experiment verifies the hypothesis that stronger causal interactions measured by the indicator make the indicator behave more like an inverse Wishart distribution.

5.4.   Causal liquidity risk indicator for brokerage/banking firms

For the brokerage dataset, the daily brokerage activities, with time series data of 15,000 clients approximately, is analyzed. The goal is to investigate the power of the structural temporal (causal) interactions between the cash-balance of the company and other related variables from the business as a function of time. Experiments are performed with cause-and-effect pairs of variables, with the effect being the cash-balance. For illustration, the results are plotted in thematic buckets, such that only the cause variables are displayed and the effect variable is understood to be the same in all cases.

The buckets consist of:

● The account status, which includes the daily amount in account currency of open positions in different products, including shares, bonds, mutual funds, CFDs, derivatives, FX Spot, and ETO.

● Bookkeeping cash, which includes all of the daily cash movements in the account currency, both internal and external cash transactions, and the financial product which is the reason for the transaction.

● Trades executed, which is focused on the daily buying-and-selling transaction activities in the account currency for all clients and in all broker products.

A description of the causal candidates' variables is shown in Table 3. The cash-balance of the company is the effect variable for all cases and figures, and the plots display all indicators' time series with their respective causal counterpart as a label.

Figures 15–17 show area plots in which the values of the indicator for each causal candidate variable are compared, so that all values for the indicators amount to $100\%$ for each date, and the time series is displayed with higher areas indicating causal interactions between them and the cash-balance. From the experiments, it can be seen that the account status bucket has more mixed results, with mutual funds from November 2022 until March 2023 comprising the majority. After that, the majority of cash movements can be explained by stocks (shares), as can be seen in Figure 15.

In the Bookkeeping cash bucket, shown in Figure 16, a sustained pattern of the majority of cash transactions coming from shares activities is presented. Until March 2023, this majority was shared with mutual funds, however this ceased from March onwards, and the derivatives and FX products have started to build importance since then. The relationship with daily variations of cash-balance gross amount (net of outflows and inflows for each closing day) is studied, with respect to the causal candidates' variables, each in terms of net movements of cash for a closing day (all outflows plus inflows).

Finally, the Trades executed bucket shows very similar behavior to the Bookkeeping cash bucket, which serves as a validation point for our results, as can be seen in Figure 17. The reason why the mutual funds product is not in this figure is because its execution process is different, as much of the operations do not need to be converted to cash (funds transfer). This could mean that, although mutual funds has been a product that, at least historically until March 2023, was responsible for the movement of cash-balance from the clients (seen in the Bookkeeping bucket in Figure 16), this was not due to daily executions in mutual funds with cash, but in the form of internal and external transfer funds. With regards to Trades executed, we can see that from June 2022 until December 2022, the relationships were mixed, with shares not holding the majority. From December 2022 until now, shares have held the majority with some derivatives gaining importance since March 2023.

With greater lags (five or more) values only get smoother but patterns look similar (in Figures 18 and 19, results for 10 lags are displayed). This verifies the theoretical setup presented in this document in that, for greater lags, the standard deviation of their largest eigenvalue varies more. This implies a higher probability that the two variables deviate from independence toward a structured (causal) relationship. This is reflected by the fact that the indicator becomes less volatile in time.

All of the findings from the Figures 15–17 are aligned with the rationale and causal behavior of the cash-flows, given the brokerage business of the dataset. The majority of the client operations were made on stocks, so it makes sense that their executions and cash transactions have more causal interactions with the cash-balance of the company (Figures 15 and 16). Periods of weak shares business operations (executions) are compensated by the second and third most traded products, the bonds and CFDs. This is true for executable products, in the case of mutual funds where the relationship is only via cash transaction or fund transfer (fund transfers do not affect the cash-balance so they are omitted). The fact that the Account status bucket gives mixed results in Figure 15 can be explained because this bucket consists of balances (levels), and increments in cash-balance are not related by causal interactions with other balances' levels. In contrast, increments of cash-balance must be related to other cash-flows within the system by causal interactions, which is well-captured in the other two buckets (cash transactions and executions). Useful findings from the experiments for brokerage business for retail and institutional clients (ie., banks) include:

● Liquidity risk can be monitored by analyzing interactions between operational flows (executions and trading activities) and increments in the cash-balance.

● As most brokerage businesses have stocks as the main traded asset, their flows should be monitored first and are highly correlated to the equities market performance (crisis and rallies would have an impact on liquidity risk).

● Mutual funds do not have an impact on liquidity as most operations are being made via fund transfers, which do not convert the asset to cash.

Finally, with regard to the limitations of the indicator, it can be said that it works better with data of higher frequencies than daily, as it can deal with a higher number of lags, in contrast to daily time series, in which more than ten lags is redundant. The fact that it only works for pairs of time series is limiting for cases in which more simultaneous relationships need to be detected.

6.   Conclusions

A method to measure causal interactions between pairs of time series using the variability in explanatory power of the largest eigenvalue is presented. The method is only applied and can deliver the expected results with no more than a pair of time series, one cause and one effect candidate. Using the homology of the interactions of two walls pushing and pulling a Coulomb gas, which can be modelled by the largest eigenvalue of a random matrix following a Tracy-Widom distribution, the standard deviation of the explanatory power of the largest eigenvalue is used to measure a similar interaction in a system consisting of two time series. The causality component of the interaction is enforced by the direction of time, thanks to lagging the correlation matrices from which the variations of the explanatory power are being measured. To the authors' knowledge, it is the first time random matrix theory and eigenvalues distributions have been applied to measure causal interactions, and also that an experimental system in statistical physics has been used as homology for this purpose. Experiments were conducted with synthetic and real time series and scenarios, including known causal relationships. The method can capture these interactions more accurately than other traditional methods such as the Granger causality test. The indicator can detect structural breaks in time series as shown by experiments with different ARMA models. It can also detect widely known causal variables' candidates for a given set of effect variables among more than one hundred causal candidates (for example, the CDS vs credit as in Figure 10). Two datasets are used for market and liquidity risk analysis based on the indicator. The simplicity, dynamism, and the fact that it does not rely on predictability like other methods, but on spectral theory, random matrix theory, and the homology with statistical and experimental physics properties of causality, make the method extremely useful for risk management with time series for financial institutions and regulators.

For future work, it would be interesting to investigate indicators that can detect causal interactions among more than two time series. The use of random matrix theory seems promising in adding structure to the problem of causal interactions detection. It would be interesting to study the connection between the inverse Wishart distribution, which the causal interactions indicator follows, and the Coulomb gas analogy. It would also be interesting to study configurations of the Coulomb gas analogy for the case of multiple connected gas lines (as in a causal graph of interactions). Finally, it would be interesting to test the indicator as a source of alpha in trading strategies in which information from options derivatives is used to predict their underlying stock price future movements. Also, in the study of prepayments behavior of loans or mortgages, it can be used to detect which variables cause these prepayments, with strong implications in household economies.

Acknowledgements

The authors wish to thank Miralta Finance Bank S.A. for providing all of the data for the experiments. They also express their gratitude to the reviewers from the journal for their valuable feedback and comments.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

All authors declare no conflicts of interest in this paper.