Multivariate Time Series

Order-Constrained Spectral Causality in Multivariate Time Series ArXiv ID: 2601.01216 “View on arXiv” Authors: Alejandro Rodriguez Dominguez Abstract We introduce an operator-theoretic framework for causal analysis in multivariate time series based on order-constrained spectral non-invariance. Directional influence is defined as sensitivity of second-order dependence operators to admissible, order-preserving temporal deformations of a designated source component, yielding an intrinsically multivariate causal notion summarized through orthogonally invariant spectral functionals. Under linear Gaussian assumptions, the criterion coincides with linear Granger causality, while beyond this regime it captures collective and nonlinear directional dependence not reflected in pairwise predictability. We establish existence, uniform consistency, and valid inference for the resulting non-smooth supremum–infimum statistics using shift-based randomization that exploits order-induced group invariance, yielding finite-sample exactness under exact invariance and asymptotic validity under weak dependence without parametric assumptions. Simulations demonstrate correct size and strong power against distributed and bulk-dominated alternatives, including nonlinear dependence missed by linear Granger tests with appropriate feature embeddings. An empirical application to a high-dimensional panel of daily financial return series spanning major asset classes illustrates system-level causal monitoring in practice. Directional organization is episodic and stress-dependent, causal propagation strengthens while remaining multi-channel, dominant causal hubs reallocate rapidly, and statistically robust transmission channels are sparse and horizon-heterogeneous even when aggregate lead–lag asymmetry is weak. The framework provides a scalable and interpretable complement to correlation-, factor-, and pairwise Granger-style analyses for complex systems. ...

Analysis of Contagion in China’s Stock Market: A Hawkes Process Perspective ArXiv ID: 2512.08000 “View on arXiv” Authors: Junwei Yang Abstract This study explores contagion in the Chinese stock market using Hawkes processes to analyze autocorrelation and cross-correlation in multivariate time series data. We examine whether market indices exhibit trending behavior and whether sector indices influence one another. By fitting self-exciting and inhibitory Hawkes processes to daily returns of indices like the Shanghai Composite, Shenzhen Component, and ChiNext, as well as sector indices (CSI Consumer, Healthcare, and Financial), we identify long-term dependencies and trending patterns, including upward, downward, and oversold rebound trends. Results show that during high trading activity, sector indices tend to sustain their trends, while low activity periods exhibit strong sector rotation. This research models stock price movements using spatiotemporal Hawkes processes, leveraging conditional intensity functions to explain sector rotation, advancing the understanding of financial contagion. ...

Causal Regime Detection in Energy Markets With Augmented Time Series Structural Causal Models ArXiv ID: 2511.04361 “View on arXiv” Authors: Dennis Thumm Abstract Energy markets exhibit complex causal relationships between weather patterns, generation technologies, and price formation, with regime changes occurring continuously rather than at discrete break points. Current approaches model electricity prices without explicit causal interpretation or counterfactual reasoning capabilities. We introduce Augmented Time Series Causal Models (ATSCM) for energy markets, extending counterfactual reasoning frameworks to multivariate temporal data with learned causal structure. Our approach models energy systems through interpretable factors (weather, generation mix, demand patterns), rich grid dynamics, and observable market variables. We integrate neural causal discovery to learn time-varying causal graphs without requiring ground truth DAGs. Applied to real-world electricity price data, ATSCM enables novel counterfactual queries such as “What would prices be under different renewable generation scenarios?”. ...

Convolutional Attention in Betting Exchange Markets ArXiv ID: 2510.16008 “View on arXiv” Authors: Rui Gonçalves, Vitor Miguel Ribeiro, Roman Chertovskih, António Pedro Aguiar Abstract This study presents the implementation of a short-term forecasting system for price movements in exchange markets, using market depth data and a systematic procedure to enable a fully automated trading system. The case study focuses on the UK to Win Horse Racing market during the pre-live stage on the world’s leading betting exchange, Betfair. Innovative convolutional attention mechanisms are introduced and applied to multiple recurrent neural networks and bi-dimensional convolutional recurrent neural network layers. Additionally, a novel padding method for convolutional layers is proposed, specifically designed for multivariate time series processing. These innovations are thoroughly detailed, along with their execution process. The proposed architectures follow a standard supervised learning approach, involving model training and subsequent testing on new data, which requires extensive pre-processing and data analysis. The study also presents a complete end-to-end framework for automated feature engineering and market interactions using the developed models in production. The key finding of this research is that all proposed innovations positively impact the performance metrics of the classification task under examination, thereby advancing the current state-of-the-art in convolutional attention mechanisms and padding methods applied to multivariate time series problems. ...

Higher Order Transformers: Enhancing Stock Movement Prediction On Multimodal Time-Series Data ArXiv ID: 2412.10540 “View on arXiv” Authors: Unknown Abstract In this paper, we tackle the challenge of predicting stock movements in financial markets by introducing Higher Order Transformers, a novel architecture designed for processing multivariate time-series data. We extend the self-attention mechanism and the transformer architecture to a higher order, effectively capturing complex market dynamics across time and variables. To manage computational complexity, we propose a low-rank approximation of the potentially large attention tensor using tensor decomposition and employ kernel attention, reducing complexity to linear with respect to the data size. Additionally, we present an encoder-decoder model that integrates technical and fundamental analysis, utilizing multimodal signals from historical prices and related tweets. Our experiments on the Stocknet dataset demonstrate the effectiveness of our method, highlighting its potential for enhancing stock movement prediction in financial markets. ...

Identifying Extreme Events in the Stock Market: A Topological Data Analysis ArXiv ID: 2405.16052 “View on arXiv” Authors: Unknown Abstract This paper employs Topological Data Analysis (TDA) to detect extreme events (EEs) in the stock market at a continental level. Previous approaches, which analyzed stock indices separately, could not detect EEs for multiple time series in one go. TDA provides a robust framework for such analysis and identifies the EEs during the crashes for different indices. The TDA analysis shows that $L^1$, $L^2$ norms and Wasserstein distance ($W_D$) of the world leading indices rise abruptly during the crashes, surpassing a threshold of $μ+4σ$ where $μ$ and $σ$ are the mean and the standard deviation of norm or $W_D$, respectively. Our study identified the stock index crashes of the 2008 financial crisis and the COVID-19 pandemic across continents as EEs. Given that different sectors in an index behave differently, a sector-wise analysis was conducted during the COVID-19 pandemic for the Indian stock market. The sector-wise results show that after the occurrence of EE, we have observed strong crashes surpassing $μ+2σ$ for an extended period for the banking sector. While for the pharmaceutical sector, no significant spikes were noted. Hence, TDA also proves successful in identifying the duration of shocks after the occurrence of EEs. This also indicates that the Banking sector continued to face stress and remained volatile even after the crash. This study gives us the applicability of TDA as a powerful analytical tool to study EEs in various fields. ...

Dynamic Time Warping for Lead-Lag Relationships in Lagged Multi-Factor Models ArXiv ID: 2309.08800 “View on arXiv” Authors: Unknown Abstract In multivariate time series systems, lead-lag relationships reveal dependencies between time series when they are shifted in time relative to each other. Uncovering such relationships is valuable in downstream tasks, such as control, forecasting, and clustering. By understanding the temporal dependencies between different time series, one can better comprehend the complex interactions and patterns within the system. We develop a cluster-driven methodology based on dynamic time warping for robust detection of lead-lag relationships in lagged multi-factor models. We establish connections to the multireference alignment problem for both the homogeneous and heterogeneous settings. Since multivariate time series are ubiquitous in a wide range of domains, we demonstrate that our algorithm is able to robustly detect lead-lag relationships in financial markets, which can be subsequently leveraged in trading strategies with significant economic benefits. ...

Portfolio Selection via Topological Data Analysis ArXiv ID: 2308.07944 “View on arXiv” Authors: Unknown Abstract Portfolio management is an essential part of investment decision-making. However, traditional methods often fail to deliver reasonable performance. This problem stems from the inability of these methods to account for the unique characteristics of multivariate time series data from stock markets. We present a two-stage method for constructing an investment portfolio of common stocks. The method involves the generation of time series representations followed by their subsequent clustering. Our approach utilizes features based on Topological Data Analysis (TDA) for the generation of representations, allowing us to elucidate the topological structure within the data. Experimental results show that our proposed system outperforms other methods. This superior performance is consistent over different time frames, suggesting the viability of TDA as a powerful tool for portfolio selection. ...

Graph Neural Networks for Forecasting Multivariate Realized Volatility with Spillover Effects ArXiv ID: 2308.01419 “View on arXiv” Authors: Unknown Abstract We present a novel methodology for modeling and forecasting multivariate realized volatilities using customized graph neural networks to incorporate spillover effects across stocks. The proposed model offers the benefits of incorporating spillover effects from multi-hop neighbors, capturing nonlinear relationships, and flexible training with different loss functions. Our empirical findings provide compelling evidence that incorporating spillover effects from multi-hop neighbors alone does not yield a clear advantage in terms of predictive accuracy. However, modeling nonlinear spillover effects enhances the forecasting accuracy of realized volatilities, particularly for short-term horizons of up to one week. Moreover, our results consistently indicate that training with the Quasi-likelihood loss leads to substantial improvements in model performance compared to the commonly-used mean squared error. A comprehensive series of empirical evaluations in alternative settings confirm the robustness of our results. ...

Bayesian Forecasting of Stock Returns on the JSE using Simultaneous Graphical Dynamic Linear Models ArXiv ID: 2307.08665 “View on arXiv” Authors: Unknown Abstract Cross-series dependencies are crucial in obtaining accurate forecasts when forecasting a multivariate time series. Simultaneous Graphical Dynamic Linear Models (SGDLMs) are Bayesian models that elegantly capture cross-series dependencies. This study forecasts returns of a 40-dimensional time series of stock data from the Johannesburg Stock Exchange (JSE) using SGDLMs. The SGDLM approach involves constructing a customised dynamic linear model (DLM) for each univariate time series. At each time point, the DLMs are recoupled using importance sampling and decoupled using mean-field variational Bayes. Our results suggest that SGDLMs forecast stock data on the JSE accurately and respond to market gyrations effectively. ...