High Dimensional Spatio-Temporal Autoregressive Models for Matrix-Valued Time Series

Modelling of time series data holds significant importance in practice, as such data are observed in a multitude of fields including finance, economics, and social networks, among others. In the era of big data, increasingly complex types of data and varied sources of information can be more readily obtained. For instance, certain financial firms are analysing alternative data sets, including satellite data, to generate their trading signals, in addition to traditional financial metrics such as price and trading volumes. On one hand, the abundance of data provides a wealth of information that researchers and practitioners across societal sectors can utilise more effectively. On the other hand, capturing the intricate dependency structures within various types of data presents an ever-increasing challenge.This project models a specific type of time series data: matrix-valued time series. The new class of models proposed aims to capture complex interactions within the data. Matrix-valued data is commonly observed in practice, and we initiate our research with a practical problem—the prediction of trading volume curves, which are key inputs for all major execution service brokers in the financial industry—as our motivating example to illustrate the necessity of modelling matrixvalued data. This necessity stems from both practical and econometric concerns. Preliminary data analyses on several liquid crypto-assets traded on Binance will be presented to confirm the existence of more intricate dependency structures within matrix-valued data. Bolstered by these empirical observations, new models and estimation methods will be proposed, and their theoretical underpinnings, such as large-sample properties, will be established. Comprehensive simulations will be conducted to validate the proposed methods. Importantly, we aim to seek collaborations with institutional equity execution service desks at CLSA, which is based in HK and owned by CITIC Securities, so that our proposed methods may be applied to their actual trading data to enhance their volume curve prediction models. An initial connection with CLSA has been established, and further details of collaboration will be elaborated upon as the project progresses.We will introduce a new class of models capable of modelling not just serial dependence or temporal dynamics, but also pure spatial dependence (for example, the relationships among different locations at the same time) for matrix-valued time series. The modelling of these complex dependency structures is more challenging and valuable than that of traditional vector time series models and conventional spatial econometric models. Two subclasses of models will be considered: one assumes prior knowledge about the interactions among different columns and rows of the matrix data, which can be represented by a weight matrix; the other allows for greater modelling flexibility by assuming the weight matrices are unknown, increasing the estimation difficulty due to the so-called “curse of dimensionality”. Based on the models proposed, we will develop estimation methods for the unknown parameters and establish their asymptotic theories, both for fixed and diverging dimensions. These theories are important from both an econometric and statistical standpoint, and are particularly significant in practice, especially in financial data analysis where the sample size is often limited.This project is poised to have a substantial impact on both academia and industry. Various possible extensions based on our proposal, such as time-varying models, structural break detection, and new models incorporating more practical considerations, provide future research directions. The industrial impact, especially for execution brokers, is evident as our model can capture additional information not accounted for by the simple moving average methods currently employed by major execution desks. This project also serves as a starting point for forging connections with industry partners, with an aim to establish consistent ongoing collaborations in the future.