Tensor Models

QFFE Spring School 2025 (link)

7th Quantitative Finance and Financial Econometrics International Conference

3rd June, 2025, Marseille, France

Instructor

Roberto Casarin (Ca' Foscari University of Venice)

Contents

The tutorial provides students with models and tools from tensor valued data analysis (response tensors and covariate tensors). One of the main applications regards modelling and forecasting Granger financial networks, which are used in measuring financial interconnectedness.

The organisation of the course can be split into three main parts. The first introduces some background in graph theory, multilinear tensor algebra and Bayesian inference. In the second part, linear models such as tensor autoregressions and a static matrix de-noising model, are presented. Multi-shrinkage Lasso-type estimation is used within a Bayesian inference framework. The third part deals with nonlinear modelling, and includes Markov-switching for binary tensor-on-tensor logit models and smooth transition models for matrix data.

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Slides and codes

1. Background

1.1 Networks: Extraction, Graph Theory, Connectivity, Example (.zip slides)

1.2 Temporal Networks (.pdf slides, .zip MATLAB code)

1.3 Tensor Algebra (.pdf slides, .zip MATLAB code)

1.4 Bayesian Inference (.pdf slides, .pdf notes, .zip MATLAB code)

2. Linear Tensor Models

2.1 Dynamic Tensor Models (.pdf slides, .zip MATLAB code)

2.2 Matrix t Model (.pdf slides, .zip MATLAB code)

3. Non-linear Tensor Models

3.1 Markov-switching tensor logit models (.pdf slides)

3.2 Smooth transition tensor models (.pdf slides, .zip MATLAB code)

QFFE Presentation ---> "Dynamic Tensor Models" (.pdf slides)

Some references

Casarin, R., Radu, C., Wang, Q. (2025), Markov Switching Multiple-equation Tensor Regressions, Journal of Multivariate Analysis, 208, 105427 (article, supplement)
Billio, M., Casarin, R., Iacopini, M. (2024), Bayesian Markov-Switching Tensor Regression for Time-varying Networks, Journal of the American Statistical Association (Theory & Methods), 119(545), 109-121 (article, supplement)
Billio, M., Casarin, R., Kaufmann, S., Iacopini, M. (2023), Bayesian Dynamic Tensor Regression Models, Journal of Business & Economic Statistics, 41(2), 429-439 (article, supplement)
Billio, M., Casarin, R., Costola, M., Iacopini, M. (2021). A matrix-variate t model for networks, Frontiers in Artificial Intelligence, 4, 49 (article, supplement)
Billio, M., Casarin, R., Costola, M. and Iacopini, M. (2022), Matrix-variate Smooth Transition Models for Temporal Networks. In Andriette Bekker, Mohammad Arashi, and Din Chen, Johan Ferreira (eds) Innovations in Multivariate Statistical Modelling: Navigating Theoretical and Multidisciplinary Domains, Springer Emerging Topics in Statistics and Biostatistics, Springer Verlag. (article)

Google Scholar: "bayesian tensor regression", "multilinear algebra", "tensor models tucker"