Bayesian Tensor Quantile Regression

Abstract

In several fields, including finance and economics, observations with inherently multiple dimensions (i.e., tensors) are often generated over time. For instance, a dynamic network encoding multiple types of connections over time can be represented by a 3-dim tensor. Another example is a time series of returns from a portfolio of stocks formed by applying multiple classifications. Reshaping the multidimensional observations into long vectors results in losing the information contained in the natural structure and dependence of each dimension of the data. This paper proposes a novel regression model in multilinear form to jointly estimate marginal conditional quantiles of tensor-valued response variables. It utilizes the tensor structure to achieve dimensional reduction and enhance the interpretability. In particular, the tensor asymmetric Laplace distribution is exploited under a Bayesian approach to form the likelihood, and it is proven its connection to a quantile regression model where the response variable is a multidimensional array.

We design a Markov chain Monte Carlo algorithm that relies on a reparameterisation of the tensor asymmetric Laplace distribution and exploit its location-scale mixture representation to make inference on the model parameters. The validity of our approach is analysed in a simulation study, then a the model is applied to investigate the dynamics of the conditional quantiles of a time series of multidimensional financial portfolios.