[Paper Review] Generalised Wishart Processes

Andrew Gordon Wilson, Zoubin Ghahramani|arXiv (Cornell University)|Dec 31, 2010

Advanced Statistical Methods and Models23 references21 citations

TL;DR

This paper introduces the Generalised Wishart Process (GWP), a stochastic process with Wishart-distributed marginals that models time-varying covariance matrices using arbitrary dependent variables, including time and covariates. It outperforms multivariate GARCH and standard Wishart processes in capturing complex covariance structures, especially off-diagonal elements, on financial data with missing data and high-dimensional scalability, while requiring no free parameters and enabling interpretable Bayesian inference.

ABSTRACT

We introduce a stochastic process with Wishart marginals: the generalised Wishart process (GWP). It is a collection of positive semi-definite random matrices indexed by any arbitrary dependent variable. We use it to model dynamic (e.g. time varying) covariance matrices. Unlike existing models, it can capture a diverse class of covariance structures, it can easily handle missing data, the dependent variable can readily include covariates other than time, and it scales well with dimension; there is no need for free parameters, and optional parameters are easy to interpret. We describe how to construct the GWP, introduce general procedures for inference and predictions, and show that it outperforms its main competitor, multivariate GARCH, even on financial data that especially suits GARCH. We also show how to predict the mean of a multivariate process while accounting for dynamic correlations.

Motivation & Objective

• Address the limitations of existing multivariate volatility models, such as multivariate GARCH and Wishart processes, which suffer from high parameter complexity, poor scalability, and restricted dependency structures.
• Develop a unified framework that models dynamic covariance matrices Σ(t) as a stochastic process indexed by arbitrary dependent variables, including time and covariates like interest rates.
• Enable robust inference and prediction under missing data and complex covariance structures (e.g., periodic, smooth, or non-Markovian dependencies) without free parameters.
• Provide a general, interpretable, and scalable alternative to multivariate GARCH and standard Wishart processes for high-dimensional time-series modeling in econometrics and machine learning.
• Demonstrate the model's superiority in capturing co-movements (off-diagonal covariances) and handling real-world data challenges such as the financial crisis period and missing observations.

Proposed method

• Define the Generalised Wishart Process (GWP) as a stochastic process with Wishart-distributed marginals, where the covariance matrix Σ(t) evolves over an arbitrary index set (e.g., time, covariates).
• Construct the GWP using a kernel-based covariance structure (e.g., squared exponential kernel) to model dependence between Σ(t) at different index points, enabling flexible modeling of temporal and covariate-driven dynamics.
• Use conjugate Bayesian inference with Wishart and inverse-Wishart priors to enable efficient posterior updates and predictive distributions over Σ(t), including for missing data.
• Introduce a non-Markovian, non-autoregressive framework that allows complex dependencies beyond the Ornstein-Uhlenbeck structure of prior Wishart processes.
• Implement predictive procedures for both one-step-ahead forecasts and historical reconstructions of Σ(t), using empirical data and synthetic data with known true covariance matrices.
• Apply the GWP to real financial data (EXCHANGE) and synthetic data (EQUITY) generated from empirical covariance matrices of equity indices, enabling ground-truth comparison via known Σ(t).

Experimental results

Research questions

• RQ1 Can a stochastic process with Wishart marginals be generalized to model dynamic covariance matrices indexed by arbitrary dependent variables, including time and external covariates?
• RQ2 Does the GWP outperform multivariate GARCH and standard Wishart processes in capturing complex, time-varying covariance structures, especially off-diagonal elements, on financial data?
• RQ3 Can the GWP handle missing data and high-dimensional settings without requiring free parameters or complex parameter tuning?
• RQ4 How well does the GWP reconstruct true covariance dynamics when the underlying Σ(t) is known, compared to existing models?
• RQ5 To what extent can the GWP model diverse covariance patterns—such as periodicity, smoothness, or non-Markovian dependencies—beyond the fixed Ornstein-Uhlenbeck structure of prior models?

Key findings

• The GWP outperforms multivariate GARCH on both the EXCHANGE and EQUITY datasets, even though GARCH is especially suited to financial return data.
• The GWP significantly outperforms the standard Wishart process (WP), which is restricted to an Ornstein-Uhlenbeck covariance structure, indicating the GWP’s ability to capture more complex interdependencies.
• The GWP achieves superior performance in predicting off-diagonal covariance elements (co-movements), suggesting it better models dynamic correlations between financial assets.
• The model effectively reconstructs historical covariance matrices, as shown in Figure 3, where GWP predictions closely track the true underlying Σ(t) across diagonal variances and off-diagonal covariances.
• The GWP enables accurate one-step-ahead forecasts and historical predictions using only 200 training points, demonstrating scalability and robustness on real-world financial data.
• The absence of free parameters and the interpretability of optional hyperparameters (e.g., kernel length scales) make the GWP both practical and theoretically sound for high-dimensional applications.