[Paper Review] Large Order-Invariant Bayesian VARs with Stochastic Volatility

Joshua C. C. Chan, Gary Koop+1 more|arXiv (Cornell University)|Nov 14, 2021

Forecasting Techniques and ApplicationsDecision Sciences18 citations

TL;DR

This paper proposes a large-order-invariant Bayesian VAR with stochastic volatility that avoids the Cholesky decomposition, ensuring identification through time-varying variances. The method achieves order invariance, improves forecasting accuracy in high-dimensional macroeconomic models, and outperforms conventional approaches sensitive to variable ordering.

ABSTRACT

Many popular specifications for Vector Autoregressions (VARs) with multivariate stochastic volatility are not invariant to the way the variables are ordered due to the use of a Cholesky decomposition for the error covariance matrix. We show that the order invariance problem in existing approaches is likely to become more serious in large VARs. We propose the use of a specification which avoids the use of this Cholesky decomposition. We show that the presence of multivariate stochastic volatility allows for identification of the proposed model and prove that it is invariant to ordering. We develop a Markov Chain Monte Carlo algorithm which allows for Bayesian estimation and prediction. In exercises involving artificial and real macroeconomic data, we demonstrate that the choice of variable ordering can have non-negligible effects on empirical results. In a macroeconomic forecasting exercise involving VARs with 20 variables we find that our order-invariant approach leads to the best forecasts and that some choices of variable ordering can lead to poor forecasts using a conventional, non-order invariant, approach.

Motivation & Objective

• Address the lack of order invariance in conventional large Bayesian VARs with stochastic volatility, which stems from Cholesky decompositions of the error covariance matrix.
• Develop a model specification that remains invariant to variable ordering while maintaining identification through stochastic volatility.
• Enable scalable Bayesian estimation and forecasting in high-dimensional VARs using an efficient MCMC algorithm.
• Demonstrate empirically that variable ordering significantly affects forecast performance in non-order-invariant models, especially in large systems.

Proposed method

• Replace the Cholesky decomposition of the error covariance matrix with an unrestricted impact matrix, avoiding ordering dependence.
• Use multivariate stochastic volatility to identify the model, as the homoscedastic version would be unidentified.
• Employ a Wishart-type prior for the time-varying covariance matrix to ensure mathematical tractability and order invariance.
• Develop a computationally efficient MCMC algorithm that allows for equation-by-equation estimation, reducing complexity from O(n⁶) to O(n⁴) in high dimensions.
• Incorporate shrinkage priors on VAR coefficients to handle high-dimensional estimation and improve forecast performance.
• Use a Gibbs sampling scheme with data augmentation to jointly estimate the VAR coefficients, stochastic volatility, and error covariance structure.

Experimental results

Research questions

• RQ1 Does the use of a Cholesky decomposition in conventional large Bayesian VARs with stochastic volatility lead to significant sensitivity to variable ordering in empirical results?
• RQ2 Can a large Bayesian VAR with stochastic volatility be made order-invariant while retaining identification and computational feasibility?
• RQ3 How does the forecasting performance of the proposed order-invariant model compare to conventional non-order-invariant models across different variable orderings?
• RQ4 What are the practical implications of variable ordering in large VAR forecasting exercises, particularly in macroeconomic datasets with 20 or more variables?

Key findings

• The choice of variable ordering has non-negligible effects on empirical results in conventional VAR-SV models, with some orderings producing significantly worse forecasts than others.
• In a 20-variable macroeconomic forecasting exercise, the proposed order-invariant model (OI-VAR-SV) produced the best forecasts across all horizons and variables, outperforming all non-order-invariant alternatives.
• The conventional Cogley-Sargent (2005) approach (CS-VAR-SV) produced highly variable forecast performance depending on ordering: for example, one ordering led to RMSFE 0.007∗∗∗ for PCE inflation at h=12, while the order-invariant model achieved 0.002∗∗, indicating a substantial improvement.
• The order-invariant model achieved superior predictive performance in terms of both RMSFE and ALPL, with the best model achieving 1.522∗∗∗ ALPL for industrial production at h=6, compared to 2.266 for the worst-performing ordering.
• The OI-VAR-SV model maintained consistent performance across all variables and forecast horizons, with no single ordering dominating, confirming its robustness to variable reordering.
• The MCMC algorithm enabled efficient estimation and prediction in high-dimensional VARs, with computational complexity reduced to O(n⁴), making it scalable for large systems.