Large Bayesian VARs: A Flexible Kronecker Error Covariance Structure

ABSTRACT

We introduce a class of large Bayesian vector autoregressions (BVARs) that allows for non-Gaussian, heteroscedastic, and serially dependent innovations. To make estimation computationally tractable, we exploit a certain Kronecker structure of the likelihood implied by this class of models. We propose a unified approach for estimating these models using Markov chain Monte Carlo (MCMC) methods. In an application that involves 20 macroeconomic variables, we find that these BVARs with more flexible covariance structures outperform the standard variant with independent, homoscedastic Gaussian innovations in both in-sample model-fit and out-of-sample forecast performance.

KEY WORDS:

• ARMA
• Forecasting
• Non-Gaussian
• Stochastic volatility.

Acknowledgments

Financial support by the Australian Research Council via a Discovery Early Career Researcher Award (DE150100795) is gratefully acknowledged. I would also like to thank seminar and conference participants at University of Technology Sydney, Deakin University, University of Sydney, Purdue University, Macquarie University, Melbourne Bayesian Econometrics Workshop, the 10th Rimini Bayesian Econometrics Workshop, the 10th International Conference on Computational and Financial Econometrics, and the 4th Meeting of the Sydney Econometrics Research Group. In particular, this article has benefited from helpful comments from Gary Koop, Todd Clark, and James Morley.

Notes

1 Following Carriero, Clark, and Marcellino (2016), the unconditional mean of h_t is assumed to be zero for identification purposes. In fact, the marginal distribution of h_t is $\mathcal{N}(0,{\sigma }_h^2/(1-{\rho }^2))$. This prior implies that the standard deviation exp (h_t/2) is log-normally distributed with mean exp (σ²_h/(8(1 − ρ²))). For example, if ρ = 0.98 and σ²_h = 0.1, the prior mean of exp (h_t/2) is about 1.37.

2 In the online Appendix A, we discuss how one can construct $\Omega$ efficiently from elementary matrices.

3 This algorithm is well-known and is described in the textbook by Bauwens, Lubrano, and Richard (1999, p. 320). More recently, this algorithm was used to estimate dynamic matrix-variate graphical models by Carvalho and West (2007) and Wang, Reeson, and Carvalho (2011).

4 Forward and backward substitutions are implemented in standard packages such as Matlab, Gauss, and R. In Matlab, for example, it is done by mldivide(B,c) or simply B\c.

5 Similar computational savings can be generated for operations such as multiplication, forward, and backward substitution by using band matrix routines, which are implemented in standard packages such as Matlab, Gauss, and R.

6 The authors have indicated that if some supplementary calculations—such as storage and computing quantities for the marginal likelihood—are removed, the computation time can be substantially reduced.

7 If we fit a univariate MA(1) model to each of the 20 residual series, only 2 MA estimates are larger than 0.1.

8 The marginal likelihood has a built-in penalty for complexity. In fact, it trade-offs between model fit vs model complexity and it does not always favor the most general model. For more details, see, for example, the discussion in Koop (2003).

9 Even though some of the full conditional densities are non-standard, they can still be quickly evaluated. For instance, the full conditional density of the MA coefficient ψ can be evaluated using the Monte Carlo methods described in Chan (2013).

10 In the related context of computing the deviance information criterion, Li, Zeng, and Yu (2012) gave theoretical arguments why the conditional likelihood should not be used.

11 Since the Fed funds rate essentially hits the zero lower bound after 2007, in Online Appendix D we present results for a sample ending in 2007Q4. The main results and conclusions remain the same.