Large Order-Invariant Bayesian VARs with Stochastic Volatility

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 lower triangular parameterization of 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 lower triangular parameterization. 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 when using the nonorder invariant approach. 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.

Keywords:

• Large vector autoregression
• Order invariance
• Shrinkage prior
• Stochastic volatility

Supplementary Materials

The supplementary materials contain a description of the dataset, technical details on the estimation procedures, and additional analytical and simulation results.

Acknowledgments

We would like to thank Christiane Baumeister, Robin Braun, Drew Creal, Gergely Gánics, Chenghan Hou, Elmar Mertens, Ulrich Müller, Ivan Petrella, Mikkel Plagborg-Møller, Minchul Shin and Christopher Sims for their constructive comments and suggestions.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

1 In their classic study of the effects of monetary policy, Leeper, Sims, and Zha (1996) estimate a few SVARs of various sizes, including one with 18 variables.

2 In small VARs, results for a small number of different orderings can be presented as a robustness check. In addition, methods have been proposed for searching over different orderings in papers such as Levy and Lopes (2021) and Wu and Koop (2022). But with a large number of variables a logical ordering of the variables may not present itself and it is not practical to consider every possible ordering.

3 It is worth noting that one major reason for triangularizing the system is to allow for equation by equation estimation of the model. Any approach which allows for equation by equation estimation can be shown to lead to large computational benefits since there is no need to manipulate the (potentially enormous) posterior covariance matrix of the VAR coefficients in a single block. Triangularization is the most popular equation by equation method and Carriero, Clark, and Marcellino (2019) show how, relative to full system estimation, triangularization can reduce computational complexity of Bayesian estimation of large VARs from $O(n^6)$ to $O(n^4)$ where n is VAR dimension. This highlights the importance of development of VAR specifications, such as the one in this article, which allow for equation by equation estimation.

4 Following the bulk of the literature on large Bayesian VARs, we focus on the case of constant VAR coefficients. The model can be readily extended to accommodate time-varying VAR coefficients, but with additional computational costs.

5 There are many different versions of “the Minnesota prior.” The version that we are using is an independent normal and inverse-Wishart prior that is applicable to VARs with time-varying volatility. See, for example, eq. (14) in Karlsson (2013) or eq. (13) in Carriero, Clark, and Marcellino (2015).

6 In the simulations and the empirical application, we set ${\mathbf{b}}_{0,i}={\mathbf{e}}_i$ and ${\mathbf{V}}_{{\mathbf{b}}_i}={\mathbf{I}}_n$, where ${\mathbf{e}}_i$ is the ith column of the identity matrix ${\mathbf{I}}_n$. This choice of hyperparameters implies an order invariant prior in the sense that if we permute the endogenous variables ${\tilde{\mathbf{y}}}_t=\mathbf{P}{\mathbf{y}}_t$, the prior mean of $\mathbf{P}{\mathbf{B}}_0\mathbf{P}\prime$ is also an identity matrix (and all prior variances of the elements of $\mathbf{P}{\mathbf{B}}_0\mathbf{P}\prime$ are also 1). Since ${\mathbf{B}}_0$ is identified up to permutations and sign normalization, this relatively weak prior works well in a range of simulations.

7 In MATLAB, the orthogonal complement of ${\mathbf{v}}_1$ can be obtained using null $({\mathbf{v}}_1^{\prime })$.

8 In the original model of Cogley and Sargent (2005), the log-volatility follows a unit root process. Here we consider a version in which the log-volatility follows a stationary AR(1) process (with a nonzero mean) so that it is directly comparable to the OI-VAR-SV.

9 Note that we are comparing the two models under the first ordering of the variables. The comparable figure using the second ordering reveals similar patterns.

10 For m < n, the n × n diagonal matrix ${\mathbf{D}}_t$ in (2) is set to be ${\mathbf{D}}_t=\text{diag}({\text{e}}^{h_{1,t}},\dots ,{\text{e}}^{h_{m,t}},1,\dots ,1)$.

Additional information

Funding

Yu is supported by Shanghai Sailing Program No. 23YF1402000 and National Natural Science Foundation of China (72192845).