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 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.

Real‐time forecasting of the Australian macroeconomy using flexible Bayesian VARs

“…To capture the potential heavy-tailed feature exhibited in the data, our second model is a t-distributed error version of the VAR-SV model in which the time-varying variance-covariance matrix is given by 5 Note that one conventional modeling strategy for multivariate stochastic volatility models in the macroeconomic forecasting literature is to assume B 0 to be a lower triangular matrix. However, recent studies of Arias et al (2021) and Chan et al (2021) have pointed out that such a recursive structure in B 0 is not invariant to the ordering of the VAR variables which would become more serious in large dimensional models. To this end, we specify B 0 to be a full unrestricted matrix to avoid this ordering issue resulting from the conventional recursive structure.…”

“…For VAR models with independent prior, the details of the estimation procedure for the VAR-FSV can be found in Chan (2021). For the VAR-SV model, we follow the recently developed approach of Chan et al (2021) to sample the model parameters. To be specific, the posterior draws of B 0 are obtained by using the method of Waggoner and Zha (2003) and Villani (2009), and the posterior draws of the VAR coefficients are obtained by using the equation-by-equation approach of developed by Carriero et al (2022).…”

Nowcasting in a pandemic using non-parametric mixed frequency VARs

“…posterior and predictive densities depend on the manner in which the variables are ordered in the VAR). The importance of order dependence, and in particular its impact on predictive variances in larger VARs, is discussed in papers such as Arias et al (forthcoming) and Chan et al (2021). There have been some order invariant approaches proposed which do allow for equation-by-equation estimation, including Chan et al (2021) and Wu and Koop (2023) but these assume Gaussian errors and the former relies on the presence of stochastic volatility to identify the model.…”

“…The importance of order dependence, and in particular its impact on predictive variances in larger VARs, is discussed in papers such as Arias et al (forthcoming) and Chan et al (2021). There have been some order invariant approaches proposed which do allow for equation-by-equation estimation, including Chan et al (2021) and Wu and Koop (2023) but these assume Gaussian errors and the former relies on the presence of stochastic volatility to identify the model. However, the presence of large VAR shocks which imply sudden shifts in variances and/or asymmetries in predictive densities means more flexibility is required.…”

Large Bayesian VARs with Factor Stochastic Volatility: Identification, Order Invariance and Structural Analysis

“…The cost for this tractability, however, is that they are generally too tightly parameterized, and consequently, they tend to underperform in forecasting macroeconomic variables relative to standard stochastic volatility models such as Cogley and Sargent (2005) and Primiceri (2005) (see Arias, Rubio-Ramirez, and Shin, 2021, for an example). Lastly, the recent paper Chan, Koop, and Yu (2021) extends the stochastic volatility model of Cogley and Sargent (2005) by avoiding the use of Cholesky decomposition so that the extension is order-invariant. So far this reduced-form VAR is used for forecasting, and further research is needed to incorporate identification restrictions for structural analysis.…”