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

Abstract

Vector autoregressions (VARs) with multivariate stochastic volatility are widely used for structural analysis. Often the structural model identified through economically meaningful restrictions-e.g., sign restrictions-is supposed to be independent of how the dependent variables are ordered. But since the reduced-form model is not order invariant, results from the structural analysis depend on the order of the variables. We consider a VAR based on the factor stochastic volatility that is constructed to be order invariant. We show that the presence of multivariate stochastic volatility allows for statistical identification of the model. We further prove that, with a suitable set of sign restrictions, the corresponding structural model is point-identified. An additional appeal of the proposed approach is that it can easily handle a large number of dependent variables as well as sign restrictions.We demonstrate the methodology through a structural analysis in which we use a 20-variable VAR with sign restrictions to identify 5 structural shocks.

Bayesian modeling of time-varying parameters using regression trees

“…One key observation is that, conditional on the factors q t , the shocks ε t are independent and equation-by-equation estimation is possible. This factor structure has been used in other papers to facilitate estimation of large VARs (Kastner and Huber, 2020;Chan, Forthcoming;Clark et al, Forthcoming) and, in addition, this structure can also facilitate identification of the factors as structural VAR disturbances (Korobilis, 2022;Chan et al, 2022). Moreover, in contrast to VAR-based estimation using a Cholesky decomposition of the error covariances, another convenient feature of our model is that it is invariant to how the variables are ordered in y t (for a formal argument, see, for example, Chan et al, 2022).…”

“…We do not impose identification restrictions on our factor model during MCMC estimation. Results in Chan et al (2022) suggest that the decomposition in Eq. ( 2) is identified up to column and sign switching.…”

“…In this application, we focus on the question of how adverse business cycle shocks impact a set of inflation measures. To do so, we exploit the factor structure on the reduced-form VAR shocks to identify a business cycle shock (for related identification approaches, see Korobilis, 2022;Chan et al, 2022). As emphasized by Gorodnichenko (2005), in VAR models like ours where the number of variables is relatively large (we have M = 12) having fewer structural shocks than M can facilitate structural interpretation.…”

“…10 Absent heteroscedasticity, the resulting factors still have no economic interpretation and thus additional structure is required to identify the shocks, given that the factors and the factor loadings can be rotated by any random orthogonal matrix. But, given the heteroscedasticity in R t , we can follow Chan et al (2022) and identify, up to sign and scale, a business cycle shock as that factor (shock) that explains the largest amount of variation in innovations to output and unemployment variations during recessionary periods (as identified by the NBER). A related identification strategy has been used in Stock and Watson (2005) to identify a G7 (or global) business cycle shock, as that shock to GDP that impacts all countries contemporaneously.…”

“…These steps yield partial identification, implying that the business cycle shock is uniquely identified, whereas the remaining Q q − 1 factors (and the associated columns in Γ) are left unrestricted. This identification approach is related to ones developed in recent papers (Korobilis, 2022;Chan et al, 2022) that advocate using sign restrictions on the factor loadings to pin down a shock of interest. But our approach differs in the sense that we solve the We then refer to the factor that accounts for the maximal volatility of real output growth and unemployment as our business cycle shock.…”