Large Bayesian vector autoregressions with stochastic volatility and non-conjugate priors

Abstract

Recent research has shown that a reliable vector autoregression (VAR) for forecasting and structural analysis of macroeconomic data requires a large set of variables and modeling time variation in their volatilities. Yet, there are no papers that provide a general solution for combining these features, due to computational complexity. Moreover, homoskedastic Bayesian VARs for large datasets so far restrict substantially the allowed prior distributions on the parameters. In this paper we propose a new Bayesian estimation procedure for (possibly very large) VARs featuring time-varying volatilities and general priors. We show that indeed empirically the new estimation procedure performs well in applications to both structural analysis and out-of-sample forecasting.

Bayesian Estimation of Large Dimensional Time Varying VARs Using Copulas

“…This paper proposes a copula-based Bayesian estimation methodology for large TVP-VARs with heteroskedasticity. Similarly to Carriero et al (2019), our estimators are fully Bayesian, thus allowing for the computation of the uncertainty around all estimators.…”

“…Following seminal papers by Uhlig (1997), Cogley and Sargent (2001) and Primiceri (2005), recent examples where the assumption of homoskedasticity has been relaxed include Koop and Korobilis (2013) and Koop et al (2019), who attempt to reduce the dimensionality issue essentially by imposing a factor structure onto the volatilities -see also Clark (2011), Carriero et al (2015), Clark and Ravazzolo (2015) and Carriero et al (2016). In a recent landmark paper, Carriero et al (2019) propose a far less restrictive set-up, which allows for fully Bayesian inference without imposing restrictions on the form of the heteroskedasticity.…”

“…Conversely, in the case of a heteroskedastic VAR, this is no longer possible, and the computational burden cannot be resolved through the choice of an appropriate prior. As explained in Carriero et al (2019), heteroskedasticity invalidates the so-called "symmetry"across equations that characterises homoskedastic VARs. Such property entails that a homoskedastic VAR is a SUR model where the regressors are the same across all equations; in turn, this entails that the covariance matrix of the VAR coefficients has a Kronecker structure, which makes estimation much simpler than if one had to deal with large matrices that do not have a simplifying structure.…”

Forecasting with High‐Dimensional Panel VARs

“…In this case, the state-space system is nonlinear and multivariate estimation would need to rely on computationally intensive simulation methods. A potential solution to this problem would be to follow Carriero, Clark and Marcellino (2019) and estimate the model equation-byequation: the first equation does not contain any contemporaneous information on the right-hand side so can be estimated independently of other equations using a linear filter; the second equation is dependent on " 1t which can be replaced by residuals from the first equation; the third equation is dependent on " 1t , " 2t which can also be replaced by residuals, and so on until equation NG which also depends on residuals available from the previous NG − 1 equations. 6 However, such an option is not available to us, since the pooling prior we adopt clusters coefficients among different equations.…”

Large Hybrid Time-Varying Parameter VARs

“…2 Since there is a large body of empirical evidence that demonstrates the importance of accommodating time-varying structures in small systems, there has been much interest in recent years to build TVP-VARs for large datasets. While there are a few proposals to build large constant-coefficient VARs with stochastic volatility (see, e.g., Carriero, Clark, andMarcellino, 2016, 2019;Chan, 2018Chan, , 2019Kastner and Huber, 2018), the literature on large VARs with time-varying coefficients remains relatively scarce. 3 We propose a class of models we call hybrid TVP-VARs-VARs in which some equations have time-varying coefficients and/or stochastic volatility, whereas others have constant coefficients and/or homoscedastic errors.…”