Carriero-Clark-Marcellino (2016) Common Drifting Volatility in Large Bayesian VARs

Summary

Estimating large VARs with stochastic volatility by standard methods (an independent volatility process for every variable) is computationally very demanding. Carriero, Clark, and Marcellino instead model the conditional volatilities as driven by a single common unobserved factor — motivated by the empirical observation that estimated volatility patterns are very similar across variables. Combining a standard natural-conjugate prior for the VAR coefficients with an independent prior on a common stochastic-volatility factor, they derive the posterior for a BVAR with common stochastic volatility (BVAR-CSV). Crucially, under this prior the conditional posterior of the VAR coefficients retains a Kronecker structure, allowing fast estimation even in large systems. On US and UK data, the common-volatility model significantly improves fit and forecast accuracy over a constant-volatility BVAR — with gains comparable to a conventional independent-volatility specification, but at a fraction of the computational cost.

Key Claims

• Volatility co-moves across variables. Empirically, the estimated stochastic volatilities of many macro series rise and fall together (e.g., the Great Moderation and its reversals), which justifies a single common volatility factor rather than $n$ separate ones.
• Common stochastic volatility (CSV) model. The error covariance is scaled by one time-varying factor: $\operatorname{Var}(\varepsilon_t) = h_t\,\Sigma$, where $h_t$ follows a (log) stochastic-volatility process shared by all equations, and $\Sigma$ is a constant covariance matrix.
• Conjugacy is preserved. With a natural-conjugate prior on the VAR coefficients and an independent prior on $\{h_t\}$, the coefficient posterior conditional on the volatility path keeps the Kronecker (variance ⊗ prior) structure — so the coefficients can be drawn quickly even in a large VAR, and the volatility path is sampled with standard SV methods.
• Big computational saving. Because only one volatility process must be sampled and the Kronecker structure is retained, BVAR-CSV scales to large systems where a full independent-SV BVAR would be infeasible.
• Forecasting and fit gains. On US and UK macro data, allowing common drifting volatility markedly improves in-sample fit and out-of-sample forecast accuracy relative to constant volatility, matching the gains of the far more expensive independent-SV model.

Concepts Introduced or Extended

• Bayesian VAR
• Stochastic Volatility
• Forecasting

Entities Mentioned

• Andrea Carriero
• Todd E. Clark
• Massimiliano Marcellino

Quotes

"We propose to model conditional volatilities as driven by a single common unobserved factor... Under the chosen prior the conditional posterior of the VAR coefficients features a Kronecker structure that allows for fast estimation, even in a large system."

My Take

This is the pragmatic answer to "how do you put stochastic volatility in a 20- or 100-variable BVAR without waiting all week." The independent-SV route destroys the Kronecker structure that makes the natural conjugate BVAR fast, so CCM buy back the speed by assuming a single common volatility factor — a strong restriction, but one the data largely support because macro volatilities move together. The trade is explicit and well-measured: you give up variable-specific volatility dynamics but keep almost all of the forecast gain of full stochastic volatility at a tiny fraction of the cost. It complements the same authors' specification-choices study and Chan's asymmetric-conjugate work — three angles (default choices, cross-variable shrinkage, common volatility) on making large BVARs both flexible and fast.