Chan (2022) Asymmetric Conjugate Priors for Large Bayesian VARs

Summary

Chan develops a shrinkage prior for large Bayesian VARs that combines the analytical convenience of the natural conjugate prior with the modeling flexibility of cross-variable (asymmetric) shrinkage. The natural conjugate (Normal–Inverted-Wishart) prior is popular for large BVARs because it yields fast posterior simulation and closed-form results — but its Kronecker structure forces the same shrinkage on every equation, ruling out the Minnesota idea of shrinking coefficients on other variables' lags more aggressively than own lags. The proposed asymmetric conjugate prior accommodates cross-variable shrinkage while retaining a closed-form marginal likelihood and enabling fast equation-by-equation posterior simulation (for a 100-variable, 4-lag BVAR, 10,000 draws take under half a minute). In a forecasting exercise, a data-driven asymmetric prior outperforms both a data-driven symmetric prior and a subjective asymmetric prior. (CAMA Working Paper 51/2019; published in Quantitative Economics 13(3): 1145–1169.)

Key Claims

• The conjugacy–flexibility tension. The natural conjugate prior gives large BVARs their tractability (closed-form posterior and marginal likelihood, fast simulation), but the Kronecker (variance ⊗ prior) structure it requires ties the prior covariance across equations together, so it cannot express cross-variable shrinkage.
• Asymmetric conjugate prior. Chan reparameterizes the VAR in a triangular/structural (recursive) form in which each equation has its own error variance and its own prior. This breaks the Kronecker restriction — allowing equation-specific, asymmetric (own-lag vs. other-lag) shrinkage — while the prior remains conjugate equation-by-equation.
• Analytical results preserved. The marginal likelihood is available in closed form, so hyperparameters (including the degree of cross-variable shrinkage) can be chosen by maximizing the marginal likelihood — a fully data-driven, "empirical Bayes" selection.
• Fast and scalable. Because the equations are conditionally independent under the triangular parameterization, posterior draws are obtained equation-by-equation, making even 100-variable BVARs cheap to estimate.
• Forecasting gains. The data-driven asymmetric prior forecasts better than a data-driven symmetric prior and than a subjective (Minnesota-style fixed) asymmetric prior, showing that both asymmetry and data-based hyperparameter selection matter.

Concepts Introduced or Extended

• Bayesian VAR
• Minnesota Prior
• Marginal Data Density

Entities Mentioned

• Joshua C. C. Chan

Quotes

"We develop a prior that has the best of both worlds: it can accommodate cross-variable shrinkage, while maintaining many useful analytical results, such as a closed-form expression of the marginal likelihood."

My Take

This paper resolves the specific limitation that Carriero–Clark–Marcellino flagged as the price of the convenient conjugate setup: the Kronecker structure that makes the natural conjugate prior fast is exactly what forbids cross-variable shrinkage. Chan's trick — move to a triangular structural form so each equation carries its own variance and prior — is elegant because it keeps everything the conjugate prior gave you (closed-form marginal likelihood, equation-by-equation speed) while letting the Minnesota asymmetry back in and letting the data pick the shrinkage. That the marginal likelihood stays analytical is the crucial part: it turns hyperparameter choice into an optimization rather than a guess, which is why the data-driven asymmetric prior wins. It has become one of the standard large-BVAR priors, alongside the stochastic-volatility extensions.