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

This 2019 Journal of Econometrics paper by Andrea Carriero, Todd Clark, and Massimiliano Marcellino develops a computationally efficient Markov Chain Monte Carlo (MCMC) algorithm for estimating large Bayesian vector autoregressions (VARs) – models with many included variables – that feature stochastic volatility (time-varying error variances) together with non-conjugate priors, meaning economically motivated shrinkage priors (Minnesota, Sims-Zha, cross-variable shrinkage) whose independent Normal-Wishart form breaks the Kronecker structure that the standard closed-form conjugate posterior requires. The key device is a triangular (Cholesky-type) factorization of the time-varying error covariance matrix, used purely as a computational tool rather than as a structural identification assumption – the authors argue (Section 3.1, “The role of variable ordering”) that the draw of the VAR coefficients from their conditional posterior via this triangular recursion is invariant to the ordering of variables used to build A – which lets the VAR coefficients be drawn equation-by-equation with standard GLS formulas instead of requiring the full system-wide posterior precision matrix to be formed and inverted. This lowers the computational complexity of the coefficient draw from O(N^6) to O(N^4): for a 20-variable VAR the paper reports that estimation using the traditional system-wide algorithm was about 356 times slower than the triangular algorithm (Section 4.1), with the advantage growing further as the number of variables rises. The algorithm makes estimation of very large systems tractable: the paper estimates a 125-variable monthly VAR with 13 lags (roughly 203,250 mean coefficients) on the FRED-MD dataset, drawing 5,000 posterior draws in about seven hours, and documents substantial heterogeneity in stochastic volatility across financial, real, and price variable groups; evidence consistent with the Great Moderation, with posterior median volatilities declining from the 1970s/early 1980s into the mid-1980s; a first principal component of log-volatilities that accounts for roughly 45% of total volatility variance; and recursively identified monetary-policy-shock responses (with the federal funds rate ordered last) consistent with Banbura, Giannone, and Reichlin (2010): a contractionary shock lowers output and inflation with no price puzzle. In a second, forecasting application using four macroeconomic variables over 1960:3-2014:5 with 531 recursive out-of-sample forecasting exercises, the paper finds that a large cross-section improves point forecasts (RMSE) while stochastic volatility adds little further gain there, but stochastic volatility matters for density-forecast accuracy (log scores), and that combining a large cross-section with stochastic volatility produces gains that are super-additive – larger than the sum of the gains from each ingredient taken separately – holding across variables and forecast horizons of 1, 3, 6, and 12 months.

Massimiliano Marcellino | Macro Paper Warehouse

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