We consider $K$-dimensional Bayesian vector autoregressions (BVARs) with Cholesky stochastic volatility (SV), in which the innovation covariance matrix is a lower-triangular linear transform of $K$ independent univariate SV processes. Such models are widely used in empirical macroeconomics to capture time-varying uncertainty and improve forecast accuracy, but the cost of posterior simulation is the binding constraint on the size of the system, and a major bottleneck for empirical work. We introduce a Markov chain Monte Carlo (MCMC) kernel that mixes better than existing samplers at the same computational complexity. It rests on a reparametrisation that makes the $K$ volatility trajectories conditionally independent, and on Particle Gibbs to update each trajectory. The kernel targets the exact posterior, rather than an approximation of it, and in an application with $K = 15$ it raises the mean effective sample size per second, relative to the benchmark corrected triangular algorithm, by a factor of approximately $14$ for the VAR coefficients and $3.4$ for the volatilities. We also introduce a a Sequential Monte Carlo squared (\smcsq{}) sampler, which used our MCMC kernel as a building block, and which delivers at every $t$ the one-step-ahead predictive density and the marginal likelihood of the data up to $t$, and hence on-line forecasting and model choice. To our knowledge, this is the first algorithm that delivers sequential marginal likelihoods for Cholesky-SV BVARs with static contemporaneous coefficients and a non-conjugate prior. We illustrate both on US monthly macroeconomic data, with $K=15$ for posterior inference and $K=6$ for the sequential sampler and model choice.

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