The original algorithm contained a mistake that meant the conditional distributions used for the VAR's coefficients were missing a piece of information. We propose a new algorithm that uses the same factorization but includes the missing term. The new, correct algorithm has the same computational complexity as the old, incorrect one (i.e., O(N-4)), and therefore it still allows the estimation of large VARs. (C) 2021 Published by Elsevier B.V.