The auto-regression and the moving-average

We explore some relationships in the second-order properties of a causal auto-regression and an invertible moving-average process with the same polynomial. We reveal that the inverse variance matrix for random variables from the auto-regression is equal to a conditional variance matrix of Gaussian random variables from the moving-average and vice versa. While the inverse variance matrix for the auto-regression can be written explicitly, we manage to write down the exact Gaussian likelihood of consecutive observations from the moving-average process, by using the properties of the auto-regression.