On the inverse of the covariance matrix of a first order moving average

Three distinct expressions are noted for the elements of the inverse of the covariance matrix ΣT of T successive observations from a stationary first order moving average process. An observation of Arato (1961) shows that the problem of finding the exact inverse can be reduced to the problem of finding the covariance determinant |∑T|⁠. Approximate expressions for ∑T-1 are then obtained by approximating various forms of the determinant. The resulting approximate inverses are all the same and coincide with the expression usually employed.