  We derive a closed-form expression for the finite predictor coefficients of
multivariate ARMA (autoregressive moving-average) processes. The expression is
given in terms of several explicit matrices that are of fixed sizes independent
of the number of observations. The significance of the expression is that it
provides us with a linear-time algorithm to compute the finite predictor
coefficients. In the proof of the expression, a correspondence result between
two relevant matrix-valued outer functions plays a key role. We apply the
expression to determine the asymptotic behavior of a sum that appears in the
autoregressive model fitting and the autoregressive sieve bootstrap. The
results are new even for univariate ARMA processes.
