  We derive an explicit formula for likelihood function for Gaussian VARMA
model conditioned on initial observables where the moving-average (MA)
coefficients are scalar. For fixed MA coefficients the likelihood function is
optimized in the autoregressive variables $\Phi$'s by a closed form formula
generalizing regression calculation of the VAR model with the introduction of
an inner product defined by MA coefficients. We show the assumption of scalar
MA coefficients is not restrictive and this formulation of the VARMA model
shares many nice features of VAR and MA model. The gradient and Hessian could
be computed analytically. The likelihood function is preserved under the root
invertion maps of the MA coefficients. We discuss constraints on the gradient
of the likelihood function with moving average unit roots. With the help of FFT
the likelihood function could be computed in $O((kp+1)^2T +ckT\log(T))$ time.
Numerical calibration is required for the scalar MA variables only. The
approach can be generalized to include additional drifts as well as integrated
components. We discuss a relationship with the Borodin-Okounkov formula and the
case of infinite MA components.
