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

An Explicit Formula for Likelihood Function for Gaussian Vector  Autoregressive Moving-Average Model Conditioned on Initial Observables with  Application to Model Calibration