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

TL;DR

This paper presents an explicit likelihood formula for Gaussian VARMA models with scalar MA coefficients, enabling efficient calibration and analysis with analytical gradients and Hessians, and discusses extensions and theoretical connections.

Contribution

It derives a closed-form likelihood for Gaussian VARMA models with scalar MA coefficients, facilitating efficient calibration and theoretical analysis.

Findings

Likelihood function can be computed efficiently using FFT.

Analytical gradient and Hessian are obtainable for optimization.

Likelihood remains invariant under root inversion maps of MA coefficients.

Abstract

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.