Explicit Form of Coefficients in any MA(2) Process

TL;DR

The paper derives explicit formulas for the coefficients of the invertible version of any MA(2) process based on its autocovariances, highlighting implications for model fitting and prediction.

Contribution

It provides the first explicit analytical expressions for MA(2) coefficients in both invertible and non-invertible regions based on autocovariances.

Findings

Unique invertible MA(2) process exists for almost all autocovariance sets.

Explicit formulas for coefficients in invertible case derived.

Implications for model fitting and prediction accuracy.

Abstract

We shall show that for {\it any} $MA(2)$ process (apart from those with coefficients $\theta_1,\theta_2 $ lying on certain line-segments) there is {\it one and only one invertible} $MA(2)$ process with the {\it same} autocovariances $\gamma_0,\gamma_1,\gamma_2$. It is this invertible version which computer-packages fit, regardless, even if data came from a non-invertible $MA(2)$ process. This has consequences for prediction from a fitted process, inasmuch as such prediction would seem to be inappropriate. We express the coefficients $\theta_1,\theta_2 $ of the invertible version in terms of $\gamma_0,\gamma_1,\gamma_2$ explicitly using analytical reasoning, following a graphical approach of Sbrana (2012) which indicates this result within the invertibility region. We also express $(\theta_1,\theta_2)$ in the non-invertibility region.