Explicit Form of Coefficients in any MA(2) Process

We shall show that for {\it any} $MA(2)$ process (apart from those with coefficients $θ_1,θ_2$ lying on certain line-segments) there is {\it one and only one invertible} $MA(2)$ process with the {\it same} autocovariances $γ_0,γ_1,γ_2$. 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 $θ_1,θ_2$ of the invertible version in terms of $γ_0,γ_1,γ_2$ explicitly using analytical reasoning, following a graphical approach of Sbrana (2012) which indicates this result within the invertibility region. We also express $(θ_1,θ_2)$ in the non-invertibility region.

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