Explicit Form of Coefficients in any MA(2) Process

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.