Finite-sample innovations of an MA(1) process (${\lambda }_t={\gamma }_1/{\phi }_{t-1},{\phi }_t={\gamma }_0-{\gamma }_1{\lambda }_t$)

Starting at time one, the orthogonal residuals satisfy $U_t=X_t-{\lambda }_t{U}_{t-1}$, with ${\lambda }_t={\gamma }_1/{\phi }_{t-1}$ and ${\phi }_t={\gamma }_0-{\gamma }_1{\lambda }_t$. Earlier innovation directions have zero covariance with the new observation. Explicitly, with $S_m={\sum }_{j=0}^m{\theta }^{2j}$, ${\phi }_t={\sigma }^2{S}_t/S_{t-1}$ and ${\lambda }_t=\theta S_{t-2}/S_{t-1}$ for $t\ge 2$. These formulas follow by induction from $S_t=(1+{\theta }^2)S_{t-1}-{\theta }^2{S}_{t-2}$, with $S_0=1$ and $S_1=1+{\theta }^2$. This includes $\theta =\pm 1$, where ${\phi }_t={\sigma }^2(t+1)/t$.

For an MA(1) process, the recursion ${\lambda }_t=\theta /(1+{\theta }^2-\theta {\lambda }_{t-1})$ has candidate limits $\theta$ and $1/\theta$. The limit in $[-1,1]$ is $\theta$ when $|\theta |\le 1$ and $1/\theta$ otherwise. The limiting innovation variance is ${\sigma }^2\max (1,{\theta }^2)$, expressing the same covariance law through an invertible reciprocal representation.