Invertibility of a moving-average model

Invertibility permits reconstruction of innovations from present and past observations by a stable filter. For $Y_t-\mu ={\epsilon }_t+\theta {\epsilon }_{t-1}$, $|\theta |<1$ gives ${\epsilon }_t={\sum }_{j\ge 0}(-\theta {)}^j(Y_{t-j}-\mu )$. This is a different property from stationarity: a finite moving-average model is stationary for every finite coefficient.

Reflecting an inside-unit-circle zero across the unit circle produces an invertible moving-average model with the same spectral density of a stationary process, after rescaling the driving white noise variance. For real $|\theta |>1$, $|1-\theta e^{-i\lambda }{|}^2={\theta }^2|1-{\theta }^{-1}{e}^{-i\lambda }{|}^2$. The transformed driving sequence is a linear innovation process; without Gaussianity it need not be strong white noise.