Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 36 / 2 / c / Solution

Apply $1-\varphi B$ to the observed process and call the result $V_t$. ThenIts only nonzero covariance lags areSeek an invertible moving-average factor $V_t={\epsilon }_t+\vartheta {\epsilon }_{t-1}$ with $\mathrm{Var}({\epsilon }_t)=\nu$. Matching these covariances requires $\nu (1+{\vartheta }^2)=A$ and $\nu \vartheta =C$. Solving givesThese are the three requested parameters in the positive-sign moving-average convention. For nonzero total noise variance,so $|\vartheta |<1$ and the larger quadratic root gives the invertible factor.

Covariance matching alone would not identify arbitrary processes in distribution. To obtain an actual representation on the given space, defineThe series converges in L2. The spectrum of $V$ is $(A+2C\cos \omega )/(2\pi )=\nu |1+\vartheta e^{-i\omega }{|}^2/(2\pi )$, so this filtered process has constant spectrum $\nu /(2\pi )$ and is white noise. ThusThis is the invertible ARMA factorization of an AR(1)-plus-noise process. If ${\sigma }_w^2=0$, it reduces to AR(1); if $\varphi =0$, it reduces to white noise. If ${\sigma }_z^2=0$ and ${\sigma }_w^2>0$, then $\vartheta =-\varphi$, the common factor cancels and $Y=W$. Therefore the orders are at most (1,1); no unnecessarily minimal-order claim is made in those degenerate cases.