Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 2 / d / iv / Solution

Part iii and the stated characterization imply that $U$ has a moving-average process of order one representation $U_t={\eta }_t+\theta {\eta }_{t-1}$. Sincewhere $B$ is the backshift operator,Thus $Y$ is a causal autoregressive moving-average process of order $(1,1)$.