Part iii and the stated characterization imply that $U$ has a <moving-average process of order one> representation $U_t=\eta_t+\theta\eta_{t-1}$. Since
$$
(1-\phi B)Y_t=U_t,
$$
where $B$ is the <backshift operator>,
$$
(1-\phi B)Y_t=(1+\theta B)\eta_t.
$$
Thus $Y$ is a causal <autoregressive moving-average process> of order $(1,1)$.

Solution

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

{title2=$B$}

The backshift operator acts on a time series by $BX_t=X_{t-1}$ and hence $B^jX_t=X_{t-j}$.


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