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

The autoregressive process of order one is causal exactly whenbecause then $X_t={\sum }_{j\ge 0}{\varphi }^j{\epsilon }_{t-j}$ converges in mean square. Its autocovariance is

Under the intended assumption that the two white-noise sequences are mutually uncorrelated at every pair of times, $X$ and $W$ are uncorrelated. Their sum is therefore weakly stationary with

Strictly, the printed condition $\mathbb{E}[{\epsilon }_t{W}_t]=0$ only at equal times is insufficient. For example, $W_t=(-1{)}^t{\epsilon }_{t-1}$ is itself white noise and is contemporaneously uncorrelated with ${\epsilon }_t$, but the cross-covariance contribution can depend on $t$. The displayed answer therefore uses the standard intended cross-series white-noise assumption $\mathbb{E}[{\epsilon }_t{W}_s]=0$ for all $s,t$.

Using $X_t-\varphi X_{t-1}={\epsilon }_t$ givesUnder the cross-series uncorrelatedness used in part ii, terms in $U_t$ and $U_{t-h}$ involve disjoint white-noise times whenever $|h|>1$. HenceFor completeness,

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 $(1,1)$.