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The sample autocorrelation has one substantial positive spike at lag one and then cuts off, while the partial autocorrelation tails off with alternating signs. This is the characteristic pattern of a moving-average process of order one.

At lag $h$, the plot shows the sample autocorrelation functionUnder a white noise process, each fixed nonzero-lag sample autocorrelation is approximately $N(0,1/n)$, so the dashed pointwise $95\%$ reference lines are approximately $\pm 1.96/\sqrt{n}$.

The first nonzero-lag bar is well above the upper line, which contradicts the zero autocorrelation expected from white noise. Since the plot then largely cuts off, an moving-average process of order one is a plausible model; with the sampling interval as the time unit this is an $\mathrm{ARMA}(0,1)$ model.

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 $(1,1)$.