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

The selected zero-mean autoregressive moving-average process isThe reported maximum-likelihood estimates areUsing the displayed asymptotic standard error gives the Wald confidence intervalThis normal interval is unreliable and likely too narrow because the series has only about twenty observations, the moving-average estimate is near the noninvertibility boundary $\theta =1$, and the same data were used to select the model. The finite-sample likelihood is consequently skewed and model-selection uncertainty is omitted.