Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 218 / 6

A weakly stationary process has a constant finite mean and covariance $\mathrm{Cov}(X_t,X_{t+h})$ depending only on lag $h$. The plotted process is not stationary: it has a declining trend and a pronounced oscillation of period about $25$, so its mean depends on time.

Fit a trend ${\hat{T}}_t$, form $D_t=X_t-{\hat{T}}_t$, and estimate the period-25 seasonal effect byThe residual is $R_t=X_t-{\hat{T}}_t-{\hat{S}}_{t\mathrm{mod}25}$. This additive decomposition is sensible when seasonal amplitude does not systematically change with the level or trend; multiplicative seasonality would require a logarithmic transform or ratio decomposition.

A period-50 business cycle can be a stochastic cycle in the residual process and need not violate additive trend-plus-period-25 seasonality. Applying $1-B^{50}$ discards the first 50 of only 100 observations and introduces a noninvertible seasonal moving-average factor, creating strong artificial dependence and risking overdifferencing rather than modeling the cycle.

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.

The first line searches candidate ARIMA models and selects the one with smallest Akaike information criterion. The second constructs a one-step-ahead point forecast and nominal 95-percent prediction interval from the fitted values of the selected model.

The interval treats the selected model and estimated detrending and seasonal components as fixed, often assumes approximately Gaussian homoscedastic innovations, and ignores model-selection and parameter uncertainty. With only 100 observations these omissions can materially reduce coverage. A residual or parametric bootstrap that repeats decomposition, model selection, fitting, and forecasting can propagate those sources of uncertainty; time-series cross-validation can additionally assess empirical one-step coverage.