Residual Seasonality and a Pandemic Shock in Monthly Claim Counts

The Ljung–Box p-value of 0.02 rejects the white-noise null at the 5% level, and the lag-12 residual autocorrelation of 0.24 is large enough to indicate remaining seasonal structure. The model is therefore not adequate for inference or forecasting as it stands, although it may still be a useful baseline. The remaining seasonality could mean that the seasonal MA(1) term is insufficient, that seasonal differencing has induced overdifferencing, or that the 2020–2021 shock is contaminating the residual diagnostics. I would examine residual ACF and PACF plots, test for seasonal unit roots, and compare a SARIMA with an additional seasonal AR term, a model with Fourier seasonal terms, and an ETS(A,A,A) alternative. The choice should not be based on AIC alone; rolling-origin forecast accuracy and interval coverage are more relevant. If the residual seasonality persists after adding a seasonal AR term, the model should be rejected rather than used for 12-step forecasting.

Discarding 2020–2021 loses information, creates an artificial gap in the series, and can distort the estimated seasonal and trend components. It also makes the model unable to represent a future similar shock. A better approach is to include deterministic intervention terms: a pulse for the main lockdown months, and a ramp or level shift for the rebound, based on external event dates rather than data mining. The intervention coefficients should be tested for significance, and the model should be checked for remaining autocorrelation. A pulse/ramp intervention preserves the full time series and separates the shock from the regular seasonal dynamics. If the shock is expected to be non-repeating, the intervention is preferable for forecasting; if future shocks are plausible, scenario analysis should be added. The final model should be compared with the no-intervention ARIMA and ETS alternatives using out-of-sample rolling forecasts, not only in-sample fit.

Use rolling-origin cross-validation over the last 24–36 months, refitting each model at each origin and evaluating point forecast error and interval coverage. Compare the ARIMA baseline, a SARIMA with the additional seasonal term, the intervention model, and ETS(A,A,A). Choose the model on out-of-sample performance and on whether residuals pass white-noise checks. For the prediction interval, do not rely only on the innovation variance: simulate from the fitted model, including parameter uncertainty via bootstrap or Bayesian methods, and incorporate uncertainty about the intervention effects. Report the interval as a scenario-aware interval, and check its empirical coverage in the rolling validation. If the residuals are non-normal or the shock is not fully captured, use empirical quantiles from the rolling errors. The final recommendation should state the model, the intervention assumptions, the validation results, and the interval coverage, rather than presenting a single interval as if it were model-free.