Is Algorithmic Stability Necessary for Finite Sample Prediction from a Single Dependent Time Series?

Forecasting from a single dependent time series requires balancing finite-sample predictive validity with the amount of data available for model fitting. Split conformal prediction reserves observations for calibration, whereas Leave a Window Out (LWO) reuses nearly all historical observations but its existing finite-sample guarantee requires algorithmic stability to control changes in fitted predictions after window removal. We introduce Windowed Jackknife+ (WJ+), which removes a forward window beginning at each historical observation and uses the same window-deleted predictor to compute both the historical residual and the forecast prediction. This construction yields a finite-sample Jackknife+-type marginal miscoverage bound without a separate calibration split or an algorithmic stability assumption. The bound separates the contribution from unavailable score comparisons created by window removal from the contribution due to temporal dependence. We show that the comparison contribution cannot generally be removed, even when the predictor is independent of the training sample, and give constructions attaining the universal comparison bound. When the relative window length and dependence contributions vanish, the bound approaches the usual Jackknife+ $2\alpha$ miscoverage scale. Experiments on synthetic processes and pretrained time-series foundation models illustrate the effects of window removal, temporal dependence, and model adaptation.