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I understand the way to compute the prediction interval at 5% and 95% for one step forward forecast based on Bill's answer to the question at Bootstrap prediction interval. The idea being that based on the residuals in the time series using bootstrapping(replacement with sampling multiple times) one can construct the distribution at 5% and 95% percentiles for the first forecast point. What I am struggling to understand is how to calculate the prediction interval for the second point (and other points in the future thereafter). Based on the bootstrap distribution we get one value each of the 5% and 95% percentiles, and the residuals should be added over time

So,

e{5, t+1} = obtained from bootstrap distribution
e{5, t+2} = e{5, t+1} + e{5, t+1}
e{5, t+3} = e{5, t+1} + e{5, t+1} + e{5, t+1}

It will just end up giving a diverging cone with straight lines, and I feel I am missing something here. I am trying to find the prediction intervals solely based on the nature of residuals in historical data and the output that I have from a black-box forecast method. Would love some insights into this

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You can refer to Forecasting: Principles and Practice - 3.5 Prediction Intervals (Hyndman and Athanasopoulos, 2018). In short, for each bootstrap sequence, you can sample from residuals to generate forecast for t+1; then you treat the forecast at t+1 as groundtruth and generate forecast for t+2 in the same way; repeat until you generate forecast for t+k (essentially generating forecast in an autoregressive fashion while bootstrapping residuals). After you obtained multiple bootstrap sequences until t+k, you can compute the empirical quantiles of the bootstrap forecasts at t+k to obtain desired prediction interval (e.g., 0.025 and 0.975 quantiles for 95% prediction interval).

The described bootstrap method doesn't consider uncertainty associated with parameter estimate or model misspecification. So the prediction interval is very likely to be too narrow and fail to cover 95% predictions in practice. For a better (and more computationally expensive) way to construct bootstrap prediction interval see Forecasting: Principles and Practice - 11.4 Bootstrapping and bagging

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