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I have recently read the article by Gardner (1988) who proposes Chebyshev inequality-based prediction intervals for forecast:

suppose we have a model selected on the usual basis of one-step-ahead fit. One pass through the data is made to compute the variance of the fitted errors at one-step-ahead. A second pass is made to compute the variance at two-steps-ahead. It is important to understand that the forecasting model is not re-estimated. We simply make two-step-ahead forecasts with the same model and compute the variance of the fitted errors. This process is continued until an individual variance is computed for each desired leadtime. (...) The second step in the procedure is to compute standard errors at each leadtime. The final step is to apply a multiplier to each standard error that yields the desired prediction intervals. The multiplier is based on the Chebyshev inequality (...)

The description is pretty vague, so I would like to ask how exactly such intervals are calculated? Say I have black-box forecasting function $b_h(x_1,...,x_n)$ that uses $x_1,...,x_n$ datapoints to produce forecast $\hat x_{n+1},...,\hat x_{n+h}$, how would I apply this method to assess PI's for the forecast?


Gardner, E.S. (1988). A simple method of computing prediction intervals for time series forecasts. Management Science, 34(4), 541-546.

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