# Using Bootstrapped Residuals to Estimate Time Series Prediction Intervals

I am working with a very simple forecasting "model" which is not a standard statistical model. I am trying to use the methodology described in Hyndman's textbook under the section "Prediction intervals from bootstrapped residuals" to estimate some prediction intervals.

However, I noticed that if the model is biased over the periods where the errors are being estimated, then the forecast itself is not centered in the prediction intervals. This is because, on average, we are adding positive (or negative) values $$e_t$$ to the forecast to get our "simulated data",

$$y_t = \hat{y}_{t|t-1} + e_t.$$

This bias is not uncommon in my use case because this model is being applied to ~1000 time series, so some of them will behave in this way (e.g. if they have an unexpected jump in level during the error estimation period, or if the time series is trended but the model does not account for trend, etc).

The black line is the forecast, and the red intervals are the prediction intervals at every 5% (i.e. 5%, 10%, ..., 95%) as estimated by the method in the link above. (Training data not shown and scale has been altered for privacy reasons). This seems unacceptable to me - the point forecast is supposed to be the "best" estimate of future values, while the prediction intervals are saying that the most likely future values lie somewhere else entirely. Am I doing something wrong, or is this just an unavoidable quirk of this methodology?

## 2 Answers

The question is whether the bias seen in the historical forecasts will continue into the future. If you think it will, then you should adjust your point forecasts accordingly. If you think it won't, then you should remove the bias from your residuals.

This can happen even without bias, depending very much on the distribution of the noise.

If your data has a strongly asymmetric noise on top of the (probably time-varying) mean, then your expectation forecast may well be equal to the 10% quantile forecast, which will look exactly like the example you give. An example would be IID Beta(40,0.04) data: the expectation is 0.999, which is roughly the 0.1 quantile of the distribution. Note that there is zero bias in this situation.

However, this presupposes a strong left skew, which is quite uncommon. For instance, my answer here uses a right skewed lognormal noise distribution, where the expectation forecast is much higher than the median, so in an illustration like yours, the bold expectation forecast would be at the upper boundary of your prediction intervals for some given coverage probability.

Of course, a median forecast (which you would elicit using the MAE) will always be in the center of your fan plot.