# What do error bounds in forecasting represent?

What do error bounds actually mean in forecasting timeseries? For example, when I get a forecast I get the 85% and 95% high and low error bounds. I can also set my own error bounds to be calculated but am unsure of what an error bound in terms of timeseries forecasting means.

In terms of forecasting I think you mean a prediction interval for the forecast. It mainly depends on the mean squared error of the forecast. Of course, you can give different probability levels, mostly the PI are given for 95%. So the PI mainly tells you, that the forecast will lie with a 95% probability in this region. With a probability of 5% it will lie outside. The MSE is always the same, only the quantile changes, if another probability is choosen (since the disturbances are normal, you use the normal dsitribution). Also the variance of the error is needed for calcualtion.

So

\begin{align}y_{T+I}=\hat{y}_{T+I|T}+1.96*MSE^{1/2} * \sigma_\epsilon\end{align} gives the upper bound and

\begin{align}y_{T+I}=\hat{y}_{T+I|T}-1.96*MSE^{1/2} * \sigma_\epsilon\end{align} gives the lower bound. MSE is the mean squared error of the I-th forecast.

E.g. for an AR(1) this looks like:

• +1 - It might be good to distinguish between prediction intervals and confidence intervals, as it wasn't specified by the OP (although I agree most forecasting software will likely give prediction intervals by default). Commented May 13, 2013 at 11:54
• Here's an interesting article by Prof Hyndman explaining the difference between prediction intervals and confidence intervals, if the OP is interested.
– Vara
Commented May 13, 2013 at 12:53
• Thanks for the explanation Stat Tistician and thanks for the link from Prof Hyndman, Vara. Commented May 14, 2013 at 8:25