# How to calculate the prediction interval from two independent forecasts of time series?

Lets say I have two time series a and b and I predict the next value and a prediction interval (say lower and upper 80%) of each of these two. I end up with something like that: a: [10, 20, 30] and b: [2, 7, 12] (as [lo80%, prediction, high80%]). As these time series are independent, I see I could just add 20 + 7 to get a prediction of a + b, but what about the prediction interval? I suppose I cant just add them to [12, 27, 42], right?

(Sorry if it is too simple a question)

You need to make some assumptions about the distribution of each prediction, but since your confidence intervals are symmetric I am writing the Gaussian distribution case.

For a Gaussian RV the 80 percentile corresponds to being $\approx 0.8416 \sigma$ away from the mean, so the standard deviation for each prediction can be calculated as $\sigma = (q_{(0.80)} - \mu) / 0.8416$.

For independent RVs, $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$, therefore

\begin{align} q_{(0.80)} &= 0.8416\cdot\sigma_{X+Y} + \mu_{X+Y} \\ &= 0.8416 \cdot \sqrt{(10 / 0.8416)^2 + (5/.8416)^2} + \mu_{X+Y}\approx 11.18 + \mu_{X+Y} \end{align}

So the 80% CI and prediction for the sum is [15.82, 27, 38.18].

• +1 Did you notice that the value of $0.8416$ is irrelevant? You can obtain $11.18 = \sqrt{(10^2+5^2)}$.
– whuber
Sep 4, 2012 at 12:31
• @whuber I wish I could say the inclusion was solely pedagogical but alas I missed that fact. Of course if one wants something other than the 80th quantile of the sum, this shows how to calculate it from the 80th quantile of the parts. Sep 4, 2012 at 22:17