The "sum" of prediction intervals

Question

Suppose I have a regression model which yields a couple of predicted values with their respective prediction intervals and the random quantity that I am interested in is the sum of (some subset) of those predicted values. How do I go about building a new prediction interval for this sum? Is it as easy as summing up the limits of the respective prediction intervals?

Answer 1

In short, no, you don't just add the limits. Maybe if the predictions were perfectly correlated, but that's not usually the case at all.

Typically (if the model assumes independence) and you want an interval for a sum of predicted values, you might then think that you can treat the predictions as independent, but they generally aren't independent even when the observations are, because the predictions generally share parameter estimates.

In ordinary regression it's fairly straightforward; you can work out the mean and standard deviation of the sum and construct a t interval similar to the way you would for a single prediction.

If the model is the multiple regression model $y = X \beta + e$ with $e$ ~ $ N(0, \sigma^2 I)$, and you're predicting a vector of future values, $y_f$, where you have a set of predictors for those predictions, $X_f$.

then you want an interval for $a' y_f$ (where in your case, $a$ is a vector of $1$s)

Then $R = a' (y_f - \hat{y}_f)$ ~ $N(0, \sigma^2 m)$

where $m = a' (I + X_f (X'X)^{-1} X_f') a$

so $Q = R/(\sqrt{m}.s)$ is distributed as (standard) Student t with d.f. the d.f. in the estimate of the variance, $\sigma^2$, which for regression is normally $n-p$, where $p$ is the number of predictors including the constant. From the interval for Q, you can then back out an interval for R and then a prediction interval for $a'y_f$.

Assuming I didn't screw up along the way.

Answer 2

In order to compute the variance of a sum of forecasts you need to incorporate the covariance between these forecasts. Thus compute the varriance and covariance of the observed series for as many lags as the length of your forecast. Compute the sum of these variances and covariances in a standard manner and the use this to talk about uncertainty in the sum of your forecasts. WE use this routinely to take daily predictions and convert them to probabilities of making month-end numbers.