How do I derive a formula for the prediction intervals for the sum of responses of two independent future observations?

Question

So far I've tried using the formula for confidence intervals for a full rank model and trying to use that to get a prediction interval formula. What I don't understand is how to get a prediction interval for a sum of responses based on two predictors. Any help would be appreciated!

The main question is here

Answer 1

The variance of a future observation, $y_1$ based on $x_1$ is

$\sigma^2 \frac{1}{x_{1} x_{1}^{\prime}} + \sigma^2$.

The first component is the variance due to $\beta$ and the second component is the variance of the noise term which comes with a new observation.

Next, the The variance of a future observation, $y_2$ based on $x_2$ is

$\sigma^2 \frac{1} {x_{2} x_{2}^{\prime}} + \sigma^2$.

Next, the variance of the sum is just the sum of the variances so you can add those two to get the variance of the sum of the $y_{1}$ and $y_{2}$ predictions.

Then, depending on whether $\sigma^2$ is known or not, you can use the z tables or the t-tables to get the CI for the sum of the predictions:

It will be $\hat{\beta} x_{1} + \hat{\beta} x_{2} \pm z_{\frac{\alpha}{2}} \sqrt{\sigma^2 \frac{1}{x_{1} x_{1}^{\prime}} + \sigma^2 + \sigma^2 \frac{1}{x_{2} x_{2}^{\prime}} + \sigma^2 } $

NOTE: Check the variance due to $\beta$ to make sure that the formula I used for that component is correct.