Interpretation of (simultaneous) confidence band against fitted values in multiple regression

Question

In a homework question, I am asked to interpret a figure of the confidence band and simultaneous confidence band of 95% confidence level plotted against predicted values. The confidence bands are produced from linear regression.

I did some searching, but the explanation by Wikipedia only covers the situation where we have only one predictor, which is hard to generalize to multiple regression.

Here is my own attempt: since the fitted value by a linear model is simply a linear combination of predictors and 1, confidence bands against fitted values can be thought of as confidence bands against different levels of predictors. On the one hand, at each level of predictors, the probability of a point falling in the point-wise confidence band is 95%; on the other hand, the probability for a line representing the real relationship between predictors and the response to fall into the simultaneous confidence band across all level of predictors is 95%.

However, I'm not satisfactory about the answer, because "levels of predictors" is too vague and sounds like word-playing. In addition, the coefficients used to calculate the predicted values are random variables and thus prone to statistical error, which means the whole X-axis is somewhat random.

How do I interpret it?

Answer 1

You have two options: Option A) Try to figure out a confidence band for each observation in y, given its corresponding value of x (or x's, in case of multiple regressors).Keep on doing this independently for each observation, as if the other predictions did not exist, and you get your red-band.

Note, the band has some non-zero width since the predicted value of y is uncertain, and that's because the estimated regression betas are themselves uncertain (the estimates are function of what the sampled observations of (x,y) are - different samples would, of course, lead to different betas, hence the beta estimate is a random variable on its own, having some non-zero variance, leading to the aforementioned uncertainty).

Option B) Here, you want to find out a confidence band for the entire set of y-predictions. So, it's the confidence interval of the whole n*1 vector of y =(y_1, y_2,...y_n) which again, comes from the uncertainty of the beta estimates themselves. So, this gives you a n*1 upper bound and a n*1 lower bound , at one go, instead of computing similar bounds one by one, independently for each y value. This would be different from the case A because in A, the prediction of y_i and that of y_j, ignore the correlation induced by their common dependence on the beta estimate.