Under what condition (or conditions if you think it necessary) would one observe no change in the regression coefficient (e.g., b-hat y on X1) for some variable when another variable is added to the regression equation?
Under what condition (or conditions if you think it necessary) would one observe no change in the regression coefficient (e.g., b-hat y on X1) for some variable when another variable is added to the regression equation?
If you have k predictors, X_1, X_2, ...., X_k that are mutually orthogonal and you add X_{k + 1} that is also orthogonal. My intuition says (i.e. I haven't worked it out myself) that this is sufficient and easy to show.
You may be able to get away with a less stringent condition that is the same in spirit (i.e. rooted in orthogonality) but the problem seems more difficult then.