Timeline for Correlation of residuals and explanatory variables
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Dec 13, 2023 at 8:21 | history | bounty ended | Marlon Brando | ||
| S Dec 13, 2023 at 8:21 | history | notice removed | Marlon Brando | ||
| Dec 13, 2023 at 8:20 | vote | accept | Marlon Brando | ||
| Dec 12, 2023 at 9:53 | answer | added | Sextus Empiricus | timeline score: 1 | |
| Dec 9, 2023 at 16:24 | comment | added | Marlon Brando | I just realized that the original projection matrix P = X (X'X)^-1 X' does not involve any assumption about the error term. Is this what you mean? But that automatically implies the fact $x_i û_i = 0$ right? | |
| Dec 9, 2023 at 16:13 | comment | added | Marlon Brando | So in other words, orthogonality holds always? But $xi' ûi = 0$ only holds if we have a constant in the model? Is that not a contradiction? | |
| Dec 5, 2023 at 14:21 | comment | added | whuber♦ | Since you understand the geometry, you don't need to ask about orthogonality: you know the residuals are orthogonal to all vectors in the space spanned by the regressor variables. (That's called the Normal Equations.) What might be confusing is that when that space contains no nonzero constant vectors, orthogonality does not imply uncorrelated. | |
| S Dec 5, 2023 at 13:58 | history | bounty started | Marlon Brando | ||
| S Dec 5, 2023 at 13:58 | history | notice added | Marlon Brando | Improve details | |
| Dec 2, 2023 at 15:29 | comment | added | Marlon Brando | stats.stackexchange.com/questions/474102/… The first answer also tells that the fact is only true, when we have a constant in the model.... not sure what to think right now.... | |
| Dec 2, 2023 at 15:28 | history | edited | Marlon Brando | CC BY-SA 4.0 |
added 2 characters in body
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| Dec 2, 2023 at 14:13 | comment | added | Marlon Brando | Hi Nick, Thanks! Not sure if I really get you right. So if I regress y on x without a constant, the projection will still be orthogonal? And what about the other two facts? I assume that exogeneity does only matter for causal interpretation, but not in terms of the construction. And the third one I am still not sure. I think applying the law of iterated expectation yields always 0, if x is exogenous. So that would bring me back to the second statement. Long story short: Is û always uncorrelated with the explanatory variables? | |
| Nov 25, 2023 at 12:18 | comment | added | Nick Cox | There are sometimes substantive grounds to force a regression through the origin. That can make sense as a reason for insisting on $y = bx$. After all, the laws of physics met at introductory level often have this form. I would need a stronger story for insisting on $y = b_1 x_1 + b_2 x_2 + \cdots + b_p x_p$ and omitting a $b_0$. | |
| Nov 25, 2023 at 11:41 | history | asked | Marlon Brando | CC BY-SA 4.0 |