Timeline for Mean adjusted $R^2$ for linear regression with gaussian noise covariates
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2018 at 21:44 | comment | added | Ben | Okay, in that case I think I may have misinterpreted your question. Hopefully some of what I've written is helpful to you anyway. | |
| Feb 7, 2018 at 13:44 | comment | added | Olivier | My question isn't assuming that $\omega$ has no relationship with $Y$ ($\omega$ is a random variable and, in fact, the only source of randomness relevant in my problem). | |
| Feb 7, 2018 at 4:19 | comment | added | Ben | If your question is assuming that $\omega$ has no relationship with $Y$ then you are effectively assuming that $\beta_\omega = 0$. Under this condition the equation for the new model is $\boldsymbol{Y} = \boldsymbol{X} \beta + \boldsymbol{\omega} \beta_\omega + \varepsilon = X \beta + \varepsilon$. Hence, although the new variable changes the design matrix, this does not affect the underlying model form. | |
| Feb 7, 2018 at 3:40 | comment | added | Olivier | While the noise variables have no direct relationship with $Y$, they are covariates and hence adding them changes the dimensions of the design matrix (as well as the dimension of the parameters). The equation $Y = X\beta + \varepsilon$ changes with the model. | |
| Feb 7, 2018 at 3:27 | comment | added | Ben | Not sure if I interpreted your question correctly. I assume you're saying that the newly added noise variable does not have any relationship with Y? Is that correct? If so, then it would not appear in the model equation, so both models would have the same underlying form. | |
| Feb 7, 2018 at 2:26 | comment | added | Olivier | What you mean by "both model have the same underlying equation"? I don't see why this is the case and how you conclude. | |
| Feb 6, 2018 at 6:02 | history | answered | Ben | CC BY-SA 3.0 |