Skip to main content
Cross Validated

Return to Question

Notice removed Improve details by Marlon Brando
Bounty Ended with Sextus Empiricus's answer chosen by Marlon Brando
Notice added Improve details by Marlon Brando
Bounty Started worth 50 reputation by Marlon Brando
added 2 characters in body
Source Link
Marlon Brando
Marlon Brando
  • 623
  • 8

We say that in "standard" OLS regression the residuals $û$ are uncorrelated with $k$-th explanatory variable $x_k$. I know the argument can be intuitively derived from geometry of OLS. There are a lot of interesting articles in this forum on that topic with the bottom line, that the projection is orthogonal.

I am only interested, if that is also true in the following twothree cases:

  • we do a regression without a constant
  • $x_k$ is not exogenous
  • $E(x_{ik}' u_i) = 0$

I am not quite sure about the first one. I think I've heard, that we need the constant in the model for it to be true. The exogeneity should not matter, as far as I know, i.e. the projection is still orthogonal. And I think the third one does not matter for the question.

We say that in "standard" OLS regression the residuals $û$ are uncorrelated with $k$-th explanatory variable $x_k$. I know the argument can be intuitively derived from geometry of OLS. There are a lot of interesting articles in this forum on that topic with the bottom line, that the projection is orthogonal.

I am only interested, if that is also true in the following two cases:

  • we do a regression without a constant
  • $x_k$ is not exogenous
  • $E(x_{ik}' u_i) = 0$

I am not quite sure about the first one. I think I've heard, that we need the constant in the model for it to be true. The exogeneity should not matter, as far as I know, i.e. the projection is still orthogonal. And I think the third one does not matter for the question.

We say that in "standard" OLS regression the residuals $û$ are uncorrelated with $k$-th explanatory variable $x_k$. I know the argument can be intuitively derived from geometry of OLS. There are a lot of interesting articles in this forum on that topic with the bottom line, that the projection is orthogonal.

I am only interested, if that is also true in the following three cases:

  • we do a regression without a constant
  • $x_k$ is not exogenous
  • $E(x_{ik}' u_i) = 0$

I am not quite sure about the first one. I think I've heard, that we need the constant in the model for it to be true. The exogeneity should not matter, as far as I know, i.e. the projection is still orthogonal. And I think the third one does not matter for the question.

Source Link
Marlon Brando
Marlon Brando
  • 623
  • 8

Correlation of residuals and explanatory variables

We say that in "standard" OLS regression the residuals $û$ are uncorrelated with $k$-th explanatory variable $x_k$. I know the argument can be intuitively derived from geometry of OLS. There are a lot of interesting articles in this forum on that topic with the bottom line, that the projection is orthogonal.

I am only interested, if that is also true in the following two cases:

  • we do a regression without a constant
  • $x_k$ is not exogenous
  • $E(x_{ik}' u_i) = 0$

I am not quite sure about the first one. I think I've heard, that we need the constant in the model for it to be true. The exogeneity should not matter, as far as I know, i.e. the projection is still orthogonal. And I think the third one does not matter for the question.