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If we have regressors $X_1, X_2$, then if they are orthogonal, then that means $\hat{\beta}_1$ determined from the multiple regression is identical to $\hat{\beta}_1$ determined from regression $y$ on $X_1$.

I was wondering if there are cases where the estimator is equal when $X_1, X_2$ are not orthogonal?

user5965026
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  • The answers at https://stats.stackexchange.com/questions/17336 will help you see that equality implies orthogonality. – whuber Jul 27 '20 at 13:56
  • @whuber Interesting, so is it an if and only if $\iff$ condition that $\hat{\beta}_1$ from multiple regression equals $\hat{\beta}_1$ from single regression IFF $X_1$ is orthogonal to all the other regressors? – user5965026 Jul 27 '20 at 19:32
  • It depends a little on your assumptions. For instance, when $X_2=X_1,$ you can take the new $\hat\beta_1$ to equal the old $\hat\beta_1$ and $\hat\beta_2=0,$ even though $X_1$ and $X_2$ are not orthogonal. – whuber Jul 27 '20 at 19:49
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    Oh I see, but if you had perfect multicollinearity like that, then technically your $\hat{\beta}$ estimators can have an infinite number of solutions since the normal equations are now singular? – user5965026 Jul 27 '20 at 19:53

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