Let $\hat{\beta}=(\hat{\beta}_1,\hat{\beta}_2)^T$ be the least squares estimator in the regression model $Y=X_1\beta_1+X_2\beta_2+u$. Let $\hat{\delta}_1$ be the least squares estimator of the regression of $Y$ on $X_1$. Show that $$ \hat{\delta}_1=\hat{\beta}_1+(X_1^T X_1)^{-1} X_1^TX_2\hat{\beta}_2. $$

Hello, to be honest I do not understand what do to here resp. what is meant with $\hat{\delta}_1$.

To my knowledge it is $$ \hat{\beta}=(X^TX)^{-1}X^TY. $$

Is here meant that $X=(X_1,X_2)$ with $X_1=(1,...,1)^T$?

And what is meant with $\hat{\delta}_1$? Maybe $Y=\delta_1 X_1+u$ and $$ \hat{\delta}_1=(X_1^TX_1)^{-1}X_1^TY? $$


Yes you are right in understanding what is $\hat \delta_1$. So now it is rather simple. Just replace in the last formula $Y=X_1\hat \beta_1 +X_2\hat \beta_2 + u$ and take in account that $X_1^T u=0$,

| cite | improve this answer | |
  • $\begingroup$ But it is $Y=X_1\beta_1+X_2\beta_2+u$ and not $Y=X_1\hat{\beta}_1+X_2\hat{\beta}_2+u$ - or what do you mean? $\endgroup$ – math12 May 28 '14 at 20:27
  • $\begingroup$ After estimation $\beta_{1,2}$ you get $u$ which is unobservable but derived from least squares such that $u$ is orthogonal to $X_{1,2}$ $\endgroup$ – Alexander Vigodner May 28 '14 at 21:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.