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 $X_1^T u=0$,