Im studying for an exam, i don't have the solution, so I hope some of you guys can help me. I have tried a lot but i can't do this proof.

Here is the task:

Suoppose we have the linear regression model $U_i=\beta_kX_{k,i}+u_i,$ $i=1,2 .... , n,$

where $k$ is an index which we will use later ( it will be the k´th covariate in a more general linear regression model). The least squares estimator of $\beta_k$, which we call for $\hat{b}_k$ is the minimizer of

$L_k(b_k)= \sum_{i=1}^{n}(Y_i-b_kX_{k,i})^2$.

Show that the first order equation $\frac{d}{db_k}L_k(b_k)=0$ has the single solution $\hat{b}=\frac{\sum ^n_{i=1}X_{k,i}Y_k }{\sum ^n_{i=1}X^{2}_{k,i}}$

You may without justification conclude that this is the minimizer of $L_k(b_k)$

  • $\begingroup$ @OP: Did you try taking the derivative: $\frac{\mathrm{d}}{\mathrm{d}b_k}\sum_{i=1}^{n}(Y_i-b_kX_{k,i})^2$? $\endgroup$ – Winter Soldier May 9 at 19:43
  • $\begingroup$ @Vision Yes sir, but i feel like it something I'm doing wrong. I haven't had math for years. Is it possible to show me a solution? $\endgroup$ – Analyist May 10 at 8:38

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