Recent questions in Residuals

Residuals
Answered

Landon Wolf
2022-08-09

Sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also we assume that E(ϵ)=0. Why doesn't it imply errors are also not independent?

Residuals
Answered

Ronnie Rojas
2022-08-09

$y={p}_{0}+{p}_{1}{x}_{1}+{p}_{2}{x}_{2}+...{p}_{M}{x}_{M}=\xi P$

I know that a property of the least squares estimator is that the sum of the residuals, ${r}_{i}={\hat{y}}_{i}-{y}_{i}$, is equal to zero. However, what I am finding is that the sum of the residuals multiplied by one of the regressors, ${\xi}_{m}\in (\mathbf{1},{x}_{1},{x}_{2},...,{x}_{M})$, is also equal to zero. I have found this to be true in all cases in simulation, but I am not sure how to prove it.

$\sum _{i=0}^{N}{r}_{i}{\xi}_{m}=0$

Residuals
Answered

Mariah Sparks
2022-07-22

Residuals
Answered

Pierre Holmes
2022-07-21

$\hat{Y}=X\hat{\beta}$ and

$E[e]=E(Y-\hat{Y})=E[X{\beta}_{0}-X\hat{\beta}]=X({\beta}_{0}-E(\hat{\beta}))=0$

since $\hat{\beta}$ is an unbiased estimator of ${\beta}_{0}$.

Am I making a mistake?