Recent questions in Residuals
Landon Wolf 2022-08-09

### Why are errors independent but residuals dependent?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?

Ronnie Rojas 2022-08-09

### Approximating the following system using ordinary least squares regression$y={p}_{0}+{p}_{1}{x}_{1}+{p}_{2}{x}_{2}+...{p}_{M}{x}_{M}=\xi P$I know that a property of the least squares estimator is that the sum of the residuals, ${r}_{i}={\stackrel{^}{y}}_{i}-{y}_{i}$, is equal to zero. However, what I am finding is that the sum of the residuals multiplied by one of the regressors, ${\xi }_{m}\in \left(\mathbf{1},{x}_{1},{x}_{2},...,{x}_{M}\right)$, 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.$\sum _{i=0}^{N}{r}_{i}{\xi }_{m}=0$

Mariah Sparks 2022-07-22