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Approximating the following system using ordinary least squares regression
$y = p_{0} + p_{1} x_{1} + p_{2} x_{2} + . . . p_{M} x_{M} = ξ P$
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, $ξ_{m} \in (1, x_{1}, x_{2}, . . ., x_{M})$ , 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.
$\sum_{i = 0}^{N} r_{i} ξ_{m} = 0$

Mariah Sparks 2022-07-22

Show that: $\sum x_{i} e_{i} = 0$ and also show that $\sum {\hat{y}}_{i} e_{i} = 0$ . Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that $\sum e_{i} = 0$ .

Pierre Holmes 2022-07-21

Assume $Y = X β_{0} + ϵ$ where $ϵ$ is zero mean and $X$ is fixed. I know that under certain conditions on the design matrix $X$ in OLS, the sample mean of the residuals $\bar{e}$ is $0.$ Can we say the same for the true population mean of residuals as well?
$\hat{Y} = X \hat{β}$ and
$E [e] = E (Y - \hat{Y}) = E [X β_{0} - X \hat{β}] = X (β_{0} - E (\hat{β})) = 0$
since $\hat{β}$ is an unbiased estimator of $β_{0}$ .
Am I making a mistake?

Tamara Bryan 2022-07-20

Given a linear regression model obtained by ordinary least squares, prove that the sample covariance between the fitted values and the residuals is zero.

anudoneddbv 2022-07-15

Assume a linear regression model: $y_{i}$ = $β_{0}$ $β_{1} x_{i 1}$ +...+ $β_{p} x_{i p}$ + $ϵ_{i}$
$C o v (e, \hat{Y})$
where: $e$ = the residuals vector
$\hat{Y}$ = the predicted vector of $Y$
Use the fact that $X^{T} e$

Math can be tricky, but understanding residuals can make it easier. Residuals are a way to measure the difference between a predicted value and the actual value in a regression analysis. Residuals are calculated by subtracting the predicted value from the observed value. They can be used to tell if a model is accurate or not. A negative residual indicates that the observed value is larger than the predicted value, and a positive residual indicates that the predicted value is larger than the observed value. When the residuals are small, it means that the model is accurate. Understanding how residuals work can give you a better understanding of how your data is being used. With the right knowledge, you can make the most of your data and make better decisions.

High school statistics

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Show that: ∑ x i e i = 0 and also show that ∑ y ^ i e i = 0. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that ∑ e i = 0.

Given a linear regression model obtained by ordinary least squares, prove that the sample covariance between the fitted values and the residuals is zero.

Assume a linear regression model: y i = β 0 β 1 x i 1 +...+ β p x i p + ϵ i C o v ( e , Y ^ )where: e = the residuals vector Y ^ = the predicted vector of YUse the fact that X T e

Show that: $\sum x_{i} e_{i} = 0$ and also show that $\sum {\hat{y}}_{i} e_{i} = 0$ . Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that $\sum e_{i} = 0$ .

Assume a linear regression model: $y_{i}$ = $β_{0}$ $β_{1} x_{i 1}$ +...+ $β_{p} x_{i p}$ + $ϵ_{i}$
$C o v (e, \hat{Y})$
where: $e$ = the residuals vector
$\hat{Y}$ = the predicted vector of $Y$
Use the fact that $X^{T} e$