The regression residuals e are sample counterparts of the po

The regression residuals, e, are sample counterparts of the population:

The regression residuals, e, are sample counterparts of the population: (a) regression functions intercept. (c) Errors. (b) Regression function slope. (d) Regression function\'s predicted values.

Solution

The regression residuals, e, are sample counterparts of the population:

Errors

The error term is the random disturbance, which disturbs the stable relationship.

Residuals are the observed differences between predicted and observed values in the sample. Residuals and errors are different, but the residuals can be used as estimates for the errors.

Let’s use a linear regression model

Y = X*beta + er,

er = error term

Y is also the fitted value + the residual

Y = X*beta_et + rs

beta_et = estimation of beta

Therefore

rs= Y-X*beta_et

   =X*beta + er - X*beta_et

   =X* (beta-beta_et) +er.

Thus, we realize that residual (rs) is not the same as the error (er), but the difference between them DOES have an expected value of zero, because the expected value of beta_et equals beta if the necessary assumptions of the classical regression model are fulfilled.

Therefore, we can use residuals to estimate the standard error of the regression model.