Relationship between the standard error of the estimators and the standard error of the error term

Question

I'm trying to solve the following problem:

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My thinking is that If the Standard error of the error term would then the error term of the regression would also fall (as the regression would start performing better).

Answer 1

You are right, e.g., for the slope $$ Var(\hat{\beta}_1) =\frac{1}{(\sum(x_i - \bar{x})^2)^2}Var\left( \sum (x_i - \bar{x})y_i \right) = \frac{\sigma^2}{\sum(x_i - \bar{x})^2}, $$ where $Var(\epsilon_i) = \sigma^2$. For the intercept you have $$ Var(\hat{\beta}_0) =\frac{\sigma^2}{n} + \frac{\sigma^2\bar{x}^2}{\sum(x_i - \bar{x})^2}, $$ thus reducing the variance of $\epsilon_i$ will reduce the standard error of the OLS estimators.