Question

How would including irrelevant variables in the multiple regression model affect the properties of the estimator?

What would happen to the estimator if relevant variables have in fact been omitted?

Answer 1

In a multiple regression, presence of irrelevant variables increases the standard error for the estimator and decreases the slope coefficient. So, we can say that the irrelevant variables make the multiple regression worst.

If we omit the relevant variables, then we will get reduced slope coefficient with lower coefficient of determination and increased standard error for the estimator as we know that removing relevant variables will result in a bad multiplie regression results.