Averages vs Totals in Multiple Regression - Simple Question

#1
Say in a multiple regression, I have three independent variables, each of which is the sum of 10 values (for ex: annual scores each year for a decade).

In a second multiple regression, I have three independent variables which are just the averages of the scores instead of the total sum

Why is it that if I perform OLS on a dependent variable, the beta coefficients remain the same, regardless if I took the averages or just used the totals?
 

Dragan

Super Moderator
#2
Say in a multiple regression, I have three independent variables, each of which is the sum of 10 values (for ex: annual scores each year for a decade).

In a second multiple regression, I have three independent variables which are just the averages of the scores instead of the total sum

Why is it that if I perform OLS on a dependent variable, the beta coefficients remain the same, regardless if I took the averages or just used the totals?
Because you're just dividing all the independent variables by a Constant (N).

Ask yourself what the interpretation is of a Partial Regression Coefficient.

For example, in simple linear regression:

b = r*(sy/sx)

Dividing by a constant will neither change the standard deviations of X and Y nor the correlation between X and Y.