Linear regression. Lowering response maintaining equal independent variable.

Question

I have put some data together and modelled the behaviour of the response ($y$) as function of three independent variables $x_1$, $x_2$ and $x_3$. A simple multi-linear regression. The model looks like:

$y = k + a*x_1 + b*x_2 + c*x_3 + e$

Up to this point everything is OK. But now I want to lower the response by a $15%$. Like reducing some commissions or costs. The only idea I came across is to multiply the responses by $0.85$ and readjust the whole model. Recalculate $a$, $b$ and $c$ with the new values. I have been trying to find another way of doing this without touching the data samples. Just changing and adjusting the coefficients $a$, $b$ and $c$. Does anybody know how this should be done? An idea you come across with would be okay.

Answer 1

The regression coefficients are given by:

$\hat{\beta} = (X'X)^{-1}X'Y$

Thus, if you scale $Y$ values by $0.85$ then this is effectively equivalent to the following:

$Y_{\text{new}} = 0.85e' Y $

where,

$e$ is a vector of ones.

Thus, the new estimate is given by

$\hat{\beta}_{\text{new}} = (X'X)^{-1}X' (0.85 e'Y)$

Thus, you get the following relationshiop:

$\hat{\beta}_{\text{new}} = 0.85 \hat{\beta}$

Answer 2

Just subtract 15 from all the y-values, then run the regression. You are using linear regression, and therefore applying an additive "translation" (i.e., subtraction) form the y-values before regression will simply result in coefficients that will predict values whose mean is less 15 units.