Q:

Linear regression. Lowering response maintaining equal independent variable.

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.

A:

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

