A friend of mine was telling me today about the need to mean adjust input variables to zero in order to "get rid of implicit intercept" or scale terms in the slope coefficients and to make their interpretation easier. He was stressing its use and even standardization by the weighted mean in a weighted regression. Both of these I had never quite heard of other than when the inputs are of a very different scale (like 6 digit dollar values and single digit times or something) for stability of the estimation. Is it sensible what I am hearing? And that this "mean shifting" increases correlation between the slope coefficients? If so where can I read more about this?
The argument I was given goes something like this:
If i have the model
$$y = ax + b + \epsilon$$
and then I shift $x$ by $D$, I will have $y = a' + b(x+D) + \epsilon$ where the estimate for $b$ does not change (and neither do the residuals) but the estimate for the intercept should be shifted accordingly. Thus we can equate $y - \epsilon$ to see that $a' = a - bD$ suggesting that shifting $x$ makes estimates of the intercept more correlated with $b$ than before. It feels to me like no new information has been added at all and there is nothing better about the coefficient estimates with shifts than with no shifts.