Adjusting for zero mean (standardizing) in a multiple regression model

Question

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.

Answer 1

I don't know how representative my experience is, but I've met several people with somewhat strong opinions on centering. (Which is weird, really, because who cares?) It may be that you have run into this yourself. Often, centering is discussed in the context of multicollinearity. In general, I don't think centering really does much; if $x_1$ & $x_2$ are correlated, so will the centered versions of them, as @Henry points out. The one case I know of where it can make a difference is when fitting polynomial terms. For example, if your predictor variable, $x$, ranges from 2 to 4, and you want to add an $x^2$ term to capture a possible curvilinear trend, $x$ & $x^2$ can be awfully hard to distinguish. On the other hand, if you center $x$ first (and the quadratic was concave up) $x^2$ would be going down on the left and up on the right, which makes them much more distinguishable. I'm not aware of much beyond that. There are what I would call 'cosmetic' changes, though. For instance, sometimes it just doesn't make much sense to talk about the effect when $x$ is 0, but you are interested in the predicted $y$ value at the mean of $x$. In such a case, centering causes the regression intercept to reflect that value. Note that this hasn't actually changed anything, however; it is just as possible to get a predicted value at $-\bar{x}$ with your new model, and you could have easily gotten the predicted value at $\bar{x}$ with your old one.