I am interested in understanding the different options for gauging relative importance of variables in the results of a linear model. One way I've done this is multiplying the raw variable coefficients by the standard deviation of the variable. I realize that there is no reason why a 1 SE change in one variable should be comparable to a 1 SE change in another... however, for similar variables, this is a decent approximation.
My question is, do any manipulations with the standard error (as opposed to the standard deviation) ever figure into determining the relative importance of variables? I think my question reveals a misunderstanding of the true meaning / importance of standard errors.