Dealing with correlated regressors In a multiple linear regression with  highly correlated regressors, what is the best strategy to use? Is it a legitimate approach to add the product of all the correlated regressors?
 A: Here is another thought that is inspired by Stephan's answer:
If some of your correlated regressors are meaningfully related (e.g., they are different measures of intelligence i.e., verbal, math etc) then you can create a single variable that measures the same variable using one of the following techniques:


*

*Sum the regressors (appropriate if the regressors are components of a whole, e.g., verbal IQ + math IQ = Overall IQ)

*Average of the regressors (appropriate if the regressors are measuring the same underlying construct e.g., size of left shoe, size of right shoe to measure length of feet)

*Factor analysis (to account for errors in measurements and to extract a latent factor)
You can then drop all the correlated regressors and replace them with the one variable that emerges from the above analysis. 
A: I was about to say much the same thing as Stephan Kolassa above (so have upvoted his answer). I'd only add that sometimes multicollinearity can be due to using extensive variables which are all highly correlated with some measure of size, and things can be improved by using intensive variables, i.e. dividing everything through by some measure of size. E.g. if your units are countries, you might divide by population, area, or GNP, depending on context. 
Oh - and to answer the second part of the original question: I can't think of any situation when adding the product of all the correlated regressors would be a good idea. How would it help? What would it mean? 
A: Principal components make a lot of sense... mathematically. However, I'd be wary of simply using some mathematical trick in this case and hoping that I don't need to think about my problem.
I'd recommend thinking a little about what kind of predictors I have, what the independent variable is, why my predictors are correlated, whether some of my predictors are actually measuring the same underlying reality (if so, whether I can just work with a single measurement and which of my predictors would be best for this), what I am doing the analysis for - if I'm not interested in inference, only in prediction, then I could actually leave things just as they are, as long as future predictor values are similar to past ones.
A: You can use principal components or ridge regression to deal with this problem.  On the other hand, if you have two variables that are correlated highly enough to cause problems with parameter estimation, then you could almost certainly drop either one of the two without losing much in terms of prediction--because the two variables carry the same information.  Of course, that only works when the problem is due to two highly correlated independents.  When the problem involves more than two variables that are together nearly collinear (any two of which may have only moderate correlations), you'll probably need one of the other methods.
A: I'm no expert on this, but my first thought would be to run a principal component analysis on the predictor variables, then use the resulting principal components to predict your dependent variable.
A: One of the ways to reduce the effects of correlation is to standardize the regressors. In standardizing, all the regressors are subtracted by their respective means and divided by their respective standard deviations. Specifically, if $X$ is the regression matrix:
$$x_{ij}^{standardized}=\frac {x_{ij}-\overline{x_{.j}}} {s_{j}}$$
This is not a remedy, but definitely a step in the right direction.
