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Stuart_Armstrong comments on Why the tails come apart - LessWrong
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Another approach seems to be stepwise regression: http://en.wikipedia.org/wiki/Stepwise_regression
I see a lot of stepwise regression being used by non-statisticians, but I think statisticians themselves think its something of a joke. If you have more predictors than you can fit coefficients for, and want an understandable linear model you are better off with something like LASSO.
Edit: Don't just take my word for it, google found this blog post for me: http://andrewgelman.com/2014/06/02/hate-stepwise-regression/
I concur. Stepwise regression is a very crude technique.
I find it useful as an initial filter if I have to dig through a LOT of potential predictors, but you can't rely on it to produce a decent model.
So it wasn't as clear with the previous link, but it seems to me that the nth step of this method doesn't condition on the fact that the last n-1 steps failed.