Dear Daniele,

    I don't think that you should be hoping to eliminate this issue. These variables are important, so why are you letting the limitations of statistical technique drive your question?

   I will support the recommendation of Rimantas in that many researchers will map these predictors onto a latent variable and work from there. The difference between this and simply averaging together the scores depends more or less on the structure of covariance between these predictors; you only work this out by trying it both ways.

   Given that these variables are arising from quantitative analyses you may also decide to simply dichotomize them and build a predictor which is the number of these indicators which is present. This might be more honest then the latent variable approach because the latter requires (but this is rarely done) some verification about what is actually common between a set of variables.

Dear Daniele, let me ask you - what is the final objective of your project? Will you focus on outcome (prediction) or on process itself (for example studying impact of individual variables)? Because for prediction, multicollinearity does not have to be such big issue. A nice comment has been recently done in this study - https://www.toi.no/getfile.php/Publikasjoner/T%C3%98I%20rapporter/2015/1403-2015/1403-2015-elektronisk.pdf (chapter 13 - Don’t worry about multicollinearity).

I read your question one week ago. I was writing a paper related to your questions. I finish to wright a paper and spend many hours to look for your questions again. RG should offer the function to find by key-word.  

Below is the conclusion. This paper will be upload next March. But, you can download about CPD data after 2014. I compare three approach such to analyze the original data, analyze after PCA (approach 1) and delete some variables relate to collinearity (approach 2). The best result is approach 2 and the worst one is approach 1. See my papers about CPD data after 2014.

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5.       Conclusion

In this research, we discuss the new method and model selection procedure of the discriminant analysis using the CPD data with collinearities. Our examination tells us that the second approach (to exclude the variables relate to collinearities) is the best, and the first approach (bundle of PCA and regression analysis) is the worst. The common consensus that we had better regressed the data with collinearity using new variables by PCA may be wrong. We must examine this fact in other cases if we can access other data with collinearities.

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See my new uploaded " How to .....".

If you can't delete the one of the predictors (which shows collinearity with other predictor), you can try to use combination of these variables, if you objective is to find the impact of predictors on response. As colllinearity reduces power of the test. For prediction purpose, collinearity is not a big issue.

There are may ways to deal with multi-colinearity, you need to wisely choose the best option.

For variable selection, automated processes (like step-wise method) should not be used, as stepwise regressions are controversial and might lead to model misspecification. [Paper by Peter Flom: http://www.lexjansen.com/pnwsug/2008/DavidCassell-StoppingStepwise.pdf]

 You can use PCA to reduce dimensions of the model or use can use regularization/ penalization (L1/L2 norm or combination of both) or least angle regression (LAR). I personally prefer using ridge/lasso or elastic net for dealing with the issue.

If you are using R for your analysis, it's easy to tune the parameters of the model by using cross-validation.