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Linear Regression


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Looking for someone to point me in the right direction so I can start researching  a solution to the problem below: 


Linear regression with high dimensionality - p=29, n = 5000ish, input variables are generally quite highly correlated

When using the model for prediction, data sets regularly have a missing input parameter(s). At the moment I just refit a LSQ solution from the training data with that input deleted. This seems to lead to quite unstable results. Stability is important for my application, more so than absolute accuracy in some senses.  

Regularisation (e.g. Ridge) feels like it should help, but (and I'm not formally train in stats) as I understand that will reduce the variance of the model,  with all input variables - and doesn't necessarily achieve anything for model stability where input parameters are deleted.

Thanks in advance. 

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