title: Analyzing correlated data in R: Linear, Ridge regression, PCR
body: <p>I've got a time series of observations of 5 variables <code>y, x_1, x_2, x_3, x_4</code> and the task is to find which of the <code>x</code>es are responsible for the changes in <code>y</code>. Now the problem is that all of them are strongly cross-correlated and exhibit collinearity. <code>x_1, x_2, x_3, x_4</code> don't have hidden components inside which are common to them pairwise or in any other way - they are just naturally correlated.</p>
  
  <p>Predictably, linear regression gives unreasonable results with coefficients changing wildly after removing one of the variables, which is a normal picture for highly-collinear data. </p>
  
  <p>As advised on <a href="http://en.wikipedia.org/wiki/Multicollinearity#Remedies_for_multicollinearity" rel="nofollow">wiki</a>, some of the remedies for multicollinearity are using ridge regression and principal component regression. However, when I use <code>lm.ridge</code> method, it gives me exactly the same coefficients as <code>lm</code>. </p>
  
  <p>Can PCR help in such a case and if so, what is an easy way in R to retrieve the coefficients and p-values from it? Something like a summary table for <code>lm</code> function.</p>