# How to select significant variables for multi-factor regression?

I have a set of data $Y \in \mathbb{R}^n$ and a set of $k$ factors $\{F_1, ..., F_K\}$ with $F_i \in \mathbb{R}^n ~ \forall i$.

I would like to perform a factor analysis which consits in finding $\beta \in \mathbb{R}^k$ such that:

$$Y_t = \alpha + \sum_{i=1}^k \beta_i {F_i}_t + \varepsilon_t$$

I believe that I would need to make sure that the factors are independent from each other, and I wonder what process (statistical test) I should use to determine which factors I shall remove from the framework because they are too correlated.

There are two main questions:

1. How do I find and remove the factors that are not linearly independent?
2. How do I determine which of the remaining factors are significant?
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I think you have to conduct model selection to check which variables-factors can be removed. For this purpose, you have an ocean of possibilities. –  user10525 May 23 '12 at 13:47
Probably worth plotting factor pairwise to check dependence visually - not all important dependencies are linear. R's pairs() works well for this. –  naught101 May 27 '12 at 8:58