# Multiple linear regression: does BIC drop (vaguely) collinear variables?

Say I have the following multiple linear regression:

Y ~ X1 + X2 + X3 + X4


All X variables are independent, but X1 and X2 look kind of linearly related when plotted against one another (e.g., R-squared ~0.5, if I fit a simple linear regression to X1 ~ X2). Will BIC or AIC throw out one of the variables because this? How much collinearity is too much? Is there a formal procedure for identifying and dealing with such collinearity?

A2 & A3: There is no unique answer to this question. Most of the time we will look at Variance inflation factor i.e. $VIF_j=\dfrac{1}{1-R^2_{j}}$, where $R^2_{j}$ is the R-squared of the model with $X_j$ as the response over all other remaining independent variables. As a rule of thumb, if the $VIF>10$ then there is a serious multicollinearity problem. Although, in some text books they consider cut-off value of 5.