I have a linear least squares model of the form $\hat{y}=\hat{\beta}X$ with multiple predictors in $X$ and I want to know if, based on $\hat{\beta}$, I can determine the relative importance of each predictor. This question suggests that if all predictors are independent and my data are standardized, I can use the magnitude of the coefficients as the correlation of each predictor. My question is what are the coefficients estimates of if the predictors aren't independent (i.e. can they still be used to signify relative importance), and is there a computationally quick way to estimate the relative importance of each predictor?
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