# What is the practical significance of a linear regression, lasso regression, and ridge regression outputting the same coefficients? [closed]

The lasso and ridge regression are tuned to the same alpha parameter. No matter what I tune the parameter to [0,1], the results of all three regressions are always the same (linear, ridge, lasso), which is a bit befuddling because I had previously suspected that there was some multicollinearity in my data due to high variance inflation factors. I thought that either the lasso or ridge would address the issue of multicollinearity, but this does not seem to be the case. If anyone can provide some insight that would be great.

Thanks.

## closed as unclear what you're asking by gung♦, Momo, Nick Cox, Andy, chlSep 13 '14 at 10:38

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• Could you precise something about your data ? What is the dimension of the data ? What about the goodness of fit of the OLS regression ? – Stéphane Laurent Aug 21 '14 at 21:34
• There's not enough information here to discern what's happening. A reproducible minimal example (i.e. the smallest example that shows the problem you describe) might throw light on what's happening. – Glen_b Aug 21 '14 at 23:49

$\min \| X \beta - y \|_{2}^{2} + \lambda \| \beta \|_{2}^{2}$
with $\lambda=1$, and that in the optimal solution, $\| \beta \|_{2}=10$, but $\| X \beta - y \|_{2}=10000000$. In this case, the ridge regression penalty term is so small as to be insignificant in comparison to the linear regression term and you'll end up with essentially the same answer with or without the penalty term.
So, in your examples, what is the size of $\| X \beta -y \|_{2}$? What about the ridge regression and LASSO penalty terms?