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Data consists of 40 observations with 4 dimensions and a response-variable.

When doing a ridge regression on my data and plotting the coefficients and coefficient errors (MSE of the ridge coefficients vs. normal linear regression coefficients) as functions of the regularization parameter 'alpha' I get the follow plots:

enter image description here

Can this really be the case? That the coefficient error decreases as the regularization parameter increases? Wouldn't that imply that coefficients with value ~0 is preferable?

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Just realized my mistake, I calculated the ridge regression without fitting the intercept fit_intercept = False which obviously wont work in this comparison.

For comparison:

enter image description here

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