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Suppose I run a regularized regression model such as Lasso. For simplicity let's say it's a linear model. After using cross-validation to find the $\lambda$ parameter, the model is refit (without regularization) using only the variables with non-zero coefficients. This will return, along with the fit, some confidence intervals for the coefficients.

Can these be used for inference in any way? I understand that they can't be used in hypothesis testing because of selection bias but do they have any other uses? For example I think that if the interval contains $0$ then that variable, although possibly helpful for prediction, might not be very important in the underlying process. This is because after accounting for selection bias the interval can only get wider, not narrower.

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  • $\begingroup$ Confidence intervals are inversions of hypothesis tests, so what you’re doing is failing to reject a null hypothesis of $\beta_k=0$ and accepting that $\beta_k=0$. $\endgroup$
    – Dave
    Commented Mar 6, 2021 at 2:10
  • $\begingroup$ Yes but these tests aren't valid because of the previous selection step. So I'm wondering what other uses these intervals can have $\endgroup$
    – badmax
    Commented Mar 6, 2021 at 3:09
  • $\begingroup$ You’re doing those tests when you check if a confidence interval contains zero, so if the test is invalid, so is the confidence interval. $\endgroup$
    – Dave
    Commented Mar 6, 2021 at 3:25
  • $\begingroup$ Right, in the sense that the intervals are too narrow. Can they also be too wide? $\endgroup$
    – badmax
    Commented Mar 6, 2021 at 3:46

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