I'm looking at a regression model where a very large number of possible explanatory variables are being evaluated, and a small number are finally chosen via the lasso method of variable selection. The $\lambda$ tuning parameter in the lasso is chosen by looking at cross-validation forecast performance, which is pretty standard.
However, when I take the list of chosen variables and just run OLS on them, many turn out to be statistically insignificant. That may be perfectly fine if they are jointly significant and the forecast performance is superior to other models (in addition, there would be a question of what the t-test means when you have already screened the variables in a separate step, but I'm leaving that aside).
I'm curious though whether it makes sense to look at statistical significance of individual variables in a model chosen by lasso using CV forecast performance to select the tuning parameter. The problem is that lasso ends up selecting various dummy variables that are only true on small segments of the population and which are insignificant in OLS, and there is a natural question as to whether the model should be judgmentally simplified.