I have a question on testing the hypothesis that a particular regression coefficient in a simple OLS scheme with all the good assumptions is zero or not.In particular eq 3.12, in the book by Tibshirani and coll., Elements of Statistical Learning defines the z-score asz=β/σ?vjMy question is that given the regression coefficients are all jointly normal, how can we separate one coefficient out like that? Is that a conditional z-score assuming all others betas have a particular value?How about independence, I guess under classical OLS it is fair to say that the betas are independent, is that the case?. If yes, then the multivariate normal degenerates to individual normal distributions and then I can build any test on any single coefficient. If no, then how does one take into account the values other coefficients are taking. We cannot test a condition on a dependent variable (beta) without thinking what the other variable (beta) is doing. What happens to a more complex case of lasso, where we know that betas are dependent?