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?

Quotemultivariate normal degenerates to individual normal distributionsThis is right. However joint distribution of the estimated beta coef. are not independent normal. This is just 1 way to make inference about a particular beta coef. The thought process behind that is:"Okay, if null was true then, then all OLS estimated beta coef. will be multivariate normally distributed. Therefore there would be some Max. and Min. limit for all estimated beta (with some probability.) But as my estimated beta coef. falls outside that range, there must be something going wrong in my data in the sense that, that data might not come from my hypothetical DGP. Hence, null is false)".Otherwise you may be interested on jointly testing many estimated betas using appropriate Ch-Square statistic, which takes care of the correlation structure among the estimated betas.HTH

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