# What are the relative merits of testing significance of a regression coefficient

If I have several hundred coefficients generated by running multiple variable regression model (keeping it as broad as possible by not specifying the nature of the predictors and outcome variable), it would appear to me that I have two options for assessing the significance of any one result.

1. Identify the largest standardized predictors by calculating z-scores for regression coefficients and assessing how many exceed a fixed threshold (e.g. 2 Z-scores from 0).

2. Use permutation testing to simulate the distribution of covariates under a null hypothesis (of no covariates being associated with the outcome) and calculate a p-value by comparing the actual results, ranked by their order of significance calculated from Z-scores, to the distributions of ranked coefficients in the permutation tests

My question is: What are the advantages and disadvantages of each approach?

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The biggest problem here, especially if $p \gg n$ is that you waste a lot actually estimating values for coefficients you will not ultimately care about. Why not use a LASSO to constrain variables that are very close to zero to actually take 0 as a value? Then you can spend your data estimating a smaller model more reliably. The LASSO is a fantastic method for feature selection and prediction.