  Many statistical methods have been proposed for variable selection in the
past century, but few balance inference and prediction tasks well. Here we
report on a novel variable selection approach called Penalized regression with
Second-Generation P-Values (ProSGPV). It captures the true model at the best
rate achieved by current standards, is easy to implement in practice, and often
yields the smallest parameter estimation error. The idea is to use an l0
penalization scheme with second-generation p-values (SGPV), instead of
traditional ones, to determine which variables remain in a model. The approach
yields tangible advantages for balancing support recovery, parameter
estimation, and prediction tasks. The ProSGPV algorithm can maintain its good
performance even when there is strong collinearity among features or when a
high dimensional feature space with p > n is considered. We present extensive
simulations and a real-world application comparing the ProSGPV approach with
smoothly clipped absolute deviation (SCAD), adaptive lasso (AL), and mini-max
concave penalty with penalized linear unbiased selection (MC+). While the last
three algorithms are among the current standards for variable selection,
ProSGPV has superior inference performance and comparable prediction
performance in certain scenarios. Supplementary materials are available online.
