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In this paper we give a completely new approach to the problem of covariateselection in linear regression. A covariate or a set of covariates is includedonly if it is better in the sense of least squares than the same number ofGaussian covariates consisting of i.i.d. $N(0,1)$ random variables. TheGaussian P-value is defined as the probability that the Gaussian covariates arebetter. It is given in terms of the Beta distribution, it is exact and it holdsfor all data. The covariate selection procedures based on this require only acut-off value $\alpha$ for the Gaussian P-value: the default value in thispaper is $\alpha=0.01$. The resulting procedures are very simple, very fast, donot overfit and require only least squares. In particular there is noregularization parameter, no data splitting, no use of simulations, noshrinkage and no post selection inference is required. The paper includes theresults of simulations, applications to real data sets and theorems on theasymptotic behaviour under the standard linear model. Here the stepwiseprocedure performs overwhelmingly better than any other procedure we are awareof. An R-package {\it gausscov} is available.

Covariate Selection Based on a Assumpton-free Approach to Linear Regression with Exact Probabilities