  The computational complexity of simultaneous inference methods in
high-dimensional linear regression models quickly increases with the number
variables. This paper proposes a computationally efficient method based on the
Moore-Penrose pseudoinverse. Under a symmetry assumption on the available
regressors, the estimators are normally distributed and accompanied by a
closed-form expression for the standard errors that is free of tuning
parameters. We study the numerical performance in Monte Carlo experiments that
mimic the size of modern applications for which existing methods are
computationally infeasible. We find close to nominal coverage, even in settings
where the imposed symmetry assumption does not hold. Regularization of the
pseudoinverse via a ridge adjustment is shown to yield possible efficiency
gains.
