Title: Inference in high-dimensional linear regression models,
Abstract: We introduce an asymptotically unbiased estimator for the full
high-dimensional parameter vector in linear regression models where the number
of variables exceeds the number of available observations. The estimator is
accompanied by a closed-form expression for the covariance matrix of the
estimates that is free of tuning parameters. This enables the construction of
confidence intervals that are valid uniformly over the parameter vector.
Estimates are obtained by using a scaled Moore-Penrose pseudoinverse as an
approximate inverse of the singular empirical covariance matrix of the
regressors. The approximation induces a bias, which is then corrected for using
the lasso. Regularization of the pseudoinverse is shown to yield narrower
confidence intervals under a suitable choice of the regularization parameter.
The methods are illustrated in Monte Carlo experiments and in an empirical
example where gross domestic product is explained by a large number of
macroeconomic and financial indicators.