  We propose a method for constructing p-values for general hypotheses in a
high-dimensional linear model. The hypotheses can be local for testing a single
regression parameter or they may be more global involving several up to all
parameters. Furthermore, when considering many hypotheses, we show how to
adjust for multiple testing taking dependence among the p-values into account.
Our technique is based on Ridge estimation with an additional correction term
due to a substantial projection bias in high dimensions. We prove strong error
control for our p-values and provide sufficient conditions for detection: for
the former, we do not make any assumption on the size of the true underlying
regression coefficients while regarding the latter, our procedure might not be
optimal in terms of power. We demonstrate the method in simulated examples and
a real data application.
