  We develop novel LASSO-based methods for coefficient testing and confidence
interval construction in the Gaussian linear model with $n\ge d$. Our methods'
finite-sample guarantees are identical to those of their ubiquitous
ordinary-least-squares-$t$-test-based analogues, yet have substantially higher
power when the true coefficient vector is sparse. In particular, our
coefficient test, which we call the $\ell$-test, performs like the one-sided
$t$-test (despite not being given any information about the sign) under
sparsity, and the corresponding confidence intervals are more than 10% shorter
than the standard $t$-test based intervals. The nature of the $\ell$-test
directly provides a novel exact adjustment conditional on LASSO selection for
post-selection inference, allowing for the construction of post-selection
p-values and confidence intervals. None of our methods require resampling or
Monte Carlo estimation. We perform a variety of simulations and a real data
analysis on an HIV drug resistance data set to demonstrate the benefits of the
$\ell$-test. We end with a discussion of how the $\ell$-test may asymptotically
apply to a much more general class of parametric models.
