  A great deal of interest has recently focused on conducting inference on the
parameters in a high-dimensional linear model.
  In this paper, we consider a simple and very na\"{i}ve two-step procedure for
this task, in which we (i) fit a lasso model in order to obtain a subset of the
variables, and (ii) fit a least squares model on the lasso-selected set.
Conventional statistical wisdom tells us that we cannot make use of the
standard statistical inference tools for the resulting least squares model
(such as confidence intervals and $p$-values), since we peeked at the data
twice: once in running the lasso, and again in fitting the least squares model.
However, in this paper, we show that under a certain set of assumptions, with
high probability, the set of variables selected by the lasso is identical to
the one selected by the noiseless lasso and is hence deterministic.
Consequently, the na\"{i}ve two-step approach can yield asymptotically valid
inference. We utilize this finding to develop the \emph{na\"ive confidence
interval}, which can be used to draw inference on the regression coefficients
of the model selected by the lasso, as well as the \emph{na\"ive score test},
which can be used to test the hypotheses regarding the full-model regression
coefficients.
