  This paper explores the following question: what kind of statistical
guarantees can be given when doing variable selection in high-dimensional
models? In particular, we look at the error rates and power of some multi-stage
regression methods. In the first stage we fit a set of candidate models. In the
second stage we select one model by cross-validation. In the third stage we use
hypothesis testing to eliminate some variables. We refer to the first two
stages as "screening" and the last stage as "cleaning." We consider three
screening methods: the lasso, marginal regression, and forward stepwise
regression. Our method gives consistent variable selection under certain
conditions.
