On the number of variables to use in principal component regression

We study least squares linear regression over $N$ uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features $p$ is at most the sample size $n$, the estimator under consideration coincides with the principal component regression estimator; when $p> n$, the estimator is the least $\ell_2$ norm solution over the selected features.