  Over the last two decades, many exciting variable selection methods have been
developed for finding a small group of covariates that are associated with the
response from a large pool. Can the discoveries from these data mining
approaches be spurious due to high dimensionality and limited sample size? Can
our fundamental assumptions about the exogeneity of the covariates needed for
such variable selection be validated with the data? To answer these questions,
we need to derive the distributions of the maximum spurious correlations given
a certain number of predictors, namely, the distribution of the correlation of
a response variable $Y$ with the best $s$ linear combinations of $p$ covariates
$\mathbf{X}$, even when $\mathbf{X}$ and $Y$ are independent. When the
covariance matrix of $\mathbf{X}$ possesses the restricted eigenvalue property,
we derive such distributions for both a finite $s$ and a diverging $s$, using
Gaussian approximation and empirical process techniques. However, such a
distribution depends on the unknown covariance matrix of $\mathbf{X}$. Hence,
we use the multiplier bootstrap procedure to approximate the unknown
distributions and establish the consistency of such a simple bootstrap
approach. The results are further extended to the situation where the residuals
are from regularized fits. Our approach is then used to construct the upper
confidence limit for the maximum spurious correlation and to test the
exogeneity of the covariates. The former provides a baseline for guarding
against false discoveries and the latter tests whether our fundamental
assumptions for high-dimensional model selection are statistically valid. Our
techniques and results are illustrated with both numerical examples and real
data analysis.
