Spurious discoveries and endogeneity for Big Data

                         Jianqing Fan
                        Princeton University
         (Joint work with Qiman Shao and Wenxin Zhou)
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 by such data mining approaches be spurious?  Can our fundamental assumptions on exogeneity of 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 certain number of predictors. When the covariance matrix of covariates possesses the restricted eigenvalue property, we derive such distributions, using Gaussian approximation and empirical process techniques.  However, such a distribution depends on the unknown covariance matrix of the covariate. Hence, we propose a multiplier bootstrap method to approximate the unknown distributions and establish the consistency of such a simple bootstrap approach. The results are further extended to the situation where residuals are from regularized fits.  Our approach is then applied to construct the upper confidence limit for the maximum spurious correlation and testing exogeneity of covariates.   The former provides a baseline for guiding false discoveries due to data mining and the latter tests whether our fundamental assumptions for high-dimensional model selection are statistically valid.  Our techniques and results are
illustrated by both numerical examples.