Most powerful test against a sequence of high dimensional local alternatives

We develop a powerful quadratic test for the overall significance of many covariates in a dense regression model in the presence of nuisance parameters. By equally weighting the sample moments, the test is asymptotically correct in high dimensions even when the number of coefficients is larger than the sample size. Our theory allows a non-parametric error distribution and weakly exogenous nuisance variables, in particular autoregressors in many applications. Using random matrix theory, we show that the test has the optimal asymptotic testing power among a large class of competitors against local alternatives whose coordinates are dense in the eigenbasis of the high dimensional sample covariance matrix among regressors. The asymptotic results are adaptive to the covariates’ cross-sectional and temporal dependence structure and do not require a limiting spectral law of their sample covariance matrix. In the most general case, the nuisance estimation may play a role in the asymptotic limit and we give a robust modification for these irregular scenarios. Monte Carlo studies suggest a good power performance of our proposed test against high dimensional dense alternative for various data generating processes. We apply the test to detect the significance of over one hundred exogenous variables in the FRED-MD database for predicting the monthly growth in the US industrial production index.