Most Powerful Test against High Dimensional Free Alternatives

We propose a powerful quadratic test for the overall significance of many weak exogenous variables in a dense autoregressive model. By shrinking the classical weighting matrix on the sample moments to be identity, 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 estimation of the autoregressive coefficients. Using random matrix theory, we show that the test has the optimal asymptotic testing power among a large class of competitors against local dense alternatives whose direction is free in the eigenbasis of the sample covariance matrix among regressors. The asymptotic results are adaptive to the predictors' cross-sectional and temporal dependence structure, and do not require a limiting spectral law of their sample covariance matrix. The method extends beyond autoregressive models, and allows more general nuisance parameters. Monte Carlo studies suggest a good power performance of our proposed test against high dimensional dense alternative for various data generating processes. We apply our tests to detect the overall significance of over one hundred exogenous variables in the latest FRED-MD database for predicting the monthly growth in the US industrial production index.