F statistics for high-dimensional inference of linear model

We aim to estimate and conduct inference for the effects of multiple covariates of interest simultaneously after adjusting the impact of high-dimensional control variables under a linear model. An F-type statistic is proposed based on the residuals from the regularized fitting of the response variable and the target covariates on the control covariates. A testing procedure and confidence interval are constructed. The proposed procedures reduce the impact of potential over-fitting errors from the regularized regression on the inference of the target parameters. The essence is to eliminate the prediction errors in the direction of the actual regression error and achieve a more accurate size and coverage rate. Expansions of the proposed statistics are derived, showing the proposed method's error reduction property. Simulation studies verify the theoretical results and demonstrate the proposed method has better performance than the existing methods. A real data analysis for S&P 500 stock returns is conducted to show the utility of the proposed method in practice.