Testing covariates in high dimension linear regression with latent factors

We propose here both F-test and zz-test (or tt-test) for testing global significance and individual effect of each single predictor respectively in high dimension regression model when the explanatory variables follow a latent factor structure (Wang, 2012). Under the null hypothesis, together with fairly mild conditions on the explanatory variables and latent factors, we show that the proposed F-test and tt-test are asymptotically distributed as weighted chi-square and standard normal distribution respectively. That leads to quite different test statistics and inference procedures, as compared with that of Zhong and Chen (2011) when the explanatory variables are weakly dependent. Moreover, based on the pp-value of each predictor, the method of Storey et al. (2004) can be used to implement the multiple testing procedure, and we can achieve consistent model selection as long as we can select the threshold value appropriately. All the results are further supported by extensive Monte Carlo simulation studies. The practical utility of the two proposed tests are illustrated via a real data example for index funds tracking in China stock market.