High Dimensional Beta Test with High Frequency Data

This is the first paper about the high dimensional beta tests with high frequency financial data, which allowing that the number of regressors can be larger than the number of observations within each estimation block and can also grow to infinity in asymptotics. In this paper, the sum-type test and max-type test have been proposed, where the sum-type test is suitable for the dense alternative and the max-type test is suitable for the sparse alternative. By showing the asymptotic independence between the sum-type test and max-type test, a Fisher’s combination test is proposed, which is robust to both dense and sparse alternatives. The limiting null distributions of the three proposed tests are derived and the asymptotic behavior of their powers are also analyzed. Monte Carlo simulations demonstrate the validity of the theoretical results developed in this paper. Empirical study with real high frequency financial data shows the robustness of the proposed Fisher’s combination test under both dense and sparse alternatives