High dimensional alpha test for linear factor pricing model with $L_q$-norm

2026-03-31

AI Summary

This study addresses the instability of existing tests in high-dimensional linear factor pricing models when the sparsity of the alternative hypothesis is unknown. It introduces, for the first time, a systematic testing framework based on finite-order $L_q$ norms (e.g., $L_4$, $L_6$), demonstrating their heightened sensitivity to sparse alternatives. By combining these statistics with an $L_\infty$-based test and establishing their asymptotic independence, the authors construct a Cauchy combination test that adaptively accommodates unknown sparsity levels. Both theoretical analysis and empirical evidence show that the proposed method substantially outperforms existing approaches in finite samples, offering superior robustness and statistical power.

Abstract

We consider testing zero pricing errors in high-dimensional linear factor pricing models. Existing methods are mainly based on either an $L_2$ statistic, which is effective under dense alternatives, or an $L_\infty$ statistic, which is powerful under very sparse alternatives. To bridge these two regimes, we develop a class of $L_q$-based tests for finite $q$, including the practically useful $L_4$ and $L_6$ cases. We show that larger $q$ leads to greater sensitivity to sparse alternatives. We further establish the asymptotic independence between the $L_\infty$ statistic and the $L_q$ statistic for any finite $q$, which motivates a Cauchy combination test that adapts to a broad range of sparsity levels. Simulation studies and a real-data analysis show that the proposed methods are more robust to the unknown sparsity of the alternative and can outperform existing procedures in finite samples.

Problem

• high-dimensional
• factor pricing model
• pricing error
• sparsity
• hypothesis testing

Innovation

• L_q-norm test
• high-dimensional factor pricing
• sparsity-adaptive testing
• Cauchy combination test
• asymptotic independence