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

TL;DR

This paper introduces $L_q$-norm based tests for high-dimensional linear factor pricing models, bridging the gap between dense and sparse alternative detection, and demonstrating improved robustness and performance.

Contribution

It develops a new class of $L_q$-based tests, including $L_4$ and $L_6$, with an asymptotic independence property enabling a combined adaptive testing procedure.

Findings

The $L_q$ tests are more robust to unknown sparsity levels.

Simulation results show the proposed methods outperform existing tests.

Real-data analysis confirms practical effectiveness.

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.