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

Ping Zhao, Huifang Ma, Long Feng

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.

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

Abstract

We consider testing zero pricing errors in high-dimensional linear factor pricing models. Existing methods are mainly based on either an statistic, which is effective under dense alternatives, or an statistic, which is powerful under very sparse alternatives. To bridge these two regimes, we develop a class of -based tests for finite , including the practically useful and cases. We show that larger leads to greater sensitivity to sparse alternatives. We further establish the asymptotic independence between the statistic and the statistic for any finite , 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.

Paper Structure

This paper contains 13 sections, 15 theorems, 275 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Single $L_q$-norm Test
  4. Cauchy Combination Test
  5. Simulation
  6. Real Data Application
  7. Conclusion
  8. Appendix
  9. Useful Lemmas
  10. Proof of Theorem \ref{['thm:null-joint']}
  11. Proof of Theorem \ref{['thm:Bhat']}
  12. Proof of Theorem \ref{['thm:power']}
  13. Proof of Theorem \ref{['thm:indep']}

Key Result

Theorem 2.1

Suppose Assumptions ass:factor--ass:growth hold and $H_0$ is true. Then Equivalently, whenever $\mathbf{B}_N\to \mathbf{B}$ for some positive definite matrix $\mathbf{B}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Power curves of six methods with $\delta_\gamma=0$ and $(T,N)=(120,200)$.
  • Figure 2: Power curves of six methods with $\delta_\gamma=1/4$ and $(T,N)=(120,200)$.
  • Figure 3: Power curves of six methods with $\delta_\gamma=1/2$ and $(T,N)=(120,200)$.
  • Figure 4: p-value paths of the six tests over rolling windows.

Theorems & Definitions (27)

  • Remark 2.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.1
  • Lemma 6.1: Powers of the weight vector
  • proof
  • Lemma 6.2: Cumulants of the projected components
  • ...and 17 more