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 $L_2$ 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.

• The paper introduces a continuum of Lq-norm tests that detect nonzero alphas in high-dimensional pricing models, blending performance between dense and sparse alternatives.
• It develops explicit asymptotic variance expressions and finite-sample corrections to ensure accurate calibration even under weak cross-sectional dependencies.
• An adaptive Cauchy combination test is proposed, demonstrating near-optimal power across different sparsity regimes in both simulation studies and empirical CAPM analysis.

High-Dimensional Alpha Testing for Linear Factor Pricing Models Using $L_q$-Norms

Introduction and Motivation

This work introduces an adaptive framework for hypothesis testing in high-dimensional linear factor pricing models, focusing on the presence of nonzero pricing errors (alphas) when the cross-sectional asset dimension $N$ is large and potentially exceeds the time dimension $T$—a regime characteristic of modern empirical finance. The paper reformulates the classical alpha test as a global hypothesis test on the intercept vector $\boldsymbol{\alpha}$ in models such as the CAPM and APT, with $H_0: \boldsymbol{\alpha} = 0$.

Conventional methodologies, including the GRS test and average $t$-statistics, become unreliable as $N$ increases due to covariance matrix estimation instability. Recent approaches tailored for high-dimensions typically deploy $L_2$-type (quadratic/sum-of-squares) or $L_\infty$-type (maximum) test statistics, each being optimal in distinct sparsity regimes. However, the actual sparsity of mispricing signals in empirical settings is unknown, motivating robust tests capable of power enhancement across a continuum of alternatives.

Methodological Innovations

$L_q$-Norm Test Family

The central innovation is a continuum of tests indexed by an even integer $N$0, interpolating between dense ($N$1) and ultra-sparse ($N$2) alternatives. The $N$3-statistic is defined via standardized OLS intercept estimates: $N$4 where $N$5 is the studentized intercept and $N$6 is the $N$7-th central moment of a $N$8-distribution. For $N$9, the corresponding $T$0, $T$1, and $T$2 statistics capture deviations due to alternative sparsity levels.

The authors derive asymptotic normality and explicit variance expressions for these statistics under general weak cross-sectional dependence, leveraging independent component models (ICM) for idiosyncratic errors. Finite-sample variance corrections are provided to ensure calibration accuracy.

Adaptive Test Combination via Cauchy Aggregation

A key theoretical result is the asymptotic independence between $T$3 (max-type extreme value) and the finite-$T$4 $T$5 statistics under the null, enabling the construction of combination rules. The paper proposes a four-way Cauchy combination test (CC), which aggregates $T$6-values from $T$7, $T$8, $T$9, and $\boldsymbol{\alpha}$0, producing a test with null distribution approximated by the standard Cauchy law: $\boldsymbol{\alpha}$1 This mechanism adaptively borrows power from whichever underlying test is optimal for the unknown sparsity pattern.

Power Analysis

Analytic power calculations show that, for dense signal alternatives, the $\boldsymbol{\alpha}$2 statistic is optimal; as sparsity increases, $\boldsymbol{\alpha}$3 and then $\boldsymbol{\alpha}$4 increase in relative power, overtaking $\boldsymbol{\alpha}$5 in extreme sparsity. The Cauchy combination is numerically demonstrated to maintain maximal or near-maximal power across all regimes.

Simulation Results

A comprehensive simulation study considers realistic factor structures, latent omitted factors, heavy-tailed idiosyncratic errors, and spatial cross-sectional dependence. Nominal Type I error rates are well-maintained across all six methods ($\boldsymbol{\alpha}$6, $\boldsymbol{\alpha}$7, $\boldsymbol{\alpha}$8, $\boldsymbol{\alpha}$9, minP, and CC), and the power investigation substantiates the theoretical power ordering:

• Very sparse alternatives: $H_0: \boldsymbol{\alpha} = 0$0 and $H_0: \boldsymbol{\alpha} = 0$1 deliver highest power.
• Moderate sparsity: $H_0: \boldsymbol{\alpha} = 0$2 and the Cauchy combination are most competitive.
• Dense alternatives: $H_0: \boldsymbol{\alpha} = 0$3 dominates.

Notably, the adaptive Cauchy combination test achieves close-to-optimal power throughout, with a marked advantage over the minP rule in moderately sparse regimes.

Figure 1: Power curves of six methods with $H_0: \boldsymbol{\alpha} = 0$4 and $H_0: \boldsymbol{\alpha} = 0$5.

Figure 2: Power curves of six methods with $H_0: \boldsymbol{\alpha} = 0$6 and $H_0: \boldsymbol{\alpha} = 0$7.

Figure 3: Power curves of six methods with $H_0: \boldsymbol{\alpha} = 0$8 and $H_0: \boldsymbol{\alpha} = 0$9.

Empirical Application: CAPM on S&P 500

The empirical study applies these tests to time-rolling CAPM regressions for S&P 500 constituents, using a roughly 20-year monthly panel. Rolling-window $t$0-value paths reveal:

• Persistent and substantial departures from zero-alpha, indicating CAPM mispricing is neither rare nor concentrated in a small number of assets.
• $t$1, $t$2, and CC dominate in power, with $t$3 consistently less sensitive. This suggests cross-sectional mispricing is moderately sparse or diffuse rather than featuring a few extreme outliers.
• The CC and minP combination rules closely track the most powerful individual test.

  Figure 4: $t$4-value paths of the six tests over rolling windows.

Implications and Future Directions

The main practical implication is that practitioners in high-dimensional asset pricing environments should favor $t$5-based or combination procedures over single-regime statistics. The $t$6-test, in particular, emerges as a robust, non-combinatorial option.

From a theoretical perspective, the framework suggests that critical value calibration for global testing in high dimensions can be greatly enhanced by leveraging norm-indexed test statistics with adaptive combination methodology, without intensive resampling or reliance on parametric sparsity assumptions.

Future research avenues include extending this framework to temporally dependent observations and conditional factor models with time-varying loadings and risk premia, which are characteristic of dynamic financial markets.

Conclusion

This paper provides a comprehensive, rigorously justified, and empirically validated foundation for high-dimensional alpha testing in linear factor pricing models using a unified $t$7-norm-based methodology and adaptive combination tests. The approach offers substantial improvements in robustness and power across unknown alternative regimes, establishing new best practices for econometric asset pricing hypothesis testing (2603.29764).