Testing Alpha in High-Dimensional Conditional Time-Varying Factor Models with Dependent Observations

TL;DR

This paper develops new statistical tests for alpha in high-dimensional, time-varying factor models with dependent data, combining sum, max, and Cauchy methods for robust inference.

Contribution

It introduces a comprehensive testing framework using B-spline sieve methods, with theoretical guarantees and practical validation for dependent, high-dimensional data.

Findings

Proposed tests control size well and have high power.

Derived asymptotic distributions for sum and max statistics.

Validated methods with simulations and an asset-pricing application.

Abstract

This paper studies alpha testing in a high-dimensional conditional time-varying factor model with temporally dependent observations. Both factor loadings and alpha processes are allowed to vary smoothly over time, and the cross-sectional dimension may be comparable to or larger than the sample size. Using a B-spline sieve method, we develop a sum-type test for dense alternatives, a max-type test for sparse alternatives, and a Cauchy combination test for adaptive inference. On the theoretical side, we derive explicit stochastic expansions for the estimated average alphas, establish asymptotic normality of the sum statistic, and develop the extreme-value limit theory for the max statistic by showing its Gumbel convergence under temporal dependence together with the validity of block-bootstrap calibration. We further prove asymptotic independence between the sum and max statistics and thereby justify the Cauchy combination test. Simulation results demonstrate that the proposed procedures achieve satisfactory size control and competitive power across a wide range of dense and sparse alternatives. An empirical application further illustrates the usefulness of the proposed methods in testing asset-pricing models with time-varying structure.