Test of conditional independence in factor models via Hilbert–Schmidt independence criterion

Abstract

This work is concerned with testing conditional independence under a factor model setting. We propose a novel multivariate test for non-Gaussian data based on the Hilbert–Schmidt independence criterion (HSIC). Theoretically, we investigate the convergence of our test statistic under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution, showing that its consistency is justified. Methodologically, we generalize the HSIC-based independence test approach to a situation where data follow a factor model structure. Our test requires no nonparametric smoothing estimation of functional forms including conditional probability density functions, conditional cumulative distribution functions and conditional characteristic functions under the null or alternative, is computationally efficient and is dimension-free in the sense that the dimension of the conditioning variable is allowed to be large but finite. Further extension to nonlinear, non-Gaussian structure equation models is also described in detail and asymptotic properties are rigorously justified. Numerical studies demonstrate the effectiveness of our proposed test relative to that of several existing tests.