Encompassing and Progression Testing in High-Dimensions

This paper introduces the Subseries-based Cauchy Combination Test (SCT), a novel procedure for testing encompassing relationships or model validity using identification conditions formulated as multiple moment restrictions. SCT applies to weakly or short-range-dependent data and eliminates the need to estimate high-dimensional covariance matrices. Unlike Wald- or J-type tests, it remains reliable in both low- and high-dimensional settings. The test is asymptotically unbiased and near-minimax-rate optimal, with asymptotic power no less than that of an oracle max-type test under alternatives in which the selected model fails to encompass the valid model. SCT accommodates redundancy, progression, and nonlinearity testing in rank-deficient systems. As an empirical illustration, we apply SCT to the U.S. factor zoo and show how a handful of factors effectively span the country-level factors over the period 1964–2022.