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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.

The least squares estimator can be cast as depending only on the precision matrix of the data, similar to the weights of a global minimum variance portfolio. We give conditions under which any plug-in precision matrix estimator produces an unbiased and consistent least squares estimator for stationary time series regressions, in both low- and high-dimensional settings. Such conditions define a class of "Precision Least Squares" (PrLS) estimators, which are shown to be approximately Gaussian, efficient, and to provide automatic family-wise error control in large samples. For estimating high-dimensional sparse regression models, we propose a LASSO Cholesky estimator of the plug-in precision matrix. We show its consistency and how to properly bias correct it, thereby obtaining a LASSO Cholesky-based PrLS (LC-PrLS) estimator. LC-PrLS performs well in finite samples and better than state-of-the-art high-dimensional estimators. We employ LC-PrLS to investigate the dynamic network of predictive connections among a large set of global bank stock returns. We find that crisis years correspond to a collapse of predictive linkages.