Variable selection in linear regressions with possibly all strongly correlated covariates

Penalized regression methods, particularly LASSO, require the irrepresentable condition for variable selection consistency. This condition places an upper bound on the magnitudes of correlation between the signals and the rest of the covariates under consideration. Chudik, Kapetanios, and Pesaran (2018) proposed an alternative procedure called the one covariate at a time multiple testing (OCMT) that does not impose restrictions on the magnitude of these correlations. However, it requires that the number of covariates correlated with the signals grows at a rate less than the square root of the number of observations, denoted by T. Notably, the required conditions for both LASSO and OCMT can be challenged when possibly all the covariates under consideration are correlated with the signals. In this article, we follow the ideas from latent factor literature to adapt OCMT to allow for such scenarios. We refer to our proposed method as generalized one covariate at a time multiple testing (GOCMT). We establish that GOCMT selects a model that contains all the signals and none of the noise variables asymptotically. We also show that the least squares estimator of the post GOCMT selected model is T-consistent. The proposed method demonstrates promising finite-sample performance in our Monte Carlo experiments. An empirical application in risk factor selection within asset pricing further underscores the utility of our method.惩罚回归方法（尤其是套索回归(LASSO)）要实现变量选择一致性，需满足不可表示条件。该条件对目标信号与其余待考虑协变量间的相关系数幅值设定了上限。Chudik、Kapetanios与Pesaran（2018）提出了一种名为每次单协变量多重检验(OCMT)的替代方法，该方法无需对上述相关系数的幅值施加限制。但该方法要求与信号相关的协变量数量的增长速率，需低于以T表示的观测数的平方根。值得注意的是，当所有待考虑协变量均与信号存在相关性时，套索回归与OCMT的适用条件均可能不再成立。本文借鉴潜在因子领域的研究思路，对OCMT进行改进以适配此类场景。我们将所提出的方法命名为广义每次单协变量多重检验(GOCMT)。我们证明了GOCMT能够在渐近意义下选出包含所有信号变量且不包含任何噪声变量的模型。此外，我们证明了GOCMT选择模型后的最小二乘估计量满足T相合性。在我们的蒙特卡洛(Monte Carlo)实验中，所提方法展现出了优异的有限样本表现。在资产定价领域的风险因子选择实证应用中，本文方法的实用性得到了进一步验证。