This paper examines LASSO, a widely-used $L_{1}$-penalized regression method, in high dimensional linear predictive regressions, particularly when the number of potential predictors exceeds the sample size and numerous unit root regressors are present. The consistency of LASSO is contingent upon two key components: the deviation bound of the cross product of the regressors and the error term, and the restricted eigenvalue of the Gram matrix. We present new probabilistic bounds for these components, suggesting that LASSO's rates of convergence are different from those typically observed in cross-sectional cases. When applied to a mixture of stationary, nonstationary, and cointegrated predictors, LASSO maintains its asymptotic guarantee if predictors are scale-standardized. Leveraging machine learning and macroeconomic domain expertise, LASSO demonstrates strong performance in forecasting the unemployment rate, as evidenced by its application to the FRED-MD database.

翻译：本文研究LASSO（一种广泛使用的$L_{1}$惩罚回归方法）在高维线性预测回归中的应用，特别是在潜在预测变量数量超过样本量且存在大量单位根回归变量的情况下。LASSO的一致性取决于两个关键要素：回归变量与误差项交叉乘积的偏差界，以及Gram矩阵的受限特征值。我们为这些要素提出了新的概率界，表明LASSO的收敛速率与横截面情形下通常观察到的结果有所不同。当应用于包含平稳、非平稳和协整预测变量的混合场景时，若预测变量经过尺度标准化，LASSO仍能保持渐近性能保证。结合机器学习和宏观经济领域专业知识，LASSO在预测失业率方面表现出强大的性能，其在FRED-MD数据库中的应用结果也验证了这一点。