For the last two decades, high-dimensional data and methods have proliferated throughout the literature. Yet, the classical technique of linear regression has not lost its usefulness in applications. In fact, many high-dimensional estimation techniques can be seen as variable selection that leads to a smaller set of variables (a “submodel”) where classical linear regression applies. We analyze linear regression estimators resulting from model selection by proving estimation error and linear representation bounds uniformly over sets of submodels. Based on deterministic inequalities, our results provide “good” rates when applied to both independent and dependent data. These results are useful in meaningfully interpreting the linear regression estimator obtained after exploring and reducing the variables and also in justifying post-model-selection inference. All results are derived under no model assumptions and are nonasymptotic in nature.

在过去的二十年里，高维数据和方法在文献中激增。然而，线性回归的经典技术并没有失去其在应用中的有用性。事实上，许多高维估计技术可以被视为变量选择，导致应用经典线性回归的较小变量集（“子模型”）。我们通过在子模型集上统一证明估计误差和线性表示边界来分析由模型选择产生的线性回归估计量。基于确定性不平等，我们的结果在应用于独立数据和相关数据时提供“良好”的比率。这些结果对于有意义地解释在探索和减少变量后获得的线性回归估计量以及证明模型选择后推断的合理性很有用。所有结果都是在没有模型假设的情况下得出的，并且本质上是非渐近的。