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For the last two decades, high-dimensional data and methods have proliferatedthroughout the literature. Yet, the classical technique of linear regressionhas not lost its usefulness in applications. In fact, many high-dimensionalestimation techniques can be seen as variable selection that leads to a smallerset of variables (a ``sub-model'') where classical linear regression applies.We analyze linear regression estimators resulting from model-selection byproving estimation error and linear representation bounds uniformly over setsof submodels. Based on deterministic inequalities, our results provide ``good''rates when applied to both independent and dependent data. These results areuseful in meaningfully interpreting the linear regression estimator obtainedafter exploring and reducing the variables and also in justifying postmodel-selection inference. All results are derived under no model assumptionsand are non-asymptotic in nature.

Uniform-in-Submodel Bounds for Linear Regression in a Model Free Framework