Algebraic and Statistical Properties of the Ordinary Least Squares
  Interpolator
Dennis Shen Dogyoon Song Peng Ding Jasjeet S. Sekhon
Deep learning research has uncovered the phenomenon of benign overfitting for
over-parameterized statistical models, which has drawn significant theoretical
interest in recent years. Given its simplicity and practicality, the ordinary
least squares (OLS) interpolator has become essential to gain foundational
insights into this phenomenon. While properties of OLS are well established in
classical settings, its behavior in high-dimensional settings is less explored
(unlike for ridge or lasso regression) though significant progress has been
made of late. We contribute to this growing literature by providing fundamental
algebraic and statistical results for the minimum $\ell_2$-norm OLS
interpolator. In particular, we provide high-dimensional algebraic equivalents
of (i) the leave-$k$-out residual formula, (ii) Cochran's formula, and (iii)
the Frisch-Waugh-Lovell theorem. These results aid in understanding the OLS
interpolator's ability to generalize and have substantive implications for
causal inference. Additionally, under the Gauss-Markov model, we present
statistical results such as a high-dimensional extension of the Gauss-Markov
theorem and an analysis of variance estimation under homoskedastic errors. To
substantiate our theoretical contributions, we conduct simulation studies that
further explore the stochastic properties of the OLS interpolator.