Conjecture on the absorption coefficient c(Δ, p) in LOCO

Determine whether the proposed absorption coefficient c(Δ, p) = (2Δ + (p−2)Δ^2) / (1 − 4Δ + 2pΔ + p^2Δ^2 − 3pΔ^2 + 3Δ^2) correctly characterizes how the removed covariate’s effect is redistributed among the remaining predictors when retraining a linear regression model in the Leave-One-Covariate-Out (LOCO) procedure under the latent variable framework X = Z(ΔJ + (1 − Δ)I). Prove or refute this conjecture by deriving c as an explicit function of Δ and p from first principles.

Background

In the LOCO derivation, the authors model the effect of permuting a training covariate by refitting the linear regression and assuming that the removed covariate’s coefficient is absorbed uniformly by the remaining correlated predictors. This redistribution is parameterized by an absorption coefficient c, which enters the closed-form LOCO expression and depends on the correlation parameter Δ and the dimension p.

While LOCO admits an exact form contingent on c, the authors propose a conjectured analytical formula for c(Δ, p) based on empirical observations. They note that, if true, this conjecture yields a simplified closed-form LOCO expression and a precise connection between LOCO and the t-statistic. Monte Carlo experiments provide empirical support, but a formal proof remains to be established.