Estimating causal vs non-causal deviations and optimal source-domain selection

Determine practical procedures to estimate the Causal Deviation Δ_c and the Non-Causal Deviation Δ_{nc} in the constrained optimization-based causal discovery framework for stock prediction, where Δ_c and Δ_{nc} are defined respectively as the minimal average squared deviation across source domains of the domain-wise least-squares regression coefficients for features φ(X) that correspond to causal features and for features φ(X) that correspond to non-causal features; and identify the optimal selection and partitioning of source domains (i.e., training horizon and domain construction) that yields favorable values of Δ_c and Δ_{nc} to enable reliable discovery of causal features.

Background

In the causal discovery section, the authors introduce a constrained optimization approach to learn invariant feature representations φ(X) that maintain a stable linear relationship with standardized returns across domains. They define two key quantities: the Causal Deviation Δc and the Non-Causal Deviation Δ{nc}, which quantify, respectively, the variance across source domains of the least-squares coefficients associated with causal and non-causal (spurious) features.

Proposition 4.1 shows that if Δc < Δ{nc}, the optimization discovers generalizable causal features under suitable bounds. However, the authors note that both Δc and Δ{nc} depend on the choice of training horizon and how source domains are constructed, and that estimating these quantities or finding the best domain configuration is non-trivial. They explicitly mark this as an open question, highlighting the need for methodological development to estimate these deviations and design domain selection strategies that facilitate correct causal discovery.