Self-Normalizing Denominators in Rational Causal Estimation

Abstract: Rational causal estimators in linear structural equation models take the form of one covariance polynomial divided by another, and a small denominator is commonly interpreted as weak identification. We show that, under Gaussian sampling, some denominators cannot enter this regime at first order. Their sampling variation is exactly proportional to their magnitude, so the standardized denominator is constant in every sample. Products of powers of nested covariance minors have this property in every dimension and admit an exact Wishart pivot. The converse is complete in dimension two. In dimension three, one mixed family remains open, while a factor-and-rank criterion classifies all denominators with linear or quadratic determinant-free factors and covers instrumental-variable, front-door and proximal formulas. For linear front-door adjustment, Wald inference remains asymptotically valid even as the mediator residual variance vanishes at an arbitrary rate, provided the treatment--mediator coefficient is nonzero. In simulations, proximal Wald coverage fell as a naive treatment--proxy diagnostic strengthened, while front-door coverage stayed nominal, and right-heart-catheterization data distinguished naive from denominator-relevant diagnostics.