Is price-impact decay statistically identifiable by propagator estimation? Diffusive prices force the inferred kernel to be transient and fine-tuned

Market-order flow has long memory, yet prices are close to diffusive. The propagator model reconciles the two by letting price impact decay at a rate finely balanced against the order-flow autocorrelation, and kernels estimated from data have seemed to confirm that balance.

The authors prove that the estimate cannot tell. Propagators are inferred by Wiener–Hopf deconvolution, and under long-memory flow the inferred kernel is necessarily transient even when the true impact is permanent; under a mild condition it decays with the fine-balance exponent, which depends only on the order-flow autocorrelation. An exactly solvable model with order splitting, the square-root law and strictly permanent impact still produces an apparent fine-tuned decay. Nonlinear propagator estimators of Wiener–Hopf type are subject to the same theorems.