Bootstrap validity after autoregressive order and innovation estimation

Establish the full statistic-specific validity of the prewhitened recursive-sign bootstrap after autoregressive order selection and innovation estimation.

Background

The paper develops a prewhitened recursive Rademacher bootstrap for serial observations. The procedure estimates a finite-order autoregression, selects its order using BIC, estimates the autoregressive coefficients and innovations, randomly changes the signs of the estimated dated innovations, recursively reconstructs bootstrap outcomes, and recomputes the complete finite-grid MBCMI statistic.

The paper proves validity for a restricted class of stable finite-order autoregressions with conditionally sign-symmetric innovations, and separately establishes oracle sign-randomisation validity and asymptotic replacement results under stated assumptions. However, the authors explicitly identify the broader proof of validity for the complete statistic after both order selection and innovation estimation as unresolved. This problem concerns closing that remaining theoretical gap for the implemented serial bootstrap construction.