A Predictable E-Process for Anytime-Valid Overidentification Monitoring in GMM

Classical Hansen–Sargan overidentification testing is not robust to optional stopping and repeated “peeking” in real-time monitoring. This manuscript intro- duces the S M Nazmuz Sakib eJ-Test and the associated Sakib Index SIt, an anytime-valid evidence stream for overidentified moment restrictions. The key idea is a predictable orthogonalization of GMM moments that removes parameter directions online, combined with a Gaussian mixture supermartingale to produce an e-process Et that is valid under optional continuation. All empirical illustrations are dataset-based and use open World Bank data on U.S. inflation and unemployment (1975–2024). In our inflation illustration, the resulting e-process crosses the 1/α threshold for α = 0.05 in year 2023, while an expanding-window J-path crosses much earlier, illustrating how repeated monitoring can distort classical fixed-sample testing.