A Characterization of the $M$-tests Under Nearly Integrated Nearly White Noise

2026-09-24

AI Summary

This study addresses the limiting distribution of M-tests for nearly integrated near-white noise (NINW) processes under unknown linear trends, alongside the inherent difficulties in handling sequences with large negative moving average coefficients. By leveraging unit root statistical theory and quasi-differencing techniques, the authors derive the asymptotic distributions of the M-test family within the NINW framework. Finite-sample simulations are further employed to systematically examine the discrepancies between GLS and OLS detrending and the impact of long-run variance estimation. The results demonstrate that the Gaussian power envelope is asymptotically equivalent to the standard benchmark, thereby revealing the power inefficiency of the oracle M-test. Moreover, this work highlights the absence of a uniformly optimal solution among existing methods and clarifies their fundamental limitations when applied to sequences characterized by large negative moving average coefficients.

Abstract

We derive the limiting distributions of the $M$-test family of unit root statistics in the nearly integrated nearly white noise (NINW) framework introduced by Nabeya and Perron (1994) in the case of an unknown linear time trend. In the case of known long run variance (LRV), the limiting distributions of the $M^{GLS}$ tests are contaminated by additional noise terms as a result of quasi differencing whereas these terms are less present in the $M^{OLS}$ limiting distributions, both of which display conservative properties under conventional critical values. Furthermore, we prove the Gaussian power envelope in the NINW model is asymptotically equivalent to the standard envelope of Elliott, Rothenberg, and Stock (1996), and that the oracle $M$-tests have inefficient power relative to this benchmark. We then derive the limiting distributions of the feasible statistics and show that the autoregressive estimate of the LRV commonly used overestimates the LRV, creating altered limiting distributions. Finally, finite sample simulations illustrate that, of the procedures considered, no uniformly satisfactory solution exists for handling a series with a large negative moving average coefficient.

Problem

• M-tests
• unit root
• nearly integrated nearly white noise
• long run variance
• limiting distribution

Innovation

• Nearly integrated nearly white noise
• M-tests
• Unit root statistics
• Long run variance
• Limiting distributions