The role of standard likelihood based measures of information and efficiency is unclear when regressions involve nonstationary data. Typically the standardized score is not asymptotically Gaussian and the standardized Hessian has a stochastic, rather than deterministic limit. Here we consider a time series regression involving a deterministic covariate which can be evaporating, slowly evolving or nonstationary. It is shown that conditional information, or equivalently, profile Kullback-Leibler and Fisher Information remain informative about both the accuracy, i.e. asymptotic variance, of profile maximum likelihood estimators, as well as the power of point optimal invariant tests for a unit root. Specifically these information measures indicate fractional, rather than linear trends may minimize inferential accuracy. Such is confirmed in numerical experiment.
Marsh, P. (2019). The role of information in nonstationary regression. Statistics, 53(3), 656-672. https://doi.org/10.1080/02331888.2019.1605516