Robust inference for spurious regressions and cointegrations involving processes moderately deviated from a unit root

This paper studies spurious regressions involving processes moderately deviated from a unit root (PMDURs), and establishes the limiting distributions for the least squares estimator, the associated t-statistic, the coefficient of determination R2 and the Durbin–Watson statistic. We find that these limiting distributions critically depend on nuisance parameters that characterize the local deviations from unity, making inference for spurious regressions practically impossible using the conventional t-statistic. As a cure, we propose robust inference based on the balanced regression model, where the lagged regressor and the lagged dependent variable are augmented to the original regression. The induced t-statistic via such an augmentation is shown to be asymptotically standard normal and is therefore free of nuisance parameters, which turns out to be a robust and simple-to-implement tool for spurious regressions inference. Moreover, the limiting properties of other statistics are investigated. The balanced regression based inference is further shown to continue to work for cointegration models with PMDURs, which is therefore robust to whether the PMDURs are spuriously related or cointegrated. Finally, the finite sample properties of the robust method are demonstrated through both Monte Carlo and real data examples.