Abstract

It has been well recognized that regulation in dynamics may be tested within the threshold autoregressive setting, with the null hypothesis being a global non-stationary process and the basic idea has been implemented via several approaches. Nonetheless, the practical usefulness of such a setting is debatable since the data are often corrupted by measurement errors. Thus, a more appropriate setting is to consider a threshold autoregressive moving-average model as the general hypothesis. We implement this new setting with the integrated moving-average model of order one as the null hypothesis. We derive a Lagrange multiplier test which is shown to admit an asymptotically similar null distribution and provide the first rigorous proof of tightness pertaining to testing for threshold nonlinearity against difference stationarity, which is of independent interest. Simulation studies show that the proposed approach enjoys significantly less bias and higher power in detecting threshold regulation than existing tests which do not account for measurement errors. We apply the new approach to the daily real exchange rates of the eurozone countries and find support for the PPP hypothesis, governed by a nonlinear mean-reversion mechanism triggered upon crossing a threshold located in the extreme upper tail of the distribution. The adequacy of the TARMA fit can shed new light on the purchasing power parity debate.