The purpose of this paper is to characterize three commonly used double unit root tests in terms of their asymptotic local power. To this end, we study a class of nearly doubly integrated processes which in the limit will behave as a weighted integral of a double indexed Ornstein-Uhlenbeck process. Based on a numerical examination of the analytical distributions, a comparison of the tests is made via their asymptotic local power functions.

We examine some of the consequences on commonly used unit root tests when the underlying series is integrated of order two rather than of order one. It turns out that standard augmented Dickey-Fuller type of tests for a single unit root have excessive density in the explosive region of the distribution. The lower (stationary) tail, however, will be virtually unaffected in the presence of double unit roots. On the other hand, the Phillips-Perron class of semi-parametric tests is shown to diverge to plus infinity asymptotically and thus favoring the explosive alternative. Numerical simulations are used to demonstrate the analytical results and some of the implications in finite samples.

This paper introduces a representation of an integrated vector time series in which the coefficient of multiple correlation computed from the long-run covariance matrix of the innovation sequences is a primitive parameter of the model. Based on this representation, we propose a notion of near cointegration, which helps bridging the gap between the polar cases of spurious regression and cointegration. Two applications of the model of near cointegration are provided. As a first application, the properties of conventional cointegration methods under near cointegration are characterized, hereby investigating the robustness of cointegration methods. Secondly, we illustrate how to obtain local power functions of cointegration tests that take cointegration as the null hypothesis.

Frequently, seasonal and non-seasonal data (especially macro time series) are observed with noise. For instance, the time series can have irregular abrupt changes and interruptions following as a result of additive or temporary change outliers caused by external circumstances which are irrelevant for the series of interest. Equally, the time series can have measurement errors. In this paper we analyse the above types of data irregularities on the behaviour of seasonal unit roots. It occurs that in most cases outliers and measurement errors can seriously affect inference towards the rejection of seasonal unit roots. It is shown how the distortion of the tests will depend upon the frequency, magnitude, and persistence of the outliers as well as on the signal to noise ratio associated with measurement errors. Some solutions to the implied inference problems are suggested.