On the Robustness of Unit Root Tests in the Presence of Double Unit Roots

Abstract

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 favouring the explosive alternative. Numerical simulations are used to demonstrate the analytical results and some of the implications in finite samples.

Human activities and global warming: a cointegration analysis

“…Others are not close to the unit circle. This information indicates that there is no I(2) trend in this system, whereas it seems to verify that explosive roots mimic the behavior of I(2) trends, as reported by Haldrup and Lildholdt (2002). Therefore in this case we conclude with r=1, s 1 =1 and s 2 =0.…”

“…Stern and Kaufmann (2000) proceeded using three different univariate unit root tests. However, as documented by Haldrup and Lildholdt (2002), these statistics are incorrect. Because when testing for I(2) and the underlying series is indeed integrated of order two, these statistics give rise to an excessive rejection of the null hypothesis of a unit root in favor of the stationary and explosive alternatives.…”

“…This size distortion is caused by the fact that the test statistics have a different distribution originated by one additional unit root. In consequence, the recommendation of Haldrup and Lildholdt (2002) is to test I(2) against I(1) prior to testing I(1) against I(0). The authors conclude that all basic univariate unit root tests suffer from this issue.…”

“…The last result is not totally clear after the application of all statistics. The difficulty with this case is the fact that explosive roots mimic the behavior of I(2) processes, see Haldrup and Lildholdt (2002).…”

On the Behaviour of Phillips–Perron Tests in the Presence of Persistent Cycles

“…An important point to note in this analysis is that as ϕ →0 the cyclical component becomes indistinguishable from an I (2) component, as also observed by Bierens (). Haldrup and Lildholdt () provide a discussion of the limit and finite‐sample behaviour of the ADF and PP unit root tests when the data generating mechanism is an I (2) process. Their main conclusions regarding the PP test are consistent with the results observed in Figures and (they only considered kernel‐based long‐run variance estimators in their analysis) when ϕ is in the neighbourhood of zero.…”