Abstract
In this paper we analyse the impact of non-stationary volatility on the recently developed unit
root tests which allow for a possible break in trend occurring at an unknown point in the sample,
considered in Harris, Harvey, Leybourne and Taylor (2009) [HHLT]. HHLT’s analysis hinges on
a new break fraction estimator which, when a break in trend occurs, is consistent for the true
break fraction at rate Op(T��1). Unlike other available estimators, however, when there is no
trend break HHLT’s estimator converges to zero at rate Op(T��1=2). In their analysis HHLT
assume the shocks to follow a linear process driven by IID innovations. Our rst contribution
is to show that HHLT’s break fraction estimator retains the same consistency properties as
demonstrated by HHLT for the IID case when the innovations display non-stationary behaviour
of a quite general form, including, for example, the case of a single break in the volatility of the
innovations which may or may not occur at the same time as a break in trend. However, as
we subsequently demonstrate, the limiting null distribution of unit root statistics based around
this estimator are not pivotal in the presence of non-stationary volatility. Associated Monte
Carlo evidence is presented to quantify the impact of a one-time change in volatility on both
the asymptotic and nite sample behaviour of such tests. A solution to the identi ed inference
problem is then provided by considering wild bootstrap-based implementations of the HHLT
tests, using the trend break estimator from the original sample data. The proposed bootstrap
method does not require the practitioner to specify a parametric model for volatility, and is
shown to perform very well in practice across a range of models.
Keywords: Unit root tests; quasi di erence de-trending; trend break; non-stationary volatility;
wild bootstrap.