Level Shift Estimation in the Presence of Non-stationary Volatility with an Application to the Unit Root Testing Problem

In this paper we contribute to two separate literatures. Our principal contribution is made to the literature on break fraction estimation. Here we investigate the properties of a class of weighted residual sum of squares estimators for the location of a level break in time series whose shocks display non-stationary volatility (permanent changes in unconditional volatility). This class contains the ordinary least squares (OLS) and weighted least squares (WLS) estimators, the latter based on the true volatility process. For fixed magnitude breaks we show that the estimator attains the same consistency rate under non-stationary volatility as under homoskedasticity. We also provide local limiting distribution theory for the estimator when the break magnitude is either local-to-zero at some rate in the sample size or exactly zero. The former includes the Pitman drift rate which is shown via Monte Carlo experiments to predict well the key features of the finite sample behaviour of the OLS estimator and a feasible version of the WLS estimator based on an adaptive estimate of the volatility path of the shocks. The simulations highlight the importance of the break location, break magnitude, and the form of non-stationary volatility for the finite sample performance of these estimators, and show that the feasible WLS estimator can deliver significant improvements over the OLS estimator in certain heteroskedastic environments. We also contribute to the unit root testing literature. We demonstrate how the results in the first part of the paper can be applied, by using level break fraction estimators on the first differences of the data, when testing for a unit root in the presence of trend breaks and/or non-stationary volatility. In practice it will be unknown whether a trend break is present and so we also discuss methods to select between the break and no break cases, considering both standard information criteria and feasible weighted information criteria based on our adaptive volatility estimator. Simulation evidence suggests that the use of these feasible weighted estimators and information criteria can deliver unit root tests with significantly improved finite sample behaviour under heteroskedasticity relative to their unweighted counterparts.