Abstract
This paper focuses on the estimation of the location of level breaks in time series whose shocks display non-stationary volatility (permanent changes in unconditional volatility). We propose a new feasible weighted least squares (WLS) estimator, based on an adaptive estimate of the volatility path of the shocks. We show that this estimator belongs to a generic class of weighted residual sum of squares which also contains the ordinary least squares (OLS) and WLS estimators, the latter based on the true volatility process. For fixed magnitude breaks we show that the consistency rate of the generic estimator is unaffected by non-stationary volatility. We also provide local limiting distribution theory for cases where the break magnitude is either local-to-zero at some polynomial rate in the sample size or is 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 both the OLS and our feasible WLS estimators. 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 our proposed feasible WLS estimator can deliver significant improvements over the OLS estimator under heteroskedasticity. We discuss how these results 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. Methods to select between the break and no break cases, using standard information criteria and feasible weighted information criteria based on our adaptive volatility estimator, are also discussed. Simulation evidence suggests that unit root tests based on these weighted quantities can display significantly improved finite sample behaviour under heteroskedasticity relative to their unweighted counterparts. An empirical illustration to U.S. and U.K. real GDP is also considered.
Highlights
Breaks in the deterministic trend function appear prevalent in macroeconomic series; see, inter alia, Stock and Watson (1996, 1999, 2005) and Perron and Zhu (2005)
The results show that the feasible weighted least squares (WLS) estimator can deliver substantial improvements over the ordinary least squares (OLS) estimator in certain heteroskedastic environments, most notably where the level break occurs in a low volatility regime
Noting that ω increases as the magnitude of the linear trend increases and is higher for τOLS than for τFWLS 3 and that the level break occurs near the start of the series, we clearly see that the efficacy of the estimators in finite samples is related to the average volatility across the whole sample rather than just to the volatility level near the level break, and to the weighting function used in constructing the level break fraction estimator, in each case as Theorem 2 predicts
Summary
Breaks in the deterministic trend function appear prevalent in macroeconomic series; see, inter alia, Stock and Watson (1996, 1999, 2005) and Perron and Zhu (2005). Cavaliere et al (2011) refine the approach of Harris et al (2009) to use wild bootstrap unit root tests Their procedures are still based around applying the OLS level break fraction estimator of Bai (1994) to the first differences and trend break pre-test, each of which were developed for homoskedastic innovations. While they show that both of these are asymptotically robust to time-varying volatility, their finite sample efficacy will clearly have important forward implications for the behaviour of the resulting unit root tests. D := D[0, 1] denotes the space of right continuous with left limit (càdlàg) processes on [0, 1]
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