Abstract: This paper is devoted to testing for the explosive bubble under time-varying non-stationary volatility. Because the limiting distribution of the seminal Phillips et al. (2011) test depends on the variance function and usually requires a bootstrap implementation under heteroskedasticity, we construct the test based on a deformation of the time domain. The proposed test is asymptotically pivotal under the null hypothesis and its limiting distribution coincides with that of the standard test under homoskedasticity, so that the test does not require computationally extensive methods for inference. Appealing finite sample properties are demonstrated through Monte-Carlo simulations. An empirical application demonstrates that the upsurge behavior of cryptocurrency time series in the middle of the sample is partially explained by the volatility change.
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Abstract: This paper provides a simple approach for robust testing for the trend function in the time series under uncertainty over the order of integration of the error term. The proposed approach relies on the asymptotic normality of the trend coefficient estimator and utilizes t-statistic approach of Ibragimov and Müller (2010) based on splitting the sample. The Monte-Carlo results demonstrate that the approach has the correct finite sample size and favorable finite sample power properties for all data generating processes considered. The proposed approach is robust to very general assumptions on the error term including various forms of non-stationary volatility and heavy tails.
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