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This paper studies the unit-root testing with unspecified and heavy-tailed heteroscedastic noises. A new weighted least squares estimation (WLSE) is designed to be used in the Dickey-Fuller (DF) test, of which the asymptotic normality is verified. However, the performance of the DF test strongly relies on the estimation accuracy of the asymptotic variance, which is not stable for dependent time series. To overcome this issue, we develop two novel unit-root tests by applying the empirical likelihood technique to the WLSE score equation. It is shown that both of the empirical likelihood-based tests converge weakly to a chi-squared distribution with one degree of freedom. Furthermore, the limiting theory is extended to the weighted M -estimation score equation. In contrast to existing unit-root tests for heavy-tailed time series, the empirical likelihood tests do not involve any estimators of the unknown parameters or any restrictions on the tail index of noise, which is of more practical appealing, and thus can be widely used in finance and econometrics. Extensive simulation studies are conducted to examine the effectiveness of the proposed methods.

We propose a location-adaptive self-normalization (SN) based test for change points in time series. The SN technique has been extensively used in changepoint detection for its capability to avoid direct estimation of nuisance parameters. However, we find that the power of the SN-based test is susceptible to the location of the break and may suffer from a severe power loss, especially when the change occurs at the early or late stage of the sequence. This phenomenon is essentially caused by the unbalance of the data used before and after the change point when one is building a test statistic based on the cumulative sum (CUSUM) process. Hence, we consider leaving out the samples far away from the potential locations of change points and propose an optimal data selection scheme. Based on this scheme, a new SN-based test statistic adaptive to the locations of breaks is established. The new test can significantly improve the p ower of the existing SN-based tests while maintaining a satisfactory size. It is a unified treatment that can be readily extended to tests for general quantities of interest, such as the median and the model parameters. The derived optimal subsample selection strategy is not specific to the SN-based tests but is applicable to any method that relies on the CUSUM process, which may provide new insights in the area for future research.